Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 913458 14 pageshttpdxdoiorg1011552013913458
Research ArticleEffective and Robust Generalized Predictive SpeedControl of Induction Motor
Patxi Alkorta1 Oscar Barambones2 Asier Zubizarreta3 and Joseacute Antonio Cortajarena14
1 Department of Automatic Control and Systems Engineering EUITI de Eibar University of the Basque Country (UPVEHU)Otaola Hiribidea 29 20600 Eibar Spain
2Department of Automatic Control and Systems Engineering EUI de Vitoria-Gasteiz University of the Basque Country (UPVEHU)Nieves Cano 12 01006 Vitoria-Gasteiz Spain
3 Department of Automatic Control and Systems Engineering ETSI de Bilbao University of the Basque Country (UPVEHU)Alameda Urquijo sn 48013 Bilbao Spain
4Department of Electronics Technology EUITI de Eibar University of The Basque Country (UPVEHU) Otaola Hiribidea 2920600 Eibar Spain
Correspondence should be addressed to Jose Antonio Cortajarena joseancortajarenaehues
Received 20 June 2013 Revised 30 September 2013 Accepted 16 October 2013
Academic Editor Bo-Chao Zheng
Copyright copy 2013 Patxi Alkorta et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents and validates a new proposal for effective speed vector control of inductionmotors based on linear GeneralizedPredictive Control (GPC) lawThe presented GPC-PI cascade configuration simplifies the design with regard to GPC-GPC cascadeconfiguration maintaining the advantages of the predictive control algorithm The robust stability of the closed loop system isdemonstrated by the poles placement method for several typical cases of uncertainties in inductionmotorsThe controller has beentested using several simulations and experiments and has been compared with Proportional Integral Derivative (PID) and SlidingMode (SM) control schemes obtaining outstanding results in speed tracking even in the presence of parameter uncertaintiesunknown load disturbance and measurement noise in the loop signals suggesting its use in industrial applications
1 Introduction
The Model Predictive Control (MPC) groups a set of con-trollers which are based on the model of the system and theknown future reference for optimal control signal calculationThe operational principle of predictive control is to calculatein advance the control signal required by the system whenthe future input reference that will be applied is knownbeforehand [1] In this sense the system is able to react tothe input reference anticipating its changes and avoiding theeffects of delay in system response [2] There are countlessapplications in industry where the input reference is knownbeforehand such as robotic systems and machine toolsTherefore in all these systems predictive control algorithmscan be implemented Since Clarke et al proposed the designprinciples of Generalized Predictive Control [3 4] many
authors have used this advanced technique for inductionmotor control in the last two decades There is extensiveresearch related to the application of predictive controllersin electric drives and for this reason predictive algorithmscompete with other advanced control techniques such asfuzzy control [5] sliding mode control [6 7] and nonlinear119867
inf control [8]Predictive algorithms are often implemented using two or
more GPC blocks to control several loops of the electricalmachine and usually they are connected in cascade form[9ndash11] Frequently only one predictive regulator for thecontrol of the main variable of the machine is implementedsuch as speed [12ndash15] or position [16] while the rest ofthe variables are controlled with classical algorithms usuallyPIPID and hysteresis comparators Some authors use morecomplex formulations in order to control several variables of
2 Mathematical Problems in Engineering
the engine such as the multivariable GPC [17] and othersimplement a predictive algorithm in the current loops [18 19]using the classic PI regulators for the main variables such asspeed and rotor flux
All predictive control schemes are based on the mini-mization of a cost function In the GPC this implies solv-ing a quadratic programming problem in the case wherephysical constraints are introduced in the optimization Ifno constraints are considered an analytical solution can beobtained In this sense it is known that all real systems haveconstraints such as saturation values frequencies and timelimits of the actuators [13 16 20] Hence a numerical methodis required to solve the quadratic programming problemwhich implies a high computation cost for the processorwhere the controller is implemented For this reason thesample period of a predictive controller is usually greater thanother types of controllers [9] which limit its applicability toquick response systems Due to this issue some works do notconsider constraints implementing the analytical solutionwith acceptable results [15 17] Moreover constraints can beconsidered after the predictive control law is obtained [15]
In addition even if the delay time of the electric motorsystems is usually small sometimes it can be long enoughso that its compensation improves significantly the systembehaviour [21] and in this way it can be used in precisionapplications In this sense the predictive algorithms allow tocompensate easily the delay time of the controlled systembecause this aspect is included in the implementation of thesealgorithms
Finally the robustness of the GPC regulator is anotheraspect included by some authors in the controller design[16 20] obtaining relevant results but with an arduouscontroller design process In this sense it is known that allclosed loop controlled systems have inherent robustness [22]that in the presented GPC speed controller is enough toovercome the typical uncertainty cases of induction motorwithout having to design an explicit robust controller norto include any method for the adaptive identification of themotor parameters
After all these considerations and taking into accountthe complexity in the design of predictive controllers andtheir important computational costs this paper presents aninduction motor speed indirect vector control that combinesthe GPC algorithm with PI regulators proposing a simplerobust and effective design which provides better dynamicalbehaviour than other speed regulators such as other GPCPID and Sliding Mode (SM) The rest of the paper isorganized as follows In Section 2 the design of the proposedGPC speed regulator is presented detailing the objectives thedynamics of the induction motor the design of the controllerand its tuning and finally its robust stability Section 3contains a brief description of the used experimental platformand the simulation and experimental tests carried out byimplementing the proposed regulator Comparative resultsare given of the presented speed GPC regulator with otherGPC PIPID and SM speed controllers Finally Section 4summarizes the most important ideas
2 Linear GPC Speed Controller forInduction Motor
21 Objective and Description The objective of this paper isto demonstrate experimentally that theGPCalgorithmcan beused in speed regulation of induction motors in an effectiveway with a simple robust and stable design offering fasterspeed tracking than other algorithms such as PID or SMallowing being implemented in industrial applications
The proposed speed regulator combines a GPC schemewith twoPI current regulatorsThedynamics of the inductionmotor is regulated using a distributed control in cascadeform the stator (1) (2) is regulated with the PI currentregulators and the rotor (4) (5) with a GPC speed controllerThe PI current regulators are very effective simple andprovide fast response offering similar results to the onesobtained by implementing a GPC-based current regulatorThese are combined with a Space Vector Pulse Width Mod-ulation (SVPWM) which is a standard modulator that isimplemented in many commercial Digital Signal Process(DSP) processors even in many low-cost processors Theproposed GPC controller is designed taking into accountthe first order transfer function (mechanical equation (4)(5)) of the induction motor considering the delay time butwithout considering the constraints Therefore it is a simplerapproach to GPC-based controllers than previous works
22 Induction Motor Dynamics The dynamics of the motorcan be described by the stator voltage equations and therotor flux equation expressed all in the 119889-119902 synchronousrotating reference frame [23] assuming that the quadraturecomponent of rotor flux is null120595
119903119902asymp 0 and consequently the
rotor flux is formed only by its direct component 120595119903asymp 120595119903119889
V119904119889
= 119877119904119894119904119889
+ 120590119871119904
119889119894119904119889
119889119905+
119871119898
119871119903
119889120595119903119889
119889119905minus 120596119890120590119871119904119894119904119902 (1)
V119904119902
= 119877119904119894119904119902
+ 120590119871119904
119889119894119904119902
119889119905+ 120596119890
119871119898
119871119903
120595119903119889
+ 120596119890120590119871119904119894119904119889 (2)
119889120595119903119889
119889119905=
119877119903
119871119903
119871119898119894119904119889
minus119877119903
119871119903
120595119903119889 (3)
119879119890minus 119879119871= 119869
119889120596119898
119889119905+ 119861V120596119898 (4)
119879119890=
3
4119901119871119898
119871119903
120595119903119889119894119904119902 (5)
The employed symbols are described as follows
119861V viscous friction coefficient119869 moment of inertia119871119898 magnetizing inductance
119871119904 stator inductance
119871119903 rotor inductance
119877119903 rotor resistance
119877119904 stator resistance
Mathematical Problems in Engineering 3
120596m
iA
iC
iB
GPC120596
3sim
RECT
VSI
IM
Load
SVPWMand
drivers
PI
PI
+ +_
+
+
+
+
+
disturb
minus
minus
disturb Calc 120579e
120579e
R S T380Vsim
+540V minus
dq
dq
ABC
ABC
VDCilowastsq lowastsq
ilowastsd
998400sq
998400sd
lowastsd
lowastABC
sq
sd
120596lowastm
Figure 1 Diagram of GPC speed control of induction motor with the PI current control and SVPWM
119901 number of poles120590 leakage factor119879119890 electromagnetic torque
119879119871 load or disturbance torque
120596119890 synchronous rotating speed
120596119898 mechanical rotor speed
120596119903 electrical rotor speed
120595119903 rotor flux vector
120595119903119889 120595119903119902 direct and quadrature components of the
rotor flux vector119894119904 stator current vector
119894119904119889 119894119904119902 direct and quadrature components of the stator
current vectorV119904 stator voltage vector
V119904119889 V119904119902 direct and quadrature components of the
stator voltage vector
23 GPC Speed Controller Design Figure 1 shows the speedvector control diagram of an induction motor where theGPC120596block is the proposed GPC controller for the speed
loop whose algorithmwill be detailed laterThe two PI blocksare a pair of PI controllers for the two current loops Theirfunction is to convert the 119894
119904119889and 119894119904119902
current commandsin their respective V
119904119889and V
119904119902voltage commands This
conversion is necessary because the inverter needs voltagecommands instead of current commands The VSI blockis the three-phase Voltage Sourced Inverter 119860119861119862 rarr 119889119902
block gets the 119894119904space vector in the 119889-119902 synchronous rotating
