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Research ArticleOptimal Robust Motion Controller Design UsingMultiobjective Genetic Algorithm
Andrej Sarjaš1 Rajko SveIko1 and Amor Chowdhury12
1 Faculty of Electrical Engineering and Computer Science University of Maribor Smetanova 17 2000 Maribor Slovenia2Margento BV Crystal Tower Orlyplein 10 1043 DP Amsterdam The Netherlands
Correspondence should be addressed to Andrej Sarjas andrejsarjasumsi
Received 28 February 2014 Accepted 18 March 2014 Published 8 May 2014
Academic Editors N Barsoum V N Dieu P Vasant and G-W Weber
Copyright copy 2014 Andrej Sarjas et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper describes the use of a multiobjective genetic algorithm for robust motion controller design Motion controller structureis based on a disturbance observer in an RIC framework The RIC approach is presented in the form with internal and externalfeedback loops in which an internal disturbance rejection controller and an external performance controller must be synthesisedThis paper involves novel objectives for robustness and performance assessments for such an approach Objective functions forthe robustness property of RIC are based on simple even polynomials with nonnegativity conditions Regional pole placementmethod is presented with the aims of controllersrsquo structures simplification and their additional arbitrary selection Regional poleplacement involves arbitrary selection of central polynomials for both loops with additional admissible region of the optimizedpole location Polynomial deviation between selected and optimized polynomials is measured with derived performance objectivefunctions A multiobjective function is composed of different unrelated criteria such as robust stability controllersrsquo stability andtime-performance indexes of closed loopsThe design of controllers and multiobjective optimization procedure involve a set of theobjectives which are optimized simultaneously with a genetic algorithmmdashdifferential evolution
1 Introduction
Modern positioning systems require high performance andcomplex operation which demand highly efficient and com-plex controller structures Complex controller algorithmsand their designs are mostly related to very complex designprocedures and optimisation techniques with which design-ers try to achieve the desired performance specification Todate many advanced design methods have been developedbased on different structures of closed-loop systems andparadigmsThemost widely used designs are based on distur-bance observers [1ndash3] internal model control [4] and modelbased disturbance attenuation [5] The disturbance observerdesigns have been mostly used in industrial environmentsbecause of its simple transparent structure and capability toensure proper tradeoff between robustness and performanceproperties Tradeoffs between criteria can be handled withheuristic optimization approaches ormathematical program-ming optimization techniques abbreviated as LP QP SDPNP and so forth [6 7] Optimization-based methods in
computer aided design (CAD) have proved to be a valuabletool in engineering practice with which one can achieveproper performance and suitable controllers for closed-loopsystems In general the optimization procedure can bedivided into convex or nonconvex optimization problems
Mathematical programming approaches such as LP QPand SDP are very efficient for convex problems whichcover many modern robust controller designs in state spaceapproach [8ndash10] However there are also many controlproblems where the evaluation of the objective is not strictlyconvex or where the desired criteria cannot be combinedinto a set of convex objective functions For example poly-nomial synthesis with fixed order controller in a transferfunction form and controller structure optimization is abasically nonconvex problem [11 12] One way to over-come this problem is by combining heuristic optimizationmethods with new or conventional controller designs [13ndash15] This combination provides a set of efficient tools toaddress complex multivariable problems with performanceconstraints [13]There is much evidence which indicates high
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 978167 15 pageshttpdxdoiorg1011552014978167
2 The Scientific World Journal
rF(s)
e
minusC(s)
cPm(s)
Tin(s)K(s)
K(s)u i
din dout
P0(s) P998400(s)
120585998400
y
y
120585
minus
y
y
Figure 1 Disturbance observer in an RIC framework
efficiency of the heuristic optimization technique especiallywith genetic algorithms (GAs) The GA optimization tech-nique with well-formulated objective functions can preservesatisfactory results of controlled systems while overcomingsome problems and limitations of conventional designs [1316ndash18]
This paper considers a robust disturbance observer(DOB) in a robust internal compensator framework (RIC)with multiobjective optimization of different robustnessand performance criteria The DOB structure has alreadybeen studied in detail and has several different structuralapproaches [1 19] The more convenient and straightforwardmethods of DOB are based on Q-filter and inverse nominalmodels within the internal feedback branch [1 20 21] Themore advanced approaches are based on internal referencemodels in RIC framework [2 20] Bothmethods Q-filter andRIC deal well with disturbance attention and have similarapproaches and structures [2 20 22] However the RICapproach is more transparent and reliable in comparisonwith the Q-filter design The main difference between bothapproaches is in the design of their internal loops [5 22]Thesignificant disadvantage of the DOB with Q-filter design isthe usage of an inverse nominal model within the internalfeedback loop Most mechanical systems are presented asstrict proper functions which prevents direct usage of themodel inverse The second disadvantage is the internal-loopstructure with a low-pass Q-filter within a partially positivefeedback loopThe property of the Q-filter lowers the closed-looprsquos stability and the selection is strictly empirical withsome vague guidance for the first- second- and third-orderfilters [5 21 23 24] Each selection of approximated inversefunction and Q-filter requires additional assessment of theclosed-looprsquos performance and robustness The RIC on theother hand has better structural transparency and can bemuch easily incorporated into the optimization procedureInternal and external controllers can be acquired from theoptimization procedure so that the robustness of the RICsystem can be ensured [25 26]
This paper describes the design of an RIC disturbanceobserver with optimization of robustness and performancecriteria with multiobjective differential evolution (DE) [2728] The main contribution of the presented paper is amultiobjective optimization approach with simultaneousoptimization of a set of criteria for inner and outer loopsof the RIC structure The used robustness and performancecriteria are derived directly from the property of the norm119867infin
and uncertainty models [29 30] The criteria have theformof even polynomials where simple quasi-convex control
problems arise Robust stability of the system can be quitestraightforwardly assessed if the nonnegativity conditionof the polynomial is provided Even polynomials can bedirectly used in the optimization technique within geneticalgorithm without any other additional transformationThe second contribution of this paper is the design ofa simple parameterized controller structure with selectedcentral characteristic polynomial The central characteristicpolynomial has an admissible region in the stable half plainprescribed so that it ensures the expected performance andstability of the RIC system The basic idea comes fromthe pole-colouring technique and regional pole assignmentmethod [31 32] In comparison to similar methods thepresented approach uses objective functions of the regionalpole placement technique and can be used for nonparametricuncertainties The presented objectives are optimized with agenetic algorithm differential evolution (DE)DEhas evidentadvantages over other similar techniques simple structurefast convergence lower space complexity adaptive parametersettings and so forth [33] All design criteria within the DEalgorithm are evaluated over the nonnegativity property ofrobustness criteria and roots calculations of the closed-loopcharacteristic polynomial for regional pole placement
This paper is divided into seven sections In Section 2we describe the RIC disturbance observer for a positioningsystem Section 3 describes the regional pole placement forinternal and external loops of the RIC system and appliedparameterization of the controller structure Thereafter inSection 4 we describe performance and robustness proper-ties with the metric119867
infin based on nonnegativity assessments
of the even polynomial Section 5 describes a multiobjectiveoptimization approach with DE and a composite multiobjec-tive function After Section 5 a design example of an RICsystem with multiobjective optimization results is providedin Section 6The conclusions of the paper are summarized inSection 7
2 RIC Disturbance Observer forPositioning Systems
A disturbance observer in the RIC framework is composed ofinternal and external loops [34ndash36]Thedesign of the internalloop is based on input disturbance attention where mostoften appearing disturbances are reaction and load torquefriction and unmodeled dynamics [20 34]The external loopis designed so that it ensures the overall performance of theclosed-loop system The RIC structure is shown in Figure 1where119870(119904) 119875
0(119904) 1198751015840(119904) 119875
119898(119904) 119862(119904) and 119865(119904) are the internal
The Scientific World Journal 3
controller plant reference model and 2 DOF structureswith an external controller and a prefilter respectively Theinternal loopwith119875
0(119904) covers the angular velocity in relation
to input torques and disturbance rejection 119889in where 1198751015840
(119904)
is the transfer function of the rotary encoder The transferfunction of the rotary encoder is mostly treated as a systemwith pure integral behavior The RIC transfer functions are
1198750(119904) =
1198610(119904)
1198600(119904) 119875
1015840
(119904) =1198611015840
(119904)
1198601015840 (119904) 119870 (119904) =
119871119870(119904)
119877119870(119904)
119875119898(119904) =
119861119898(119904)
119860119898(119904) 119862 (119904) =
119871119862(119904)
119877119862(119904) 119865 (119904) =
119871119865(119904)
119877119865(119904)
(1)
The matrix form of the RIC system in Figure 1 is
[[[[[
[
119865minus1
0 0 119865minus1
0
minus119862 (119875119898119870 + 1) 1 0 0 119870
0 minus11987501198751015840
1 0 0
0 0 minus1 1 0
0 minus1198750
0 0 1
]]]]]
]
(
119890
119894
119910
119910
119910
)
=(
119903
119889in119889out120585
1205851015840
)
(2)
where 119903 119889in 119889out 120585 and 1205851015840 are reference signal input
disturbance output disturbance internal noise and externalnoise respectively
The closed-loop transfer function is
(
119890
119894
119910
119910
119910
) =1
1 + 1198701198750+ 119862119875
01198751015840 + 119870119862119875
11989811987501198751015840
times
[[[[[
[
119865 (1198701198750+ 1) minus119875
01198751015840
minus (1 + 1198701198750) minus (1 + 119870119875
0) 119875
01198751015840
119870
119865119862 (119875119898119870 + 1) 1 minus119862 (119875
119898119870 + 1) minus119862 (119875
119898119870 + 1) minus119870
11987501198751015840
119865119862 (119875119898119870 + 1) 119875
01198751015840
(1 + 1198701198750) minus119875
01198751015840
119862 (119875119898119870 + 1) minus119875
01198751015840
119865
11987501198751015840
119865119862 (119875119898119870 + 1) 119875
01198751015840
(1 + 1198701198750) (1 + 119870119875
0) minus119875
01198751015840
119865
1198750119865119862 (119875
119898119870 + 1) 119875
0minus119862119875
0(119875119898119870 + 1) minus119875
0119862 (119875
119898119870 + 1) (119875
01198751015840
119862 + 11987501198751015840
119862119875119898119870 + 1)
]]]]]
]
times (119903 119889in 119889out 120585 1205851015840
)119879
(3)
The RIC system in Figure 1 is slightly a modified clas-sic structure The internal loop in the modified structureembraces only the angular velocity while the classical internalloop uses for the feedback branch the same output as forthe external loop [21 22] The modified structure providesindirect decupling of the internal and external loops wherethe poles of the internal loop do not directly influencethe zeroes of the external loop Most RIC designs includestandard controller structures in internal loops [20 24]Standard controllers like PID or PI with integral behaviourimprove disturbance rejection for low-frequency disturbance119889in On the other hand internal integral behaviour causesa double-integrator effect on the external loop which sig-nificantly lowers the stability domain and generates unde-sired responses on the output disturbances and referencesignals This paper will consider an RIC structure shownin Figure 1 where the internal integral behaviour of thecontroller 119870(119904) does not directly influence the externalloop For the sake of the RIC structure simplification weassume that the prefilter function is constant and is equal to119865(119904) = 1
3 Regional Pole Placement of the RIC System
The design procedure of the RIC system can be addressed intwo steps the first step covers the internal loop controllerdesign and the second step the external loop controllerdesign Both controllers are designed with the regionalpole placement technique based on the selected centralpolynomial in the expected stable area The expected areais freely chosen by the designer and exhibits closed-loopdynamic performanceThe parameterization of controllers isapplied to ensure good disturbance rejection and robustnessproperties
Let us consider the internal loop of the system in Figure 1The closed-loop system is
(119910
119894) =
1
1 + 1198701198750
[1198750(1 + 119870119875
119898) 119875
0
1 + 119870119875119898
1](
119888
119889in)
(119910
119894) = 119878in [
1198750(1 + 119870119875
119898) 119875
0
1 + 119870119875119898
1](
119888
119889in)
(4)
4 The Scientific World Journal
Desired region of the poles
120590ts high
120590ts low
p desired locationp optimized location
j120596
p1
p3
p2
p2
p3
p1
120590
Figure 2 Settling-time objective function with settling-timeboundaries 120590
119905119904 low and 120590119905119904 high
where 119878in is the sensitivity function of the internal closed-loop The characteristic polynomial is defined as
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) (5)
Equation (5) is the standard starting point of pole place-ment design The solution of the equation is provided understrict polynomial degree conditions [37] The presentedapproach uses only the conditions when an exact solutiondoes not exist deg119877
119896le deg119860
0minus2 In this case the controller
structure does not depend on plant order like in the classicpole placement design and is arbitrarily selected by thedesigner The only way to satisfy expression (5) is regionalpole placementwith a prescribed region and deviation assess-ment as described in [32] Deviation is assessed betweenthe given polynomial 119862in(119904) and the candidate polynomialobtained with optimization The term prescribed region ofthe closed polynomial is similar to the term domain stability[29 38] The prescribed region with the selected centralpolynomial offers more leeway for further multiobjectiveoptimization of robustness and performance criteria Thepolynomial equation for regional pole placement is
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)
(6)
where 119862in(119904) is the candidate polynomial obtained with theoptimization with controller coefficient 119870(119904) The prescribedregion of the desired pole location can be derived from theproperties of the Laplace stability domain119863in = 119904 = 120590+119895120596 isin
C | 120590 isin minusR119890 119895120596 isin I119898 where 119862in isin 119863in holds true Thepossible objective functions of the prescribed region for con-trollers119870(119904) 119862(119904) are described in the following subsections
31 Settling-Time Objective Function Roughly speaking thesettling time of the closed-loop system is inversely propor-tional to the real part of dominant poles in the complexdomain119863in The objective function can be composed sothat all closed-loop poles lie in a vertical stripe boundedby parameters120590
119905119904 high and 120590119905119904 low The vertical stripe belongs
to domain119863in Parameters120590119905119904 high and 120590
119905119904 low specify theobjective function criteria of the closed-loop settling time(see Figure 2)
The proposed objective function is
1198691= min
119870
(
1minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low
1003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
forall120590119905119904 high lt R119890 119862in lt 120590119905119904 low and
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (7)
where 120590119905119904 high and 120590
119905119904 low indicate the location of the stripeboundary and 119899 indicates the number of closed-looppoles The right boundary 120590
119905119904 high of the stripe preventsthe possibility of obtaining poles with large negativereal part values Large negative real part pole values cancause a too high dynamic of the closed-loop system Ahigher dynamic of the closed-loop can render properreal-time implementation of the discrete controller on theembedded system or cause improper behaviour of controlleroutputs Improper higher dynamic of the controller outputmostly manifests itself as tempestuous responses whichindicate nonoptimal energy behaviour of the closed-loopsystem
The first term of the 1198691indicates a normalized value of
the objective if all poles lie on the left of the vertical line120590119905119904 lowThe second term assesses the deviation of the real part
pole value between the desired 119901 and the optimized 119901 polelocation where 119901 is 119901 isin 119862in119901 isin 119862in
32 Rise-TimeObjective Function Therise time of the closed-loop system is roughly proportional to the inverse of thedominant poles imaginary values andmay be upper-boundedby the parameter120596
119905119903Theobjective function can be composed
so that all closed-loop poles lie on the horizontally boundedstripe with boundsplusmn119895120596
119905119903(see Figure 3) The selected stripe
belongs to domain119863in
The Scientific World Journal 5
The proposed rise-time objective function is
1198692= min
119870
(
1minus sum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903
1003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816120596119905119903
1003816100381610038161003816
forall10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903
1003816100381610038161003816
