Research Article Feedback Linearisation for Nonlinear ...Research Article Feedback Linearisation for Nonlinear Vibration Problems S.Jiffri, 1 P.Paoletti, 1 J.E.Cooper, 2 andJ.E.Mottershead

Research ArticleFeedback Linearisation for Nonlinear Vibration Problems

S Jiffri1 P Paoletti1 J E Cooper2 and J E Mottershead1

1 Centre for Engineering Dynamics School of Engineering The University of Liverpool Brownlow Hill Liverpool L69 3GH UK2Department of Aerospace Engineering University of Bristol Queens Building University Walk Bristol BS8 1TR UK

Correspondence should be addressed to S Jiffri sjiffrilivacuk

Received 4 July 2013 Accepted 7 January 2014 Published 5 June 2014

Academic Editor Nuno Maia

Copyright copy 2014 S Jiffri et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Feedback linearisation is a well-known technique in the controls community but has not been widely taken up in the vibrationscommunity It has the advantage of linearising nonlinear system models thereby enabling the avoidance of the complicatedmathematics associated with nonlinear problems A particular and common class of problems is considered where the nonlinearityis present in a system parameter and a formulation in terms of the usual second-order matrix differential equation is presentedTheclassical texts all cast the feedback linearisation problem in first-order form requiring repeated differentiation of the output usuallypresented in the Lie algebra notation This becomes unnecessary when using second-order matrix equations of the problem classconsidered herein Analysis is presented for the general multidegree of freedom system for those cases when a full set of sensors andactuators is available at every degree of freedom and when the number of sensors and actuators is fewer than the number of degreesof freedom Adaptive feedback linearisation is used to address the problem of nonlinearity that is not known precisely The theoryis illustrated by means of a three-degree-of-freedom nonlinear aeroelastic model with results demonstrating the effectiveness ofthe method in suppressing flutter

1 Introduction

The effects of nonlinearity are everywhere to be seen innature and it is true to say that many problems in structuraldynamics are nonlinear but in the past it has been convenientand easier to assume linearity This linear analysis approachhas necessarily led to conservative design of engineeringstructures that operate dynamically In the last few decadesthe need to assume linearity has been increasingly challengedmainly because of the high cost of fuel and materials and theneed to preserve the earthrsquos valuable natural resourcesThis isparticularly true of the aerospace industries where the needto produce lightweight fuel efficient aircraft is an unremittingpressure on design engineers One problem is of course thatnonlinear dynamic analysis is complicated and the usefullinear-analysis methods such as modal decomposition arenot applicable

Inevitably active control will increasingly be seen as asolution for problems of nonlinearity in elastomechanicsand aeroelasticity One option is to design lightweight fuelefficient aircraft and use active control to nullify the effects

of the nonlinearity By this approach well-understood linearanalysis methods can still be used Feedback linearisationdescribed for example by Isidori [1] is a technique nowwell established in the active control community Dependingupon the implementation it is able to completely or partiallylinearise the nonlinear system In the latter case there remainsa (generally nonlinear) subsystem with dynamics that mustbe investigated to ensure stability Of course the linear partcan be used for the attainment of a chosen control objectiveusing conventional linear time invariant (LTI) techniquesAn important aspect of feedback linearisation is that for thedesired control objectives to be met precisely the nonlinearitymust be known in terms of its physical location its type (egcubic quartic free-play etc) and the numerical values of itsparameters This seems initially to pose a serious restrictionon its application but fortunately there exist techniques suchas adaptive feedback linearisation that are able to account forincorrectly estimated nonlinear terms

It is apparent from the literature that the feedbacklinearisation method which holds much promise has foundonly very limited application in active vibration suppression

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 106531 16 pageshttpdxdoiorg1011552014106531

2 Shock and Vibration

Examples include Fossen and Paulsen [2] who applied adap-tive feedback linearisation to the automatic steering of shipsIn aeroelasticity Ko and his colleagues [3ndash5] used feedbacklinearisation methods (including adaptation) and carried outa series of wind-tunnel tests on a two degree of freedom aero-foil with either one or two control surfaces They found thatwith an erroneous nonlinear parameter but without adapta-tion their system reached a nonzero equilibrium rather thanthe zero equilibrium that is usually sought Monahemi andKrstic [6] employed adaptive feedback linearisation to updatethe aerodynamic parameters in their nonlinear model andthereby suppress wing-rockmotion a phenomenon triggeredprimarily by aerodynamic nonlinearities Poursamad [7]implemented a hybrid neural-network controller for antilockbraking with adaptive feedback linearisation to handle non-linear and time-varying brake parameters Bechlioulis andRovithakis [8] developed a multiple-input multiple-outputtracking controller with adaptive feedback linearisation andShojaei et al [9] demonstrated the ability of adaptive feedbacklinearisation in aiding effective trajectory tracking in thepresence of both parametric andnonparametric uncertaintiesin wheeled robots Tuan et al [10] designed a controller basedon partial feedback linearisation of the nonlinear dynamics ofa 3D overhead crane

In this paper the problem of active control of nonlinearsystems of the form

A1x + A2x + A3x + fnl (x x) = Bu (1)

representative of nonlinear vibration problems in elastome-chanics and aeroelasticity is consideredThe vectors x x andu typically contain the state variables and inputs respectivelyThe nonlinearity fnl is given as a function of x and x and thematricesA

1A2A3B represent the usual system parameters

This class of problem characterised by the second-ordermatrix differential equation with additional nonlinearityconfined to the left-hand side of (1) prevails to a very largeextent in engineering mechanics and is therefore worthy ofthe special attention devoted to it in this paper The classicaloutput feedback linearisation [1] may be greatly simplified inthe case of elastomechanical or aeroelastic systems describedby (1) In particular

(i) the essential theory is carried out entirely using thesecond-order matrix differential equation familiarto structural and aero-structural dynamicists withthe result that the need for repeated differentiationusually described using the Lie-algebra notation isrendered unnecessary

(ii) a linear transformation applies between the statevariable in (1) and the coordinates of the linearisedsystem

(iii) cancellation of the whole of the open-loop systemdynamics (not just the nonlinear terms) results in a setof independent linear single degree of freedom sys-tems for the application of conventional LTI controlmethods

(iv) complete linearisation may be achieved with an equalnumber of actuators and sensors If the number of

actuators and sensors is less than the dimension of thesystem then there will remain a nonlinear subsystemof dimension equal to the difference between thedimension of the full system and the number ofsensors (and actuators) This subsystem can be madeindependent of the linearised part and methods aredescribed to check its stability

(v) Adaptive feedback linearisation is described for thetreatment of an incorrect estimate of the nonlinearityThis makes use of Lyapunov stability criteria andresults in a parameter update rule that evolves withtime to ensure stability of the system

The method is illustrated by means of a three degree offreedom aeroelastic system consisting of a flexible wingdescribed in terms of two assumed modes in bending andtorsion and a third degree of freedom that describes theangularmotion of an underslung pylon-engine assemblyTheparameters and dimensions of the system are carefully chosento have realistic values

2 Active Feedback Linearisation

Feedback linearisation [1] is a process whereby a nonlinearsystem is rendered linear by virtue of active control UnlikeJacobian linearisation it is exact and does not entail anapproximation at any stage The method is implemented bytransforming a nonlinear system into a linear one For theclass of second-order systems given by (1) considered in thispaper

x = f (x x) + Gu f (x x) = Ψx +Φx +Ωfnl

Ψ = minusAminus11A3 Φ = minusAminus1

1A2 Ω = minusAminus1

1 G = Aminus1

1B(2)

becomes

z = Ψz +Φz + Gu (3)

In these equations u and u are respectively the actual (orphysical) input applied to the nonlinear system and the so-called ldquoartificialrdquo input to the linear system The matrices ΨΦ G are dependent on the chosen inputs u The mappingfrom the nonlinear domain to the linearised domain isachieved through a nonsingular linear coordinate transfor-mation The actual input is designed to neutralise the effectof the nonlinearity a procedure which can sometimes beachieved in full and sometimes partially as will be explainedtheoretically and by means of illustrative examples

The process is quite straightforward for elastomechanicaland aeroelastic systems described by second-order matrixdifferential equations with nonlinearity in the system param-eters The method described by Isidori [1] allows for greatergenerality including nonlinearity in the input and outputas well as in the system parameters which is not requiredhere and its omission leads to simplifications which aid theunderstanding of an otherwise fairly complicated procedure

The feedback linearisation procedure classically using theLie algebra entails repeated differentiation of each of the