reference frame from the 119894119860 119894119861 and 119894
119862motor stator three-
phase currentsmeasuredwithHall effect sensors using Parkrsquostransformation [23 24] while the 119889119902 rarr 119860119861119862 block makesthe reverse Parkrsquos transformation It should be noted that thistransformation uses the rotor flux angular position 120579
119890 and
therefore this angle is calculated using the indirect methodobtained by integrating the 120596
119890synchronous speed
Next the dynamics of the inductionmotor system will becalculated in order to design the GPC
As it is known that the synchronous speed can beexpressed as follows
120596119890= 120596119904+ 120596119903 (6)
where 120596119904is the slip speed and 120596
119903is the rotor speed As it is
assumed that the 119889 direct and 119902 quadrature components ofthe rotor flux are decoupled then 120595
119903119902asymp 0 and 119889120595
119903119902119889119905 asymp 0
and consequently the rotor flux is formed only by the directcomponent [23 24] In this context the slip speed is obtainedfrom the following
120596119904=
119871119898
120595119903
119877119903
119871119903
119894119904119902 (7)
where the rotor flux is calculated by (3)
119877119903
119871119903
119889120595119903
119889119905+ 120595119903= 119871119898119894119904119889 (8)
while the rotor speed is proportional to the 120596119898mechanical
rotor speed which is measured using a incremental encoder
120596119903= 120596119898
119901
2 (9)
From diagram of Figure 1 it is possible to obtain the externalrepresentation of the induction motor (Figure 2)
Usually the rotor flux is held constant fixing the rotor fluxcurrent command (119894lowast
119904119902) to a constant value that is 119868
119904119889 see
Figures 1 and 2Then the pole associated to the electrical timeconstant can be neglected and consequently (3) in a steadystate takes the following expression
120595119903= 119871119898119868119904119889 (10)
4 Mathematical Problems in Engineering
_+PI
SVPWMVSI
PISVPWM
VSI
120596mTe
TL
1B1 + s(JB)
Lm1 + s(LrRr)
ilowastsq
ilowastsd
isq
isd
K998400T
120595rd
Figure 2 External representation of the induction motor with the PI current control and SVPWM
The electromagnetic torque of the induction motor insteady state taking into account that torque and rotorflux current components are decoupled in the 119889-119902 rotatingreference frame (5) also can be expressed as follows [24]
119879119890=
3119901
4
119871119898
119871119903
120595119903119889119894119904119902
= 119870119879119894119904119902 (11)
where119870119879is the torque constant (12) and 120595
119903119889is the rotor flux
119870119879=
3119901
4
119871119898
119871119903
120595119903119889
= 1198701015840
119879120595119903119889
= 1198701015840
119879119871119898119868119904119889 (12)
Considering that the SVPWM and the VSI modules haveneither dynamics nor gain in diagramof Figure 1 it is possibleto obtain the following transfer function
120596119898(119904)
119868lowast119904119902(119904)
=1198701015840
119879120595119903119889119861V
(1 + 119904120591119903) (1 + 119904120591
119898)
times (119870119901119894119904 + 119870119894
119894
1205901198711199041199042 + (119877
119904+ 119870119901119894) 119904 + 119870119894
119894
)
2
(13)
where the mechanical time constant and the rotor electricaltime constant are respectively
120591119898
=119869
119861V
120591119903=
119871119903
119877119903
(14)
Now if the dynamics associated to the two current loopsare neglected because they are much faster than the restthe following second order transfer function of the inductionmotor is obtained as follows
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898) (1 + 119904120591
119903)119890minus120591119889 (15)
where 120591119889is the delay time of the induction motor
Taking into account the consideration to obtain (10)from (3) Figure 2 and also considering that the dynamicsassociated to the rotor electrical time constant is much fasterthan the mechanical time constant then the electrical pole isneglected Thus the following first order transfer function ofthe induction motor is obtained
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898)119890minus120591119889 (16)
The design of the GPC controller is carried out usingthe first order transfer function of the motor As the GPCcontroller is defined in discrete time the transfer functionmust be transformed into a discrete time transfer functionThen using the ZOH (Zero-Order Hold) discretizationmethod it is obtained that
120596119898(119911minus1
)
119868lowast119904119902(119911minus1)
=
119861 (119911minus1
)
119860 (119911minus1)119911minus119889
=1198870
(1 minus 119886119911minus1)119911minus119889
(17)
Taking into account GPC theory and employing theCARIMA model [1] it is possible to obtain the followingequation of the system in which the output 120596
119898is replaced by
119910 the input 119894lowast119904119902is replaced by 119906 and white noise is included
in the previous transfer function (17)
119860(119911minus1
) 119910 (119905) = 119861 (119911minus1
) 119911minus119889
119906 (119905 minus 1) + 119862 (119911minus1
)120576 (119905)
Δ (18)
where 119889 is the delay time of the system and it ismultiple of thesample time chosen 120576 is the white noise with null averageΔ = 1 minus 119911
minus1 and the 119862(119911minus1
) is the noise polynomial that ischosen to be 1 for simplicity in this design [1]
The GPC algorithm involves applying a control sequencethat minimizes a multistage cost function of the form
119869119888=
1198732
sum
119895=1198731
120575 (119895) [119910 (119905 + 119895 | 119905) minus 119908 (119905 + 119895)]2
+
119873119906
sum
119895=1
120582 (119895) [Δ119906 (119905 + 119895 minus 1)]2
(19)
where 119910(119905 + 119895 | 119905) is the predicted output 119895 step ahead ofactual time 119905 119873
1is the minimum cost horizon 119873
2is the
maximum cost horizon 119873119906is the control horizon 120575(119905) and
120582(119905) are the weighting sequences and 119908(119905 + 119895) is the futurereference trajectory The GPC regulator design requires tun-ing the prediction horizons and the two weighting factorsIf one takes into account the possibility that the plant has adelay time then the minimum and maximum horizons arerespectively 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 [1] where 119873
is the prediction horizon Since a high value for the controlhorizon produces an undesirable oscillation in the controlsignal [3 4] which could cause the chattering phenomenonin the induction motor then it is assumed that119873
119906= 1
Mathematical Problems in Engineering 5
The prediction of the future output is as follows
119910 (119905 + 119895 + 119889 | 119905) = 119865119895+119889
(119911minus1
) 119910 (119905)
+ 119864119895+119889
(119911minus1
) 119861 (119911minus1
) Δ119906 (119905 + 119895 minus 1)
(20)
where 119865(119911minus1
) and 119864(119911minus1
) polynomials are calculated byoperating 1119860(119911
minus1
)1198732times where 119860(119911
minus1
) = 119860(119911minus1
)Δaccording to the following Diophantine identity
1 = 119864119895(119911minus1
)119860 (119911minus1
) + 119911minus119895
119865119895(119911minus1
) (21)
Taking into account that
119866119895(119911minus1
) = 119864119895(119911minus1
) 119861 (119911minus1
) (22)
the following set of 119895 ahead optimal predictions for the systemexpressed in matrix form (23) can be obtained
119910 = 119866u + 119865 (119911minus1
) 119910 (119905)
+ 1198661015840
(119911minus1
) Δ119906 (119905 + 119895 minus 1) = 119866u + f (23)
where for an119873 prediction horizon one obtains the119866matrixwhich contains values of the step sequence of the plant(induction motor) with119873 times 119873 dimension
119866 =
[[[[[
[
1198921198731
0 sdot sdot sdot 0
1198921198731+1
1198921198731
d
1198921198731+1
d 0
1198921198732
sdot sdot sdot sdot sdot sdot 1198921198731
]]]]]
]
=
[[[[[[
[
1198661198731
(119911minus1
)
1198661198732
(119911minus1
)
]]]]]]
]
(24)
Thus from (22) and (24) we can see that the 119866 matrixdepends only on the process parameters while the f term isthe free response of the system and it is easy to deduce thatit is a vector of119873 elements
f =[[
[
11986511987310
+ 1198651198731 1
119911minus1
11986511987320
+ 1198651198732 1
119911minus1
]]
]
119910 (119905)
+
[[[
[
1198921198731+1
+ 1198921198731+2
119911minus1
+ sdot sdot sdot + 1198921198731+119889
119911minus(119889minus1)
1198921198732+1
+ 1198921198732+2
119911minus1
+ sdot sdot sdot + 1198921198732+119889
119911minus(119889minus1)
]]]
]
Δ119906 (119905 minus 1)
f =
[[[
[
1198651198731
(119911minus1
)
1198651198732
(119911minus1
)
]]]
]
119910 (119905) +
[[[
[
1198661015840
1198731
(119911minus1
)
1198661015840
1198731
(119911minus1
)
]]]
]
Δ119906 (119905 minus 1)
(25)
Assuming that the 120575(119905) control weighting sequence isconstant and equal to 1 then the cost function (19) can bewritten in the following form
119869119888= (119866u + f minus w)
119879
(119866u + f minus w) + 120582u119879u (26)
minus
+K
120596m
GPC120596
Freeresponsecalculation
b01 minus azminus1
zminusdilowastsq120596lowastm
119943
Figure 3 Diagram of the induction motor speed control schemeusing GPC linear regulator
The minimum of the cost function (26) when no con-straints are considered can be calculated by equalling 0 thegradient of 119869
119888 obtaining the following analytical expression
u = (119866119879
119866 + 120582119868)minus1
119866119879
(w minus f) (27)
where u and w are two vectors of119873 elements
u =[[
[
Δ119906 (119905)
Δ119906 (119905 + 119873 minus 1)
]]
]
w =[[
[
119908 (119905 + 1198731)
119908 (119905 + 119873
2)
]]
]
(28)
As the receding horizon strategy is used the control signalapplied to the process is obtained from the first element of uwhere the following expression must be employed
Δ119906 (119905) = 119870 (w minus f) (29)
where 119870 = [11987011198702sdot sdot sdot 119870119873] that is the first row of the
(119866119879
119866 + 120582119868)minus1
119866119879 matrix in (27)
Now replacing the variables 119910(119905) and 119906(119905) by 120596lowast
119898and 119894lowast
119904119902
respectively it is obtained that
Δ119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) (30)
where finally the control law for the induction motor GPCspeed regulation (31) is obtained Figure 3 as follows
119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) + 119894
lowast
119904119902(119905 minus 1) (31)
The analytical solution of the cost function minimizationis possible only if the control signal (31) is not restrictedor limited [1] As induction motors can support statorovercurrents up to 25 times the rated value in short periodsthis GPC design does not need to limit the stator currentsThat is the 119894
lowast
119904119902control signal is not limited allowing the
implementation of the more simple low computational costanalytical solution With regard to the 119894
lowast
119904119889current it is fixed
to a value to produce the nominal rotor flux 119868119904119889 Finally the
calculation of the 119870 119866 119865 and 1198661015840 parameters is carried out
off-line without using any identification algorithm becausethe usual variation of the main parameters of the motorshould be overcome in an effective form by the inherentrobustness of any closed loop controller [22]
6 Mathematical Problems in Engineering