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816I119898119862in1003816100381610038161003816 minus
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (8)
The objective function 1198692describes the desired stripe in
the complex plain where the first term indicates the positioninside the horizontal stripe and the second the imaginarydeviation between the desired 119901 and the optimized 119901 poles
33 Damping-RatioObjective Function Damping ratio is alsoan important design criterion of the closed-loop dynamicperformance and is related to the angle of the vectors betweenthe negative real and the imaginary axis of the dominantpoles [32] The damping ratio is determined with the angleboundary plusmn120593
120577(see Figure 4)
The proposed damping-ratio objective function is
1198693= min
119870
(1 minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816
forall10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )
119862in isin 119863in
(9)
The proposed objective function 1198693represents the cone
region of the desired poles All objective functions 1198691ndash1198693
represent min-optimization procedure where all objectivesare normalized on the interval [0-1]
34 The Composite Objective Function of the Dynamic Crite-ria The composite objective function represents the inter-section of objectives 119869
1ndash1198693 A graphical presentation of the
composite objective function is shown in Figure 5The composite objective function can be presented as
multiobjective criteria for DE where the expected solutionarea is equal to
Ψin = (1198691 cap 1198692 cap 1198693) andmin (dev (119901in 119901in))
forallΨin isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
(10)
The multiobjective optimization approach will be dis-cussed in the following sections
35 Internal Controller Parameterization 119870(119904) Controllerparameterization is an applied technique which ensuressatisfied properties of the closed-loop system The internal
controllers 119870 and 119862 can be parameterized so as to ensuregood input disturbance rejection and robustness propertiesInmany applications the input disturbance 119889in in positioningsystems represents the load torque and Coulombrsquos frictionwith a typical low-frequency characteristic [20] The inter-nal system can provide good elimination of low-frequencydisturbances 119889in and reference tracking 119888 if the sensitivityfunction |119878in| ≪ 1 has a proper damping effect on thefrequency span 119861 = [120596
119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under
this assumption the controller119870 can be parameterized usingdifferent polynomial structures with a known effect on thesensitivity function |119878in| Controller parameterization withintegral action 119877
119870(119904) = 119877
1015840
119870(119904)119904 has a well-known influence
on the sensitivity function at low-frequency characteristicssuch as the known simple structures PI and PID For thesake of operational safety the integral action can in manycases be approximated with the stable pole (119904 + 120575) 120575 isin R |
0 lt 120575 ≪ 1 where 119870(119904) denominator polynomial is equalto 1198771015840
119870(119904)(119904 + 120575) In this case the damping of the sensitivity
function 119878in is lowered by parameter 120575 The lowered dampingvalue of 119878in is not noticeable in real-time operation especiallyif the absolute damping value is lower than the absolutemeasurement accuracy
A minimized tracking error 119890 and good low-frequencyinput disturbance rejection 119889in can be achieved if the follow-ing holds true
lim120596rarr119861
|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)
where 119861 is the low-frequency span of the sensitivity func-tion Sensitivity function 119878in for span 119861 with controllerparameterization (119904 + 120575) is
lim120596rarr119861
1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim
120596rarr119861
(1003816100381610038161003816119878in (120596)
1003816100381610038161003816radic1205962 + 1205752)
lim120596rarr119861
|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0
(12)
Parameter 120575 determines the closed-loop tracking accu-racy and the capability of disturbance rejection with charac-teristic polynomial119862in(119904) Controller structure 119870 is parame-terized as
119877119870(119904) =
1198771015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896)
1198771015840
119870(119904) 119877
119870par (119904)
(13)
6 The Scientific World Journal
Parameter 120575119896represents an additional parameter for
further optimization of performance and robustness criteriaThe polynomial equation with parametric solutions is
1198600(119904) 119877
1015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
1198600(119904) 119877
119870(119904 120575) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
(14)
The solution of the optimization procedure is a properset of polynomial coefficients where the internal polynomial119862in(119904) belongs to the Ψin and a strong proximity condition119862in(119904) asymp 119862in(119904) holds true
36 The Design of the External Controller 119862(119904) Controller119862(119904)design technique is the same as for the internal controller119870(119904) where the external characteristic polynomial 119862out is
1198600(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119870(119904) 119877
119862(119904)
+ 1198610(119904) 119871
119870(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119862(119904)
+ 1198610(119904) 119861
1015840
(119904) 119871119862(119904) 119860
119898(119904) 119877
119870(119904)
+ 1198610(119904) 119861
1015840
(119904) 119861119898(119904) 119871
119870(119904) 119871
119862(119904) = 119862out (119904)
(15)
The possible controller parameterization 119862(119904) is
Γ (119904) 1198771015840
119862(119904)
119901
prod
119896=1
(119904 + 120578119896) + Υ (119904) 119871
119862(119904) = 119862out (119904)
Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871
119862(119904) = 119862out (119904)
(16)
where polynomials Γ(119904) and Υ(119904) are
Γ (119904) = 1198601015840
(119904) 119860119898(119904) (119860
0(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904))
Υ (119904) = 1198610(119904) 119861
1015840
(119904) (119860119898(119904) 119877
119870(119904) + 119861
119898(119904) 119871
119870(119904))
(17)
Coefficient 120578 can also be used as an additional parameterfor criteria optimization in the same sense as parameter 120575
The controller119862(119904) is the proper solution of the optimiza-tion problem for external loop if the following conditions aresatisfied
Ψout = (119869out 1 cap 119869out 2 cap 119869out 3) andmin (dev (119901out 119901out))
forallΨout isin 119863out
forall119901out = 119901out isin 119862out | 119862out isin 119863out
forall119901out = 119901out isin 119862in | 119862out isin 119863out
(18)
Objective functions 119869out 1ndash119869out 3 have the same propertiesand meanings as objective functions 119869
1ndash1198693
4 Nonparametric Uncertainty andRobustness Criterion of RIC
The robustness of the RIC system is assessed using uncer-tainty models Δ119875 and a stable proper input weight 1198811015840noise
120590
p1
p3
p2
p2
p3
p1
j120596
Frequency boundj120596tr
minusj120596tr
Figure 3 Rise-time objective function with a horizontal bound-aryplusmn119895120596
119905119903
120590
p1
p3
p2
p2
p3
p1
j120596Damping line
120593120577
minus120593120577
120577
120593p1
120593p2
Figure 4 Damping-ratio objective function with parameter plusmn120593120577
The uncertainty models describe model deviation within thegiven frequency space and are represented with a nominalmodel and stable uncertainty weights Δ119882 [30 39] Weights1198811015840
noise representmeasured noise spectrumof the signal11990810158401 For
RIC design we assume that the reference model is 119875119898asymp 119875
0isin
119875119867infin The internal loop of the RIC with uncertainty models
is shown in Figure 6The robustness assessment of the internal loop should
be considered with multiplicative and inverse uncertaintyThe multiplicative uncertainty represents uncertainties bylower frequencies while the inverse uncertainty representsuncertainties by higher frequencies [39]
The Scientific World Journal 7
σ
Stable area
Intersection
120590ts low120590ts high j120596
p2
p1
p3
120593120577
120577
j120596tr
minusj120596tr
Ψ
Figure 5 Intersection of objectives 1198691ndash1198693
c iPm(s) K(s 120590)
K(s 120590)
din
ΔP(s)y
120585998400w998400
1V998400noise(s)minus
Figure 6 Internal loop with uncertainty models Δ119875
Let us consider the multiplicative uncertainty modelΔ119875
119872= 119875
0(1 +Δ119882
119872)with nominal plant 119875
0 The closed-loop
characteristic using the multiplicative uncertainty modelΔ119875
119872is
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119872))minus1
times [1198750(1 + Δ119882
119872) (1 + 119870119875
119898) 119875
0(1 + Δ119882
119872)
1 + 119870119875119898
1](
119888
119889in)
(19)
Robust stability for the multiplicative uncertainty is pre-served if the following holds true
10038171003817100381710038171003817100381710038171003817
Δ119882119872
1198701198750
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin
1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin
lt 1
(20)
where 119879in is a complementary sensitivity function of theinternal loop
+e c y
z
Ws(s)
C(s 120578) TP119898 in(s)y
P998400(s)
++minus
w1
dout
Vout(s)
120585Vnoise(s)
w2
Figure 7 External loop optimization structure
The internal loop with inverse uncertainty Δ119875119868
=
1198750(1 + Δ119882
119868)minus1 is defined as
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119868)minus1
)minus1
times [1198750(1 + Δ119882
119868)minus1
(1 + 119870119875119898) 119875
0(1 + Δ119882
119868)minus1
1 + 119870119875119898
1]
times (119888
119889in)
(21)
The robustness for inverse models is satisfied if the followingholds true
10038171003817100381710038171003817100381710038171003817
Δ119882119868
1
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)
where 119878in is the sensitivity function of the internal loopThe noise suppression 120585
1015840 of the internal loop can beassessed with the following criterion
1198791199081015840
1
= (1 + 119870 (120575) Δ119875)minus1
1198811015840
noise (23)
The robustness criterion of noise suppression with theuncertainty model is
1198791199081015840
1
= min119870
10038171003817100381710038171003817100381710038171003817
[119882119872
119882119868] [119879in 0
0 119878in]119881
1015840
noise
10038171003817100381710038171003817100381710038171003817
(24)
After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7
Selected weights119882119904 119881out 119881noise belong to the PH
infin
domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882
119904is selected to ensure a smooth frequency
characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
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Electrical and Computer Engineering
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
2 The Scientific World Journal
rF(s)
e
minusC(s)
cPm(s)
Tin(s)K(s)
K(s)u i
din dout
P0(s) P998400(s)
120585998400
y
y
120585
minus
y
y
Figure 1 Disturbance observer in an RIC framework
efficiency of the heuristic optimization technique especiallywith genetic algorithms (GAs) The GA optimization tech-nique with well-formulated objective functions can preservesatisfactory results of controlled systems while overcomingsome problems and limitations of conventional designs [1316ndash18]
This paper considers a robust disturbance observer(DOB) in a robust internal compensator framework (RIC)with multiobjective optimization of different robustnessand performance criteria The DOB structure has alreadybeen studied in detail and has several different structuralapproaches [1 19] The more convenient and straightforwardmethods of DOB are based on Q-filter and inverse nominalmodels within the internal feedback branch [1 20 21] Themore advanced approaches are based on internal referencemodels in RIC framework [2 20] Bothmethods Q-filter andRIC deal well with disturbance attention and have similarapproaches and structures [2 20 22] However the RICapproach is more transparent and reliable in comparisonwith the Q-filter design The main difference between bothapproaches is in the design of their internal loops [5 22]Thesignificant disadvantage of the DOB with Q-filter design isthe usage of an inverse nominal model within the internalfeedback loop Most mechanical systems are presented asstrict proper functions which prevents direct usage of themodel inverse The second disadvantage is the internal-loopstructure with a low-pass Q-filter within a partially positivefeedback loopThe property of the Q-filter lowers the closed-looprsquos stability and the selection is strictly empirical withsome vague guidance for the first- second- and third-orderfilters [5 21 23 24] Each selection of approximated inversefunction and Q-filter requires additional assessment of theclosed-looprsquos performance and robustness The RIC on theother hand has better structural transparency and can bemuch easily incorporated into the optimization procedureInternal and external controllers can be acquired from theoptimization procedure so that the robustness of the RICsystem can be ensured [25 26]
This paper describes the design of an RIC disturbanceobserver with optimization of robustness and performancecriteria with multiobjective differential evolution (DE) [2728] The main contribution of the presented paper is amultiobjective optimization approach with simultaneousoptimization of a set of criteria for inner and outer loopsof the RIC structure The used robustness and performancecriteria are derived directly from the property of the norm119867infin
and uncertainty models [29 30] The criteria have theformof even polynomials where simple quasi-convex control
problems arise Robust stability of the system can be quitestraightforwardly assessed if the nonnegativity conditionof the polynomial is provided Even polynomials can bedirectly used in the optimization technique within geneticalgorithm without any other additional transformationThe second contribution of this paper is the design ofa simple parameterized controller structure with selectedcentral characteristic polynomial The central characteristicpolynomial has an admissible region in the stable half plainprescribed so that it ensures the expected performance andstability of the RIC system The basic idea comes fromthe pole-colouring technique and regional pole assignmentmethod [31 32] In comparison to similar methods thepresented approach uses objective functions of the regionalpole placement technique and can be used for nonparametricuncertainties The presented objectives are optimized with agenetic algorithm differential evolution (DE)DEhas evidentadvantages over other similar techniques simple structurefast convergence lower space complexity adaptive parametersettings and so forth [33] All design criteria within the DEalgorithm are evaluated over the nonnegativity property ofrobustness criteria and roots calculations of the closed-loopcharacteristic polynomial for regional pole placement
This paper is divided into seven sections In Section 2we describe the RIC disturbance observer for a positioningsystem Section 3 describes the regional pole placement forinternal and external loops of the RIC system and appliedparameterization of the controller structure Thereafter inSection 4 we describe performance and robustness proper-ties with the metric119867
infin based on nonnegativity assessments
of the even polynomial Section 5 describes a multiobjectiveoptimization approach with DE and a composite multiobjec-tive function After Section 5 a design example of an RICsystem with multiobjective optimization results is providedin Section 6The conclusions of the paper are summarized inSection 7
2 RIC Disturbance Observer forPositioning Systems
A disturbance observer in the RIC framework is composed ofinternal and external loops [34ndash36]Thedesign of the internalloop is based on input disturbance attention where mostoften appearing disturbances are reaction and load torquefriction and unmodeled dynamics [20 34]The external loopis designed so that it ensures the overall performance of theclosed-loop system The RIC structure is shown in Figure 1where119870(119904) 119875
0(119904) 1198751015840(119904) 119875
119898(119904) 119862(119904) and 119865(119904) are the internal
The Scientific World Journal 3
controller plant reference model and 2 DOF structureswith an external controller and a prefilter respectively Theinternal loopwith119875
0(119904) covers the angular velocity in relation
to input torques and disturbance rejection 119889in where 1198751015840
(119904)
is the transfer function of the rotary encoder The transferfunction of the rotary encoder is mostly treated as a systemwith pure integral behavior The RIC transfer functions are
1198750(119904) =
1198610(119904)
1198600(119904) 119875
1015840
(119904) =1198611015840
(119904)
1198601015840 (119904) 119870 (119904) =
119871119870(119904)
119877119870(119904)
119875119898(119904) =
119861119898(119904)
119860119898(119904) 119862 (119904) =
119871119862(119904)
119877119862(119904) 119865 (119904) =
119871119865(119904)
119877119865(119904)
(1)
The matrix form of the RIC system in Figure 1 is
[[[[[
[
119865minus1
0 0 119865minus1
0
minus119862 (119875119898119870 + 1) 1 0 0 119870
0 minus11987501198751015840
1 0 0
0 0 minus1 1 0
0 minus1198750
0 0 1
]]]]]
]
(
119890
119894
119910
119910
119910
)
=(
119903
119889in119889out120585
1205851015840
)
(2)
where 119903 119889in 119889out 120585 and 1205851015840 are reference signal input
disturbance output disturbance internal noise and externalnoise respectively
The closed-loop transfer function is
(
119890
119894
119910
119910
119910
) =1
1 + 1198701198750+ 119862119875
01198751015840 + 119870119862119875
11989811987501198751015840
times
[[[[[
[
119865 (1198701198750+ 1) minus119875
01198751015840
minus (1 + 1198701198750) minus (1 + 119870119875
0) 119875
01198751015840
119870
119865119862 (119875119898119870 + 1) 1 minus119862 (119875
119898119870 + 1) minus119862 (119875
119898119870 + 1) minus119870
11987501198751015840
119865119862 (119875119898119870 + 1) 119875
01198751015840
(1 + 1198701198750) minus119875
01198751015840
119862 (119875119898119870 + 1) minus119875
01198751015840
119865
11987501198751015840
119865119862 (119875119898119870 + 1) 119875
01198751015840
(1 + 1198701198750) (1 + 119870119875
0) minus119875
01198751015840
119865
1198750119865119862 (119875
119898119870 + 1) 119875
0minus119862119875
0(119875119898119870 + 1) minus119875
0119862 (119875
119898119870 + 1) (119875
01198751015840
119862 + 11987501198751015840
119862119875119898119870 + 1)
]]]]]
]
times (119903 119889in 119889out 120585 1205851015840
)119879
(3)