Shock and Vibration 3

outputs with respect to time until the input terms appearThe classical procedure is greatly simplified in the case ofsecond-order matrix systems such as those in elastomechan-ics or aeroelasticity as explained in the sequel The presentwork addresses two cases The first case is that of completeinput-output linearisation meaning a full complement ofoutputs and inputs at every degree of freedom of the systemIn this case since the number of outputs is equal to thedimension of the system it is possible to linearise the entirenonlinear system The complete dynamics of the originalsystem are preserved during the transformation In thesecond case an incomplete systemof equal numbers of inputsand outputs is assumed In this case as the number of inputsand outputs is less than the dimension of the overall modelonly a partial linearisation of the system can be achievedTheportion that remains untransformed will contribute to whatis known as the internal dynamics whose stability must beensured for stability of the overall closed-loop system Thisis achieved by examining the stability of the zero dynamics[1] which is obtained by setting all coordinates of the lin-earised subsystem to zero in the expressions for the internaldynamics Expressions for the latter are obtained such thattheir time-derivatives are orthogonal to the inputs renderingthe zero dynamics uncontrollableThe zero dynamics may beeither linear or nonlinear

21 Complete Input-Output Linearisation (119899-Inputs 119899-Outputs) In the present case the number of inputs andoutputs is equal to the dimension of the systemThematricesand vectors of the original nonlinear system given in (2)have the dimensions Ψ Φ Ω G isin R119899times119899 x u f isin R119899times1 andthose given by the desired linearised equation (3) by Ψ ΦG isin R119899times119899 z u isin R119899times1 The first step is to choose the vectorof actual inputs that cancels the nonlinearity

u = Gminus1 (u minus f (x x)) (4)

It can be seen that the nonlinearity is indeed cancelled bysubstituting (4) into (2) In fact this is a special case wherenot only the nonlinearity but also the complete open loopdynamics is cancelled by the choice of actual inputThe resultis the linearised system of independent second-order singledegree of freedom equations

(

1

2

119899

) =(

119906

1

119906

2

119906

119899

) (5)

In fact these equations are a special case of single degree offreedom equations where each equation is simply a doubleintegrator Equation (5) is a particular form of (3) wherealso it is seen particularly that x = z The simplicity of (5)is an advantage of the complete cancellation of the open-loop dynamics in (4) The choice of the artificial input unecessarily depends upon the control objective for examplethe assignment of a pair of complex conjugate poles in each ofthe systems in (5) to avoid resonances Whatever the control

objective is it will result in the determination of gains definedhere in terms of negative feedback as

119906

1= minus [1205891

]1]

119909

1

1

119906

2= minus [1205892

]2]

119909

2

2

119906

119899= minus [120589119899

]119899]

119909

119899

119899

(6)

In this special case where the entire open loop dynamics arecancelled the control results in a closed-loop system that iscomprised of 119899 decoupled single degree of freedom subsys-tems Then having defined the artificial inputs the actualinputs that provide the desired linearisation are determinedfrom (4) It is seen that the nonlinearity in (2) is neutralisedand the closed-loop system is indeed linear with the requireddynamics

If the actual input were chosen to cancel the nonlinearityalone (not the complete open-loop dynamics) then (4)wouldbe replaced by

u = Gminus1 (u minusΩfnl) (7)

and the linearised equations would remain coupled (unlike(5)) and consequently the gains in (6) would take a differentform as

119906

1

119906

2

119906

119899

= minus

[

[

[

[

[

120589

11]11120589

12]12sdot sdot sdot 120589

1119899]1119899

120589

21]21120589


2119899]2119899

d

120589

1198991]1198991120589

1198992]1198992sdot sdot sdot 120589

119899119899]119899119899

]

]

]

]

]

times

119909

1

1

119909

2

2

119909

119899

119899

(8)

Clearly there are a greater number of control gains in (8)than in (6) whichmeans that there is more control flexibilitywhich might be used for example to assign the eigenvectorsas well as the eigenvaluesThis may be readily achieved usingmethods such as that presented in [11] and is illustratedthrough a numerical example later on

22 Partial Input-Output Linearisation (119898-Inputs119898-Outputs119898 lt 119899) The inputs and outputs (actuators and sensors)u x(1119898)

isin R119898times1 in equal numbers are now fewer thanthe dimension of the system Linearisation results in similar


expressions to those obtained for the complete input-outputcase presented above Equation (2) is now rewritten as

1

119898

119898+1

119899

=

119891

1(x x)

119891

119898(x x)

119891

119898+1(x x)

119891

119899(x x)

+

[

[

[

[

[

[

[

[

[

[

119892

11sdot sdot sdot 119892

1119898

d

119892


119898119898

119892

119898+11sdot sdot sdot 119892

119898+1119898

d

119892


119899119898

]

]

]

]

]

]

]

]

]

]

119906

1

119906

119898

(9)

and the coordinate systemwhichmaps the original nonlinearsystem to the partially linearised system may be expressed as

(1199111119911


119898)

119879

= (1199091119909


119898)

119879

(10)

which is identical to the full output feedback case exceptof course that there are now only 119898 outputs Further (119899-119898)coordinates are needed and are chosen as coefficients of theorthonormal basis of the null space of G119879

(11198991119898)so that

(

119909

1

119909

119899

) = V(119911

119898+1

119911

119899

)

V119879V = I(119899minus119898)times(119899minus119898)

V119879G(11198991119898)

= 0

V isin R119899times(119899minus119898)

(11)

As with the full output feedback case it is now necessary tochoose actual inputs so that the nonlinearity is eliminatedThis is achieved by

u = [G11198981119898

]

minus1

(u minus f(x x)(11198981)

)

f(x x)(11198981)

= Ψ(1119898)

x +Φ(1119898)

x +Ω(1119898)

fnl(12)

and substitution of (12) into the upper partition of (9) leadsto119898 independent linear second-order systems with artificialinputs u = 119906


119898

119879 expressed as

(

1

2

119898

) =(

1

2

119898

) =(

119906

1

119906

2

119906

119898

) (13)

Then by combining (9) (10) and (11) it is found that

119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

+ V119879G(11198991119898)

119906

1

119906

119898

(14)

so that from (11)

119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

(15)

which ensures uncontrollability of the nonlinear internaldynamics (15)

The stability of the complete system is then determinedby the zero dynamics which are generally nonlinear obtainedby setting to zero in (15) the external coordinates (119911

1 119911

119898)

of the partially linearised system in (13) The equations ofthe zero dynamics and their stability will be addressed for aspecific aeroelastic example in the sequel

As before the artificial inputs in (13) may be chosen asa linear combination of the instantaneous displacement andvelocity to fulfil a control objective When the zero dynamicsare found to be globally stable then the desired controlbehaviour is unaffected by the nonlinearity confined to theinternal dynamics

3 Aeroservoelastic Model

The governing equation of aeroservoelastic systems takes theusual form [12] given by

Aq + (120588119881B +D) q + (1205881198812C + E) q + fnl (q) = fext (16)

where AD E are the inertia structural damping and struc-tural stiffnessmatrices respectivelyBC are the aerodynamicdamping and aerodynamic stiffness matrices respectivelyand 120588 119881 are air density and velocity respectively (thepresent B is different from the input distribution matrix alsodenoted byB in Section 1)The vector q contains generalisedcoordinates describing the motion of the system whereasthe vector fext contains externally applied generalised forcingterms including control forces and gusts The nonlinearityis confined to fnl(q) Modified aerodynamic strip theory isused to compute the lift and pitch moment and an additionalunsteady aerodynamic derivative term is included to accountfor significant unsteady effects [12] which appears in thematrixB In practice the aerodynamicmatrices are frequencydependent [12] and the time domain model would need toinclude aerodynamic states to account for this The approachused is perfectly adequate for low speed high aspect ratiowings The system consists of a wing with an underslungengine attached by a pylon Aerodynamic forcesmomentsacting on the pylon-engine arrangement are assumed to benegligible compared to those acting on the wing

In the analysis presented here the wing deflection isdescribed in terms of two assumed modes and a further gen-eralised coordinate representing the angular displacement ofthe engine about the axis of the wingThe assumedmodes aredepicted in Figure 1 which also shows the coordinate systemwith its origin at the root leading edge of the wing

The vertical deflection of the wing 120577(119909 119910) is then given by

120577 = 119910

2119902

1+ 119910 (119909 minus 119909

119891) 119902

2 (17)