Table 1 Induction motor parameters (manufacturer)
Parameter Symbol ValueStator resistance 119877
119904081Ω
Rotor resistance 119877119903
057ΩMagnetizing inductance 119871
1198980117774H
Stator inductance 119871119904
0120416HRotor inductance 119871
1199030121498H
Nominal rotor flux 120595119903
101WbNumber of poles 119901 4Nominal torque 119879
119873493Nm
Moment of inertia 119869 0057 kgm2
Viscous friction coefficient 119861V 0015Nm(rads)Temperature coeff AlCu 120572 00039Kminus1
24 Regulator Tuning
241 PID and PI Controllers The two current PI regulatorsuse the same tuning valuesThe higher the bandwidth chosenfor these controllers the faster the current loops dynamicsare However in practice any real systemrsquos bandwidth islimited physically In the employed experimental platform(Section 31) which uses the induction motor described inTable 1 this limit is located at 4000 rads up to this value theplatform produces the undesirable chattering phenomenonand dangerous mechanical vibrations that can damage themachine
As to the speed regulation in the first case a PID speedcontroller has been used to measure the delay time of thesystem This PID speed regulator has also been designed inthe frequency domain using a bandwidth which is 10 timessmaller than the PI current loop regulators [24] Then theoptimal tuning of the PID speed regulator has been obtainedtaking into account this ratio of bandwidths and the physicallimit of the experiment platform Thus the most efficientPID-PI (speed currents) controller system has been designedwith the following parameter values 120596
119888120596= 300 rads with a
margin phase of PM120596= 14311 rad (82∘) and120596
119888119894= 3000 rads
with a margin phase of PM119894= 15707 rad (90∘) The D action
of the speed loop is added after PI is tuned and a small valueis enough to do faster the response but without any overshotwhere 119870
119889= 002 and the PID-PI controller is obtained The
delay time measured in the experiment platform using thisPID-PI speed regulator tuning is 700120583s using the sampletime of 100 120583s (Section 32)
242 SM Controller A sliding-mode speed regulator forinductionmotor is also implemented in order to be comparedwith the proposed GPC regulator This advanced speedregulator was presented by the authors in [7] where theparameters used in its design for the induction motor(Table 1) are obtained considering an uncertainty around 50in the system parameters 119886 = 03947 119887 = 77601 119891 =
17544 lowast119879119871 119896 = 400 120574 = 13 and 120577 = 25 This speed scheme
Table 2 Several stable cases of GPC speed regulator designs
GPC speed regulator designD1 (efficient 1)
Nominal mech parameters120591119898= 38 s
119879119904= 100120583s N = 5 d = 7
N1 = 8 N2 = 12119873119906= 1
120582 = 017
GM120596= 15 dB PM
120596= 8∘
D2 (efficient 2)Nominal mech parameters120591119898= 19 s (119869 = 00285 kgm2)
119879119904= 700120583s N = 5 d= 1
N1 = 2 N2 = 6119873119906= 1 120582 = 1
GM120596= 12 dB PM
120596= 35∘
U1IM 120591119898higher
120591119898= 76 s (J = 0114 kgm2)
GM120596= 25 dB PM
120596= 5∘
U1IM 120591119898higher
120591119898= 38 s (J = 0057 kgm2)
GM120596= 25 dB PM
120596= 25∘
U2IM 120591119898smaller
120591119898= 038 s (119861V = 015Nms)
GM120596= 15 dB PM
120596= 9∘
U2IM 120591119898smaller
120591119898= 019 s (119861V = 015Nms)
GM120596= 12 dB PM
120596= 35∘
PI currents regulators designStator nominal electr parameters 120591
119904= 01487 s (119879 = 20∘C)
119879119904= 100120583s 120596
119888119894= 3000 rads PM
119894= 15707 (90∘)
Us1 120591119904higher 120591
119904= 01612 s (119879 = 0∘C)
PM119894= 90∘
Us2 120591119904smaller 120591
119904= 01040 s (T = 130∘C)
PM119894= 90∘
uses the same two current PI regulators employed inGPC andPI speed schemes and detailed in the previous subsection
243 GPC Controller The tuning of the GPC regulatorrequires choosing the values of two horizons and two weight-ing factors The control horizon 119873
119906 and the output error
weighting factor 120575 both have been fixed to 1 (Section 21)The prediction horizon 119873 determines the size of the 119866
matrix and the f vector and as a consequence the numberof the GPC controllerrsquos coefficients A larger value for 119873
increases the anticipative effect involving a better controland performance of the system However this increases thenumber of the coefficients and the computational cost of thecontrol law which requires increasing the sample time So119873 should be selected to ensure proper dynamic behaviourand low computational cost In this design it has been setto 5 From Section 21 it is known that the minimum andmaximum horizons take into account the delay time of thesystem 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 Considering two GPC
speed regulator designs D1 and D2 using sample times of100 120583s and 700120583s respectively and taking into account thatthe measured experimental delay time is 700 120583s (Section 32)then the delay values are 119889 = 7 for D1 and 119889 = 1 for D2(Table 2)
The control weighting factor 120582 has a direct impact onthe response of the controlled system Hence the higher itsvalue the slower the resulting controlled system On theother hand if its value is too small it can produce thechattering phenomenon that can damage the motor in a realcase Then it is desirable to find a value for 120582 to provide
Mathematical Problems in Engineering 7
+
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
minus180
minus175
minus170
minus165
minus150
minus145
minus140
minus130
minus120
minus100
minus30
minus20
minus10
0
10
20
30
40
6dB
3dB
1dB
05dB
025 dB
GPC D1GPC D1U1GPC D1U2GPC D2
GPC D2U1GPC D2U2PID
Figure 4 Nichols diagram of the GPC and PID speed loops
an effective response of the controlled system Taking intoaccount the recommendation in [15] an easy way to obtain aneffective and stable value for 120582 factor is applying the followingempirical expression
120582opt = 119898 trace (119866119879119866) (32)
where 119898 takes a value inversely proportional to the sampletime For D1 design the 119898 parameter results in a value of 60and for D2119898 equals 2 (tens for 100 120583s units for 700 120583s)
25 Robust Stability Analysis The closed loop stability of themotor 119889-119902 currents with the PI controllers is guaranteed if thePM119894phasemargins are positive and sufficiently highAs to the
speed loop the stability of the controlled system is analysedusing the classic RST poles placement scheme [25] where thecontrol law is obtained as
119894lowast
119904119902(119905) =
1
1199030
[119879 (119911minus1
) 120596lowast
119898(119905) 119879
minus119878 (119911minus1
) 120596119898(119905) minus 119894
lowast
119904119902(119905 minus 1) 119877
1015840
(119911minus1
)]
(33)
minus180 minus135 minus90 minus45 0minus20
minus15
minus10
minus5
0
5 6dB
minus6
minus12
minus20
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
PIPI Us1PI Us2
(dB)
Figure 5 Nichols diagram of the PI current loops
In this sense it is necessary to translate the GPC con-trolled system parameters to its equivalent RST controlledsystem parameters [1] to check the stability of the systemIdentifying the terms between (31) and (33) we can obtainthe general expression of equivalence in both RST and GPCalgorithms for any first order plant with delay time
119877 (119911minus1
) = 1199030+ 1199031119911minus1
+ sdot sdot sdot + 119903119889+1
119911minus(119889+1)
= 1199030+ 119911minus1
1198771015840
(119911minus1
)
119878 (119911minus1
) = 1199040+ 1199041119911minus1
119879 (119911minus1
) = 1199050119911minus1198731 + 119905
1119911minus(1198731+1) + sdot sdot sdot + 119905
119873minus1119911minus1198732
(34)
1199030= 1
1199031= 119870[119892
1198731+1sdot sdot sdot 1198921198732+1
]119879
minus 1
1199032= 119870([119892
1198731+2sdot sdot sdot 1198921198732+2
]119879
minus[1198921198731+1
sdot sdot sdot 1198921198732+1
]119879
)
119903119889= 119870([119892
1198731+119889sdot sdot sdot 1198921198732+119889
]119879
minus[1198921198731+119889minus1
sdot sdot sdot 1198921198732+119889minus1
]119879
)
119903119889+1
= minus119870[1198921198731+119889
sdot sdot sdot 1198921198732+119889
]119879
1199040= 119870[119865
1198731 0sdot sdot sdot 1198651198732 0
]119879
1199041= 119870[119865
1198731 1sdot sdot sdot 1198651198732 1
]119879
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
the engine such as the multivariable GPC [17] and othersimplement a predictive algorithm in the current loops [18 19]using the classic PI regulators for the main variables such asspeed and rotor flux
All predictive control schemes are based on the mini-mization of a cost function In the GPC this implies solv-ing a quadratic programming problem in the case wherephysical constraints are introduced in the optimization Ifno constraints are considered an analytical solution can beobtained In this sense it is known that all real systems haveconstraints such as saturation values frequencies and timelimits of the actuators [13 16 20] Hence a numerical methodis required to solve the quadratic programming problemwhich implies a high computation cost for the processorwhere the controller is implemented For this reason thesample period of a predictive controller is usually greater thanother types of controllers [9] which limit its applicability toquick response systems Due to this issue some works do notconsider constraints implementing the analytical solutionwith acceptable results [15 17] Moreover constraints can beconsidered after the predictive control law is obtained [15]
In addition even if the delay time of the electric motorsystems is usually small sometimes it can be long enoughso that its compensation improves significantly the systembehaviour [21] and in this way it can be used in precisionapplications In this sense the predictive algorithms allow tocompensate easily the delay time of the controlled systembecause this aspect is included in the implementation of thesealgorithms
Finally the robustness of the GPC regulator is anotheraspect included by some authors in the controller design[16 20] obtaining relevant results but with an arduouscontroller design process In this sense it is known that allclosed loop controlled systems have inherent robustness [22]that in the presented GPC speed controller is enough toovercome the typical uncertainty cases of induction motorwithout having to design an explicit robust controller norto include any method for the adaptive identification of themotor parameters
After all these considerations and taking into accountthe complexity in the design of predictive controllers andtheir important computational costs this paper presents aninduction motor speed indirect vector control that combinesthe GPC algorithm with PI regulators proposing a simplerobust and effective design which provides better dynamicalbehaviour than other speed regulators such as other GPCPID and Sliding Mode (SM) The rest of the paper isorganized as follows In Section 2 the design of the proposedGPC speed regulator is presented detailing the objectives thedynamics of the induction motor the design of the controllerand its tuning and finally its robust stability Section 3contains a brief description of the used experimental platformand the simulation and experimental tests carried out byimplementing the proposed regulator Comparative resultsare given of the presented speed GPC regulator with otherGPC PIPID and SM speed controllers Finally Section 4summarizes the most important ideas