The RIC system in Figure 1 is slightly a modified clas-sic structure The internal loop in the modified structureembraces only the angular velocity while the classical internalloop uses for the feedback branch the same output as forthe external loop [21 22] The modified structure providesindirect decupling of the internal and external loops wherethe poles of the internal loop do not directly influencethe zeroes of the external loop Most RIC designs includestandard controller structures in internal loops [20 24]Standard controllers like PID or PI with integral behaviourimprove disturbance rejection for low-frequency disturbance119889in On the other hand internal integral behaviour causesa double-integrator effect on the external loop which sig-nificantly lowers the stability domain and generates unde-sired responses on the output disturbances and referencesignals This paper will consider an RIC structure shownin Figure 1 where the internal integral behaviour of thecontroller 119870(119904) does not directly influence the externalloop For the sake of the RIC structure simplification weassume that the prefilter function is constant and is equal to119865(119904) = 1
3 Regional Pole Placement of the RIC System
The design procedure of the RIC system can be addressed intwo steps the first step covers the internal loop controllerdesign and the second step the external loop controllerdesign Both controllers are designed with the regionalpole placement technique based on the selected centralpolynomial in the expected stable area The expected areais freely chosen by the designer and exhibits closed-loopdynamic performanceThe parameterization of controllers isapplied to ensure good disturbance rejection and robustnessproperties
Let us consider the internal loop of the system in Figure 1The closed-loop system is
(119910
119894) =
1
1 + 1198701198750
[1198750(1 + 119870119875
119898) 119875
0
1 + 119870119875119898
1](
119888
119889in)
(119910
119894) = 119878in [
1198750(1 + 119870119875
119898) 119875
0
1 + 119870119875119898
1](
119888
119889in)
(4)
4 The Scientific World Journal
Desired region of the poles
120590ts high
120590ts low
p desired locationp optimized location
j120596
p1
p3
p2
p2
p3
p1
120590
Figure 2 Settling-time objective function with settling-timeboundaries 120590
119905119904 low and 120590119905119904 high
where 119878in is the sensitivity function of the internal closed-loop The characteristic polynomial is defined as
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) (5)
Equation (5) is the standard starting point of pole place-ment design The solution of the equation is provided understrict polynomial degree conditions [37] The presentedapproach uses only the conditions when an exact solutiondoes not exist deg119877
119896le deg119860
0minus2 In this case the controller
structure does not depend on plant order like in the classicpole placement design and is arbitrarily selected by thedesigner The only way to satisfy expression (5) is regionalpole placementwith a prescribed region and deviation assess-ment as described in [32] Deviation is assessed betweenthe given polynomial 119862in(119904) and the candidate polynomialobtained with optimization The term prescribed region ofthe closed polynomial is similar to the term domain stability[29 38] The prescribed region with the selected centralpolynomial offers more leeway for further multiobjectiveoptimization of robustness and performance criteria Thepolynomial equation for regional pole placement is
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)
(6)
where 119862in(119904) is the candidate polynomial obtained with theoptimization with controller coefficient 119870(119904) The prescribedregion of the desired pole location can be derived from theproperties of the Laplace stability domain119863in = 119904 = 120590+119895120596 isin
C | 120590 isin minusR119890 119895120596 isin I119898 where 119862in isin 119863in holds true Thepossible objective functions of the prescribed region for con-trollers119870(119904) 119862(119904) are described in the following subsections
31 Settling-Time Objective Function Roughly speaking thesettling time of the closed-loop system is inversely propor-tional to the real part of dominant poles in the complexdomain119863in The objective function can be composed sothat all closed-loop poles lie in a vertical stripe boundedby parameters120590
119905119904 high and 120590119905119904 low The vertical stripe belongs
to domain119863in Parameters120590119905119904 high and 120590
119905119904 low specify theobjective function criteria of the closed-loop settling time(see Figure 2)
The proposed objective function is
1198691= min
119870
(
1minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low
1003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
forall120590119905119904 high lt R119890 119862in lt 120590119905119904 low and
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (7)
where 120590119905119904 high and 120590
119905119904 low indicate the location of the stripeboundary and 119899 indicates the number of closed-looppoles The right boundary 120590
119905119904 high of the stripe preventsthe possibility of obtaining poles with large negativereal part values Large negative real part pole values cancause a too high dynamic of the closed-loop system Ahigher dynamic of the closed-loop can render properreal-time implementation of the discrete controller on theembedded system or cause improper behaviour of controlleroutputs Improper higher dynamic of the controller outputmostly manifests itself as tempestuous responses whichindicate nonoptimal energy behaviour of the closed-loopsystem
The first term of the 1198691indicates a normalized value of
the objective if all poles lie on the left of the vertical line120590119905119904 lowThe second term assesses the deviation of the real part
pole value between the desired 119901 and the optimized 119901 polelocation where 119901 is 119901 isin 119862in119901 isin 119862in
32 Rise-TimeObjective Function Therise time of the closed-loop system is roughly proportional to the inverse of thedominant poles imaginary values andmay be upper-boundedby the parameter120596
119905119903Theobjective function can be composed
so that all closed-loop poles lie on the horizontally boundedstripe with boundsplusmn119895120596
119905119903(see Figure 3) The selected stripe
belongs to domain119863in
The Scientific World Journal 5
The proposed rise-time objective function is
1198692= min
119870
(
1minus sum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903
1003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816120596119905119903
1003816100381610038161003816
forall10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903
1003816100381610038161003816
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816I119898119862in1003816100381610038161003816 minus
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (8)
The objective function 1198692describes the desired stripe in
the complex plain where the first term indicates the positioninside the horizontal stripe and the second the imaginarydeviation between the desired 119901 and the optimized 119901 poles
33 Damping-RatioObjective Function Damping ratio is alsoan important design criterion of the closed-loop dynamicperformance and is related to the angle of the vectors betweenthe negative real and the imaginary axis of the dominantpoles [32] The damping ratio is determined with the angleboundary plusmn120593
120577(see Figure 4)
The proposed damping-ratio objective function is
1198693= min
119870
(1 minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816
forall10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )
119862in isin 119863in
(9)
The proposed objective function 1198693represents the cone
region of the desired poles All objective functions 1198691ndash1198693
represent min-optimization procedure where all objectivesare normalized on the interval [0-1]
34 The Composite Objective Function of the Dynamic Crite-ria The composite objective function represents the inter-section of objectives 119869
1ndash1198693 A graphical presentation of the
composite objective function is shown in Figure 5The composite objective function can be presented as
multiobjective criteria for DE where the expected solutionarea is equal to
Ψin = (1198691 cap 1198692 cap 1198693) andmin (dev (119901in 119901in))
forallΨin isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
(10)
The multiobjective optimization approach will be dis-cussed in the following sections
35 Internal Controller Parameterization 119870(119904) Controllerparameterization is an applied technique which ensuressatisfied properties of the closed-loop system The internal
controllers 119870 and 119862 can be parameterized so as to ensuregood input disturbance rejection and robustness propertiesInmany applications the input disturbance 119889in in positioningsystems represents the load torque and Coulombrsquos frictionwith a typical low-frequency characteristic [20] The inter-nal system can provide good elimination of low-frequencydisturbances 119889in and reference tracking 119888 if the sensitivityfunction |119878in| ≪ 1 has a proper damping effect on thefrequency span 119861 = [120596
119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under
this assumption the controller119870 can be parameterized usingdifferent polynomial structures with a known effect on thesensitivity function |119878in| Controller parameterization withintegral action 119877
119870(119904) = 119877
1015840
119870(119904)119904 has a well-known influence
on the sensitivity function at low-frequency characteristicssuch as the known simple structures PI and PID For thesake of operational safety the integral action can in manycases be approximated with the stable pole (119904 + 120575) 120575 isin R |
0 lt 120575 ≪ 1 where 119870(119904) denominator polynomial is equalto 1198771015840
119870(119904)(119904 + 120575) In this case the damping of the sensitivity
function 119878in is lowered by parameter 120575 The lowered dampingvalue of 119878in is not noticeable in real-time operation especiallyif the absolute damping value is lower than the absolutemeasurement accuracy
A minimized tracking error 119890 and good low-frequencyinput disturbance rejection 119889in can be achieved if the follow-ing holds true
lim120596rarr119861
|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)
where 119861 is the low-frequency span of the sensitivity func-tion Sensitivity function 119878in for span 119861 with controllerparameterization (119904 + 120575) is
lim120596rarr119861
1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim
120596rarr119861
(1003816100381610038161003816119878in (120596)
1003816100381610038161003816radic1205962 + 1205752)
lim120596rarr119861
|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0
(12)
Parameter 120575 determines the closed-loop tracking accu-racy and the capability of disturbance rejection with charac-teristic polynomial119862in(119904) Controller structure 119870 is parame-terized as
119877119870(119904) =
1198771015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896)
1198771015840
119870(119904) 119877
119870par (119904)
(13)
6 The Scientific World Journal
Parameter 120575119896represents an additional parameter for
further optimization of performance and robustness criteriaThe polynomial equation with parametric solutions is
1198600(119904) 119877
1015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
1198600(119904) 119877
119870(119904 120575) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
(14)
The solution of the optimization procedure is a properset of polynomial coefficients where the internal polynomial119862in(119904) belongs to the Ψin and a strong proximity condition119862in(119904) asymp 119862in(119904) holds true
36 The Design of the External Controller 119862(119904) Controller119862(119904)design technique is the same as for the internal controller119870(119904) where the external characteristic polynomial 119862out is
1198600(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119870(119904) 119877
119862(119904)
+ 1198610(119904) 119871
119870(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119862(119904)
+ 1198610(119904) 119861
1015840
(119904) 119871119862(119904) 119860
119898(119904) 119877
119870(119904)
+ 1198610(119904) 119861
1015840
(119904) 119861119898(119904) 119871
119870(119904) 119871
119862(119904) = 119862out (119904)
(15)
The possible controller parameterization 119862(119904) is
Γ (119904) 1198771015840
119862(119904)
119901
prod
119896=1
(119904 + 120578119896) + Υ (119904) 119871
119862(119904) = 119862out (119904)
Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871
119862(119904) = 119862out (119904)
(16)
where polynomials Γ(119904) and Υ(119904) are
Γ (119904) = 1198601015840
(119904) 119860119898(119904) (119860
0(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904))
Υ (119904) = 1198610(119904) 119861
1015840
(119904) (119860119898(119904) 119877
119870(119904) + 119861
119898(119904) 119871
119870(119904))
(17)
Coefficient 120578 can also be used as an additional parameterfor criteria optimization in the same sense as parameter 120575
The controller119862(119904) is the proper solution of the optimiza-tion problem for external loop if the following conditions aresatisfied
Ψout = (119869out 1 cap 119869out 2 cap 119869out 3) andmin (dev (119901out 119901out))
forallΨout isin 119863out
forall119901out = 119901out isin 119862out | 119862out isin 119863out
forall119901out = 119901out isin 119862in | 119862out isin 119863out
(18)
Objective functions 119869out 1ndash119869out 3 have the same propertiesand meanings as objective functions 119869
1ndash1198693
4 Nonparametric Uncertainty andRobustness Criterion of RIC
The robustness of the RIC system is assessed using uncer-tainty models Δ119875 and a stable proper input weight 1198811015840noise
120590
p1
p3
p2
p2
p3
p1
j120596
Frequency boundj120596tr
minusj120596tr
Figure 3 Rise-time objective function with a horizontal bound-aryplusmn119895120596
119905119903
120590
p1
p3
p2
p2
p3
p1
j120596Damping line
120593120577
minus120593120577
120577
120593p1
120593p2
Figure 4 Damping-ratio objective function with parameter plusmn120593120577
The uncertainty models describe model deviation within thegiven frequency space and are represented with a nominalmodel and stable uncertainty weights Δ119882 [30 39] Weights1198811015840
noise representmeasured noise spectrumof the signal11990810158401 For
RIC design we assume that the reference model is 119875119898asymp 119875
0isin
119875119867infin The internal loop of the RIC with uncertainty models
is shown in Figure 6The robustness assessment of the internal loop should
be considered with multiplicative and inverse uncertaintyThe multiplicative uncertainty represents uncertainties bylower frequencies while the inverse uncertainty representsuncertainties by higher frequencies [39]
The Scientific World Journal 7
σ
Stable area
Intersection
120590ts low120590ts high j120596
p2
p1
p3
120593120577
120577
j120596tr
minusj120596tr
Ψ
Figure 5 Intersection of objectives 1198691ndash1198693
c iPm(s) K(s 120590)
K(s 120590)
din
ΔP(s)y
120585998400w998400
1V998400noise(s)minus
Figure 6 Internal loop with uncertainty models Δ119875
Let us consider the multiplicative uncertainty modelΔ119875
119872= 119875
0(1 +Δ119882
119872)with nominal plant 119875
0 The closed-loop
characteristic using the multiplicative uncertainty modelΔ119875
119872is
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119872))minus1
times [1198750(1 + Δ119882
119872) (1 + 119870119875
119898) 119875
0(1 + Δ119882
119872)
1 + 119870119875119898
1](
119888
119889in)
(19)
Robust stability for the multiplicative uncertainty is pre-served if the following holds true
10038171003817100381710038171003817100381710038171003817
Δ119882119872
1198701198750
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin
1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin
lt 1
(20)
where 119879in is a complementary sensitivity function of theinternal loop
+e c y
z
Ws(s)
C(s 120578) TP119898 in(s)y
P998400(s)
++minus
w1
dout
Vout(s)
120585Vnoise(s)
w2
Figure 7 External loop optimization structure
The internal loop with inverse uncertainty Δ119875119868
=
1198750(1 + Δ119882
119868)minus1 is defined as
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119868)minus1
)minus1
times [1198750(1 + Δ119882
119868)minus1
(1 + 119870119875119898) 119875
0(1 + Δ119882
119868)minus1
1 + 119870119875119898
1]
times (119888
119889in)
(21)
The robustness for inverse models is satisfied if the followingholds true
10038171003817100381710038171003817100381710038171003817
Δ119882119868
1
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)
where 119878in is the sensitivity function of the internal loopThe noise suppression 120585
1015840 of the internal loop can beassessed with the following criterion
1198791199081015840
1
= (1 + 119870 (120575) Δ119875)minus1
1198811015840
noise (23)
The robustness criterion of noise suppression with theuncertainty model is
1198791199081015840
1
= min119870
10038171003817100381710038171003817100381710038171003817
[119882119872
119882119868] [119879in 0
0 119878in]119881
1015840
noise
10038171003817100381710038171003817100381710038171003817
(24)
After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7
Selected weights119882119904 119881out 119881noise belong to the PH
infin
domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882
119904is selected to ensure a smooth frequency
characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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Distributed Sensor Networks
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
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ReconfigurableComputing
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The Scientific World Journal 3
controller plant reference model and 2 DOF structureswith an external controller and a prefilter respectively Theinternal loopwith119875
0(119904) covers the angular velocity in relation
to input torques and disturbance rejection 119889in where 1198751015840
(119904)
is the transfer function of the rotary encoder The transferfunction of the rotary encoder is mostly treated as a systemwith pure integral behavior The RIC transfer functions are
1198750(119904) =
1198610(119904)
1198600(119904) 119875
1015840
(119904) =1198611015840
(119904)
1198601015840 (119904) 119870 (119904) =
119871119870(119904)
119877119870(119904)
119875119898(119904) =
119861119898(119904)
119860119898(119904) 119862 (119904) =
119871119862(119904)
119877119862(119904) 119865 (119904) =
119871119865(119904)
119877119865(119904)
(1)
The matrix form of the RIC system in Figure 1 is
[[[[[
[
119865minus1
0 0 119865minus1
0
minus119862 (119875119898119870 + 1) 1 0 0 119870