Mode 1 (bending) Mode 2 (torsion)

x

y

120577

Figure 1 The two deflection patterns assumed for the flexible wing

where 1199021 1199022are generalised coordinates that quantify the

amount of bending and torsion modes present in the overalldeflection and 119909

119891is the 119909-coordinate of the wing flexural

axisThe pylon-engine is modelled as a rigid body connectedto the wing by a nonlinear stiffness A sketch of the pylon-engine and also the complete wing-pylon-engine systemmaybe found in [13]

The pylon-engine rotational degree of freedom 120599pe isthe deflection of the pylon-engine relative to the wing Theabsolute rotation 120579pe may be obtained by adding the wingtwist angle at the engine attachment location 120594

2 to 120599pe

The absolute pylon-engine rotation becomes a coordinatein the assumed-modes domain 119902pe = 120579pe The coordinatetransformation matrix T is defined by

120577

1

120594

2

120599pe

=

[

[

119910

2

10 0

0 119910

20

0 minus119910

21

]

]

119902

1

119902

2

119902pe

(18)

where 1205771and 120594

2denote the vertical displacement at point

ldquo1rdquo (the intersection of the wing flexural axis with the local119909-axis at the point of attachment of the pylon to the wing)and the angular wing twist at point ldquo2rdquo (the intersectionof the quarter-semi-span with the quarter-chord) The 119910-coordinates of points 1 and 2 are identical (119910

1= 119910

2) The

system matrices A B C and E are given in terms of theparameters of the wing and pylon-engine system in [13]whereas vectors f

119888and fnl may be found in Appendix A in

this paper Note that in the present text 119896119879 119870119879are used to

describe the coefficients of the linear and cubic componentsof coupling stiffness respectively which is different to thenotation used in [13]

4 Numerical Simulation

The dimensions and parameters chosen for the model aregiven in [13] with 119870

119879= 300 times 119896

119879 For the flexible wing the

values chosen are based on those used in a numerical examplegiven in [12] The dimensions and mass of the pylon-enginearrangement are chosen to be representative of a real aircraft(eg [14])

The three degree of freedom aeroservoelastic model isused initially to determine the flutter speed of the linearsystem Then the cubic hardening term 119870

119879 is included in

the torsional spring connecting the wing to the pylon-engine

and the nonlinear time-domain response to initial conditionsis produced at an air speed just above the flutter speed

The aeroservoelastic matrices are given in general termsin [13] and expressed here in terms of specific parametervalues in consistent units (to three significant figures) as

A119908= 10

5[

[

949 00633 0

00633 00942 0

0 0 0

]

]

E119908= 10

8[

[

110 0 0

0 0142 0

0 0 0

]

]

Ape = 103[

[

476 0 151

0 0 0

151 0 0647

]

]

Epe = 106[

[

0 0 0

0 190 minus0958

0 minus0958 0511

]

]

B = 104 [[

298 0 0

minus0229 00169 0

0 0 0

]

]

C = 103 [[

0 497 0

0 minus0406 0

0 0 0

]

]

(19)

A simplification in thematrix subscripts previously defined in[13] has beenmade hereThe control force distributionmatrixis given (as a function of air speed) by

f119888= 10

2[

[

minus319119881

2minus289119881

20

minus0302119881

2minus0120119881

2minus00188

0 0 001

]

]

times

120573

1

120573

2

119879pe

(20)

where 1205731and 120573

2are control surface (flap) angles and 119879pe is a

control torque applied directly to the pylon-engine assemblyThe nonlinear internal force is given by

fnl = 108

0

minus287

153

120599

3 119910

2= 119910

1= 1875 (21)


0 20 40 60 80 100 120 1402468

Air speed (ms)Nat

ural

freq

uenc

y (H

z)

Mode 1 (bending)Mode 2 (engine)

Mode 3-(torsion)

(a)

012


Mode 3-(torsion)

Dam

ping

ratio

()

0 20 40 60 80 100 120 140Air speed (ms)

(b)

Figure 2 119881-120596 and 119881-120577 plots for the wing-pylon-engine model

87 88 89 90

0

1

2

Time (s)

minus2

minus1

120599pe

(deg

)

Figure 3 Steady-state LCO in 120599pe response

Further information on the general form of the control andnonlinear spring forces can be found in Appendix A

41 Uncontrolled Linear System Natural Frequencies Damp-ing and Flutter Speed The structural modes in the linearmodel (nonlinearity neglected) occur at 171Hz (bending)406Hz (pylon-engine mode) and 683Hz (torsional) It isevident from Figure 2 that flutter occurs at an airspeed of776msminus1 involving coupling of the pylon-engine mode andwing bending modes

42 Nonlinear Time-Domain Response Thenonlinear systemis simulated at an airspeed of 80msminus1 just above the flutterpoint under the application of the initial conditions 120577

1=

0333mm 1205942= 000333 rad and 120599pe = 005 rad These values

have been chosen as they are representative of typical physicaldisplacements one might expect in practice for the wing-pylon-engine system specified in [13] The resulting responseof the system clearly exhibits limit cycle oscillation (LCO)A sample of the response for the 120599pe coordinate is shown inFigure 3

For comparison the simulated response for the 120599pecoordinate just below the linear flutter speed at an airspeedof 75msminus1 is shown in Figure 4 As expected the responsecontinues to decay and converges to the origin

In the following section linearising feedback controlis applied to the nonlinear model in the assumed-modesdomain In the examples considered the control objectiveis the assignment of the poles of the system Judicious

0 20 40 60 80 100 120

0

2

4

Time (s)

minus2

minus4

120599pe

(deg

)

Figure 4 Decaying response just below linear flutter speed

placement of the poles can be used to increase the flutterspeed of the system thereby increasing the flight envelopeOf course other control objectives could be used insteadTheresulting linearised system consists of the uncoupled singledegree of freedom subsystems referred to previously in (5)Two cases are considered those of 3 inputs and 3 outputs(denoted 3I3O) and of 2 inputs and 2 outputs (denoted 2I2O)which are representative of the analysis in Section 2

43 Feedback Linearisation 3I3O Case The poles of thesystem are assigned for the uncoupled linearised system in (5)(by cancellation of the entire open-loop dynamics) and withfeedback law (6) so that the natural frequencies and dampingvalues are given by

120596

1= 093Hz 120585

1= 001

120596

2= 29Hz 120585

2= 001

120596

3= 495Hz 120585

3= 001

(22)

at an air speed of 80msminus1 The first mode is predomi-nantly wing bending the second is pylon-engine defor-mation and the third is mainly the twisting motion ofthe wing With initial conditions the same as the uncon-trolled case (Section 42) namely 120577

1= 0333mm 120594

2=

000333 rad and 120599pe = 005 rad and in the absenceof any nonlinear parameter error the poles of the lin-earised system are assigned exactly The system responsenow decays to zero as shown in Figure 5 whereas the


0 20 40 60

0

05

1

Time (s)

minus 05

minus1

q1

times10minus4

(a)

0 5 10 15

0

1

2

Time (s)

minus1

minus 2

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus 005

minus 01

qpe

(c)

Figure 5 Feedback linearisation response at 80msminus1 (assumed-modes coordinates)

uncontrolled nonlinear system exhibited a limit cycleThe required control surface and actuator inputs are shownin Figure 6 where it can be seen that the input magnitudesare feasible in practice

Now suppose the linearising control is chosen to cancelthe nonlinearity alone in which case the linearised equations

0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20Time (s)

1205732

(deg

)

0

20

minus20

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)

Figure 6 Control surface deflection angles and actuator torques forexact feedback linearization

generally remain coupled resulting in a greater number ofcontrol gains in (8) and greater control flexibility as explainedin Section 21 One is then able to assign eigenvectors cor-responding to the assigned poles The same pole-placementabove is now implemented with the following respectiveeigenvectors assigned

k1= 1 1 0

119879

k2= 1 minus1 0

119879

k3= 0 0 1

119879

(23)

Thus it is desired to deliberately couple the bending andtorsion in the first twomodes whilst leaving the pylon-enginemode decoupled The resulting closed-loop responses andcontrol inputs are shown in Figures 7 and 8 respectively

It is evident from the first two plots in Figure 7 that amixing of modes has occurred as these two responses nowcontain multiple harmonics The third coordinate howeverremains unchanged This is expected as the eigenvectorassignment commanded that the pylon-engine mode shouldremain uncoupled It can also be seen from the first twoplots in Figure 8 that the required control surface deflectionshave increased it is easier to control the uncoupled modesthan the deliberately coupled ones Thus in the case of


0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)