2 Linear GPC Speed Controller forInduction Motor
21 Objective and Description The objective of this paper isto demonstrate experimentally that theGPCalgorithmcan beused in speed regulation of induction motors in an effectiveway with a simple robust and stable design offering fasterspeed tracking than other algorithms such as PID or SMallowing being implemented in industrial applications
The proposed speed regulator combines a GPC schemewith twoPI current regulatorsThedynamics of the inductionmotor is regulated using a distributed control in cascadeform the stator (1) (2) is regulated with the PI currentregulators and the rotor (4) (5) with a GPC speed controllerThe PI current regulators are very effective simple andprovide fast response offering similar results to the onesobtained by implementing a GPC-based current regulatorThese are combined with a Space Vector Pulse Width Mod-ulation (SVPWM) which is a standard modulator that isimplemented in many commercial Digital Signal Process(DSP) processors even in many low-cost processors Theproposed GPC controller is designed taking into accountthe first order transfer function (mechanical equation (4)(5)) of the induction motor considering the delay time butwithout considering the constraints Therefore it is a simplerapproach to GPC-based controllers than previous works
22 Induction Motor Dynamics The dynamics of the motorcan be described by the stator voltage equations and therotor flux equation expressed all in the 119889-119902 synchronousrotating reference frame [23] assuming that the quadraturecomponent of rotor flux is null120595
119903119902asymp 0 and consequently the
rotor flux is formed only by its direct component 120595119903asymp 120595119903119889
V119904119889
= 119877119904119894119904119889
+ 120590119871119904
119889119894119904119889
119889119905+
119871119898
119871119903
119889120595119903119889
119889119905minus 120596119890120590119871119904119894119904119902 (1)
V119904119902
= 119877119904119894119904119902
+ 120590119871119904
119889119894119904119902
119889119905+ 120596119890
119871119898
119871119903
120595119903119889
+ 120596119890120590119871119904119894119904119889 (2)
119889120595119903119889
119889119905=
119877119903
119871119903
119871119898119894119904119889
minus119877119903
119871119903
120595119903119889 (3)
119879119890minus 119879119871= 119869
119889120596119898
119889119905+ 119861V120596119898 (4)
119879119890=
3
4119901119871119898
119871119903
120595119903119889119894119904119902 (5)
The employed symbols are described as follows
119861V viscous friction coefficient119869 moment of inertia119871119898 magnetizing inductance
119871119904 stator inductance
119871119903 rotor inductance
119877119903 rotor resistance
119877119904 stator resistance
Mathematical Problems in Engineering 3
120596m
iA
iC
iB
GPC120596
3sim
RECT
VSI
IM
Load
SVPWMand
drivers
PI
PI
+ +_
+
+
+
+
+
disturb
minus
minus
disturb Calc 120579e
120579e
R S T380Vsim
+540V minus
dq
dq
ABC
ABC
VDCilowastsq lowastsq
ilowastsd
998400sq
998400sd
lowastsd
lowastABC
sq
sd
120596lowastm
Figure 1 Diagram of GPC speed control of induction motor with the PI current control and SVPWM
119901 number of poles120590 leakage factor119879119890 electromagnetic torque
119879119871 load or disturbance torque
120596119890 synchronous rotating speed
120596119898 mechanical rotor speed
120596119903 electrical rotor speed
120595119903 rotor flux vector
120595119903119889 120595119903119902 direct and quadrature components of the
rotor flux vector119894119904 stator current vector
119894119904119889 119894119904119902 direct and quadrature components of the stator
current vectorV119904 stator voltage vector
V119904119889 V119904119902 direct and quadrature components of the
stator voltage vector
23 GPC Speed Controller Design Figure 1 shows the speedvector control diagram of an induction motor where theGPC120596block is the proposed GPC controller for the speed
loop whose algorithmwill be detailed laterThe two PI blocksare a pair of PI controllers for the two current loops Theirfunction is to convert the 119894
119904119889and 119894119904119902
current commandsin their respective V
119904119889and V
119904119902voltage commands This
conversion is necessary because the inverter needs voltagecommands instead of current commands The VSI blockis the three-phase Voltage Sourced Inverter 119860119861119862 rarr 119889119902
block gets the 119894119904space vector in the 119889-119902 synchronous rotating
reference frame from the 119894119860 119894119861 and 119894
119862motor stator three-
phase currentsmeasuredwithHall effect sensors using Parkrsquostransformation [23 24] while the 119889119902 rarr 119860119861119862 block makesthe reverse Parkrsquos transformation It should be noted that thistransformation uses the rotor flux angular position 120579
119890 and
therefore this angle is calculated using the indirect methodobtained by integrating the 120596
119890synchronous speed
Next the dynamics of the inductionmotor system will becalculated in order to design the GPC
As it is known that the synchronous speed can beexpressed as follows
120596119890= 120596119904+ 120596119903 (6)
where 120596119904is the slip speed and 120596
119903is the rotor speed As it is
assumed that the 119889 direct and 119902 quadrature components ofthe rotor flux are decoupled then 120595
119903119902asymp 0 and 119889120595
119903119902119889119905 asymp 0
and consequently the rotor flux is formed only by the directcomponent [23 24] In this context the slip speed is obtainedfrom the following
120596119904=
119871119898
120595119903
119877119903
119871119903
119894119904119902 (7)
where the rotor flux is calculated by (3)
119877119903
119871119903
119889120595119903
119889119905+ 120595119903= 119871119898119894119904119889 (8)
while the rotor speed is proportional to the 120596119898mechanical
rotor speed which is measured using a incremental encoder
120596119903= 120596119898
119901
2 (9)
From diagram of Figure 1 it is possible to obtain the externalrepresentation of the induction motor (Figure 2)
Usually the rotor flux is held constant fixing the rotor fluxcurrent command (119894lowast
119904119902) to a constant value that is 119868
119904119889 see
Figures 1 and 2Then the pole associated to the electrical timeconstant can be neglected and consequently (3) in a steadystate takes the following expression
120595119903= 119871119898119868119904119889 (10)
4 Mathematical Problems in Engineering
_+PI
SVPWMVSI
PISVPWM
VSI
120596mTe
TL
1B1 + s(JB)
Lm1 + s(LrRr)
ilowastsq
ilowastsd
isq
isd
K998400T
120595rd
Figure 2 External representation of the induction motor with the PI current control and SVPWM
The electromagnetic torque of the induction motor insteady state taking into account that torque and rotorflux current components are decoupled in the 119889-119902 rotatingreference frame (5) also can be expressed as follows [24]
119879119890=
3119901
4
119871119898
119871119903
120595119903119889119894119904119902
= 119870119879119894119904119902 (11)
where119870119879is the torque constant (12) and 120595
119903119889is the rotor flux
119870119879=
3119901
4
119871119898
119871119903
120595119903119889
= 1198701015840
119879120595119903119889
= 1198701015840
119879119871119898119868119904119889 (12)
Considering that the SVPWM and the VSI modules haveneither dynamics nor gain in diagramof Figure 1 it is possibleto obtain the following transfer function
120596119898(119904)
119868lowast119904119902(119904)
=1198701015840
119879120595119903119889119861V
(1 + 119904120591119903) (1 + 119904120591
119898)
times (119870119901119894119904 + 119870119894
119894
1205901198711199041199042 + (119877
119904+ 119870119901119894) 119904 + 119870119894
119894
)
2
(13)
where the mechanical time constant and the rotor electricaltime constant are respectively
120591119898
=119869
119861V
120591119903=
119871119903
119877119903
(14)
Now if the dynamics associated to the two current loopsare neglected because they are much faster than the restthe following second order transfer function of the inductionmotor is obtained as follows
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898) (1 + 119904120591
119903)119890minus120591119889 (15)
where 120591119889is the delay time of the induction motor
Taking into account the consideration to obtain (10)from (3) Figure 2 and also considering that the dynamicsassociated to the rotor electrical time constant is much fasterthan the mechanical time constant then the electrical pole isneglected Thus the following first order transfer function ofthe induction motor is obtained
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898)119890minus120591119889 (16)
The design of the GPC controller is carried out usingthe first order transfer function of the motor As the GPCcontroller is defined in discrete time the transfer functionmust be transformed into a discrete time transfer functionThen using the ZOH (Zero-Order Hold) discretizationmethod it is obtained that
120596119898(119911minus1
)
119868lowast119904119902(119911minus1)
=
119861 (119911minus1
)
119860 (119911minus1)119911minus119889
=1198870
(1 minus 119886119911minus1)119911minus119889
(17)
Taking into account GPC theory and employing theCARIMA model [1] it is possible to obtain the followingequation of the system in which the output 120596
119898is replaced by
119910 the input 119894lowast119904119902is replaced by 119906 and white noise is included
in the previous transfer function (17)
119860(119911minus1
) 119910 (119905) = 119861 (119911minus1
) 119911minus119889
119906 (119905 minus 1) + 119862 (119911minus1
)120576 (119905)
Δ (18)
where 119889 is the delay time of the system and it ismultiple of thesample time chosen 120576 is the white noise with null averageΔ = 1 minus 119911
minus1 and the 119862(119911minus1
) is the noise polynomial that ischosen to be 1 for simplicity in this design [1]
The GPC algorithm involves applying a control sequencethat minimizes a multistage cost function of the form
119869119888=
1198732
sum
119895=1198731
120575 (119895) [119910 (119905 + 119895 | 119905) minus 119908 (119905 + 119895)]2
+
119873119906
sum
119895=1
120582 (119895) [Δ119906 (119905 + 119895 minus 1)]2
(19)
where 119910(119905 + 119895 | 119905) is the predicted output 119895 step ahead ofactual time 119905 119873
1is the minimum cost horizon 119873
2is the
maximum cost horizon 119873119906is the control horizon 120575(119905) and
120582(119905) are the weighting sequences and 119908(119905 + 119895) is the futurereference trajectory The GPC regulator design requires tun-ing the prediction horizons and the two weighting factorsIf one takes into account the possibility that the plant has adelay time then the minimum and maximum horizons arerespectively 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 [1] where 119873
is the prediction horizon Since a high value for the controlhorizon produces an undesirable oscillation in the controlsignal [3 4] which could cause the chattering phenomenonin the induction motor then it is assumed that119873
119906= 1
Mathematical Problems in Engineering 5
The prediction of the future output is as follows
119910 (119905 + 119895 + 119889 | 119905) = 119865119895+119889
(119911minus1