0 minus11987501198751015840
1 0 0
0 0 minus1 1 0
0 minus1198750
0 0 1
]]]]]
]
(
119890
119894
119910
119910
119910
)
=(
119903
119889in119889out120585
1205851015840
)
(2)
where 119903 119889in 119889out 120585 and 1205851015840 are reference signal input
disturbance output disturbance internal noise and externalnoise respectively
The closed-loop transfer function is
(
119890
119894
119910
119910
119910
) =1
1 + 1198701198750+ 119862119875
01198751015840 + 119870119862119875
11989811987501198751015840
times
[[[[[
[
119865 (1198701198750+ 1) minus119875
01198751015840
minus (1 + 1198701198750) minus (1 + 119870119875
0) 119875
01198751015840
119870
119865119862 (119875119898119870 + 1) 1 minus119862 (119875
119898119870 + 1) minus119862 (119875
119898119870 + 1) minus119870
11987501198751015840
119865119862 (119875119898119870 + 1) 119875
01198751015840
(1 + 1198701198750) minus119875
01198751015840
119862 (119875119898119870 + 1) minus119875
01198751015840
119865
11987501198751015840
119865119862 (119875119898119870 + 1) 119875
01198751015840
(1 + 1198701198750) (1 + 119870119875
0) minus119875
01198751015840
119865
1198750119865119862 (119875
119898119870 + 1) 119875
0minus119862119875
0(119875119898119870 + 1) minus119875
0119862 (119875
119898119870 + 1) (119875
01198751015840
119862 + 11987501198751015840
119862119875119898119870 + 1)
]]]]]
]
times (119903 119889in 119889out 120585 1205851015840
)119879
(3)
The RIC system in Figure 1 is slightly a modified clas-sic structure The internal loop in the modified structureembraces only the angular velocity while the classical internalloop uses for the feedback branch the same output as forthe external loop [21 22] The modified structure providesindirect decupling of the internal and external loops wherethe poles of the internal loop do not directly influencethe zeroes of the external loop Most RIC designs includestandard controller structures in internal loops [20 24]Standard controllers like PID or PI with integral behaviourimprove disturbance rejection for low-frequency disturbance119889in On the other hand internal integral behaviour causesa double-integrator effect on the external loop which sig-nificantly lowers the stability domain and generates unde-sired responses on the output disturbances and referencesignals This paper will consider an RIC structure shownin Figure 1 where the internal integral behaviour of thecontroller 119870(119904) does not directly influence the externalloop For the sake of the RIC structure simplification weassume that the prefilter function is constant and is equal to119865(119904) = 1
3 Regional Pole Placement of the RIC System
The design procedure of the RIC system can be addressed intwo steps the first step covers the internal loop controllerdesign and the second step the external loop controllerdesign Both controllers are designed with the regionalpole placement technique based on the selected centralpolynomial in the expected stable area The expected areais freely chosen by the designer and exhibits closed-loopdynamic performanceThe parameterization of controllers isapplied to ensure good disturbance rejection and robustnessproperties
Let us consider the internal loop of the system in Figure 1The closed-loop system is
(119910
119894) =
1
1 + 1198701198750
[1198750(1 + 119870119875
119898) 119875
0
1 + 119870119875119898
1](
119888
119889in)
(119910
119894) = 119878in [
1198750(1 + 119870119875
119898) 119875
0
1 + 119870119875119898
1](
119888
119889in)
(4)
4 The Scientific World Journal
Desired region of the poles
120590ts high
120590ts low
p desired locationp optimized location
j120596
p1
p3
p2
p2
p3
p1
120590
Figure 2 Settling-time objective function with settling-timeboundaries 120590
119905119904 low and 120590119905119904 high
where 119878in is the sensitivity function of the internal closed-loop The characteristic polynomial is defined as
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) (5)
Equation (5) is the standard starting point of pole place-ment design The solution of the equation is provided understrict polynomial degree conditions [37] The presentedapproach uses only the conditions when an exact solutiondoes not exist deg119877
119896le deg119860
0minus2 In this case the controller
structure does not depend on plant order like in the classicpole placement design and is arbitrarily selected by thedesigner The only way to satisfy expression (5) is regionalpole placementwith a prescribed region and deviation assess-ment as described in [32] Deviation is assessed betweenthe given polynomial 119862in(119904) and the candidate polynomialobtained with optimization The term prescribed region ofthe closed polynomial is similar to the term domain stability[29 38] The prescribed region with the selected centralpolynomial offers more leeway for further multiobjectiveoptimization of robustness and performance criteria Thepolynomial equation for regional pole placement is
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)
(6)
where 119862in(119904) is the candidate polynomial obtained with theoptimization with controller coefficient 119870(119904) The prescribedregion of the desired pole location can be derived from theproperties of the Laplace stability domain119863in = 119904 = 120590+119895120596 isin
C | 120590 isin minusR119890 119895120596 isin I119898 where 119862in isin 119863in holds true Thepossible objective functions of the prescribed region for con-trollers119870(119904) 119862(119904) are described in the following subsections
31 Settling-Time Objective Function Roughly speaking thesettling time of the closed-loop system is inversely propor-tional to the real part of dominant poles in the complexdomain119863in The objective function can be composed sothat all closed-loop poles lie in a vertical stripe boundedby parameters120590
119905119904 high and 120590119905119904 low The vertical stripe belongs
to domain119863in Parameters120590119905119904 high and 120590
119905119904 low specify theobjective function criteria of the closed-loop settling time(see Figure 2)
The proposed objective function is
1198691= min
119870
(
1minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low
1003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
forall120590119905119904 high lt R119890 119862in lt 120590119905119904 low and
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (7)
where 120590119905119904 high and 120590
119905119904 low indicate the location of the stripeboundary and 119899 indicates the number of closed-looppoles The right boundary 120590
119905119904 high of the stripe preventsthe possibility of obtaining poles with large negativereal part values Large negative real part pole values cancause a too high dynamic of the closed-loop system Ahigher dynamic of the closed-loop can render properreal-time implementation of the discrete controller on theembedded system or cause improper behaviour of controlleroutputs Improper higher dynamic of the controller outputmostly manifests itself as tempestuous responses whichindicate nonoptimal energy behaviour of the closed-loopsystem
The first term of the 1198691indicates a normalized value of
the objective if all poles lie on the left of the vertical line120590119905119904 lowThe second term assesses the deviation of the real part
pole value between the desired 119901 and the optimized 119901 polelocation where 119901 is 119901 isin 119862in119901 isin 119862in
32 Rise-TimeObjective Function Therise time of the closed-loop system is roughly proportional to the inverse of thedominant poles imaginary values andmay be upper-boundedby the parameter120596
119905119903Theobjective function can be composed
so that all closed-loop poles lie on the horizontally boundedstripe with boundsplusmn119895120596
119905119903(see Figure 3) The selected stripe
belongs to domain119863in
The Scientific World Journal 5
The proposed rise-time objective function is
1198692= min
119870
(
1minus sum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903
1003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816120596119905119903
1003816100381610038161003816
forall10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903
1003816100381610038161003816
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816I119898119862in1003816100381610038161003816 minus
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (8)
The objective function 1198692describes the desired stripe in
the complex plain where the first term indicates the positioninside the horizontal stripe and the second the imaginarydeviation between the desired 119901 and the optimized 119901 poles
33 Damping-RatioObjective Function Damping ratio is alsoan important design criterion of the closed-loop dynamicperformance and is related to the angle of the vectors betweenthe negative real and the imaginary axis of the dominantpoles [32] The damping ratio is determined with the angleboundary plusmn120593
120577(see Figure 4)
The proposed damping-ratio objective function is
1198693= min
119870
(1 minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816
forall10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )
119862in isin 119863in
(9)
The proposed objective function 1198693represents the cone
region of the desired poles All objective functions 1198691ndash1198693
represent min-optimization procedure where all objectivesare normalized on the interval [0-1]
34 The Composite Objective Function of the Dynamic Crite-ria The composite objective function represents the inter-section of objectives 119869
1ndash1198693 A graphical presentation of the
composite objective function is shown in Figure 5The composite objective function can be presented as
multiobjective criteria for DE where the expected solutionarea is equal to
Ψin = (1198691 cap 1198692 cap 1198693) andmin (dev (119901in 119901in))
forallΨin isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
(10)
The multiobjective optimization approach will be dis-cussed in the following sections
35 Internal Controller Parameterization 119870(119904) Controllerparameterization is an applied technique which ensuressatisfied properties of the closed-loop system The internal
controllers 119870 and 119862 can be parameterized so as to ensuregood input disturbance rejection and robustness propertiesInmany applications the input disturbance 119889in in positioningsystems represents the load torque and Coulombrsquos frictionwith a typical low-frequency characteristic [20] The inter-nal system can provide good elimination of low-frequencydisturbances 119889in and reference tracking 119888 if the sensitivityfunction |119878in| ≪ 1 has a proper damping effect on thefrequency span 119861 = [120596
119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under
this assumption the controller119870 can be parameterized usingdifferent polynomial structures with a known effect on thesensitivity function |119878in| Controller parameterization withintegral action 119877
119870(119904) = 119877
1015840
119870(119904)119904 has a well-known influence
on the sensitivity function at low-frequency characteristicssuch as the known simple structures PI and PID For thesake of operational safety the integral action can in manycases be approximated with the stable pole (119904 + 120575) 120575 isin R |
0 lt 120575 ≪ 1 where 119870(119904) denominator polynomial is equalto 1198771015840
119870(119904)(119904 + 120575) In this case the damping of the sensitivity
function 119878in is lowered by parameter 120575 The lowered dampingvalue of 119878in is not noticeable in real-time operation especiallyif the absolute damping value is lower than the absolutemeasurement accuracy
A minimized tracking error 119890 and good low-frequencyinput disturbance rejection 119889in can be achieved if the follow-ing holds true
lim120596rarr119861
|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)
where 119861 is the low-frequency span of the sensitivity func-tion Sensitivity function 119878in for span 119861 with controllerparameterization (119904 + 120575) is
lim120596rarr119861
1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim
120596rarr119861
(1003816100381610038161003816119878in (120596)
1003816100381610038161003816radic1205962 + 1205752)
lim120596rarr119861
|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0
(12)
Parameter 120575 determines the closed-loop tracking accu-racy and the capability of disturbance rejection with charac-teristic polynomial119862in(119904) Controller structure 119870 is parame-terized as
119877119870(119904) =
1198771015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896)
1198771015840
119870(119904) 119877
119870par (119904)
(13)
6 The Scientific World Journal
Parameter 120575119896represents an additional parameter for
further optimization of performance and robustness criteriaThe polynomial equation with parametric solutions is
1198600(119904) 119877
1015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
1198600(119904) 119877
119870(119904 120575) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
(14)
The solution of the optimization procedure is a properset of polynomial coefficients where the internal polynomial119862in(119904) belongs to the Ψin and a strong proximity condition119862in(119904) asymp 119862in(119904) holds true
36 The Design of the External Controller 119862(119904) Controller119862(119904)design technique is the same as for the internal controller119870(119904) where the external characteristic polynomial 119862out is
1198600(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119870(119904) 119877
119862(119904)
+ 1198610(119904) 119871
119870(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119862(119904)
+ 1198610(119904) 119861
1015840
(119904) 119871119862(119904) 119860
119898(119904) 119877
119870(119904)
+ 1198610(119904) 119861
1015840
(119904) 119861119898(119904) 119871
119870(119904) 119871
119862(119904) = 119862out (119904)
(15)
The possible controller parameterization 119862(119904) is
Γ (119904) 1198771015840
119862(119904)
119901
prod
119896=1
(119904 + 120578119896) + Υ (119904) 119871
119862(119904) = 119862out (119904)
Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871
119862(119904) = 119862out (119904)
(16)
where polynomials Γ(119904) and Υ(119904) are
Γ (119904) = 1198601015840
(119904) 119860119898(119904) (119860
0(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904))
Υ (119904) = 1198610(119904) 119861
1015840
(119904) (119860119898(119904) 119877
119870(119904) + 119861
119898(119904) 119871
119870(119904))
(17)
Coefficient 120578 can also be used as an additional parameterfor criteria optimization in the same sense as parameter 120575
The controller119862(119904) is the proper solution of the optimiza-tion problem for external loop if the following conditions aresatisfied
Ψout = (119869out 1 cap 119869out 2 cap 119869out 3) andmin (dev (119901out 119901out))
forallΨout isin 119863out
forall119901out = 119901out isin 119862out | 119862out isin 119863out
forall119901out = 119901out isin 119862in | 119862out isin 119863out
(18)
Objective functions 119869out 1ndash119869out 3 have the same propertiesand meanings as objective functions 119869
1ndash1198693
4 Nonparametric Uncertainty andRobustness Criterion of RIC
The robustness of the RIC system is assessed using uncer-tainty models Δ119875 and a stable proper input weight 1198811015840noise
120590
p1
p3
p2
p2
p3
p1
j120596
Frequency boundj120596tr
minusj120596tr
Figure 3 Rise-time objective function with a horizontal bound-aryplusmn119895120596
119905119903
120590
p1
p3
p2
p2
p3
p1
j120596Damping line
120593120577
minus120593120577
120577
120593p1
120593p2
Figure 4 Damping-ratio objective function with parameter plusmn120593120577
The uncertainty models describe model deviation within thegiven frequency space and are represented with a nominalmodel and stable uncertainty weights Δ119882 [30 39] Weights1198811015840
noise representmeasured noise spectrumof the signal11990810158401 For
RIC design we assume that the reference model is 119875119898asymp 119875
0isin
119875119867infin The internal loop of the RIC with uncertainty models
is shown in Figure 6The robustness assessment of the internal loop should
be considered with multiplicative and inverse uncertaintyThe multiplicative uncertainty represents uncertainties bylower frequencies while the inverse uncertainty representsuncertainties by higher frequencies [39]
The Scientific World Journal 7
σ
Stable area
Intersection
120590ts low120590ts high j120596
p2
p1
p3
120593120577
120577
j120596tr
minusj120596tr
Ψ
Figure 5 Intersection of objectives 1198691ndash1198693
c iPm(s) K(s 120590)
K(s 120590)
din
ΔP(s)y
120585998400w998400
1V998400noise(s)minus
Figure 6 Internal loop with uncertainty models Δ119875
Let us consider the multiplicative uncertainty modelΔ119875
119872= 119875
0(1 +Δ119882
119872)with nominal plant 119875
0 The closed-loop
characteristic using the multiplicative uncertainty modelΔ119875
119872is
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119872))minus1
times [1198750(1 + Δ119882
119872) (1 + 119870119875
119898) 119875
0(1 + Δ119882
119872)
1 + 119870119875119898
1](
119888
119889in)
(19)
Robust stability for the multiplicative uncertainty is pre-served if the following holds true
10038171003817100381710038171003817100381710038171003817
Δ119882119872
1198701198750
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin
1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin
lt 1
(20)
where 119879in is a complementary sensitivity function of theinternal loop
+e c y
z
Ws(s)
C(s 120578) TP119898 in(s)y
P998400(s)
++minus
w1
dout
Vout(s)
120585Vnoise(s)
w2
Figure 7 External loop optimization structure
The internal loop with inverse uncertainty Δ119875119868
=
1198750(1 + Δ119882
119868)minus1 is defined as
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119868)minus1
)minus1
times [1198750(1 + Δ119882
119868)minus1
(1 + 119870119875119898) 119875
0(1 + Δ119882
119868)minus1
1 + 119870119875119898
1]
times (119888
119889in)
(21)
The robustness for inverse models is satisfied if the followingholds true
10038171003817100381710038171003817100381710038171003817
Δ119882119868
1
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)
where 119878in is the sensitivity function of the internal loopThe noise suppression 120585
1015840 of the internal loop can beassessed with the following criterion
1198791199081015840
1
= (1 + 119870 (120575) Δ119875)minus1
1198811015840
noise (23)
The robustness criterion of noise suppression with theuncertainty model is
1198791199081015840
1
= min119870
10038171003817100381710038171003817100381710038171003817
[119882119872
119882119868] [119879in 0
0 119878in]119881
1015840
noise
10038171003817100381710038171003817100381710038171003817
(24)
After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7
Selected weights119882119904 119881out 119881noise belong to the PH