Figure 7 Feedback linearisation with cancellation of nonlinearity only and with eigenvector assignment response at 80msminus1 (assumed-modes coordinates)

the three degree of freedom wing-pylon-engine system thereis some merit in using feedback linearisation to cancelthe entire open loop dynamics despite the loss of controlflexibility For the remainder of this simulation and thoseappearing later on only the case where the entire open loopdynamics are cancelled will be considered

When a +40 error in 119870nl is incorporated and theclosed-loop response is simulated based on the above feed-back parameters and with the same initial conditions anunstable response sets in from the very beginning as seenin Figure 9 The nonlinear spring connected between wingtwist and rotation of the pylon-engine assembly is notcapable of suppressing the response (in the assumed-modescoordinates) into a limit cycle The problem of error in thenonlinear parameter is of practical importance since it isseldom possible to be precise in estimating the magnitude ofnonlinear terms This problem is addressed in the sequel

44 Feedback Linearisation 2I2O Case The three degree offreedom system is now considered to be instrumented with

only two inputs trailing edge control-surface (flap) angles120573

1and 120573

2 and two outputs discrete wing displacement 120577

1

and twist 1205942 In this case there is no control torque 119879pe The

measurement of 120599pe is necessary but only for determinationof f(z z) The zero dynamics of the 2I2O aeroservoelasticproblem are given in Appendix B and stabilised by theintroduction of structural damping

D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)

which corresponds to 1 of modal damping The intro-duction of structural damping increases the flutter speedof the system from 776msminus1 to 9505msminus1 and thus thesimulation is carried out above the new speed at 975msminus1Pole-placement is specified such that

120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)

Figure 8 Control surface deflection angles and actuator torques forexact feedback linearisation cancellation of nonlinearity only withassignment of eigenvectors

As before feedback linearisation successfully places the polesto the desired values and the transient response shown inFigure 10 is obtained

It is evident from the third subplot in Figure 10 thatthe coordinate 119902pe eventually stabilises because the zerodynamics of the system are stable When a +40 error in119870nl is incorporated and the closed-loop response is simulatedbased on the same feedback parameters instability does notoccur (as in the 3I3O case) but a degradation in performanceis observed in Figure 11 where it is seen that the control failsto eliminate an LCO

5 Treatment of Nonlinear Parameter ErrorAdaptive Feedback Linearisation

The treatment of error in the numerical value of a nonlinearparameter is carried out using adaptive feedback linearisa-tion which makes use of Lyapunov stability theory It resultsin an updating rule for the erroneous parameter whichevolves with time This approach ensures that the system

remains stable the original control objectives might be com-promised because of evolution of the nonlinear parameterestimate This might be acceptable in many cases dependingupon engineering judgment

An erroneous estimate f1015840nl is now assumed in place of thetrue nonlinear force vector fnl The corresponding erroneousinput is then determined by comparison with (2) and (4) as

u1015840 = Gminus1 (u minus f1015840 (x)) f1015840 (x) = Ψx +Φx +Ωf1015840nl (26)

and substituting (26) into (2) leads to

x = u +Ω120576 120576 = fnl minus f1015840nl (27)

Recall (6) from Section 21 earlierThismay be combined intoa single equation

u = Γxx Γ isin R119899times2119899 (28)

The artificial inputs may be expressed in terms of the statevariables x by using the above equation together with thecoordinate transformation

z = T119911119909x (29)

defined in (10) and (11)Thus

u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)

and the second-order equation of the closed-loop system isthen cast as

x = Γ10158401x + Γ10158402x +Ω120576 (31)

Equation (31) shows that the nonlinear parameter errorresults in an input to the closed-loop system Itmay be readilyshown for the 3I3O system that

Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)

The system is then represented in state-space as

z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)

where T is defined in (18) and where the erroneous nonlinearparameter is given by

119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)

Figure 9 Feedback-linearisation with nonlinear parameter error response at 80msminus1

It is conceivable that this additional (unknown) input maypotentially destabilise the system or at least degrade thecontrol performance This possibility may be eliminated byaccounting for the nonlinearity errors using an adaptivealgorithm Such an algorithm seeks to guarantee asymptoticstability of the closed-loop response

A scalar Lyapunov function 119881 in z Δ119870119879may be defined

such that asymptotic stability of the closed-loop system isguaranteed by ensuring that 119881 gt 0 and its time-derivative

119881 lt 0 [15] Such a function may be used as a basis forcomputing a parameter update law The Lyapunov function

119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)

is considered where P = P119879 ≻ 0 Differentiating the aboveequation with respect to time one obtains

119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)

Now by combining (33) and (37) and expanding and rear-ranging

119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)

Figure 10 Feedback-linearisation response at 975msminus1 (assumed-modes coordinates)

it becomes evident that

d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)

eliminates the second term on the right-hand-side of (38)resulting in

119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)

Now from the definition ofΔ119870119879in (34) and knowing that the

actual nonlinear parameter119870119879is constant it is seen that

d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)

Combining (41) and (39) an update law for1198701015840119879corresponding

to the latter equation is obtained as

d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)

Thus the initially assumed value1198701015840119879of the nonlinear param-

eter is continually updated at each time step Effectivelythis increases the dimension of the state vector by 1 as 1198701015840

119879

becomes part of the system state Now choosing P such thatfor arbitraryQ ≻ 0

Q = minus (A119879P + PA) (43)

it becomes clear that by substituting the above equation into(40) 119881 is rendered negative-definite as required

For the 2I2O case an identical approach may be followedto obtain similar expressions The definition of z changes toz = z

(1 2)z(1 2)

119879 and the parameter update rate becomes

d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)

The matrices A and B become

Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)


An important consideration in the 2I2O case is theasymptotic stability of the zero dynamics considered inAppendix B which is required for the adaptive scheme towork For the particular 2I2O configuration considered thepresence of structural damping ensures stability of the zerodynamics thus enabling application of the adaptive method

Example Stabilisation of the Aeroelastic System Using Adap-tive Feedback Linearisation The instability shown in Figure 9may be avoided altogether by implementing the adaptivecontroller described in Section 5 above In the 3I3O case theresulting controlled response for the same initial conditionsis shown in Figure 12

A comparison of Figure 12 with Figure 5 shows thatthe controlled responses are similar but not identicalAlthough adaptive feedback linearisation destroys the

original pole-placement it can be seen that the controlledresponse is stable In fact the adaptive controller successfullydrives the responses to zero for values of |120599pe|max (set asan initial condition) up to around 00735 rad The requiredcontrol in this case is accomplished through achievablecontrol surface deflection angles (asymp11∘) and actuator torquemagnitude (55 kNm)

In the case of the 2I2O system the system response withadaptation shown in Figure 13 should be compared to thatin Figure 11 for feedback linearisation without adaptationand the same error in the cubic stiffness parameter Thedegradation of the system to LCO is completely removed inFigure 13

It can also be seen that the controlled responses as wellas the response of the uncontrolled coordinate converge tozero when the adaptive controller is implemented In fact


0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)

Figure 12 3I3O system with adaptive feedback linearisation response at 80msminus1

this convergence takes place rapidly When converted to thephysical domain the resulting magnitudes of the coordinatesoccur within acceptable limits (maximum values of 033mm024∘ 292∘ for 120577

1 120594

2 120599pe resp) As before it is found that the

control surface deflections required to achieve the responsesshown in Figure 13 are feasible (120573

1 1205732asymp 30∘)

6 Conclusions

Nonlinear systems are ubiquitous in vibrations engineeringand aeroelasticity but the analysis is mathematically intricateand complicated The paper presents the feedback linearisa-tion methodology whereby a nonlinear system is renderedlinear bymeans of active control Having neutralised the non-linearity the system may be treated using well-understoodlinear analysis methods such as modal decomposition whichgenerally cannot be applied to nonlinear systems directly

The technique is formulated using the second-order repre-sentation of elastomechanical and aeroelastic systems withstiffness nonlinearity familiar to the vibrations communityThis has certain advantages over the conventional state-spaceformulation in that repeated output differentiation usuallydescribed using the Lie algebra notation is unnecessaryThe purpose of the linearising controller is to cancel thenonlinearity completely and therefore it results in a trulylinear system rather than linearisation about an operatingpoint limited to small perturbations or quasi-linearisation aswith describing functions The controller may be designed tocancel the nonlinearity only or to cancel the complete open-loop dynamics In the former case there is shown to be greatercontrol flexibility but the latter case is found to have somemerit in the particular example considered of a flexible wing-pylon-engine system with decoupled bending torsional andpylon-engine modes


0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)