) 119910 (119905)
+ 119864119895+119889
(119911minus1
) 119861 (119911minus1
) Δ119906 (119905 + 119895 minus 1)
(20)
where 119865(119911minus1
) and 119864(119911minus1
) polynomials are calculated byoperating 1119860(119911
minus1
)1198732times where 119860(119911
minus1
) = 119860(119911minus1
)Δaccording to the following Diophantine identity
1 = 119864119895(119911minus1
)119860 (119911minus1
) + 119911minus119895
119865119895(119911minus1
) (21)
Taking into account that
119866119895(119911minus1
) = 119864119895(119911minus1
) 119861 (119911minus1
) (22)
the following set of 119895 ahead optimal predictions for the systemexpressed in matrix form (23) can be obtained
119910 = 119866u + 119865 (119911minus1
) 119910 (119905)
+ 1198661015840
(119911minus1
) Δ119906 (119905 + 119895 minus 1) = 119866u + f (23)
where for an119873 prediction horizon one obtains the119866matrixwhich contains values of the step sequence of the plant(induction motor) with119873 times 119873 dimension
119866 =
[[[[[
[
1198921198731
0 sdot sdot sdot 0
1198921198731+1
1198921198731
d
1198921198731+1
d 0
1198921198732
sdot sdot sdot sdot sdot sdot 1198921198731
]]]]]
]
=
[[[[[[
[
1198661198731
(119911minus1
)
1198661198732
(119911minus1
)
]]]]]]
]
(24)
Thus from (22) and (24) we can see that the 119866 matrixdepends only on the process parameters while the f term isthe free response of the system and it is easy to deduce thatit is a vector of119873 elements
f =[[
[
11986511987310
+ 1198651198731 1
119911minus1
11986511987320
+ 1198651198732 1
119911minus1
]]
]
119910 (119905)
+
[[[
[
1198921198731+1
+ 1198921198731+2
119911minus1
+ sdot sdot sdot + 1198921198731+119889
119911minus(119889minus1)
1198921198732+1
+ 1198921198732+2
119911minus1
+ sdot sdot sdot + 1198921198732+119889
119911minus(119889minus1)
]]]
]
Δ119906 (119905 minus 1)
f =
[[[
[
1198651198731
(119911minus1
)
1198651198732
(119911minus1
)
]]]
]
119910 (119905) +
[[[
[
1198661015840
1198731
(119911minus1
)
1198661015840
1198731
(119911minus1
)
]]]
]
Δ119906 (119905 minus 1)
(25)
Assuming that the 120575(119905) control weighting sequence isconstant and equal to 1 then the cost function (19) can bewritten in the following form
119869119888= (119866u + f minus w)
119879
(119866u + f minus w) + 120582u119879u (26)
minus
+K
120596m
GPC120596
Freeresponsecalculation
b01 minus azminus1
zminusdilowastsq120596lowastm
119943
Figure 3 Diagram of the induction motor speed control schemeusing GPC linear regulator
The minimum of the cost function (26) when no con-straints are considered can be calculated by equalling 0 thegradient of 119869
119888 obtaining the following analytical expression
u = (119866119879
119866 + 120582119868)minus1
119866119879
(w minus f) (27)
where u and w are two vectors of119873 elements
u =[[
[
Δ119906 (119905)
Δ119906 (119905 + 119873 minus 1)
]]
]
w =[[
[
119908 (119905 + 1198731)
119908 (119905 + 119873
2)
]]
]
(28)
As the receding horizon strategy is used the control signalapplied to the process is obtained from the first element of uwhere the following expression must be employed
Δ119906 (119905) = 119870 (w minus f) (29)
where 119870 = [11987011198702sdot sdot sdot 119870119873] that is the first row of the
(119866119879
119866 + 120582119868)minus1
119866119879 matrix in (27)
Now replacing the variables 119910(119905) and 119906(119905) by 120596lowast
119898and 119894lowast
119904119902
respectively it is obtained that
Δ119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) (30)
where finally the control law for the induction motor GPCspeed regulation (31) is obtained Figure 3 as follows
119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) + 119894
lowast
119904119902(119905 minus 1) (31)
The analytical solution of the cost function minimizationis possible only if the control signal (31) is not restrictedor limited [1] As induction motors can support statorovercurrents up to 25 times the rated value in short periodsthis GPC design does not need to limit the stator currentsThat is the 119894
lowast
119904119902control signal is not limited allowing the
implementation of the more simple low computational costanalytical solution With regard to the 119894
lowast
119904119889current it is fixed
to a value to produce the nominal rotor flux 119868119904119889 Finally the
calculation of the 119870 119866 119865 and 1198661015840 parameters is carried out
off-line without using any identification algorithm becausethe usual variation of the main parameters of the motorshould be overcome in an effective form by the inherentrobustness of any closed loop controller [22]
6 Mathematical Problems in Engineering
Table 1 Induction motor parameters (manufacturer)
Parameter Symbol ValueStator resistance 119877
119904081Ω
Rotor resistance 119877119903
057ΩMagnetizing inductance 119871
1198980117774H
Stator inductance 119871119904
0120416HRotor inductance 119871
1199030121498H
Nominal rotor flux 120595119903
101WbNumber of poles 119901 4Nominal torque 119879
119873493Nm
Moment of inertia 119869 0057 kgm2
Viscous friction coefficient 119861V 0015Nm(rads)Temperature coeff AlCu 120572 00039Kminus1
24 Regulator Tuning
241 PID and PI Controllers The two current PI regulatorsuse the same tuning valuesThe higher the bandwidth chosenfor these controllers the faster the current loops dynamicsare However in practice any real systemrsquos bandwidth islimited physically In the employed experimental platform(Section 31) which uses the induction motor described inTable 1 this limit is located at 4000 rads up to this value theplatform produces the undesirable chattering phenomenonand dangerous mechanical vibrations that can damage themachine
As to the speed regulation in the first case a PID speedcontroller has been used to measure the delay time of thesystem This PID speed regulator has also been designed inthe frequency domain using a bandwidth which is 10 timessmaller than the PI current loop regulators [24] Then theoptimal tuning of the PID speed regulator has been obtainedtaking into account this ratio of bandwidths and the physicallimit of the experiment platform Thus the most efficientPID-PI (speed currents) controller system has been designedwith the following parameter values 120596
119888120596= 300 rads with a
margin phase of PM120596= 14311 rad (82∘) and120596
119888119894= 3000 rads
with a margin phase of PM119894= 15707 rad (90∘) The D action
of the speed loop is added after PI is tuned and a small valueis enough to do faster the response but without any overshotwhere 119870
119889= 002 and the PID-PI controller is obtained The
delay time measured in the experiment platform using thisPID-PI speed regulator tuning is 700120583s using the sampletime of 100 120583s (Section 32)
242 SM Controller A sliding-mode speed regulator forinductionmotor is also implemented in order to be comparedwith the proposed GPC regulator This advanced speedregulator was presented by the authors in [7] where theparameters used in its design for the induction motor(Table 1) are obtained considering an uncertainty around 50in the system parameters 119886 = 03947 119887 = 77601 119891 =
17544 lowast119879119871 119896 = 400 120574 = 13 and 120577 = 25 This speed scheme
Table 2 Several stable cases of GPC speed regulator designs
GPC speed regulator designD1 (efficient 1)
Nominal mech parameters120591119898= 38 s
119879119904= 100120583s N = 5 d = 7
N1 = 8 N2 = 12119873119906= 1
120582 = 017
GM120596= 15 dB PM
120596= 8∘
D2 (efficient 2)Nominal mech parameters120591119898= 19 s (119869 = 00285 kgm2)
119879119904= 700120583s N = 5 d= 1
N1 = 2 N2 = 6119873119906= 1 120582 = 1
GM120596= 12 dB PM
120596= 35∘
U1IM 120591119898higher
120591119898= 76 s (J = 0114 kgm2)
GM120596= 25 dB PM
120596= 5∘
U1IM 120591119898higher
120591119898= 38 s (J = 0057 kgm2)
GM120596= 25 dB PM
120596= 25∘
U2IM 120591119898smaller
120591119898= 038 s (119861V = 015Nms)
GM120596= 15 dB PM
120596= 9∘
U2IM 120591119898smaller
120591119898= 019 s (119861V = 015Nms)
GM120596= 12 dB PM
120596= 35∘
PI currents regulators designStator nominal electr parameters 120591
119904= 01487 s (119879 = 20∘C)
119879119904= 100120583s 120596
119888119894= 3000 rads PM
119894= 15707 (90∘)
Us1 120591119904higher 120591
119904= 01612 s (119879 = 0∘C)
PM119894= 90∘
Us2 120591119904smaller 120591
119904= 01040 s (T = 130∘C)
PM119894= 90∘
uses the same two current PI regulators employed inGPC andPI speed schemes and detailed in the previous subsection
243 GPC Controller The tuning of the GPC regulatorrequires choosing the values of two horizons and two weight-ing factors The control horizon 119873
119906 and the output error
weighting factor 120575 both have been fixed to 1 (Section 21)The prediction horizon 119873 determines the size of the 119866
matrix and the f vector and as a consequence the numberof the GPC controllerrsquos coefficients A larger value for 119873
increases the anticipative effect involving a better controland performance of the system However this increases thenumber of the coefficients and the computational cost of thecontrol law which requires increasing the sample time So119873 should be selected to ensure proper dynamic behaviourand low computational cost In this design it has been setto 5 From Section 21 it is known that the minimum andmaximum horizons take into account the delay time of thesystem 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 Considering two GPC
speed regulator designs D1 and D2 using sample times of100 120583s and 700120583s respectively and taking into account thatthe measured experimental delay time is 700 120583s (Section 32)then the delay values are 119889 = 7 for D1 and 119889 = 1 for D2(Table 2)
The control weighting factor 120582 has a direct impact onthe response of the controlled system Hence the higher itsvalue the slower the resulting controlled system On theother hand if its value is too small it can produce thechattering phenomenon that can damage the motor in a realcase Then it is desirable to find a value for 120582 to provide
Mathematical Problems in Engineering 7
+
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
minus180
minus175
minus170
minus165
minus150
minus145
minus140
minus130
minus120
minus100
minus30
minus20
minus10
0
10
20
30
40
6dB
3dB
1dB
05dB
025 dB
GPC D1GPC D1U1GPC D1U2GPC D2
GPC D2U1GPC D2U2PID
Figure 4 Nichols diagram of the GPC and PID speed loops
an effective response of the controlled system Taking intoaccount the recommendation in [15] an easy way to obtain aneffective and stable value for 120582 factor is applying the followingempirical expression
120582opt = 119898 trace (119866119879119866) (32)
where 119898 takes a value inversely proportional to the sampletime For D1 design the 119898 parameter results in a value of 60and for D2119898 equals 2 (tens for 100 120583s units for 700 120583s)
25 Robust Stability Analysis The closed loop stability of themotor 119889-119902 currents with the PI controllers is guaranteed if thePM119894phasemargins are positive and sufficiently highAs to the
speed loop the stability of the controlled system is analysedusing the classic RST poles placement scheme [25] where thecontrol law is obtained as
119894lowast
119904119902(119905) =
1
1199030
[119879 (119911minus1
) 120596lowast
119898(119905) 119879
minus119878 (119911minus1
) 120596119898(119905) minus 119894
lowast
119904119902(119905 minus 1) 119877
1015840
(119911minus1