infin
domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882
119904is selected to ensure a smooth frequency
characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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ArtificialNeural Systems
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
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4 The Scientific World Journal
Desired region of the poles
120590ts high
120590ts low
p desired locationp optimized location
j120596
p1
p3
p2
p2
p3
p1
120590
Figure 2 Settling-time objective function with settling-timeboundaries 120590
119905119904 low and 120590119905119904 high
where 119878in is the sensitivity function of the internal closed-loop The characteristic polynomial is defined as
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) (5)
Equation (5) is the standard starting point of pole place-ment design The solution of the equation is provided understrict polynomial degree conditions [37] The presentedapproach uses only the conditions when an exact solutiondoes not exist deg119877
119896le deg119860
0minus2 In this case the controller
structure does not depend on plant order like in the classicpole placement design and is arbitrarily selected by thedesigner The only way to satisfy expression (5) is regionalpole placementwith a prescribed region and deviation assess-ment as described in [32] Deviation is assessed betweenthe given polynomial 119862in(119904) and the candidate polynomialobtained with optimization The term prescribed region ofthe closed polynomial is similar to the term domain stability[29 38] The prescribed region with the selected centralpolynomial offers more leeway for further multiobjectiveoptimization of robustness and performance criteria Thepolynomial equation for regional pole placement is
1198600(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904) = 119862in (119904) 119862in (119904) asymp 119862in (119904)
(6)
where 119862in(119904) is the candidate polynomial obtained with theoptimization with controller coefficient 119870(119904) The prescribedregion of the desired pole location can be derived from theproperties of the Laplace stability domain119863in = 119904 = 120590+119895120596 isin
C | 120590 isin minusR119890 119895120596 isin I119898 where 119862in isin 119863in holds true Thepossible objective functions of the prescribed region for con-trollers119870(119904) 119862(119904) are described in the following subsections
31 Settling-Time Objective Function Roughly speaking thesettling time of the closed-loop system is inversely propor-tional to the real part of dominant poles in the complexdomain119863in The objective function can be composed sothat all closed-loop poles lie in a vertical stripe boundedby parameters120590
119905119904 high and 120590119905119904 low The vertical stripe belongs
to domain119863in Parameters120590119905119904 high and 120590
119905119904 low specify theobjective function criteria of the closed-loop settling time(see Figure 2)
The proposed objective function is
1198691= min
119870
(
1minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816minus1003816100381610038161003816120590119905119904 low
1003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
forall120590119905119904 high lt R119890 119862in lt 120590119905119904 low and
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816R119890 119862in1003816100381610038161003816 minus
10038161003816100381610038161003816R119890 119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816R119890 119862in1003816100381610038161003816 R119890
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (7)
where 120590119905119904 high and 120590
119905119904 low indicate the location of the stripeboundary and 119899 indicates the number of closed-looppoles The right boundary 120590
119905119904 high of the stripe preventsthe possibility of obtaining poles with large negativereal part values Large negative real part pole values cancause a too high dynamic of the closed-loop system Ahigher dynamic of the closed-loop can render properreal-time implementation of the discrete controller on theembedded system or cause improper behaviour of controlleroutputs Improper higher dynamic of the controller outputmostly manifests itself as tempestuous responses whichindicate nonoptimal energy behaviour of the closed-loopsystem
The first term of the 1198691indicates a normalized value of
the objective if all poles lie on the left of the vertical line120590119905119904 lowThe second term assesses the deviation of the real part
pole value between the desired 119901 and the optimized 119901 polelocation where 119901 is 119901 isin 119862in119901 isin 119862in
32 Rise-TimeObjective Function Therise time of the closed-loop system is roughly proportional to the inverse of thedominant poles imaginary values andmay be upper-boundedby the parameter120596
119905119903Theobjective function can be composed
so that all closed-loop poles lie on the horizontally boundedstripe with boundsplusmn119895120596
119905119903(see Figure 3) The selected stripe
belongs to domain119863in
The Scientific World Journal 5
The proposed rise-time objective function is
1198692= min
119870
(
1minus sum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903
1003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816120596119905119903
1003816100381610038161003816
forall10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903
1003816100381610038161003816
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816I119898119862in1003816100381610038161003816 minus
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (8)
The objective function 1198692describes the desired stripe in
the complex plain where the first term indicates the positioninside the horizontal stripe and the second the imaginarydeviation between the desired 119901 and the optimized 119901 poles
33 Damping-RatioObjective Function Damping ratio is alsoan important design criterion of the closed-loop dynamicperformance and is related to the angle of the vectors betweenthe negative real and the imaginary axis of the dominantpoles [32] The damping ratio is determined with the angleboundary plusmn120593
120577(see Figure 4)
The proposed damping-ratio objective function is
1198693= min
119870
(1 minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816
forall10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )
119862in isin 119863in
(9)
The proposed objective function 1198693represents the cone
region of the desired poles All objective functions 1198691ndash1198693
represent min-optimization procedure where all objectivesare normalized on the interval [0-1]
34 The Composite Objective Function of the Dynamic Crite-ria The composite objective function represents the inter-section of objectives 119869
1ndash1198693 A graphical presentation of the
composite objective function is shown in Figure 5The composite objective function can be presented as
multiobjective criteria for DE where the expected solutionarea is equal to
Ψin = (1198691 cap 1198692 cap 1198693) andmin (dev (119901in 119901in))
forallΨin isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
(10)
The multiobjective optimization approach will be dis-cussed in the following sections
35 Internal Controller Parameterization 119870(119904) Controllerparameterization is an applied technique which ensuressatisfied properties of the closed-loop system The internal
controllers 119870 and 119862 can be parameterized so as to ensuregood input disturbance rejection and robustness propertiesInmany applications the input disturbance 119889in in positioningsystems represents the load torque and Coulombrsquos frictionwith a typical low-frequency characteristic [20] The inter-nal system can provide good elimination of low-frequencydisturbances 119889in and reference tracking 119888 if the sensitivityfunction |119878in| ≪ 1 has a proper damping effect on thefrequency span 119861 = [120596
119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under
this assumption the controller119870 can be parameterized usingdifferent polynomial structures with a known effect on thesensitivity function |119878in| Controller parameterization withintegral action 119877
119870(119904) = 119877
1015840
119870(119904)119904 has a well-known influence
on the sensitivity function at low-frequency characteristicssuch as the known simple structures PI and PID For thesake of operational safety the integral action can in manycases be approximated with the stable pole (119904 + 120575) 120575 isin R |
0 lt 120575 ≪ 1 where 119870(119904) denominator polynomial is equalto 1198771015840
119870(119904)(119904 + 120575) In this case the damping of the sensitivity
function 119878in is lowered by parameter 120575 The lowered dampingvalue of 119878in is not noticeable in real-time operation especiallyif the absolute damping value is lower than the absolutemeasurement accuracy
A minimized tracking error 119890 and good low-frequencyinput disturbance rejection 119889in can be achieved if the follow-ing holds true
lim120596rarr119861
|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)
where 119861 is the low-frequency span of the sensitivity func-tion Sensitivity function 119878in for span 119861 with controllerparameterization (119904 + 120575) is
lim120596rarr119861
1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim
120596rarr119861
(1003816100381610038161003816119878in (120596)
1003816100381610038161003816radic1205962 + 1205752)
lim120596rarr119861
|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0
(12)
Parameter 120575 determines the closed-loop tracking accu-racy and the capability of disturbance rejection with charac-teristic polynomial119862in(119904) Controller structure 119870 is parame-terized as
119877119870(119904) =
1198771015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896)
1198771015840
119870(119904) 119877
119870par (119904)
(13)
6 The Scientific World Journal
Parameter 120575119896represents an additional parameter for
further optimization of performance and robustness criteriaThe polynomial equation with parametric solutions is
1198600(119904) 119877
1015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
1198600(119904) 119877
119870(119904 120575) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
(14)
The solution of the optimization procedure is a properset of polynomial coefficients where the internal polynomial119862in(119904) belongs to the Ψin and a strong proximity condition119862in(119904) asymp 119862in(119904) holds true
36 The Design of the External Controller 119862(119904) Controller119862(119904)design technique is the same as for the internal controller119870(119904) where the external characteristic polynomial 119862out is
1198600(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119870(119904) 119877
119862(119904)
+ 1198610(119904) 119871
119870(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119862(119904)
+ 1198610(119904) 119861
1015840
(119904) 119871119862(119904) 119860
119898(119904) 119877
119870(119904)
+ 1198610(119904) 119861
1015840
(119904) 119861119898(119904) 119871
119870(119904) 119871
119862(119904) = 119862out (119904)
(15)
The possible controller parameterization 119862(119904) is
Γ (119904) 1198771015840
119862(119904)
119901
prod
119896=1
(119904 + 120578119896) + Υ (119904) 119871
119862(119904) = 119862out (119904)
Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871
119862(119904) = 119862out (119904)
(16)
where polynomials Γ(119904) and Υ(119904) are
Γ (119904) = 1198601015840
(119904) 119860119898(119904) (119860
0(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904))
Υ (119904) = 1198610(119904) 119861
1015840
(119904) (119860119898(119904) 119877
119870(119904) + 119861
119898(119904) 119871
119870(119904))
(17)
Coefficient 120578 can also be used as an additional parameterfor criteria optimization in the same sense as parameter 120575
The controller119862(119904) is the proper solution of the optimiza-tion problem for external loop if the following conditions aresatisfied
Ψout = (119869out 1 cap 119869out 2 cap 119869out 3) andmin (dev (119901out 119901out))
forallΨout isin 119863out
forall119901out = 119901out isin 119862out | 119862out isin 119863out
forall119901out = 119901out isin 119862in | 119862out isin 119863out
(18)
Objective functions 119869out 1ndash119869out 3 have the same propertiesand meanings as objective functions 119869
1ndash1198693
4 Nonparametric Uncertainty andRobustness Criterion of RIC
The robustness of the RIC system is assessed using uncer-tainty models Δ119875 and a stable proper input weight 1198811015840noise
120590
p1
p3
p2
p2
p3
p1
j120596
Frequency boundj120596tr
minusj120596tr
Figure 3 Rise-time objective function with a horizontal bound-aryplusmn119895120596
119905119903
120590
p1
p3
p2
p2
p3
p1
j120596Damping line
120593120577
minus120593120577
120577
120593p1
120593p2
Figure 4 Damping-ratio objective function with parameter plusmn120593120577
The uncertainty models describe model deviation within thegiven frequency space and are represented with a nominalmodel and stable uncertainty weights Δ119882 [30 39] Weights1198811015840
noise representmeasured noise spectrumof the signal11990810158401 For
RIC design we assume that the reference model is 119875119898asymp 119875
0isin
119875119867infin The internal loop of the RIC with uncertainty models
is shown in Figure 6The robustness assessment of the internal loop should
be considered with multiplicative and inverse uncertaintyThe multiplicative uncertainty represents uncertainties bylower frequencies while the inverse uncertainty representsuncertainties by higher frequencies [39]
The Scientific World Journal 7
σ
Stable area
Intersection
120590ts low120590ts high j120596
p2
p1
p3
120593120577
120577
j120596tr
minusj120596tr
Ψ
Figure 5 Intersection of objectives 1198691ndash1198693
c iPm(s) K(s 120590)
K(s 120590)
din
ΔP(s)y
120585998400w998400
1V998400noise(s)minus
Figure 6 Internal loop with uncertainty models Δ119875
Let us consider the multiplicative uncertainty modelΔ119875
119872= 119875
0(1 +Δ119882
119872)with nominal plant 119875
0 The closed-loop
characteristic using the multiplicative uncertainty modelΔ119875
119872is
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119872))minus1
times [1198750(1 + Δ119882
119872) (1 + 119870119875
119898) 119875
0(1 + Δ119882
119872)
1 + 119870119875119898
1](
119888
119889in)
(19)
Robust stability for the multiplicative uncertainty is pre-served if the following holds true
10038171003817100381710038171003817100381710038171003817
Δ119882119872
1198701198750
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin
1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin
lt 1
(20)
where 119879in is a complementary sensitivity function of theinternal loop
+e c y
z
Ws(s)
C(s 120578) TP119898 in(s)y
P998400(s)
++minus
w1
dout
Vout(s)
120585Vnoise(s)
w2
Figure 7 External loop optimization structure
The internal loop with inverse uncertainty Δ119875119868
=
1198750(1 + Δ119882
119868)minus1 is defined as
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119868)minus1
)minus1
times [1198750(1 + Δ119882
119868)minus1
(1 + 119870119875119898) 119875
0(1 + Δ119882
119868)minus1
1 + 119870119875119898
1]
times (119888
119889in)
(21)
The robustness for inverse models is satisfied if the followingholds true
10038171003817100381710038171003817100381710038171003817
Δ119882119868
1
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)
where 119878in is the sensitivity function of the internal loopThe noise suppression 120585
1015840 of the internal loop can beassessed with the following criterion
1198791199081015840
1
= (1 + 119870 (120575) Δ119875)minus1
1198811015840
noise (23)
The robustness criterion of noise suppression with theuncertainty model is
1198791199081015840
1
= min119870
10038171003817100381710038171003817100381710038171003817
[119882119872
119882119868] [119879in 0
0 119878in]119881
1015840
noise
10038171003817100381710038171003817100381710038171003817
(24)
After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7
Selected weights119882119904 119881out 119881noise belong to the PH
infin
domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882
119904is selected to ensure a smooth frequency
characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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ArtificialNeural Systems
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
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The Scientific World Journal 5
The proposed rise-time objective function is
1198692= min
119870
(
1minus sum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816minus1003816100381610038161003816120596119905119903
1003816100381610038161003816
100381610038161003816100381610038161003816100381610038161003816120596119905119903
1003816100381610038161003816
forall10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816lt1003816100381610038161003816120596119905119903
1003816100381610038161003816
sum
119899
10038161003816100381610038161003816
1003816100381610038161003816I119898119862in1003816100381610038161003816 minus
10038161003816100381610038161003816I119898119862in
10038161003816100381610038161003816
10038161003816100381610038161003816
max (1003816100381610038161003816I119898119862in1003816100381610038161003816 I119898
10038161003816100381610038161003816119862in
10038161003816100381610038161003816)
) 119862in isin 119863in (8)
The objective function 1198692describes the desired stripe in
the complex plain where the first term indicates the positioninside the horizontal stripe and the second the imaginarydeviation between the desired 119901 and the optimized 119901 poles
33 Damping-RatioObjective Function Damping ratio is alsoan important design criterion of the closed-loop dynamicperformance and is related to the angle of the vectors betweenthe negative real and the imaginary axis of the dominantpoles [32] The damping ratio is determined with the angleboundary plusmn120593
120577(see Figure 4)
The proposed damping-ratio objective function is
1198693= min
119870
(1 minussum
119899
10038161003816100381610038161003816
10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816minus10038161003816100381610038161205931205891003816100381610038161003816
1003816100381610038161003816100381610038161003816100381610038161205931205891003816100381610038161003816
forall10038161003816100381610038161003816120593 119862in
10038161003816100381610038161003816lt10038161003816100381610038161205931205891003816100381610038161003816 )
119862in isin 119863in
(9)
The proposed objective function 1198693represents the cone
region of the desired poles All objective functions 1198691ndash1198693
represent min-optimization procedure where all objectivesare normalized on the interval [0-1]