Figure 13 2I2O system with feedback linearisation response at 975msminus1

Exact feedback linearisation requires knowledge of thenonlinearity and when every degree of freedom of thesystem is available formeasurement (and actuation) and thenlinearisation may be achieved completely When fewer thanthe full set of degrees of freedom is available formeasurementand actuation then the system can generally be partitionedinto independent linear and nonlinear subsystems with thedimension of the latter being the difference between thenumber of degrees of freedom and the number of sensorsand actuators If the nonlinear subsystem is stable thenthe dynamics of the linear subsystem may be controlled asrequired The problem of an imprecisely known nonlinearterm was addressed using adaptive feedback linearisationresulting in a parameter update rule that evolves in time toensure stability It requires an additional state variable to do

this and is likely to be more expensive than using feedbacklinearisation without adaptivity

Feedback linearisation techniques were illustrated usinga three degree of freedom aeroelastic model consisting of aflexible wing and a rigid pylon-engine system attached tothe wing via a nonlinear torsional spring with pole place-ment of the fully linearised and partially linearised systemcorresponding to the measurement (and actuation) at threedegrees of freedom (3I3O) and twodegrees of freedom (2I2O)respectively The parameters of the system were chosen care-fully to represent a real aircraft Adaptive linearisation wasapplied successfully to compensate for a nonlinear parametererror in the case of 3I3O and in the 2I2O case the nonlinearsubsystem was found to be stable The magnitudes of theaileron deflection angles were physically realisable


Appendices

A Control Forces and Nonlinear Spring Forces

Control forces are applied to the wing-pylon-engine systemby means of control surfaces It is assumed in this work thattwo control surfaces (different from [14]) are available thefirst (closest to the wing root) spanning 85 of the lengthof the wing and the second spanning the remaining length(the contribution of the control surfaces to the dynamics ofthe overall system is neglected) The widths of the first andsecond control surfaces are assumed to be 20 and 3333of the chord length respectivelyThese particular dimensionshave been chosen so as to optimise the distribution of workperformed by each control surface In addition to the aileronsit is assumed that a separate actuator is available to applya torque 119879pe directly on the engine rotational degree offreedomThe forcing vector is found to be

f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2

119908 120572 = 085 and each surface will

have its own deflection angle 1205731 1205732and set of aerodynamic

parameters 119886119862 119887119862 which are the rates of change of lift

coefficient and moment coefficient respectively with respectto control surface deflection angle

A cubic hardening nonlinearity is assumed in the tor-sional spring connecting the pylon-engine to the wing Thenonlinear force developed in the spring may be expressed as

119891nl = 1198701198791205993

pe (A2)

where119870119879is the stiffness coefficient of the cubic nonlinearity

The overall nonlinear force vector then takes the form

fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)

with T defined in (18)

B Zero Dynamics Expressions (2I2O Case)

In the 2I2O configuration and using (11)

119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)

and 119892119894119895denotes the 119894119895th term of thematrixGThen by invert-

ing the transformation matrix T119911119909 the following expressions

are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)

Then from (15) and differentiating the bottom row of (B1) itis found that

3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)

Evidently the internal dynamics expressions in (B4) arenonlinear Now the zero dynamics may be obtained bysetting to zero the coordinates corresponding to the externaldynamics (ie the partially linearised system) In this casethe external coordinates are 119911

1 1199112 1 2so that the zero

dynamics are given by

3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)

where the subscript ldquo119911119889rdquo signifies that the coordinates arespecified under zero dynamics conditions

Zero Dynamics Equilibrium Point Analysis To obtain theequilibrium points of the zero dynamics it is necessary to set

3119911119889=

3119911119889= 0 Equation (B6) then produces

(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)

which provides the two solutions

119911

3119911119889= 0 119911

3119911119889= plusmnradicminus

120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)

It is found that the term inside the square root is alwaysnegative and therefore the second solution for 119911

3119911119889is

inadmissibleThus the only possible equilibrium point of thezero dynamics is the trivial solution

119911

3

3

119911119889

=

0

0

(B9)


Thenature of the above equilibriumpointmay be determinedby examining the eigenvalues of (B6) evaluated at theequilibrium point

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)

The roots are found to be complex with negative real partsthus revealing the type of equilibrium point to be a stablefocus

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theauthorswish to acknowledge the support of EPSRCGrantno EPJ0049871 on Nonlinear Active Vibration Suppressionin Aeroelasticity

References

[1] A Isidori Nonlinear Control Systems Springer New York NYUSA 1995

[2] T I Fossen and M J Paulsen ldquoAdaptive feedback linearizationapplied to steering of shipsrdquo Modeling Identification and Con-trol vol 14 no 4 pp 229ndash237 1993

[3] J Ko A J Kurdila and T W Strganac ldquoNonlinear control ofa prototypical wing section with torsional nonlinearityrdquo Journalof Guidance Control and Dynamics vol 20 no 6 pp 1181ndash11891997

[4] J Ko A J Kurdila and T Strganac ldquoStability and control of astructurally nonlinear aeroelastic systemrdquo Journal of GuidanceControl and Dynamics vol 21 no 5 pp 718ndash725 1998

[5] J Ko T W Strganac and A J Kurdila ldquoAdaptive feedbacklinearization for the control of a typical wing section withstructural nonlinearityrdquo Nonlinear Dynamics vol 18 no 3 pp289ndash301 1999

[6] M M Monahemi and M Krstic ldquoControl of wing rock motionusing adaptive feedback linearizationrdquo Journal of GuidanceControl and Dynamics vol 19 no 4 pp 905ndash912 1996

[7] A Poursamad ldquoAdaptive feedback linearization control ofantilock braking systems using neural networksrdquoMechatronicsvol 19 no 5 pp 767ndash773 2009

[8] C P Bechlioulis and G A Rovithakis ldquoRobust adaptive controlof feedback linearizable MIMO nonlinear systems with pre-scribed performancerdquo IEEE Transactions on Automatic Controlvol 53 no 9 pp 2090ndash2099 2008

[9] K Shojaei AMohammad Shahri andA Tarakameh ldquoAdaptivefeedback linearizing control of nonholonomic wheeled mobile

robots in presence of parametric and nonparametric uncertain-tiesrdquo Robotics and Computer-Integrated Manufacturing vol 27no 1 pp 194ndash204 2011

[10] L Tuan S-G Lee V-H Dang S Moon and B Kim ldquoPartialfeedback linearization control of a three-dimensional overheadcranerdquo International Journal of Control Automation and Sys-tems vol 11 no 4 pp 718ndash727 2013

[11] Y M Ram and J E Mottershead ldquoMultiple-input activevibration control by partial pole placement using the methodof receptancesrdquo Mechanical Systems and Signal Processing vol40 no 2 pp 727ndash735 2013

[12] J R Wright and J E Cooper Introduction to Aircraft Aeroelas-ticity and Loads Wiley Chichester UK 2007

[13] S Jiffri J E Mottershead and J E Cooper ldquoAdaptive feedbacklinearisation and control of a flexible aircraft wingrdquo in Topicsin Modal Analysis Volume 7 Conference Proceedings of theSociety for Experimental Mechanics Series Springer 2013

[14] httpwwwairbuscomaircraftfamiliespassengeraircrafta330familya330-200specifications

[15] D P Atherton Stability of Nonlinear Systems John Wiley ampSons New York NY USA 1981

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Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


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Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Examples include Fossen and Paulsen [2] who applied adap-tive feedback linearisation to the automatic steering of shipsIn aeroelasticity Ko and his colleagues [3ndash5] used feedbacklinearisation methods (including adaptation) and carried outa series of wind-tunnel tests on a two degree of freedom aero-foil with either one or two control surfaces They found thatwith an erroneous nonlinear parameter but without adapta-tion their system reached a nonzero equilibrium rather thanthe zero equilibrium that is usually sought Monahemi andKrstic [6] employed adaptive feedback linearisation to updatethe aerodynamic parameters in their nonlinear model andthereby suppress wing-rockmotion a phenomenon triggeredprimarily by aerodynamic nonlinearities Poursamad [7]implemented a hybrid neural-network controller for antilockbraking with adaptive feedback linearisation to handle non-linear and time-varying brake parameters Bechlioulis andRovithakis [8] developed a multiple-input multiple-outputtracking controller with adaptive feedback linearisation andShojaei et al [9] demonstrated the ability of adaptive feedbacklinearisation in aiding effective trajectory tracking in thepresence of both parametric andnonparametric uncertaintiesin wheeled robots Tuan et al [10] designed a controller basedon partial feedback linearisation of the nonlinear dynamics ofa 3D overhead crane