)]
(33)
minus180 minus135 minus90 minus45 0minus20
minus15
minus10
minus5
0
5 6dB
minus6
minus12
minus20
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
PIPI Us1PI Us2
(dB)
Figure 5 Nichols diagram of the PI current loops
In this sense it is necessary to translate the GPC con-trolled system parameters to its equivalent RST controlledsystem parameters [1] to check the stability of the systemIdentifying the terms between (31) and (33) we can obtainthe general expression of equivalence in both RST and GPCalgorithms for any first order plant with delay time
119877 (119911minus1
) = 1199030+ 1199031119911minus1
+ sdot sdot sdot + 119903119889+1
119911minus(119889+1)
= 1199030+ 119911minus1
1198771015840
(119911minus1
)
119878 (119911minus1
) = 1199040+ 1199041119911minus1
119879 (119911minus1
) = 1199050119911minus1198731 + 119905
1119911minus(1198731+1) + sdot sdot sdot + 119905
119873minus1119911minus1198732
(34)
1199030= 1
1199031= 119870[119892
1198731+1sdot sdot sdot 1198921198732+1
]119879
minus 1
1199032= 119870([119892
1198731+2sdot sdot sdot 1198921198732+2
]119879
minus[1198921198731+1
sdot sdot sdot 1198921198732+1
]119879
)
119903119889= 119870([119892
1198731+119889sdot sdot sdot 1198921198732+119889
]119879
minus[1198921198731+119889minus1
sdot sdot sdot 1198921198732+119889minus1
]119879
)
119903119889+1
= minus119870[1198921198731+119889
sdot sdot sdot 1198921198732+119889
]119879
1199040= 119870[119865
1198731 0sdot sdot sdot 1198651198732 0
]119879
1199041= 119870[119865
1198731 1sdot sdot sdot 1198651198732 1
]119879
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
120596m
iA
iC
iB
GPC120596
3sim
RECT
VSI
IM
Load
SVPWMand
drivers
PI
PI
+ +_
+
+
+
+
+
disturb
minus
minus
disturb Calc 120579e
120579e
R S T380Vsim
+540V minus
dq
dq
ABC
ABC
VDCilowastsq lowastsq
ilowastsd
998400sq
998400sd
lowastsd
lowastABC
sq
sd
120596lowastm
Figure 1 Diagram of GPC speed control of induction motor with the PI current control and SVPWM
119901 number of poles120590 leakage factor119879119890 electromagnetic torque
119879119871 load or disturbance torque
120596119890 synchronous rotating speed
120596119898 mechanical rotor speed
120596119903 electrical rotor speed
120595119903 rotor flux vector
120595119903119889 120595119903119902 direct and quadrature components of the
rotor flux vector119894119904 stator current vector
119894119904119889 119894119904119902 direct and quadrature components of the stator
current vectorV119904 stator voltage vector
V119904119889 V119904119902 direct and quadrature components of the
stator voltage vector
23 GPC Speed Controller Design Figure 1 shows the speedvector control diagram of an induction motor where theGPC120596block is the proposed GPC controller for the speed
loop whose algorithmwill be detailed laterThe two PI blocksare a pair of PI controllers for the two current loops Theirfunction is to convert the 119894
119904119889and 119894119904119902
current commandsin their respective V
119904119889and V
119904119902voltage commands This
conversion is necessary because the inverter needs voltagecommands instead of current commands The VSI blockis the three-phase Voltage Sourced Inverter 119860119861119862 rarr 119889119902
block gets the 119894119904space vector in the 119889-119902 synchronous rotating
reference frame from the 119894119860 119894119861 and 119894
119862motor stator three-
phase currentsmeasuredwithHall effect sensors using Parkrsquostransformation [23 24] while the 119889119902 rarr 119860119861119862 block makesthe reverse Parkrsquos transformation It should be noted that thistransformation uses the rotor flux angular position 120579
119890 and
therefore this angle is calculated using the indirect methodobtained by integrating the 120596
119890synchronous speed
Next the dynamics of the inductionmotor system will becalculated in order to design the GPC
As it is known that the synchronous speed can beexpressed as follows
120596119890= 120596119904+ 120596119903 (6)
where 120596119904is the slip speed and 120596
119903is the rotor speed As it is
assumed that the 119889 direct and 119902 quadrature components ofthe rotor flux are decoupled then 120595
119903119902asymp 0 and 119889120595
119903119902119889119905 asymp 0
and consequently the rotor flux is formed only by the directcomponent [23 24] In this context the slip speed is obtainedfrom the following
120596119904=
119871119898
120595119903
119877119903
119871119903
119894119904119902 (7)
where the rotor flux is calculated by (3)
119877119903
119871119903
119889120595119903
119889119905+ 120595119903= 119871119898119894119904119889 (8)
while the rotor speed is proportional to the 120596119898mechanical
rotor speed which is measured using a incremental encoder
120596119903= 120596119898
119901
2 (9)
From diagram of Figure 1 it is possible to obtain the externalrepresentation of the induction motor (Figure 2)
Usually the rotor flux is held constant fixing the rotor fluxcurrent command (119894lowast
119904119902) to a constant value that is 119868
119904119889 see
Figures 1 and 2Then the pole associated to the electrical timeconstant can be neglected and consequently (3) in a steadystate takes the following expression
120595119903= 119871119898119868119904119889 (10)
4 Mathematical Problems in Engineering
_+PI
SVPWMVSI
PISVPWM
VSI
120596mTe
TL
1B1 + s(JB)
Lm1 + s(LrRr)
ilowastsq
ilowastsd
isq
isd
K998400T
120595rd
Figure 2 External representation of the induction motor with the PI current control and SVPWM
The electromagnetic torque of the induction motor insteady state taking into account that torque and rotorflux current components are decoupled in the 119889-119902 rotatingreference frame (5) also can be expressed as follows [24]
119879119890=
3119901
4
119871119898
119871119903
120595119903119889119894119904119902
= 119870119879119894119904119902 (11)
where119870119879is the torque constant (12) and 120595
119903119889is the rotor flux
119870119879=
3119901
4
119871119898
119871119903
120595119903119889
= 1198701015840
119879120595119903119889
= 1198701015840
119879119871119898119868119904119889 (12)
Considering that the SVPWM and the VSI modules haveneither dynamics nor gain in diagramof Figure 1 it is possibleto obtain the following transfer function
120596119898(119904)
119868lowast119904119902(119904)
=1198701015840
119879120595119903119889119861V
(1 + 119904120591119903) (1 + 119904120591
119898)
times (119870119901119894119904 + 119870119894
119894
1205901198711199041199042 + (119877
119904+ 119870119901119894) 119904 + 119870119894
119894
)
2
(13)
where the mechanical time constant and the rotor electricaltime constant are respectively
120591119898
=119869
119861V
120591119903=
119871119903
119877119903
(14)
Now if the dynamics associated to the two current loopsare neglected because they are much faster than the restthe following second order transfer function of the inductionmotor is obtained as follows
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898) (1 + 119904120591
119903)119890minus120591119889 (15)
where 120591119889is the delay time of the induction motor
Taking into account the consideration to obtain (10)from (3) Figure 2 and also considering that the dynamicsassociated to the rotor electrical time constant is much fasterthan the mechanical time constant then the electrical pole isneglected Thus the following first order transfer function ofthe induction motor is obtained
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898)119890minus120591119889 (16)
The design of the GPC controller is carried out usingthe first order transfer function of the motor As the GPCcontroller is defined in discrete time the transfer functionmust be transformed into a discrete time transfer functionThen using the ZOH (Zero-Order Hold) discretizationmethod it is obtained that
120596119898(119911minus1
)
119868lowast119904119902(119911minus1)
=
119861 (119911minus1
)
119860 (119911minus1)119911minus119889
=1198870
(1 minus 119886119911minus1)119911minus119889
(17)
Taking into account GPC theory and employing theCARIMA model [1] it is possible to obtain the followingequation of the system in which the output 120596
119898is replaced by
119910 the input 119894lowast119904119902is replaced by 119906 and white noise is included
in the previous transfer function (17)
119860(119911minus1
) 119910 (119905) = 119861 (119911minus1
) 119911minus119889
119906 (119905 minus 1) + 119862 (119911minus1
)120576 (119905)
Δ (18)
where 119889 is the delay time of the system and it ismultiple of thesample time chosen 120576 is the white noise with null averageΔ = 1 minus 119911
minus1 and the 119862(119911minus1
) is the noise polynomial that ischosen to be 1 for simplicity in this design [1]
The GPC algorithm involves applying a control sequencethat minimizes a multistage cost function of the form
119869119888=
1198732
sum
119895=1198731
120575 (119895) [119910 (119905 + 119895 | 119905) minus 119908 (119905 + 119895)]2
+
119873119906
sum
119895=1
120582 (119895) [Δ119906 (119905 + 119895 minus 1)]2
(19)
where 119910(119905 + 119895 | 119905) is the predicted output 119895 step ahead ofactual time 119905 119873
1is the minimum cost horizon 119873
2is the
maximum cost horizon 119873119906is the control horizon 120575(119905) and
120582(119905) are the weighting sequences and 119908(119905 + 119895) is the futurereference trajectory The GPC regulator design requires tun-ing the prediction horizons and the two weighting factorsIf one takes into account the possibility that the plant has adelay time then the minimum and maximum horizons arerespectively 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 [1] where 119873
is the prediction horizon Since a high value for the controlhorizon produces an undesirable oscillation in the controlsignal [3 4] which could cause the chattering phenomenonin the induction motor then it is assumed that119873
119906= 1
Mathematical Problems in Engineering 5
The prediction of the future output is as follows
119910 (119905 + 119895 + 119889 | 119905) = 119865119895+119889
(119911minus1
) 119910 (119905)
+ 119864119895+119889
(119911minus1
) 119861 (119911minus1
) Δ119906 (119905 + 119895 minus 1)
(20)
where 119865(119911minus1
) and 119864(119911minus1
) polynomials are calculated byoperating 1119860(119911
minus1
)1198732times where 119860(119911
minus1
) = 119860(119911minus1
)Δaccording to the following Diophantine identity
1 = 119864119895(119911minus1
)119860 (119911minus1
) + 119911minus119895
119865119895(119911minus1
) (21)
Taking into account that
119866119895(119911minus1
) = 119864119895(119911minus1
) 119861 (119911minus1
) (22)
the following set of 119895 ahead optimal predictions for the systemexpressed in matrix form (23) can be obtained
119910 = 119866u + 119865 (119911minus1
) 119910 (119905)
+ 1198661015840
(119911minus1
) Δ119906 (119905 + 119895 minus 1) = 119866u + f (23)
where for an119873 prediction horizon one obtains the119866matrixwhich contains values of the step sequence of the plant(induction motor) with119873 times 119873 dimension
119866 =
[[[[[
[
1198921198731
0 sdot sdot sdot 0
1198921198731+1
1198921198731
d
1198921198731+1
d 0
1198921198732
sdot sdot sdot sdot sdot sdot 1198921198731
]]]]]
]
=
[[[[[[
[
1198661198731
(119911minus1
)
1198661198732
(119911minus1
)
]]]]]]
]
(24)