34 The Composite Objective Function of the Dynamic Crite-ria The composite objective function represents the inter-section of objectives 119869
1ndash1198693 A graphical presentation of the
composite objective function is shown in Figure 5The composite objective function can be presented as
multiobjective criteria for DE where the expected solutionarea is equal to
Ψin = (1198691 cap 1198692 cap 1198693) andmin (dev (119901in 119901in))
forallΨin isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
forall119901in = 119901in isin 119862in | 119862in isin 119863in
(10)
The multiobjective optimization approach will be dis-cussed in the following sections
35 Internal Controller Parameterization 119870(119904) Controllerparameterization is an applied technique which ensuressatisfied properties of the closed-loop system The internal
controllers 119870 and 119862 can be parameterized so as to ensuregood input disturbance rejection and robustness propertiesInmany applications the input disturbance 119889in in positioningsystems represents the load torque and Coulombrsquos frictionwith a typical low-frequency characteristic [20] The inter-nal system can provide good elimination of low-frequencydisturbances 119889in and reference tracking 119888 if the sensitivityfunction |119878in| ≪ 1 has a proper damping effect on thefrequency span 119861 = [120596
119897 120596ℎ] 119888(119861) and 119889in(119861) ≫ 1 Under
this assumption the controller119870 can be parameterized usingdifferent polynomial structures with a known effect on thesensitivity function |119878in| Controller parameterization withintegral action 119877
119870(119904) = 119877
1015840
119870(119904)119904 has a well-known influence
on the sensitivity function at low-frequency characteristicssuch as the known simple structures PI and PID For thesake of operational safety the integral action can in manycases be approximated with the stable pole (119904 + 120575) 120575 isin R |
0 lt 120575 ≪ 1 where 119870(119904) denominator polynomial is equalto 1198771015840
119870(119904)(119904 + 120575) In this case the damping of the sensitivity
function 119878in is lowered by parameter 120575 The lowered dampingvalue of 119878in is not noticeable in real-time operation especiallyif the absolute damping value is lower than the absolutemeasurement accuracy
A minimized tracking error 119890 and good low-frequencyinput disturbance rejection 119889in can be achieved if the follow-ing holds true
lim120596rarr119861
|119870 (119861)| ≫ 1 119861 = 119861 isin R | 0 lt 119861 le 120596low (11)
where 119861 is the low-frequency span of the sensitivity func-tion Sensitivity function 119878in for span 119861 with controllerparameterization (119904 + 120575) is
lim120596rarr119861
1003816100381610038161003816119878in (120575 120596)1003816100381610038161003816 = lim
120596rarr119861
(1003816100381610038161003816119878in (120596)
1003816100381610038161003816radic1205962 + 1205752)
lim120596rarr119861
|119878 (119861)| asymp 120575 997904rArr |119878 (120575 119861)| asymp 0
(12)
Parameter 120575 determines the closed-loop tracking accu-racy and the capability of disturbance rejection with charac-teristic polynomial119862in(119904) Controller structure 119870 is parame-terized as
119877119870(119904) =
1198771015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896)
1198771015840
119870(119904) 119877
119870par (119904)
(13)
6 The Scientific World Journal
Parameter 120575119896represents an additional parameter for
further optimization of performance and robustness criteriaThe polynomial equation with parametric solutions is
1198600(119904) 119877
1015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
1198600(119904) 119877
119870(119904 120575) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
(14)
The solution of the optimization procedure is a properset of polynomial coefficients where the internal polynomial119862in(119904) belongs to the Ψin and a strong proximity condition119862in(119904) asymp 119862in(119904) holds true
36 The Design of the External Controller 119862(119904) Controller119862(119904)design technique is the same as for the internal controller119870(119904) where the external characteristic polynomial 119862out is
1198600(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119870(119904) 119877
119862(119904)
+ 1198610(119904) 119871
119870(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119862(119904)
+ 1198610(119904) 119861
1015840
(119904) 119871119862(119904) 119860
119898(119904) 119877
119870(119904)
+ 1198610(119904) 119861
1015840
(119904) 119861119898(119904) 119871
119870(119904) 119871
119862(119904) = 119862out (119904)
(15)
The possible controller parameterization 119862(119904) is
Γ (119904) 1198771015840
119862(119904)
119901
prod
119896=1
(119904 + 120578119896) + Υ (119904) 119871
119862(119904) = 119862out (119904)
Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871
119862(119904) = 119862out (119904)
(16)
where polynomials Γ(119904) and Υ(119904) are
Γ (119904) = 1198601015840
(119904) 119860119898(119904) (119860
0(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904))
Υ (119904) = 1198610(119904) 119861
1015840
(119904) (119860119898(119904) 119877
119870(119904) + 119861
119898(119904) 119871
119870(119904))
(17)
Coefficient 120578 can also be used as an additional parameterfor criteria optimization in the same sense as parameter 120575
The controller119862(119904) is the proper solution of the optimiza-tion problem for external loop if the following conditions aresatisfied
Ψout = (119869out 1 cap 119869out 2 cap 119869out 3) andmin (dev (119901out 119901out))
forallΨout isin 119863out
forall119901out = 119901out isin 119862out | 119862out isin 119863out
forall119901out = 119901out isin 119862in | 119862out isin 119863out
(18)
Objective functions 119869out 1ndash119869out 3 have the same propertiesand meanings as objective functions 119869
1ndash1198693
4 Nonparametric Uncertainty andRobustness Criterion of RIC
The robustness of the RIC system is assessed using uncer-tainty models Δ119875 and a stable proper input weight 1198811015840noise
120590
p1
p3
p2
p2
p3
p1
j120596
Frequency boundj120596tr
minusj120596tr
Figure 3 Rise-time objective function with a horizontal bound-aryplusmn119895120596
119905119903
120590
p1
p3
p2
p2
p3
p1
j120596Damping line
120593120577
minus120593120577
120577
120593p1
120593p2
Figure 4 Damping-ratio objective function with parameter plusmn120593120577
The uncertainty models describe model deviation within thegiven frequency space and are represented with a nominalmodel and stable uncertainty weights Δ119882 [30 39] Weights1198811015840
noise representmeasured noise spectrumof the signal11990810158401 For
RIC design we assume that the reference model is 119875119898asymp 119875
0isin
119875119867infin The internal loop of the RIC with uncertainty models
is shown in Figure 6The robustness assessment of the internal loop should
be considered with multiplicative and inverse uncertaintyThe multiplicative uncertainty represents uncertainties bylower frequencies while the inverse uncertainty representsuncertainties by higher frequencies [39]
The Scientific World Journal 7
σ
Stable area
Intersection
120590ts low120590ts high j120596
p2
p1
p3
120593120577
120577
j120596tr
minusj120596tr
Ψ
Figure 5 Intersection of objectives 1198691ndash1198693
c iPm(s) K(s 120590)
K(s 120590)
din
ΔP(s)y
120585998400w998400
1V998400noise(s)minus
Figure 6 Internal loop with uncertainty models Δ119875
Let us consider the multiplicative uncertainty modelΔ119875
119872= 119875
0(1 +Δ119882
119872)with nominal plant 119875
0 The closed-loop
characteristic using the multiplicative uncertainty modelΔ119875
119872is
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119872))minus1
times [1198750(1 + Δ119882
119872) (1 + 119870119875
119898) 119875
0(1 + Δ119882
119872)
1 + 119870119875119898
1](
119888
119889in)
(19)
Robust stability for the multiplicative uncertainty is pre-served if the following holds true
10038171003817100381710038171003817100381710038171003817
Δ119882119872
1198701198750
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin
1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin
lt 1
(20)
where 119879in is a complementary sensitivity function of theinternal loop
+e c y
z
Ws(s)
C(s 120578) TP119898 in(s)y
P998400(s)
++minus
w1
dout
Vout(s)
120585Vnoise(s)
w2
Figure 7 External loop optimization structure
The internal loop with inverse uncertainty Δ119875119868
=
1198750(1 + Δ119882
119868)minus1 is defined as
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119868)minus1
)minus1
times [1198750(1 + Δ119882
119868)minus1
(1 + 119870119875119898) 119875
0(1 + Δ119882
119868)minus1
1 + 119870119875119898
1]
times (119888
119889in)
(21)
The robustness for inverse models is satisfied if the followingholds true
10038171003817100381710038171003817100381710038171003817
Δ119882119868
1
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)
where 119878in is the sensitivity function of the internal loopThe noise suppression 120585
1015840 of the internal loop can beassessed with the following criterion
1198791199081015840
1
= (1 + 119870 (120575) Δ119875)minus1
1198811015840
noise (23)
The robustness criterion of noise suppression with theuncertainty model is
1198791199081015840
1
= min119870
10038171003817100381710038171003817100381710038171003817
[119882119872
119882119868] [119879in 0
0 119878in]119881
1015840
noise
10038171003817100381710038171003817100381710038171003817
(24)
After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7
Selected weights119882119904 119881out 119881noise belong to the PH
infin
domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882
119904is selected to ensure a smooth frequency
characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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Applied Computational Intelligence and Soft Computing
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Electrical and Computer Engineering
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Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
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ArtificialNeural Systems
Advances in
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RoboticsJournal of
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Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
6 The Scientific World Journal
Parameter 120575119896represents an additional parameter for
further optimization of performance and robustness criteriaThe polynomial equation with parametric solutions is
1198600(119904) 119877
1015840
119870(119904)
119901
prod
119896=1
(119904 + 120575119896) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
1198600(119904) 119877
119870(119904 120575) + 119861
0(119904) 119871
119870(119904) = 119862in (119904)
(14)
The solution of the optimization procedure is a properset of polynomial coefficients where the internal polynomial119862in(119904) belongs to the Ψin and a strong proximity condition119862in(119904) asymp 119862in(119904) holds true
36 The Design of the External Controller 119862(119904) Controller119862(119904)design technique is the same as for the internal controller119870(119904) where the external characteristic polynomial 119862out is
1198600(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119870(119904) 119877
119862(119904)
+ 1198610(119904) 119871
119870(119904) 119860
1015840
(119904) 119860119898(119904) 119877
119862(119904)
+ 1198610(119904) 119861
1015840
(119904) 119871119862(119904) 119860
119898(119904) 119877
119870(119904)
+ 1198610(119904) 119861
1015840
(119904) 119861119898(119904) 119871
119870(119904) 119871
119862(119904) = 119862out (119904)
(15)
The possible controller parameterization 119862(119904) is
Γ (119904) 1198771015840
119862(119904)
119901
prod
119896=1
(119904 + 120578119896) + Υ (119904) 119871
119862(119904) = 119862out (119904)
Γ (119904) 119877119862(119904 120578) + Υ (119904) 119871
119862(119904) = 119862out (119904)
(16)
where polynomials Γ(119904) and Υ(119904) are
Γ (119904) = 1198601015840
(119904) 119860119898(119904) (119860
0(119904) 119877
119870(119904) + 119861
0(119904) 119871
119870(119904))
Υ (119904) = 1198610(119904) 119861
1015840
(119904) (119860119898(119904) 119877
119870(119904) + 119861
119898(119904) 119871
119870(119904))
(17)
Coefficient 120578 can also be used as an additional parameterfor criteria optimization in the same sense as parameter 120575
The controller119862(119904) is the proper solution of the optimiza-tion problem for external loop if the following conditions aresatisfied
Ψout = (119869out 1 cap 119869out 2 cap 119869out 3) andmin (dev (119901out 119901out))
forallΨout isin 119863out
forall119901out = 119901out isin 119862out | 119862out isin 119863out
forall119901out = 119901out isin 119862in | 119862out isin 119863out
(18)
Objective functions 119869out 1ndash119869out 3 have the same propertiesand meanings as objective functions 119869
1ndash1198693
4 Nonparametric Uncertainty andRobustness Criterion of RIC
The robustness of the RIC system is assessed using uncer-tainty models Δ119875 and a stable proper input weight 1198811015840noise
120590
p1
p3
p2
p2
p3
p1
j120596
Frequency boundj120596tr
minusj120596tr
Figure 3 Rise-time objective function with a horizontal bound-aryplusmn119895120596
119905119903
120590
p1
p3
p2
p2
p3
p1
j120596Damping line
120593120577
minus120593120577
120577
120593p1
120593p2
Figure 4 Damping-ratio objective function with parameter plusmn120593120577
The uncertainty models describe model deviation within thegiven frequency space and are represented with a nominalmodel and stable uncertainty weights Δ119882 [30 39] Weights1198811015840
noise representmeasured noise spectrumof the signal11990810158401 For
RIC design we assume that the reference model is 119875119898asymp 119875
0isin
119875119867infin The internal loop of the RIC with uncertainty models
is shown in Figure 6The robustness assessment of the internal loop should
be considered with multiplicative and inverse uncertaintyThe multiplicative uncertainty represents uncertainties bylower frequencies while the inverse uncertainty representsuncertainties by higher frequencies [39]
The Scientific World Journal 7
σ
Stable area
Intersection
120590ts low120590ts high j120596
p2
p1
p3
120593120577
120577
j120596tr
minusj120596tr
Ψ
Figure 5 Intersection of objectives 1198691ndash1198693
c iPm(s) K(s 120590)
K(s 120590)
din
ΔP(s)y
120585998400w998400
1V998400noise(s)minus
Figure 6 Internal loop with uncertainty models Δ119875
Let us consider the multiplicative uncertainty modelΔ119875
119872= 119875
0(1 +Δ119882
119872)with nominal plant 119875
0 The closed-loop
characteristic using the multiplicative uncertainty modelΔ119875
119872is
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119872))minus1
times [1198750(1 + Δ119882
119872) (1 + 119870119875
119898) 119875
0(1 + Δ119882
119872)
1 + 119870119875119898
1](
119888
119889in)
(19)
Robust stability for the multiplicative uncertainty is pre-served if the following holds true
10038171003817100381710038171003817100381710038171003817
Δ119882119872
1198701198750
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin
1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin
lt 1
(20)
where 119879in is a complementary sensitivity function of theinternal loop
+e c y
z
Ws(s)
C(s 120578) TP119898 in(s)y
P998400(s)
++minus
w1
dout
Vout(s)
120585Vnoise(s)
w2
Figure 7 External loop optimization structure
The internal loop with inverse uncertainty Δ119875119868
=
1198750(1 + Δ119882
119868)minus1 is defined as
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119868)minus1
)minus1
times [1198750(1 + Δ119882
119868)minus1
(1 + 119870119875119898) 119875
0(1 + Δ119882
119868)minus1
1 + 119870119875119898
1]
times (119888
119889in)
(21)
The robustness for inverse models is satisfied if the followingholds true
10038171003817100381710038171003817100381710038171003817
Δ119882119868
1
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)
where 119878in is the sensitivity function of the internal loopThe noise suppression 120585
1015840 of the internal loop can beassessed with the following criterion
1198791199081015840
1
= (1 + 119870 (120575) Δ119875)minus1
1198811015840
noise (23)
The robustness criterion of noise suppression with theuncertainty model is
1198791199081015840
1
= min119870
10038171003817100381710038171003817100381710038171003817
[119882119872
119882119868] [119879in 0
0 119878in]119881
1015840
noise
10038171003817100381710038171003817100381710038171003817
(24)
After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7
Selected weights119882119904 119881out 119881noise belong to the PH
infin
domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882
119904is selected to ensure a smooth frequency
characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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The Scientific World Journal 7
σ
Stable area
Intersection
120590ts low120590ts high j120596
p2
p1
p3
120593120577
120577
j120596tr
minusj120596tr
Ψ
Figure 5 Intersection of objectives 1198691ndash1198693
c iPm(s) K(s 120590)
K(s 120590)
din
ΔP(s)y
120585998400w998400
1V998400noise(s)minus
Figure 6 Internal loop with uncertainty models Δ119875
Let us consider the multiplicative uncertainty modelΔ119875
119872= 119875
0(1 +Δ119882
119872)with nominal plant 119875
0 The closed-loop
characteristic using the multiplicative uncertainty modelΔ119875
119872is
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119872))minus1
times [1198750(1 + Δ119882
119872) (1 + 119870119875
119898) 119875
0(1 + Δ119882
119872)
1 + 119870119875119898
1](
119888
119889in)
(19)
Robust stability for the multiplicative uncertainty is pre-served if the following holds true
10038171003817100381710038171003817100381710038171003817
Δ119882119872
1198701198750
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin
1003817100381710038171003817Δ119882119872119879in1003817100381710038171003817infin
lt 1
(20)
where 119879in is a complementary sensitivity function of theinternal loop
+e c y
z
Ws(s)
C(s 120578) TP119898 in(s)y
P998400(s)
++minus
w1
dout
Vout(s)
120585Vnoise(s)
w2
Figure 7 External loop optimization structure
The internal loop with inverse uncertainty Δ119875119868
=
1198750(1 + Δ119882
119868)minus1 is defined as
(
119910
) = (1 + 119870119875
0(1 + Δ119882
119868)minus1
)minus1
times [1198750(1 + Δ119882
119868)minus1
(1 + 119870119875119898) 119875
0(1 + Δ119882
119868)minus1
1 + 119870119875119898
1]
times (119888
119889in)
(21)
The robustness for inverse models is satisfied if the followingholds true
10038171003817100381710038171003817100381710038171003817
Δ119882119868
1
1 + 1198701198750
10038171003817100381710038171003817100381710038171003817infin
lt 1
forall |Δ (120596)| lt 1 120596 = 120596 isin R | 0 le 120596 lt infin (22)
where 119878in is the sensitivity function of the internal loopThe noise suppression 120585