In this paper the problem of active control of nonlinearsystems of the form

A1x + A2x + A3x + fnl (x x) = Bu (1)

representative of nonlinear vibration problems in elastome-chanics and aeroelasticity is consideredThe vectors x x andu typically contain the state variables and inputs respectivelyThe nonlinearity fnl is given as a function of x and x and thematricesA

1A2A3B represent the usual system parameters

This class of problem characterised by the second-ordermatrix differential equation with additional nonlinearityconfined to the left-hand side of (1) prevails to a very largeextent in engineering mechanics and is therefore worthy ofthe special attention devoted to it in this paper The classicaloutput feedback linearisation [1] may be greatly simplified inthe case of elastomechanical or aeroelastic systems describedby (1) In particular

(i) the essential theory is carried out entirely using thesecond-order matrix differential equation familiarto structural and aero-structural dynamicists withthe result that the need for repeated differentiationusually described using the Lie-algebra notation isrendered unnecessary

(ii) a linear transformation applies between the statevariable in (1) and the coordinates of the linearisedsystem

(iii) cancellation of the whole of the open-loop systemdynamics (not just the nonlinear terms) results in a setof independent linear single degree of freedom sys-tems for the application of conventional LTI controlmethods

(iv) complete linearisation may be achieved with an equalnumber of actuators and sensors If the number of

actuators and sensors is less than the dimension of thesystem then there will remain a nonlinear subsystemof dimension equal to the difference between thedimension of the full system and the number ofsensors (and actuators) This subsystem can be madeindependent of the linearised part and methods aredescribed to check its stability

(v) Adaptive feedback linearisation is described for thetreatment of an incorrect estimate of the nonlinearityThis makes use of Lyapunov stability criteria andresults in a parameter update rule that evolves withtime to ensure stability of the system

The method is illustrated by means of a three degree offreedom aeroelastic system consisting of a flexible wingdescribed in terms of two assumed modes in bending andtorsion and a third degree of freedom that describes theangularmotion of an underslung pylon-engine assemblyTheparameters and dimensions of the system are carefully chosento have realistic values

2 Active Feedback Linearisation

Feedback linearisation [1] is a process whereby a nonlinearsystem is rendered linear by virtue of active control UnlikeJacobian linearisation it is exact and does not entail anapproximation at any stage The method is implemented bytransforming a nonlinear system into a linear one For theclass of second-order systems given by (1) considered in thispaper

x = f (x x) + Gu f (x x) = Ψx +Φx +Ωfnl

Ψ = minusAminus11A3 Φ = minusAminus1

1A2 Ω = minusAminus1

1 G = Aminus1

1B(2)

becomes

z = Ψz +Φz + Gu (3)

In these equations u and u are respectively the actual (orphysical) input applied to the nonlinear system and the so-called ldquoartificialrdquo input to the linear system The matrices ΨΦ G are dependent on the chosen inputs u The mappingfrom the nonlinear domain to the linearised domain isachieved through a nonsingular linear coordinate transfor-mation The actual input is designed to neutralise the effectof the nonlinearity a procedure which can sometimes beachieved in full and sometimes partially as will be explainedtheoretically and by means of illustrative examples

The process is quite straightforward for elastomechanicaland aeroelastic systems described by second-order matrixdifferential equations with nonlinearity in the system param-eters The method described by Isidori [1] allows for greatergenerality including nonlinearity in the input and outputas well as in the system parameters which is not requiredhere and its omission leads to simplifications which aid theunderstanding of an otherwise fairly complicated procedure

The feedback linearisation procedure classically using theLie algebra entails repeated differentiation of each of the






(

1

2

119899

) =(

119906

1

119906

2

119906

119899

) (5)



119906

1= minus [1205891

]1]

119909

1

1

119906

2= minus [1205892

]2]

119909

2

2

119906

119899= minus [120589119899

]119899]

119909

119899

119899

(6)





119906

1

119906

2

119906

119899

= minus

[

[

[

[

[

120589

11]11120589


1119899]1119899

120589

21]21120589


2119899]2119899

d

120589

1198991]1198991120589

1198992]1198992sdot sdot sdot 120589

119899119899]119899119899

]

]

]

]

]

times

119909

1

1

119909

2

2

119909

119899

119899

(8)






1

119898

119898+1

119899

=

119891

1(x x)

119891

119898(x x)

119891

119898+1(x x)

119891

119899(x x)

+

[

[

[

[

[

[

[

[

[

[

119892


1119898

d

119892


119898119898

119892


119898+1119898

d

119892


119899119898

]

]

]

]

]

]

]

]

]

]

119906

1

119906

119898

(9)


(1199111119911


119898)

119879

= (1199091119909


119898)

119879

(10)


(11198991119898)so that

(

119909

1

119909

119899

) = V(119911

119898+1

119911

119899

)


V119879G(11198991119898)

= 0


(11)


u = [G11198981119898

]

minus1

(u minus f(x x)(11198981)

)

f(x x)(11198981)

= Ψ(1119898)

x +Φ(1119898)

x +Ω(1119898)

fnl(12)



119898

119879 expressed as

(

1

2

119898

) =(

1

2

119898

) =(

119906

1

119906

2

119906

119898

) (13)


119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

+ V119879G(11198991119898)

119906

1

119906

119898

(14)

so that from (11)

119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

(15)



1 119911

119898)









120577 = 119910

2119902

1+ 119910 (119909 minus 119909

119891) 119902

2 (17)



x

y

120577







2 to 120599pe


120577

1

120594

2

120599pe

=

[

[

119910

2

10 0

0 119910

20

0 minus119910

21

]

]

119902

1

119902

2

119902pe

(18)

where 1205771and 120594



1= 119910

2) The







119879= 300 times 119896








A119908= 10

5[

[

949 00633 0

00633 00942 0

0 0 0

]

]

E119908= 10

8[

[

110 0 0

0 0142 0

0 0 0

]

]

Ape = 103[

[

476 0 151

0 0 0

151 0 0647

]

]

Epe = 106[

[

0 0 0

0 190 minus0958

0 minus0958 0511

]

]

B = 104 [[

298 0 0

minus0229 00169 0

0 0 0

]

]

C = 103 [[

0 497 0

0 minus0406 0

0 0 0

]

]

(19)


f119888= 10

2[

[

minus319119881

2minus289119881

20

minus0302119881

2minus0120119881

2minus00188

0 0 001

]

]

times

120573

1

120573

2

119879pe

(20)

where 1205731and 120573



fnl = 108

0

minus287

153

120599

3 119910

2= 119910

1= 1875 (21)


0 20 40 60 80 100 120 1402468

Air speed (ms)Nat

ural

freq

uenc

y (H

z)


Mode 3-(torsion)

(a)

012


Mode 3-(torsion)

Dam

ping

ratio

()

0 20 40 60 80 100 120 140Air speed (ms)

(b)


87 88 89 90

0

1

2

Time (s)

minus2

minus1

120599pe

(deg

)





1=





0 20 40 60 80 100 120

0

2

4

Time (s)

minus2

minus4

120599pe

(deg

)




120596

1= 093Hz 120585

1= 001

120596

2= 29Hz 120585

2= 001

120596

3= 495Hz 120585

3= 001

(22)


1= 0333mm 120594

2=



0 20 40 60

0

05

1

Time (s)

minus 05

minus1

q1

times10minus4

(a)

0 5 10 15

0

1

2

Time (s)

minus1

minus 2

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus 005

minus 01

qpe

(c)




0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20Time (s)

1205732

(deg

)

0

20

minus20

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)



k1= 1 1 0

119879

k2= 1 minus1 0

119879

k3= 0 0 1

119879

(23)




0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)






1and 120573


1



D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)


120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














(

1

2

119899

) =(

119906

1

119906

2

119906

119899

) (5)



119906

1= minus [1205891

]1]

119909

1

1

119906

2= minus [1205892

]2]

119909

2

2

119906

119899= minus [120589119899

]119899]

119909

119899

119899

(6)





119906

1

119906

2

119906

119899

= minus

[

[

[

[

[

120589

11]11120589


1119899]1119899

120589

21]21120589


2119899]2119899

d

120589

1198991]1198991120589

1198992]1198992sdot sdot sdot 120589

119899119899]119899119899

]

]

]

]

]

times

119909

1

1

119909

2

2

119909

119899

119899

(8)






1

119898

119898+1

119899

=

119891

1(x x)

119891

119898(x x)

119891

119898+1(x x)

119891

119899(x x)

+

[

[

[

[

[

[

[

[

[

[

119892


1119898

d

119892


119898119898

119892


119898+1119898

d

119892


119899119898

]

]

]

]

]

]

]

]

]

]

119906

1

119906

119898

(9)