Thus from (22) and (24) we can see that the 119866 matrixdepends only on the process parameters while the f term isthe free response of the system and it is easy to deduce thatit is a vector of119873 elements
f =[[
[
11986511987310
+ 1198651198731 1
119911minus1
11986511987320
+ 1198651198732 1
119911minus1
]]
]
119910 (119905)
+
[[[
[
1198921198731+1
+ 1198921198731+2
119911minus1
+ sdot sdot sdot + 1198921198731+119889
119911minus(119889minus1)
1198921198732+1
+ 1198921198732+2
119911minus1
+ sdot sdot sdot + 1198921198732+119889
119911minus(119889minus1)
]]]
]
Δ119906 (119905 minus 1)
f =
[[[
[
1198651198731
(119911minus1
)
1198651198732
(119911minus1
)
]]]
]
119910 (119905) +
[[[
[
1198661015840
1198731
(119911minus1
)
1198661015840
1198731
(119911minus1
)
]]]
]
Δ119906 (119905 minus 1)
(25)
Assuming that the 120575(119905) control weighting sequence isconstant and equal to 1 then the cost function (19) can bewritten in the following form
119869119888= (119866u + f minus w)
119879
(119866u + f minus w) + 120582u119879u (26)
minus
+K
120596m
GPC120596
Freeresponsecalculation
b01 minus azminus1
zminusdilowastsq120596lowastm
119943
Figure 3 Diagram of the induction motor speed control schemeusing GPC linear regulator
The minimum of the cost function (26) when no con-straints are considered can be calculated by equalling 0 thegradient of 119869
119888 obtaining the following analytical expression
u = (119866119879
119866 + 120582119868)minus1
119866119879
(w minus f) (27)
where u and w are two vectors of119873 elements
u =[[
[
Δ119906 (119905)
Δ119906 (119905 + 119873 minus 1)
]]
]
w =[[
[
119908 (119905 + 1198731)
119908 (119905 + 119873
2)
]]
]
(28)
As the receding horizon strategy is used the control signalapplied to the process is obtained from the first element of uwhere the following expression must be employed
Δ119906 (119905) = 119870 (w minus f) (29)
where 119870 = [11987011198702sdot sdot sdot 119870119873] that is the first row of the
(119866119879
119866 + 120582119868)minus1
119866119879 matrix in (27)
Now replacing the variables 119910(119905) and 119906(119905) by 120596lowast
119898and 119894lowast
119904119902
respectively it is obtained that
Δ119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) (30)
where finally the control law for the induction motor GPCspeed regulation (31) is obtained Figure 3 as follows
119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) + 119894
lowast
119904119902(119905 minus 1) (31)
The analytical solution of the cost function minimizationis possible only if the control signal (31) is not restrictedor limited [1] As induction motors can support statorovercurrents up to 25 times the rated value in short periodsthis GPC design does not need to limit the stator currentsThat is the 119894
lowast
119904119902control signal is not limited allowing the
implementation of the more simple low computational costanalytical solution With regard to the 119894
lowast
119904119889current it is fixed
to a value to produce the nominal rotor flux 119868119904119889 Finally the
calculation of the 119870 119866 119865 and 1198661015840 parameters is carried out
off-line without using any identification algorithm becausethe usual variation of the main parameters of the motorshould be overcome in an effective form by the inherentrobustness of any closed loop controller [22]
6 Mathematical Problems in Engineering
Table 1 Induction motor parameters (manufacturer)
Parameter Symbol ValueStator resistance 119877
119904081Ω
Rotor resistance 119877119903
057ΩMagnetizing inductance 119871
1198980117774H
Stator inductance 119871119904
0120416HRotor inductance 119871
1199030121498H
Nominal rotor flux 120595119903
101WbNumber of poles 119901 4Nominal torque 119879
119873493Nm
Moment of inertia 119869 0057 kgm2
Viscous friction coefficient 119861V 0015Nm(rads)Temperature coeff AlCu 120572 00039Kminus1
24 Regulator Tuning
241 PID and PI Controllers The two current PI regulatorsuse the same tuning valuesThe higher the bandwidth chosenfor these controllers the faster the current loops dynamicsare However in practice any real systemrsquos bandwidth islimited physically In the employed experimental platform(Section 31) which uses the induction motor described inTable 1 this limit is located at 4000 rads up to this value theplatform produces the undesirable chattering phenomenonand dangerous mechanical vibrations that can damage themachine
As to the speed regulation in the first case a PID speedcontroller has been used to measure the delay time of thesystem This PID speed regulator has also been designed inthe frequency domain using a bandwidth which is 10 timessmaller than the PI current loop regulators [24] Then theoptimal tuning of the PID speed regulator has been obtainedtaking into account this ratio of bandwidths and the physicallimit of the experiment platform Thus the most efficientPID-PI (speed currents) controller system has been designedwith the following parameter values 120596
119888120596= 300 rads with a
margin phase of PM120596= 14311 rad (82∘) and120596
119888119894= 3000 rads
with a margin phase of PM119894= 15707 rad (90∘) The D action
of the speed loop is added after PI is tuned and a small valueis enough to do faster the response but without any overshotwhere 119870
119889= 002 and the PID-PI controller is obtained The
delay time measured in the experiment platform using thisPID-PI speed regulator tuning is 700120583s using the sampletime of 100 120583s (Section 32)
242 SM Controller A sliding-mode speed regulator forinductionmotor is also implemented in order to be comparedwith the proposed GPC regulator This advanced speedregulator was presented by the authors in [7] where theparameters used in its design for the induction motor(Table 1) are obtained considering an uncertainty around 50in the system parameters 119886 = 03947 119887 = 77601 119891 =
17544 lowast119879119871 119896 = 400 120574 = 13 and 120577 = 25 This speed scheme
Table 2 Several stable cases of GPC speed regulator designs
GPC speed regulator designD1 (efficient 1)
Nominal mech parameters120591119898= 38 s
119879119904= 100120583s N = 5 d = 7
N1 = 8 N2 = 12119873119906= 1
120582 = 017
GM120596= 15 dB PM
120596= 8∘
D2 (efficient 2)Nominal mech parameters120591119898= 19 s (119869 = 00285 kgm2)
119879119904= 700120583s N = 5 d= 1
N1 = 2 N2 = 6119873119906= 1 120582 = 1
GM120596= 12 dB PM
120596= 35∘
U1IM 120591119898higher
120591119898= 76 s (J = 0114 kgm2)
GM120596= 25 dB PM
120596= 5∘
U1IM 120591119898higher
120591119898= 38 s (J = 0057 kgm2)
GM120596= 25 dB PM
120596= 25∘
U2IM 120591119898smaller
120591119898= 038 s (119861V = 015Nms)
GM120596= 15 dB PM
120596= 9∘
U2IM 120591119898smaller
120591119898= 019 s (119861V = 015Nms)
GM120596= 12 dB PM
120596= 35∘
PI currents regulators designStator nominal electr parameters 120591
119904= 01487 s (119879 = 20∘C)
119879119904= 100120583s 120596
119888119894= 3000 rads PM
119894= 15707 (90∘)
Us1 120591119904higher 120591
119904= 01612 s (119879 = 0∘C)
PM119894= 90∘
Us2 120591119904smaller 120591
119904= 01040 s (T = 130∘C)
PM119894= 90∘
uses the same two current PI regulators employed inGPC andPI speed schemes and detailed in the previous subsection
243 GPC Controller The tuning of the GPC regulatorrequires choosing the values of two horizons and two weight-ing factors The control horizon 119873
119906 and the output error
weighting factor 120575 both have been fixed to 1 (Section 21)The prediction horizon 119873 determines the size of the 119866
matrix and the f vector and as a consequence the numberof the GPC controllerrsquos coefficients A larger value for 119873
increases the anticipative effect involving a better controland performance of the system However this increases thenumber of the coefficients and the computational cost of thecontrol law which requires increasing the sample time So119873 should be selected to ensure proper dynamic behaviourand low computational cost In this design it has been setto 5 From Section 21 it is known that the minimum andmaximum horizons take into account the delay time of thesystem 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 Considering two GPC
speed regulator designs D1 and D2 using sample times of100 120583s and 700120583s respectively and taking into account thatthe measured experimental delay time is 700 120583s (Section 32)then the delay values are 119889 = 7 for D1 and 119889 = 1 for D2(Table 2)
The control weighting factor 120582 has a direct impact onthe response of the controlled system Hence the higher itsvalue the slower the resulting controlled system On theother hand if its value is too small it can produce thechattering phenomenon that can damage the motor in a realcase Then it is desirable to find a value for 120582 to provide
Mathematical Problems in Engineering 7
+
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
minus180
minus175
minus170
minus165
minus150
minus145
minus140
minus130
minus120
minus100
minus30
minus20
minus10
0
10
20
30
40
6dB
3dB
1dB
05dB
025 dB
GPC D1GPC D1U1GPC D1U2GPC D2
GPC D2U1GPC D2U2PID
Figure 4 Nichols diagram of the GPC and PID speed loops
an effective response of the controlled system Taking intoaccount the recommendation in [15] an easy way to obtain aneffective and stable value for 120582 factor is applying the followingempirical expression
120582opt = 119898 trace (119866119879119866) (32)
where 119898 takes a value inversely proportional to the sampletime For D1 design the 119898 parameter results in a value of 60and for D2119898 equals 2 (tens for 100 120583s units for 700 120583s)
25 Robust Stability Analysis The closed loop stability of themotor 119889-119902 currents with the PI controllers is guaranteed if thePM119894phasemargins are positive and sufficiently highAs to the
speed loop the stability of the controlled system is analysedusing the classic RST poles placement scheme [25] where thecontrol law is obtained as
119894lowast
119904119902(119905) =
1
1199030
[119879 (119911minus1
) 120596lowast
119898(119905) 119879
minus119878 (119911minus1
) 120596119898(119905) minus 119894
lowast
119904119902(119905 minus 1) 119877
1015840
(119911minus1
)]
(33)
minus180 minus135 minus90 minus45 0minus20
minus15
minus10
minus5
0
5 6dB
minus6
minus12
minus20
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
PIPI Us1PI Us2
(dB)
Figure 5 Nichols diagram of the PI current loops
In this sense it is necessary to translate the GPC con-trolled system parameters to its equivalent RST controlledsystem parameters [1] to check the stability of the systemIdentifying the terms between (31) and (33) we can obtainthe general expression of equivalence in both RST and GPCalgorithms for any first order plant with delay time
119877 (119911minus1
) = 1199030+ 1199031119911minus1
+ sdot sdot sdot + 119903119889+1
119911minus(119889+1)
= 1199030+ 119911minus1
1198771015840
(119911minus1
)
119878 (119911minus1
) = 1199040+ 1199041119911minus1
119879 (119911minus1
) = 1199050119911minus1198731 + 119905
1119911minus(1198731+1) + sdot sdot sdot + 119905
119873minus1119911minus1198732
(34)
1199030= 1
1199031= 119870[119892
1198731+1sdot sdot sdot 1198921198732+1
]119879
minus 1
1199032= 119870([119892
1198731+2sdot sdot sdot 1198921198732+2
]119879
minus[1198921198731+1
sdot sdot sdot 1198921198732+1
]119879
)
119903119889= 119870([119892
1198731+119889sdot sdot sdot 1198921198732+119889
]119879
minus[1198921198731+119889minus1
sdot sdot sdot 1198921198732+119889minus1