1015840 of the internal loop can beassessed with the following criterion
1198791199081015840
1
= (1 + 119870 (120575) Δ119875)minus1
1198811015840
noise (23)
The robustness criterion of noise suppression with theuncertainty model is
1198791199081015840
1
= min119870
10038171003817100381710038171003817100381710038171003817
[119882119872
119882119868] [119879in 0
0 119878in]119881
1015840
noise
10038171003817100381710038171003817100381710038171003817
(24)
After derivation of robust stability conditions of theinternal loop robust stability conditions for the externalloop can be presented The purpose of external controllerdesign is to ensure overall performance characteristics likeproper reference tracking 119903 output disturbance rejection119889out and good measurement noise cancellation 120585 Theoptimization structure of the external loop is shown inFigure 7
Selected weights119882119904 119881out 119881noise belong to the PH
infin
domain Input weights119881out119881noise represent the input spectralcharacteristics of disturbance 119889out and noise 120585 where the per-formance weight119882
119904is selected to ensure a smooth frequency
characteristic of the external-loop sensitivity function 119878out(119904)A smooth characteristic of the sensitivity function is alsoan additional indicator of the closed-loop time-performancecharacteristics
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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Applied Computational Intelligence and Soft Computing
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HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
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Electrical and Computer Engineering
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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httpwwwhindawicom Volume 2014
Advances in
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International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
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RoboticsJournal of
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Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
8 The Scientific World Journal
The closed-loop characteristic of the external loop withweights119882
119904 119881out 119881noise is
119879119911119908= 119882
minus1
119904[(1 + 119862 (120578) 119879in119875
1015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
(1 + 119862 (120578) 119879in1198751015840
)minus1
] [
[
1
119881out119881noise
]
]
= 119882minus1
119904[119878out 119878out 119878out] [
[
1
119881out119881noise
]
]
(25)
The optimal solution of the optimization problem is
min119862
10038171003817100381710038171198791199111199081003817100381710038171003817infin
= 120574min (26)
Themain goal of expression (26) is to minimize the influ-ences of inputs119908
1 119908
2on the sensitivity function 119878out and to
ensure final feedback performance with 119878out asymp 119882119904and |119878out| lt
|119882119904|
41 Robustness Criteria Assessment via Nonnegativity of theEven Polynomial Let us consider the property of the norm sdot
infinrelated to robust stability with uncertainty models
[30 39] The robust stability criterion is ensured withcondition sdot
infinlt 1 where the criterion is derived for a
given transfer function1198750(119904) = 119861(119904)119860
minus1
(119904) The condition1198750infin
lt 1 holds if the even polynomial 120587(1205962) is a strictlynonnegative function
119860(1205962
) minus 119861 (1205962
) = 120587 (1205962
) gt 0 (27)
The property of the function 120587(1205962) is derived from thefunctionΦ(119895120596) defined as in [30]
Φ(119895120596) = 1205742
119868 minus119861 (119895120596)
119860 (119895120596)
119861 (minus119895120596)
119860 (minus119895120596) (28)
Function Φ(119895120596) is a strictly continuous function for allisin R cup infin and has no imaginary zeroes The norm of thefunction equals 119875
0 le 120574 only if Φ(119895120596) gt 0 for all 120596 isin
R The robustness property for nonparametric uncertaintycan be assessed with even polynomial 120587(1205962) gt 0 (27)where we assume that 120574 = 1 and 119860(119895120596)119860(minus119895120596) = 119860(120596
2
)119861(119895120596)119861(minus119895120596) = 119861(120596
2
) holds true
Theorem 1 The norm sdot infin
of the system withpolynomials 119860 119861 is 119861119860
infinlt 1 only if the corresponding
polynomial120587(1205962) for all 120596 is a strictly nonnegative functionand 119861(119904)119860(119904)minus1 belongs to 119875119867
infin
Proof The simple explanation of the norm sdot infin
is119861119860
infin= sup
120596|119861(119895120596)119860
minus1
(119895120596)| where we assume that119861(119904)119860(119904)
minus1 belongs to 119875119867infin The norm of the transfer
function is 119861119860infin
lt 1 only if the ratio of polynomials|119861(119895120596)||119860(119895120596)|
minus1
lt 1 It is obvious from the above-mentioned that the difference (27) 120587(1205962) must be a strictly
nonnegative function and that 120587(1205962) has no real zeroes Thepositivity condition of the 120587(1205962) is an objective of robustnessassessment
Based on condition (27) the controllersrsquo robustnessproperty can be achieved with the assessment of the non-negativity of the even polynomial Each single robustnesscriterion ((20) (22) (24) and (26)) can be presentedin the form of an even polynomial and a nonnegativitycondition
42 Robust Stability Assessment with a NonnegativityCondition for the Internal Loop The internal-looprobustness conditions are presented with expressions(20) (22) and (24) Based on condition (27) the evenpolynomial 120587
119872(1205962
120575) for multiplicative uncertainty (20)is
120587119872(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119872 (119904) 119908
119886119872(minus119904)
minus 1198610(119904) 119861
0(minus119904) 119871
119870(119904 120575) 119871
119870(minus119904 120575) 119908
119887119872(119904)
times 119908119887119872
(minus119904)
120587119872(1205962
120575) = 119862in119908119886119872 (1205962
) minus 1198610119871119870119908119887119872
(1205962
120575) gt 0
(29)
where the weight 119882119872(119904) is 119882
119872(119904) = 119908
119887119872(119904)119908
minus1
119886119872(119904)
Even polynomial 120587119868(1205962
120575) for robust stability withinverse uncertainty model (22) with stable weight 119882
119868(119904) =
119908119887119868(119904)119908
minus1
119886119868(119904) is
120587119868(119904 120575) = 119862in (119904) 119862in (minus119904) 119908119886119868 (119904) 119908119886119868 (minus119904)
minus 1198600(119904) 119860
0(minus119904) 119877
119870(119904 120575) 119877
119870(minus119904 120575) 119908
119887119868(119904)
times 119908119887119868(minus119904)
120587119868(1205962
120575) = 119862in119908119886119868 (1205962
) minus 1198600119877119870119908119887119868(1205962
120575) gt 0
(30)
The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
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The Scientific World Journal 9
Even polynomial 1205871198791199081015840
1
(1205962
120575) for condition (24) withstability weights 119882
119872(119904) and 119882
119868(119904) and input weight
1198811015840
noise(119904) = V1015840119887noise(119904)V
1015840minus1
119886noise(119904) is
1205871198791199081015840
1119872
(1205962
120575) = 119908119886119872119862inV
1015840
119886noise (1205962
)
minus 1199081198871198721198600119860119898119877119870V1015840119887noise (120596
2
120575)
1205871198791199081015840
1119878
(1205962
120575) = 119908119886119878119862inV
1015840
119886out (1205962
)
minus 1199081198871198781198600119860119898119877119870V1015840119887out (120596
2
120575)
1205871198791199081015840
1
(1205962
120575) = 05 (1205871198791199081015840
1119872
(1205962
120575) + 1205871198791199081015840
1119878
(1205962
120575)) gt 0
(31)
To simplify the optimization procedure in comparison tothe classic approach the composed condition (24) is treatedsimilarly as the criterion in augments plant transformationin a classic 119867
infindesign wherein the condition 120587
1198791199081015840
1
(1205962
120575)
(31) is a simple even polynomial function In the multi-objective optimization approach the condition 120587
1198791199081015840
1
(1205962
120575)
(31) can be also considered as two distinct functions1205871198791199081015840
1119872
(1205962
120575) and1205871198791199081015840
1119878
(1205962
120575)
43 Robust Stability Assessment with a Nonnegativity Con-dition for the External Loop A polynomial function forrobustness and performance condition (26) for the externalloop in Figure 7 can be formulated the sameway as conditions(24) and (31) The even polynomial of condition (26) withperformance weights 119882
119878(119904) = 119908
119887119878(119904)119908
minus1
119886119878(119904) 119881out(119904) =
V119887out(119904)V
minus1
119886out(119904) and 119881noise(119904) = V119887noise(119904)V
minus1
119886noise(119904) is
1205871199081119911(1205962
120578) = 119908119886119878119862out (120596
2
) minus 11990811988711987811986001198601198981198601015840
119877119870119877119862(1205962
120578)
1205871199082119911(1205962
120578) = 119908119886119878119862outV119886out (120596
2
)
minus 11990811988711987811986001198601198981198601015840
119877119870119877119862Vbout (120596
2
120578)
120587119911119908(1205962
120578) = (1205871199081119911(1205962
120578)2
+ 1205871199082119911(1205962
120578)2
) gt 0
(32)
As with condition (31) we can use composite objectivefunction 120587
119911119908(1205962
120578) to simplify the optimization procedureAdditional criteria are imposed to achieve strong stability
of the RIC where the real-time operational safety for thecontrolled systemmust be preserved in case some parts of thefeedback system fail for example sensorsrsquo failure electronicdriver failure and faulty motor Controllersrsquo stability canbe assessed over strict positive realness (SPR) criteria [40]Controllers119870(119904) 119862(119904) are stable transfer functions if SPRrsquosconditions hold true
R119890 [119870 (119895120596) + 119870 (minus119895120596)] gt 0
R119890 [119862 (119895120596) + 119862 (minus119895120596)] gt 0(33)
where R119890[119870(119895120596)] and R119890[119862(119895120596)] are even functions Theeven polynomials of stability conditions (33) are
120587119870SPR
(1205962
120590)
= 2 (119877119870(minus119895120596 120590) 119871
119870(119895120596) + 119877
119870(119895120596 120590) 119871
119870(minus119895120596)) gt 0
120587119862SPR
(1205962
120578)
= 2 (119877119862(minus119895120596 120578) 119871
119862(119895120596) + 119877
119862(119895120596 120578) 119871
119862(minus119895120596)) gt 0
(34)
The robust conditions and the stability property of con-trollers (29)ndash(32) (34) can be directly used in amultiobjectiveoptimization algorithm with DE
5 Multiobjective Optimization withDE for RIC Design
The paper presents a multiobjective optimization procedurewith DE Genetic algorithm DE is known as a very efficientand powerful stochastic optimization procedure DE includessimilar operation steps as other GAs but in comparison withother classic GAs DE deviates the current population withscaled differences between two randomly selected members[28] There are many reasons for using the DE algorithm forsolving various optimization problems in different scientificdisciplines Compared to other similar algorithms DE has asimpler structure and is easier to implement on real problemsBecause of its simple structure it is suitable for large scaleoptimization problems The control parameters of DE havebeen well studied and their influence on the optimizationprocedure is well known
This paper presents the design of a robust efficientsimple and structured disturbance observer based on sim-ple objective functions for performance and robustnessassessment The main problem in optimal control designis formulating proper efficient objective functions whichcan be further used in a corresponding optimization algo-rithm Objective functions with their properties must ensureoptimal or suboptimal solutions of the given problem Thegiven optimization problem in RIC structure with regionalpole placement technique does not provide a convex set ofobjective functions There also do not exist general straight-forward procedures to convert such objectives to convexproblem which are mostly in conflict with each other andunrelated For example unrelated criteria are strong stabilityand robustness closed-loop pole location and controllerstability controller structure and robustness performanceand so forth In such cases where engineering simplificationsare needed it is very appropriate to use a metaheuristicoptimization tool like DE For this reason this paper doesnot deal with the efficiency of multiobjective DE algorithmsand their variants but only considers the efficiency of the usedapproach in an optimal control problem in anRIC frameworkwith formed objective functions for dynamic properties (7)ndash(9) and robustness (29)ndash(32) and (34)
Multiobjective optimization problems with DE involvemultiple objectives whichmust be optimized simultaneously
10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
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Applied Computational Intelligence and Soft Computing
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httpwwwhindawicom Volume 2014
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ArtificialNeural Systems
Advances in
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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10 The Scientific World Journal
(1) Select central polynomials 119862in(119904) 119862119900ut(119904) and weights119882(119904) 119881(119904)(2) Select controllersrsquo structure 119870119862 and parameters 120590 120578
deg119870 = deg119862in minus deg1198750
deg119877119870= deg 119871
119870
deg119862 = deg119862out minus (deg119862in + deg119875119898+ deg1198751015840)
deg119877119862= deg 119871
119862
(3) Parametersrsquo selection of the DE-optimization algorithm(Number of parents-NP differential weight-F crossover probability-CR mutation strategy)
(4) while (min119891(1205962 119883))(5) DE selects value of parameters 120590 120578(6) Solve polynomial equations (14) (16)(7) Derive convex polynomials 120587(1205962 119883)(8) Find current Pareto-front of the 119891(1205962 119883)(9) end while
Algorithm 1 Multiobjective DE algorithm
and a set of possible solutions must be obtained The setof possible solutions is evaluated based on the concept ofdominance and Pareto optimality [14 17 33] The presentedRIC design uses DEMO algorithm presented by Robic andFilipic [42] The DEMO combines the advantages of DEwith the mechanisms of Pareto-based ranking and crowdingdistance sorting The advantage of the DEMO algorithm isthat it ensures convergence to the Pareto-front and a uniformspread of individuals along the front [42]
A multiobjective function is composed of derived objec-tive functions (7)ndash(9) (29)ndash(32) and (34) and is equal to
min1205962or119883
119891 (1205962
119883) =
[[[[[[[[[[[[[[[[
[
1198691(119883) and 119869out 1 (119883)
1198692(119883) and 119869out 2 (119883)
1198693(119883) and 119869out 3 (119883)
120587119872(1205962
119883)
120587119868(1205962
119883)
1205871198791199081015840
1
(1205962
119883)
120587119911119908(1205962
119883)
120587119870SPR
(1205962
119883)
120587119862SPR
(1205962
119883)
]]]]]]]]]]]]]]]]
]
st 119891 (1205962
119883) gt 0
119883 isin 119875
(35)
where 119875 is parameter space and 119883 is decisionvector Decision vector 119883 contains the coefficientsof polynomials 119871
119870 119877
119870 119871
119862 119877
119862(1) with possible
parameterization coefficients 120575 and 120578 Figure 8 showsthe structure of the decision variable119883
NoteThe solution119883 isin 119875 is Pareto-optimal if and only if theredoes not exist119883 isin 119875 that satisfies119891(1205962 119883) lt 119891(1205962 119883)
The optimization algorithm starts with the preselectedcentral polynomials 119862in119862out where the controllersrsquo degreesare prescribed in the same way as in the classic polynomialdesign (6) (16) [37]The degree of controller 119870 is equal to thecondition deg119870 = deg119862in minus deg119875
0 where deg119862in gt deg119875
0
holds true The degree of controller 119862 can be determined
Table 1 Parameters of an electromechanical system
Motor parameters119869 62 sdot 10
minus3 kgm2
119861 42 sdot 10minus3Nms
119896119875
0081NmA119877 79Ω119871 57mH
C
LK RK
K
LC RC
Figure 8 Structure of decision variable119883
the same way as for controller119870 where deg119862 = deg119862out minus
(deg119862in+deg119875119898+deg1198751015840
) and deg119862out ge (deg119862in+deg119875119898+deg1198751015840) holds true The optimization procedure is shown inAlgorithm 1
6 The Design Procedure for RIC Controllers
The proposed robust motion controller design is demon-strated by an electromechanical positioning system with theparameters given in Table 1
The parameters 119869 119861 119896119875 119877 and 119871 are rotor inertia
viscous friction torque factor terminal resistance and rotorinduction respectively To ensure simple controller struc-tures of 119870(119904) and 119862(119904) we use a simplified model of themechanical system 119875
0(119904) The simplification is often applica-
ble if the plant has more strongly expressed dominant stablepoles in comparison to other stable poles The given model is
1198750(119904) =
120596 (119904)
119894 (119904)=
119896119875
119869119904 + 119861=
1306
119904 + 0667
1198751015840
(119904) =120593 (119904)
120596 (119904)=1
119904
(36)
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 11
Table 2 Uncertainty parameters of the positioning system
Uncertainty parametersΔ119869 [24 divide 98] sdot 10
minus3 kgm2
Δ119861 [33 divide 65] sdot 10minus3Nms
Δ119896119875
[06 divide 13] NmA
Table 3 RIC control requirements
Feedback loop RIC requirementsSettling time 119905
119904lt 14 s
Nominal plant overshooting 119872Pn lt 3Worst-case plant overshooting 119872worst lt 20Tracking accuracy 119890error lt |00015| radTracking reference frequency [0 divide 1] radsInput disturbance rejection |005| Nm [0 divide 02] radsOutput disturbance rejection |05| rad [0 divide 05] radsInput current limit plusmn15AVoltage limit plusmn148VRobust stabilizationStable and low-order controllers 119870119862
The output of the controller 119870(119904) is an armature current119894[119860] through a magnetic coil which is proportional to theelectrical torque To avoid additional nonlinearities it issensible to consider terminal voltage |119877119894 + 119896
119890120596| lt |119880limit|
especially if the system is driven conventional by anH-bridgeand PWM signal The coefficient 119896