(1199111119911


119898)

119879

= (1199091119909


119898)

119879

(10)


(11198991119898)so that

(

119909

1

119909

119899

) = V(119911

119898+1

119911

119899

)


V119879G(11198991119898)

= 0


(11)


u = [G11198981119898

]

minus1

(u minus f(x x)(11198981)

)

f(x x)(11198981)

= Ψ(1119898)

x +Φ(1119898)

x +Ω(1119898)

fnl(12)



119898

119879 expressed as

(

1

2

119898

) =(

1

2

119898

) =(

119906

1

119906

2

119906

119898

) (13)


119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

+ V119879G(11198991119898)

119906

1

119906

119898

(14)

so that from (11)

119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

(15)



1 119911

119898)









120577 = 119910

2119902

1+ 119910 (119909 minus 119909

119891) 119902

2 (17)



x

y

120577







2 to 120599pe


120577

1

120594

2

120599pe

=

[

[

119910

2

10 0

0 119910

20

0 minus119910

21

]

]

119902

1

119902

2

119902pe

(18)

where 1205771and 120594



1= 119910

2) The







119879= 300 times 119896








A119908= 10

5[

[

949 00633 0

00633 00942 0

0 0 0

]

]

E119908= 10

8[

[

110 0 0

0 0142 0

0 0 0

]

]

Ape = 103[

[

476 0 151

0 0 0

151 0 0647

]

]

Epe = 106[

[

0 0 0

0 190 minus0958

0 minus0958 0511

]

]

B = 104 [[

298 0 0

minus0229 00169 0

0 0 0

]

]

C = 103 [[

0 497 0

0 minus0406 0

0 0 0

]

]

(19)


f119888= 10

2[

[

minus319119881

2minus289119881

20

minus0302119881

2minus0120119881

2minus00188

0 0 001

]

]

times

120573

1

120573

2

119879pe

(20)

where 1205731and 120573



fnl = 108

0

minus287

153

120599

3 119910

2= 119910

1= 1875 (21)


0 20 40 60 80 100 120 1402468

Air speed (ms)Nat

ural

freq

uenc

y (H

z)


Mode 3-(torsion)

(a)

012


Mode 3-(torsion)

Dam

ping

ratio

()

0 20 40 60 80 100 120 140Air speed (ms)

(b)


87 88 89 90

0

1

2

Time (s)

minus2

minus1

120599pe

(deg

)





1=





0 20 40 60 80 100 120

0

2

4

Time (s)

minus2

minus4

120599pe

(deg

)




120596

1= 093Hz 120585

1= 001

120596

2= 29Hz 120585

2= 001

120596

3= 495Hz 120585

3= 001

(22)


1= 0333mm 120594

2=



0 20 40 60

0

05

1

Time (s)

minus 05

minus1

q1

times10minus4

(a)

0 5 10 15

0

1

2

Time (s)

minus1

minus 2

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus 005

minus 01

qpe

(c)




0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20Time (s)

1205732

(deg

)

0

20

minus20

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)



k1= 1 1 0

119879

k2= 1 minus1 0

119879

k3= 0 0 1

119879

(23)




0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)






1and 120573


1



D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)


120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1

119898

119898+1

119899

=

119891

1(x x)

119891

119898(x x)

119891

119898+1(x x)

119891

119899(x x)

+

[

[

[

[

[

[

[

[

[

[

119892


1119898

d

119892


119898119898

119892


119898+1119898

d

119892


119899119898

]

]

]

]

]

]

]

]

]

]

119906

1

119906

119898

(9)


(1199111119911


119898)

119879

= (1199091119909


119898)

119879

(10)


(11198991119898)so that

(

119909

1

119909

119899

) = V(119911

119898+1

119911

119899

)


V119879G(11198991119898)

= 0


(11)


u = [G11198981119898

]

minus1

(u minus f(x x)(11198981)

)

f(x x)(11198981)

= Ψ(1119898)

x +Φ(1119898)

x +Ω(1119898)

fnl(12)



119898

119879 expressed as

(

1

2

119898

) =(

1

2

119898

) =(

119906

1

119906

2

119906

119898

) (13)


119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

+ V119879G(11198991119898)

119906

1

119906

119898

(14)

so that from (11)

119898+1

119899

= V119879

119891

1(z z)

119891

119899(z z)

(15)



1 119911

119898)









120577 = 119910

2119902

1+ 119910 (119909 minus 119909

119891) 119902

2 (17)



x

y

120577







2 to 120599pe


120577

1

120594

2

120599pe

=

[

[

119910

2

10 0

0 119910

20

0 minus119910

21

]

]

119902

1

119902

2

119902pe

(18)

where 1205771and 120594



1= 119910

2) The







119879= 300 times 119896








A119908= 10

5[

[

949 00633 0

00633 00942 0

0 0 0

]

]

E119908= 10

8[

[

110 0 0

0 0142 0

0 0 0

]

]

Ape = 103[

[

476 0 151

0 0 0

151 0 0647

]

]

Epe = 106[

[

0 0 0

0 190 minus0958

0 minus0958 0511

]

]

B = 104 [[

298 0 0

minus0229 00169 0

0 0 0

]

]

C = 103 [[

0 497 0

0 minus0406 0

0 0 0

]

]

(19)


f119888= 10

2[

[

minus319119881

2minus289119881

20

minus0302119881

2minus0120119881

2minus00188

0 0 001

]

]

times

120573

1

120573

2

119879pe

(20)

where 1205731and 120573



fnl = 108

0

minus287

153

120599

3 119910

2= 119910

1= 1875 (21)


0 20 40 60 80 100 120 1402468

Air speed (ms)Nat

ural

freq

uenc

y (H

z)


Mode 3-(torsion)

(a)

012


Mode 3-(torsion)

Dam

ping

ratio

()

0 20 40 60 80 100 120 140Air speed (ms)

(b)


87 88 89 90

0

1

2

Time (s)

minus2

minus1

120599pe

(deg

)





1=





0 20 40 60 80 100 120

0

2

4

Time (s)

minus2

minus4

120599pe

(deg

)




120596

1= 093Hz 120585

1= 001

120596

2= 29Hz 120585

2= 001

120596

3= 495Hz 120585

3= 001

(22)


1= 0333mm 120594

2=



0 20 40 60

0

05

1

Time (s)

minus 05

minus1

q1

times10minus4

(a)

0 5 10 15

0

1

2

Time (s)

minus1

minus 2

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus 005

minus 01

qpe

(c)




0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20Time (s)

1205732

(deg

)

0

20

minus20

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)



k1= 1 1 0

119879

k2= 1 minus1 0

119879

k3= 0 0 1

119879

(23)




0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)






1and 120573


1



D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)


120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











x

y

120577







2 to 120599pe


120577

1

120594

2

120599pe

=

[

[

119910

2

10 0

0 119910

20

0 minus119910

21

]

]

119902

1

119902

2

119902pe

(18)

where 1205771and 120594



1= 119910

2) The







119879= 300 times 119896








A119908= 10

5[

[

949 00633 0

00633 00942 0

0 0 0

]

]

E119908= 10

8[

[

110 0 0

0 0142 0

0 0 0

]

]

Ape = 103[

[

476 0 151

0 0 0

151 0 0647

]

]

Epe = 106[

[

0 0 0

0 190 minus0958

0 minus0958 0511

]

]

B = 104 [[

298 0 0

minus0229 00169 0

0 0 0

]

]

C = 103 [[

0 497 0

0 minus0406 0

0 0 0

]

]

(19)


f119888= 10

2[

[

minus319119881

2minus289119881

20

minus0302119881

2minus0120119881

2minus00188

0 0 001

]

]

times

120573

1

120573

2

119879pe

(20)

where 1205731and 120573



fnl = 108

0

minus287

153

120599

3 119910

2= 119910

1= 1875 (21)


0 20 40 60 80 100 120 1402468

Air speed (ms)Nat

ural

freq

uenc

y (H

z)


Mode 3-(torsion)

(a)

012


Mode 3-(torsion)

Dam

ping

ratio

()

0 20 40 60 80 100 120 140Air speed (ms)

(b)


87 88 89 90

0

1

2

Time (s)

minus2

minus1

120599pe

(deg

)





1=





0 20 40 60 80 100 120

0

2

4

Time (s)

minus2

minus4

120599pe

(deg

)




120596

1= 093Hz 120585

1= 001

120596

2= 29Hz 120585

2= 001

120596

3= 495Hz 120585

3= 001

(22)


1= 0333mm 120594

2=



0 20 40 60

0

05

1

Time (s)

minus 05

minus1

q1

times10minus4

(a)

0 5 10 15

0

1

2

Time (s)

minus1

minus 2

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus 005

minus 01

qpe

(c)




0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20Time (s)

1205732

(deg

)

0

20

minus20

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)



k1= 1 1 0

119879

k2= 1 minus1 0

119879

k3= 0 0 1

119879

(23)




0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)






1and 120573


1



D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)


120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80 100 120 1402468

Air speed (ms)Nat

ural

freq

uenc

y (H

z)


Mode 3-(torsion)

(a)

012


Mode 3-(torsion)

Dam

ping

ratio

()

0 20 40 60 80 100 120 140Air speed (ms)

(b)


87 88 89 90

0

1

2

Time (s)

minus2

minus1

120599pe

(deg

)





1=





0 20 40 60 80 100 120

0

2

4

Time (s)

minus2

minus4

120599pe

(deg

)




120596

1= 093Hz 120585

1= 001

120596

2= 29Hz 120585

2= 001

120596

3= 495Hz 120585

3= 001

(22)


1= 0333mm 120594

2=



0 20 40 60

0

05

1

Time (s)

minus 05

minus1

q1

times10minus4

(a)

0 5 10 15

0

1

2

Time (s)

minus1

minus 2

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus 005

minus 01

qpe

(c)




0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20Time (s)

1205732

(deg

)

0

20

minus20

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)



k1= 1 1 0

119879

k2= 1 minus1 0

119879

k3= 0 0 1

119879

(23)




0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)






1and 120573


1



D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)


120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60

0

05

1

Time (s)

minus 05

minus1

q1

times10minus4

(a)

0 5 10 15

0

1

2

Time (s)

minus1

minus 2

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus 005

minus 01

qpe

(c)




0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20Time (s)

1205732

(deg

)

0

20

minus20

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)



k1= 1 1 0

119879

k2= 1 minus1 0

119879

k3= 0 0 1

119879

(23)




0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)






1and 120573


1



D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)


120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60

0

1

2

Time (s)

minus2

minus1

q1

times10minus3

(a)

times10minus3

0 20 40 60

0

1

2

Time (s)

minus2

minus1

q2

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)






1and 120573


1



D mod = 105[

[

205 00103 000220

00103 00767 minus000282

000220 minus000282 000356

]

]

(24)


120596

1= 093Hz 120585

1= 001

120596

2= 495Hz 120585

2= 001

(25)


0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20

0

20

Time (s)

minus20

1205731

(deg

)

(a)

0 5 10 15 20

0

50

Time (s)

minus50

1205732

(deg

)

(b)

0 5 10 15 20

0

50

Time (s)

minus50

Tpe

(kN

m)

(c)










x = u +Ω120576 120576 = fnl minus f1015840nl (27)




z = T119911119909x (29)


u = Γ1015840 xx = [Γ1015840

1Γ1015840

2]

xx Γ

1015840= Γ119911[

T119911119909

00 T119911119909

] (30)


x = Γ10158401x + Γ10158402x +Ω120576 (31)


Γ1015840

1119911= minus

[

[

120589

1

120589

2

120589

2

]

]

Γ1015840

2119911= minus

[

[

]1

]2

]2

]

]

(32)


z = Αz + Δ119870119879120599

3

peBΩ(T(3))119879

Α = [0 IΓ1015840

1119911Γ1015840

2119911

]

B = [0I] z = zz

(33)


119870

1015840

119879= 119870

119879+ Δ119870

119879 (34)

120576 = Δ119870119879120599

3

peΩ(T (3) )119879

(35)


0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1

0

2

minus2

minus4

minus6

q1

times10minus3

Time (s)

(a)

0 05 1

0

05

minus1

minus05

q2

Time (s)

(b)

0 05 1

0

2

4

6

8

minus2

qpe

Time (s)

(c)






119881 (z Δ119870119879) = z119879Pz + Δ1198702

119879gt 0 (36)


119881 (z Δ119870119879) =

z119879 (P + P119879) z + 2Δ119870119879

d (Δ119870119879)

d119905

(37)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z

+ 2Δ119870

119879(

d (Δ119870119879)

d119905+ 120599

3

peT(3)Ω119879B119879Pz)

(38)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 5 10 15 20

0

1

2

Time (s)

minus2

minus1

times10minus3

q2

(b)

0 10 20

0

005

01

Time (s)

minus01

minus005

qpe

(c)



d (Δ119870119879)

d119905= minus120599

3

peT(3)Ω119879B119879Pz (39)


119881 (z Δ119870119879) = z119879 (A119879P + PA) z (40)



d (Δ119870119879)

d119905=

d (119870119879)

d119905minus

d (1198701015840119879)

d119905= minus

d (1198701015840119879)

d119905

(41)



d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879B119879Pz (42)



119879


Q = minus (A119879P + PA) (43)



(1 2)z(1 2)


d (1198701015840119879)

d119905= 120599

3

peT(3)Ω119879

(12)B119879Pz (44)


Α = [0 I

2times2

Γ1015840

1119911 (1212)Γ1015840

2119911 (1 212)

] B = [ 0I2times2

] (45)


0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 10 20

0

05

1

Time (s)

minus1

minus05

q1

times10minus4

(a)

0 10 20

0

001

002

Time (s)

minus002

minus001

q2

(b)

0 10 20

0

01

02

Time (s)

minus01

minus02

qpe

(c)









0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60

0

2

4

Time (s)

minus2

minus4

q1

times10minus4

(a)

0 5 10 15

0

5

Time (s)

minus5

q2

times10minus3

(b)

0 10 20

0

005

01

Time (s)

minus005

minus01

qpe

(c)



1 120594



1 1205732asymp 30∘)

6 Conclusions




0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60

0

05

1

Time (s)

minus05

minus1

times10minus4

q1

(a)

0 20 40 60

0

2

4

Time (s)

times10minus3

minus2

minus4

q2

(b)

0 20 40 60

0

005

01

Time (s)

minus005

minus01

qpe

(c)






Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Appendices



f119888=

[

[

[

[

[

[

minus

1

6

119903120572

3119886

1198621119904

119908minus

1

6

119903 (1 minus 120572

3) 119886

1198622119904

1199080

1

4

119903120572

2119887

1198621119888

119908

1

4

119903 (1 minus 120572

2) 119887

1198622119888

119908minus119910

2

0 0 1

]

]

]

]

]

]

times

120573

1

120573

2

119879pe

= Bu

(A1)

where 119903 = 120588119881

2119888

119908119904

2






119891nl = 1198701198791205993

pe (A2)



fnl = T119879(0

0

119891nl

) = (

0

minus119910

2119891nl119891nl

) (A3)




119911

1

119911

2

119911

3

=

[

[

1 0 0

0 1 0

120590

1120590

2120590

3

]

]

119909

1

119909

2

119909

3

(B1)

where

120590

1= (119892

21119892

32minus 119892

31119892

22) 120590

2= (119892

31119892

12minus 119892

11119892

32)

120590

3= (119892

11119892

22minus 119892

21119892

12)

(B2)



are obtained

119909

1= 119911

1 119909

2= 119911

2 119909

3= minus

120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3

(B3)


3= 120590119879f (z z) 120590

119879= 1205901

120590

2120590

3 (B4)

where

f (z z) = Φ1 2 (minus120590

1

120590

3

1minus

120590

2

120590

3

2+

1

120590

3

3)

119879

+Ψ1199111 1199112 (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3)

119879

+Ω fnl (z)

fnl (z) = 119870119879(minus11991021199112 + (minus120590

1

120590

3

119911

1minus

120590

2

120590

3

119911

2+

1

120590

3

119911

3))

3

times (T(3))

119879

(B5)




3119911119889=

1

120590

3

120590119879Ψ(3)119911

3119911119889+

1

120590

3

120590119879Φ(3)

3119911119889

+

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

3

3119911119889

(B6)



3119911119889=


(120590119879Ψ(3)+ 119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889) 119911

3119911119889= 0 (B7)


119911

3119911119889= 0 119911


120590119879Ψ(3)

119870

119879120590119879Ω(T

(3))

119879

(B8)


3119911119889is


119911

3

3

119911119889

=

0

0

(B9)



119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)

minus

1

120590

3

3

119870

119879120590119879Ω(T(3))

119879

119911

2

3119911119889

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119911

3119911119889= 0

3119911119889= 0

= 0

(B10)

or

119904

2minus

1

120590

3

120590119879Φ(3)119904 minus

1

120590

3

120590119879Ψ(3)= 0 (B11)




Acknowledgment


References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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