119890is an electromotive force
constantThe uncertainty plant is given by
Δ119875 (119904) =Δ119896
119875
Δ119869119904 + Δ119861 (37)
where the parameters vary in intervals according to changedoperation points friction load gear box and so forth Theuncertainty parameter intervals are shown in Table 2
The estimated uncertainty weights for the robustnesscriteria (20) (22) and (24) are
119882119872(119904) =
134 times 1199043
+ 1156 times 1199042
+ 032 times 119904 + 0062
1199043 + 157 times 1199042 + 048 times 119904 + 0096
119882119868(119904) =
065 times 1199042
+ 039 times 119904 + 0083
1199042 + 082 times 119904 + 017
1198811015840
noise (119904) =00023 times 119904 + 12 sdot 10
minus6
119904 + 9803
(38)
The desired requirements of the RIC systems are given inTable 3
The reference model119875119898(119904) is selected so as to ensure
proper internal dynamic of the RIC structure where holds119875119898(119904) asymp 119875
0(119904) The selected 119875
119898(119904) is
119875119898(119904) =
2892
119904 + 15 (39)
Table 4 Allowed region for optimized internal-loop poles 119862in(119904) ina complex plain
Allowed poles region 119863in for 119862in
120590119905119904 low minus07
120590119905119904 high minus4
plusmn119895120596119905119903
plusmn11989510
plusmn120593120577
plusmn30∘
Table 5 Allowed region for optimized internal-loop poles 119862out(119904)
in a complex plain
Allowed poles region 119863out for 119862out
120590119905119904 low minus035
120590119905119904 high minus9
plusmn119895120596119905119903
plusmn11989514
plusmn120593120577
plusmn74∘
According to the RIC dynamic requirement for inputand output disturbance rejection and tracking property theadditional performance weights are selected as follows
119882119878(119904) =
21 times 1199042
+ 046 times 119904 + 0001
1199042 + 398 times 119904 + 099
119881out (119904) =0002 times 119904 + 00012
119904 + 00014
119881noise (119904) =29 sdot 10
minus3
times 119904 + 12 sdot 10minus3
119904 + 1001 sdot 103
(40)
The controller transfer function 119870(119904) is selected so as toensure simple structure and maintain desirable closed-loopperformance The selected low-order structure is
119870(119904 1199031198960) =
1198971198961119904 + 119897
1198960
1199031198961119904 + 120575
(41)
where the parameter 120575 represents controller parameterization(14) and the allowable desired value is given on the interval[10
minus4
minus10minus2
]The value of 120575 ensures proper input disturbancerejection for low-frequency signals and stable approximateintegral behaviour
Accordingly the following internal central polynomial119862in(119904) is chosen on the selected controller structure 119870(119904) onthe condition deg119862in = deg119870 + deg119875
0 The internal central
polynomial is
119862in (119904) = 1199042
+ 2 times 119904 + 1 (42)
The polynomial 119862in(119904) ensures proper dynamic and sta-bility of the internal system The selected allowed region 119863inof the optimized internal loop poles 119862in(119904) is presented inTable 4
The selected controller structure 119862(119904) is
119862 (119904 1199031198880) =
11989711988821199042
+ 1198971198881119904 + 119897
1198880
11990311988821199042 + 119903
1198881119904 + 120578
(43)
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
12 The Scientific World Journal
Table 6 Parameters of DE-optimization algorithm
Optimization algorithm parametersNumber of parents NP 50
Differential weight 119865 085Crossover probability CR 092
Mutation strategy DErand1bin
Table 7 Polynomials 119862in 119862in roots comparisons
Polynomials 119862in(119904) 119862in(119904)
minus1 minus0944 minus 119895004
minus1 minus0944 + 119895004
Table 8 Polynomials 119862out 119862out roots comparisons
Polynomials 119862out(119904) 119862out(119904)
minus34 minus 119895123 minus309 minus 119895102
minus34 + 119895123 minus309 + 119895102
minus25 minus 1198953 minus261 minus 119895276
minus25 + 1198953 minus261 + 119895276
minus12 minus109
minus05 minus048
The parameter 120578 represents 119862(119904) controller parameteri-zation in such a way that the double-integrator effect in theexternal loop is avoided The admissible value is 120578 gt 50According to controller structures 119870(119904) and 119862(119904) and thereference model 119875
119898(119904) the external central polynomial is
selected as
119862out (119904) = 1199046
+ 135 times 1199045
+ 234 times 1199044
+ 1286 times 1199043
+ 4171 times 1199042
+ 4773 times 119904 + 1490
(44)
where the following condition holds true deg119862out = deg119862 +(deg119862in + deg119875
119898+ deg1198751015840) The selected allowed region119863out
of the optimized external-loop poles 119862out(119904) is presented inTable 5
The structure of the decision variable119883 for the givenexample is shown in Figure 9
The selected parameters of DE algorithm are presented inTable 6
7 Results
Optimized controllers 119870(119904) and 119862(119904) after optimization withDE and objective function (35) are as follows
119870 (119904) =1356 sdot 10
minus3
times 119904 + 84 sdot 10minus3
119904 + 018 sdot 10minus3
119862 (119904) =027 times 119904
2
+ 23 times 119904 + 229
1199042 + 9324 times 119904 + 1021
(45)
The optimization results are presented below The differ-ence between the selected internal central polynomial 119862in(119904)
and the optimized polynomial 119862in(119904) is presented in Table 7
CK
lk1 lk0 rk1 120575 lC1 lC1 lC0 rC2 rC1 120578
Figure 9 The structure of the decision variable 119883 with 10 parame-ters
0 50 100 150 200 250 300Time (s)
Ang
ular
pos
ition
(rad
)
Reference signalWorst-case modelNominal model
minus4
minus2
0
2
4
Figure 10 Step-reference tracking
0 50 100 150 200 250 300Time (s)
Curr
ent (
A)
Worst-case modelNominal model
minus05
0
05
1
Figure 11 Output of the controller 119870(119904)
A comparison of polynomials 119862out(119904) and 119862out(119904) ispresented in Table 8
From the results of polynomial comparisons in Tables7 and 8 it is evident that the optimization procedurewith Pareto-optimal solution ensures a satisfactory fittingof pole positions in the dominant region The obtainedcontrollers119870(119904) and119862(119904) are stable and all poles of internaland external loops lie in the prescribed region
The final values of robust criteria (20) (22) (24) and (26)are presented in Table 9
The tracking RIC capabilities on step-reference signals forthe nominal and worst-case systems are shown in Figure 10The reference signal presents a rotation of the RIC system forhalf a turn to the left and to the right The worst-case modelis selected accordingly as a possible real operational casewith valuesmax(Δ119869) max(Δ119861) and min(Δ119896
119875) chosen from
Table 2 Controllersrsquo119870 and119862 outputs are shown in Figures 1112 and 13 respectively
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 13
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Ang
le er
ror (
rad)
minus05
0
05
Figure 12 Output of the controller 119862(119904)
Worst-case modelNominal model
0 50 100 150 200 250 300Time (s)
Volta
ge (V
)
minus20
minus10
0
10
20
Figure 13 Output-voltage of the RIC system
The disturbance rejection capability of the positioningsystem is shown in Figures 14 15 16 and 17
Figures 10ndash17 show that the robust motion controllerpresented in the RIC framework satisfies all control designcriteria Table 9 and Figures 10ndash17 provide evidence that thestability and performance conditions are preserved The sys-tem does not exceed the limit values for the operation voltageand current in the given operation interval so that the systemdoes not exhibit additional nonlinear or oscillating behaviour(Figures 11 and 13)The influence of themeasured noise is alsominimized with conditions 119879
1199081015840
1
infin
and 119879119908119911infin Table 9The
system has good reference tracking and disturbance rejectionwithin the prescribed area of system uncertainty (Figures14ndash17) and simple low-order structures of the internal andexternal controllers
8 Conclusion
This paper presented the design of a robust RIC structurefor a positioning system The proposed approach showsthe capability of robustness and performance optimizationover nonnegativity of an even polynomial where regionalpole placement is used The even polynomial can be alsoformulated for other types of uncertainty and performancecriteria Controllers 119870 and 119862 can be parameterized approx-imately by using known characteristics which allows thepossibility of preserving strong stability of the controlledsystem Optimizationwith themulticriterion algorithm such
0 20 40 60 80 100Time (s)
Ang
ular
pos
ition
(rad
)
Output disturbance
Reference
Worst-case modelNominal model
minus1
0
1
2
3
Input disturbance times01 Nm
Figure 14 Input-output disturbance rejection
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Curr
ent (
A)
minus02
minus01
0
01
02
03
Figure 15 Output of the controller 119870(119904) with disturbance attenua-tion
Table 9 Value of robust criteria
Criteria sdot infin
Value1003817100381710038171003817119879in119882119872
1003817100381710038171003817infin082
1003817100381710038171003817119878in119882119868
1003817100381710038171003817infin062
1003817100381710038171003817119879119908101584011003817100381710038171003817infin
023
1003817100381710038171003817119878out119882minus1
119904
1003817100381710038171003817infin073
1003817100381710038171003817119882minus1
119904119878out119882out
1003817100381710038171003817infin0123
1003817100381710038171003817119882minus1
119904119878out119882noise
1003817100381710038171003817infin047
as DE offers the possibility of including many criteria andan arbitrary number of free parameters as shown in the pre-sented example The criteria can include system knowledgeuncertainties and perturbation characteristics as well ascriteria related to frequency and time domain characteristicsof a closed-loop system The presented results in the designexample confirmed the validity of the proposed approachThe DE-optimization procedure with different robustnessand performance criteria can be used in a wide range ofdifferent controller and feedback structures designs Thepresented approach can be straightforward extended to the1198672or119867
infin1198672controller design Further work will be focused
on the pole placement robust state space controller designforMIMOsystemwhere the polynomial equation introducesa set of parametric solutions In MIMO case the exactsolution of the polynomial equation is limited to the number
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
14 The Scientific World Journal
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Ang
le er
ror (
rad)
minus02
minus01
0
01
02
03
Figure 16 Output of the controller 119862(119904) with disturbance attenua-tion
Worst-case modelNominal model
0 20 40 60 80 100Time (s)
Volta
ge (V
)
minus2
0
2
4
6
Figure 17 Output voltage of the RIC system with disturbanceattention
of the inputs and outputs of the system The parametricsolutions can be used as optimization parameters similar tothe presented approach
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Choi K Yang W K Chung H R Kim and I H SuhldquoOn the robustness and performance of disturbance observersfor second-order systemsrdquo IEEE Transactions on AutomaticControl vol 48 no 2 pp 315ndash320 2003
[2] M R Matausek and A I Ribic ldquoDesign and robust tuning ofcontrol scheme based on the PD controller plus disturbanceobserver and low-order integrating first-order plus dead-timemodelrdquo ISA Transactions vol 48 no 4 pp 410ndash416 2009
[3] B Yao M Al-Majed and M Tomizuka ldquoHigh-performancerobust motion control of machine tools an adaptive robustcontrol approach and comparative experimentsrdquo IEEEASMETransactions on Mechatronics vol 2 no 2 pp 63ndash76 1997
[4] Y Wang Z H Xiong and H Ding ldquoRobust internal modelcontrol with feedforward controller for a high-speed motionplatformrdquo in Proceedings of the IEEE IRSRSJ International
Conference on Intelligent Robots and Systems (IROS rsquo05) pp 187ndash192 August 2005
[5] B K Kim and W K Chung ldquoUnified analysis and designof robust disturbance attenuation algorithms using inherentstructural equivalencerdquo in Proceedings of the American ControlConference pp 4046ndash4051 June 2001
[6] P B Stephen C A Hax and T LMagnantiAppliedMathemat-ical Programming Addiosn-Wesley 1997
[7] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004
[8] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[9] T Bakka and H R Karimi ldquoRobust 119867infin
dynamic outputfeedback control synthesis with pole placement constraintsfor offshore wind turbine systemsrdquo Mathematical Problems inEngineering vol 2012 Article ID 616507 18 pages 2012
[10] P Gahinet and P Apkarian ldquoLinear matrix inequality approachto 119867
infincontrolrdquo International Journal of Robust and Nonlinear
Control vol 4 no 4 pp 421ndash448 1994[11] L Jin and Y C Kim ldquoFixed low-order controller design with
time response specifications using non-convex optimizationrdquoISA Transactions vol 47 no 4 pp 429ndash438 2008
[12] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[13] A Shenfield and P J Fleming ldquoMulti-objective evolutionarydesign of robust controllers on the Gridrdquo Engineering Applica-tions of Artificial Intelligence vol 27 pp 17ndash27 2014
[14] A A Hussein and R Sarker ldquoThe pareto differential evolutionalgorithmrdquo International Journal on Artificial Intelligence Toolsvol 11 no 4 pp 531ndash555 2002
[15] L Wang and L-P Li ldquoFixed-structure119867infincontroller synthesis
based on differential evolution with level comparisonrdquo IEEETransactions on Evolutionary Computation vol 15 no 1 pp120ndash129 2011
[16] K J Hunt ldquoPolynomial LQG and 119867infin
infinity controllersynthesis a genetic algorithm solutionrdquo inProceedings of the 31stIEEE Conference onDecision and Control vol 4 pp 3604ndash36091992
[17] A Chipperfield and P Fleming ldquoGas turbine engine controllerdesign using multiobjective genetic algorithmsrdquo in Proceed-ings of the 1st IEEIEEE International Conference on GeneticAlgorithms in Engineering Systems Innovations and Applications(GALESIA rsquo95) pp 214ndash219 September 1995
[18] G Avanzini and E A Minisci ldquoEvolutionary design of a full-envelope full-authority flight control system for an unstablehigh-performance aircraftrdquo Proceedings of the Institution ofMechanical Engineers GmdashJournal of Ae vol 225 no 10 pp1065ndash1080 2011
[19] H Kobayashi S Katsura and K Ohnishi ldquoAn analysis ofparameter variations of disturbance observer for motion con-trolrdquo IEEE Transactions on Industrial Electronics vol 54 no 6pp 3413ndash3421 2007
[20] H S Lee and M Tomizuka ldquoRobust motion controller designfor high-accuracy positioning systemsrdquo IEEE Transactions onIndustrial Electronics vol 43 no 1 pp 48ndash55 1996
[21] T Umeno and Y Hori ldquoRobust speed control of DC servomo-tors using modern two degrees-of-freedom controller designrdquoIEEE Transactions on Industrial Electronics vol 38 no 5 pp363ndash368 1991
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 15
[22] B K Kim and W K Chung ldquoAdvanced disturbance observerdesign for mechanical positioning systemsrdquo IEEE Transactionson Industrial Electronics vol 50 no 6 pp 1207ndash1216 2003
[23] K Kong and M Tomizuka ldquoNominal model manipulation forencancement of stability robustness for disturbance observerrdquoInternationl Journal of Control Automation and Systems vol11 no 1 pp 12ndash20 2013
[24] K Kong J Bae and M Tomizuka ldquoControl of rotary serieselastic actuator for ideal force-mode actuation in human-robot interaction applicationsrdquo IEEEASME Transactions onMechatronics vol 14 no 1 pp 105ndash118 2009
[25] A Sarja R Sveko and A Chowdhury ldquoStrong stabilizationservo controller with optimization of performance criteriardquo ISATransactions vol 50 no 3 pp 419ndash431 2011
[26] A Chowdhury A Sarjas P Cafuta and R Svecko ldquoRobustcontroller synthesiswith consideration of performance criteriardquoOptimal Control Applications and Methods vol 32 no 6 pp700ndash719 2011
[27] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[28] S Das and P N Suganthan ldquoDifferential evolution a surveyof the state-of-the-artrdquo IEEE Transactions on EvolutionaryComputation vol 15 no 1 pp 4ndash31 2011
[29] D Henrion M Sebek and V Kucera ldquoPositive polynomialsand robust stabilization with fixed-order controllersrdquo IEEETransactions on Automatic Control vol 48 no 7 pp 1178ndash11862003
[30] K Zhou J CDoyle andKGloverRobust andOptimal ControlPrentice-Hall New Jersey NJ USA 1997
[31] F YangM Gani andDHenrion ldquoFixed-order robust119867infincon-
troller designwith regional pole assignmentrdquo IEEETransactionson Automatic Control vol 52 no 10 pp 1959ndash1963 2007
[32] M T Soylemez and N Munro ldquoA note on pole assignment inuncertain systemrdquo International Journal of Control vol 66 no4 pp 487ndash497 1997
[33] T Tusar and B Filipic ldquoDifferential evolution versus geneticalgorithm in multiobjective optimizationrdquo in EvolutionaryMulti-Criterion Optimization vol 4403 of Lecture Notes inComputer Science pp 257ndash271 2007
[34] A Tesfaye H S Lee and M Tomizuka ldquoA sensitivity opti-mization approach to design of a disturbance observer indigital motion control systemsrdquo IEEEASME Transactions onMechatronics vol 5 no 1 pp 32ndash38 2000
[35] B K KimW K Chung H-T Choi I H Suh and Y H ChangldquoRobust internal loop compensator design formotion control ofprecision linear motorrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo99) pp 1045ndash1050July 1999
[36] M S Nasser J Poshtan M S Sadegh and F Tafazzoli ldquoNovelsystem identificationmethod andmulti-objective-optimalmul-tivariable disturbance observer for electric wheelchairrdquo ISATransactions vol 52 no 1 pp 129ndash139 2013
[37] V Kucera ldquoThe pole placement equationmdasha surveyrdquo Kyber-netika vol 30 no 6 pp 578ndash584 1994
[38] D D Siljak and D M Stipanovic ldquoRobust D-stability viapositivityrdquo Automatica vol 35 no 8 pp 1477ndash1484 1999
[39] J Doyle B A Francis and A Tannenbaum Feedback ControlTheory Macmillan Publishing New York NY USA 1990
[40] H Khatibi A Karimi and R Longchamp ldquoFixed-order con-troller design for polytopic systems using LMIsrdquo IEEE Transac-tions on Automatic Control vol 53 no 1 pp 428ndash434 2008
[41] R Storn andK Price ldquoDifferential evolutionmdasha simple and effi-cient heuristic for global optimization over continuousspacesrdquoJournal of Global Optimization vol 11 no 4 pp 341ndash359 1997
[42] T Robic and B Filipic ldquoDEMO differential evolutionalghorithm for multi-objective optimizationrdquo in Proceedingof 3rd International Conference of Evolution Multi-CriterionOptimization LNCS vol 3410 pp 520ndash533 2005
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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