]119879
)
119903119889+1
= minus119870[1198921198731+119889
sdot sdot sdot 1198921198732+119889
]119879
1199040= 119870[119865
1198731 0sdot sdot sdot 1198651198732 0
]119879
1199041= 119870[119865
1198731 1sdot sdot sdot 1198651198732 1
]119879
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
_+PI
SVPWMVSI
PISVPWM
VSI
120596mTe
TL
1B1 + s(JB)
Lm1 + s(LrRr)
ilowastsq
ilowastsd
isq
isd
K998400T
120595rd
Figure 2 External representation of the induction motor with the PI current control and SVPWM
The electromagnetic torque of the induction motor insteady state taking into account that torque and rotorflux current components are decoupled in the 119889-119902 rotatingreference frame (5) also can be expressed as follows [24]
119879119890=
3119901
4
119871119898
119871119903
120595119903119889119894119904119902
= 119870119879119894119904119902 (11)
where119870119879is the torque constant (12) and 120595
119903119889is the rotor flux
119870119879=
3119901
4
119871119898
119871119903
120595119903119889
= 1198701015840
119879120595119903119889
= 1198701015840
119879119871119898119868119904119889 (12)
Considering that the SVPWM and the VSI modules haveneither dynamics nor gain in diagramof Figure 1 it is possibleto obtain the following transfer function
120596119898(119904)
119868lowast119904119902(119904)
=1198701015840
119879120595119903119889119861V
(1 + 119904120591119903) (1 + 119904120591
119898)
times (119870119901119894119904 + 119870119894
119894
1205901198711199041199042 + (119877
119904+ 119870119901119894) 119904 + 119870119894
119894
)
2
(13)
where the mechanical time constant and the rotor electricaltime constant are respectively
120591119898
=119869
119861V
120591119903=
119871119903
119877119903
(14)
Now if the dynamics associated to the two current loopsare neglected because they are much faster than the restthe following second order transfer function of the inductionmotor is obtained as follows
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898) (1 + 119904120591
119903)119890minus120591119889 (15)
where 120591119889is the delay time of the induction motor
Taking into account the consideration to obtain (10)from (3) Figure 2 and also considering that the dynamicsassociated to the rotor electrical time constant is much fasterthan the mechanical time constant then the electrical pole isneglected Thus the following first order transfer function ofthe induction motor is obtained
120596119898(119904)
119868lowast119904119902(119904)
=119870119879119861V
(1 + 119904120591119898)119890minus120591119889 (16)
The design of the GPC controller is carried out usingthe first order transfer function of the motor As the GPCcontroller is defined in discrete time the transfer functionmust be transformed into a discrete time transfer functionThen using the ZOH (Zero-Order Hold) discretizationmethod it is obtained that
120596119898(119911minus1
)
119868lowast119904119902(119911minus1)
=
119861 (119911minus1
)
119860 (119911minus1)119911minus119889
=1198870
(1 minus 119886119911minus1)119911minus119889
(17)
Taking into account GPC theory and employing theCARIMA model [1] it is possible to obtain the followingequation of the system in which the output 120596
119898is replaced by
119910 the input 119894lowast119904119902is replaced by 119906 and white noise is included
in the previous transfer function (17)
119860(119911minus1
) 119910 (119905) = 119861 (119911minus1
) 119911minus119889
119906 (119905 minus 1) + 119862 (119911minus1
)120576 (119905)
Δ (18)
where 119889 is the delay time of the system and it ismultiple of thesample time chosen 120576 is the white noise with null averageΔ = 1 minus 119911
minus1 and the 119862(119911minus1
) is the noise polynomial that ischosen to be 1 for simplicity in this design [1]
The GPC algorithm involves applying a control sequencethat minimizes a multistage cost function of the form
119869119888=
1198732
sum
119895=1198731
120575 (119895) [119910 (119905 + 119895 | 119905) minus 119908 (119905 + 119895)]2
+
119873119906
sum
119895=1
120582 (119895) [Δ119906 (119905 + 119895 minus 1)]2
(19)
where 119910(119905 + 119895 | 119905) is the predicted output 119895 step ahead ofactual time 119905 119873
1is the minimum cost horizon 119873
2is the
maximum cost horizon 119873119906is the control horizon 120575(119905) and
120582(119905) are the weighting sequences and 119908(119905 + 119895) is the futurereference trajectory The GPC regulator design requires tun-ing the prediction horizons and the two weighting factorsIf one takes into account the possibility that the plant has adelay time then the minimum and maximum horizons arerespectively 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 [1] where 119873
is the prediction horizon Since a high value for the controlhorizon produces an undesirable oscillation in the controlsignal [3 4] which could cause the chattering phenomenonin the induction motor then it is assumed that119873
119906= 1
Mathematical Problems in Engineering 5
The prediction of the future output is as follows
119910 (119905 + 119895 + 119889 | 119905) = 119865119895+119889
(119911minus1
) 119910 (119905)
+ 119864119895+119889
(119911minus1
) 119861 (119911minus1
) Δ119906 (119905 + 119895 minus 1)
(20)
where 119865(119911minus1
) and 119864(119911minus1
) polynomials are calculated byoperating 1119860(119911
minus1
)1198732times where 119860(119911
minus1
) = 119860(119911minus1
)Δaccording to the following Diophantine identity
1 = 119864119895(119911minus1
)119860 (119911minus1
) + 119911minus119895
119865119895(119911minus1
) (21)
Taking into account that
119866119895(119911minus1
) = 119864119895(119911minus1
) 119861 (119911minus1
) (22)
the following set of 119895 ahead optimal predictions for the systemexpressed in matrix form (23) can be obtained
119910 = 119866u + 119865 (119911minus1
) 119910 (119905)
+ 1198661015840
(119911minus1
) Δ119906 (119905 + 119895 minus 1) = 119866u + f (23)
where for an119873 prediction horizon one obtains the119866matrixwhich contains values of the step sequence of the plant(induction motor) with119873 times 119873 dimension
119866 =
[[[[[
[
1198921198731
0 sdot sdot sdot 0
1198921198731+1
1198921198731
d
1198921198731+1
d 0
1198921198732
sdot sdot sdot sdot sdot sdot 1198921198731
]]]]]
]
=
[[[[[[
[
1198661198731
(119911minus1
)
1198661198732
(119911minus1
)
]]]]]]
]
(24)
Thus from (22) and (24) we can see that the 119866 matrixdepends only on the process parameters while the f term isthe free response of the system and it is easy to deduce thatit is a vector of119873 elements
f =[[
[
11986511987310
+ 1198651198731 1
119911minus1
11986511987320
+ 1198651198732 1
119911minus1
]]
]
119910 (119905)
+
[[[
[
1198921198731+1
+ 1198921198731+2
119911minus1
+ sdot sdot sdot + 1198921198731+119889
119911minus(119889minus1)
1198921198732+1
+ 1198921198732+2
119911minus1
+ sdot sdot sdot + 1198921198732+119889
119911minus(119889minus1)
]]]
]
Δ119906 (119905 minus 1)
f =
[[[
[
1198651198731
(119911minus1
)
1198651198732
(119911minus1
)
]]]
]
119910 (119905) +
[[[
[
1198661015840
1198731
(119911minus1
)
1198661015840
1198731
(119911minus1
)
]]]
]
Δ119906 (119905 minus 1)
(25)
Assuming that the 120575(119905) control weighting sequence isconstant and equal to 1 then the cost function (19) can bewritten in the following form
119869119888= (119866u + f minus w)
119879
(119866u + f minus w) + 120582u119879u (26)
minus
+K
120596m
GPC120596
Freeresponsecalculation
b01 minus azminus1
zminusdilowastsq120596lowastm
119943
Figure 3 Diagram of the induction motor speed control schemeusing GPC linear regulator
The minimum of the cost function (26) when no con-straints are considered can be calculated by equalling 0 thegradient of 119869
119888 obtaining the following analytical expression
u = (119866119879
119866 + 120582119868)minus1
119866119879
(w minus f) (27)
where u and w are two vectors of119873 elements
u =[[
[
Δ119906 (119905)
Δ119906 (119905 + 119873 minus 1)
]]
]
w =[[
[
119908 (119905 + 1198731)
119908 (119905 + 119873
2)
]]
]
(28)
As the receding horizon strategy is used the control signalapplied to the process is obtained from the first element of uwhere the following expression must be employed
Δ119906 (119905) = 119870 (w minus f) (29)
where 119870 = [11987011198702sdot sdot sdot 119870119873] that is the first row of the
(119866119879
119866 + 120582119868)minus1
119866119879 matrix in (27)
Now replacing the variables 119910(119905) and 119906(119905) by 120596lowast
119898and 119894lowast
119904119902
respectively it is obtained that
Δ119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) (30)
where finally the control law for the induction motor GPCspeed regulation (31) is obtained Figure 3 as follows
119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) + 119894
lowast
119904119902(119905 minus 1) (31)
The analytical solution of the cost function minimizationis possible only if the control signal (31) is not restrictedor limited [1] As induction motors can support statorovercurrents up to 25 times the rated value in short periodsthis GPC design does not need to limit the stator currentsThat is the 119894
lowast
119904119902control signal is not limited allowing the
implementation of the more simple low computational costanalytical solution With regard to the 119894
lowast
119904119889current it is fixed
to a value to produce the nominal rotor flux 119868119904119889 Finally the
calculation of the 119870 119866 119865 and 1198661015840 parameters is carried out
off-line without using any identification algorithm becausethe usual variation of the main parameters of the motorshould be overcome in an effective form by the inherentrobustness of any closed loop controller [22]
6 Mathematical Problems in Engineering
Table 1 Induction motor parameters (manufacturer)
Parameter Symbol ValueStator resistance 119877
119904081Ω
Rotor resistance 119877119903
057ΩMagnetizing inductance 119871
1198980117774H
Stator inductance 119871119904
0120416HRotor inductance 119871
1199030121498H
Nominal rotor flux 120595119903
101WbNumber of poles 119901 4Nominal torque 119879
119873493Nm
Moment of inertia 119869 0057 kgm2
Viscous friction coefficient 119861V 0015Nm(rads)Temperature coeff AlCu 120572 00039Kminus1
24 Regulator Tuning
241 PID and PI Controllers The two current PI regulatorsuse the same tuning valuesThe higher the bandwidth chosenfor these controllers the faster the current loops dynamicsare However in practice any real systemrsquos bandwidth islimited physically In the employed experimental platform(Section 31) which uses the induction motor described inTable 1 this limit is located at 4000 rads up to this value theplatform produces the undesirable chattering phenomenonand dangerous mechanical vibrations that can damage themachine
As to the speed regulation in the first case a PID speedcontroller has been used to measure the delay time of thesystem This PID speed regulator has also been designed inthe frequency domain using a bandwidth which is 10 timessmaller than the PI current loop regulators [24] Then theoptimal tuning of the PID speed regulator has been obtainedtaking into account this ratio of bandwidths and the physicallimit of the experiment platform Thus the most efficientPID-PI (speed currents) controller system has been designedwith the following parameter values 120596
119888120596= 300 rads with a
margin phase of PM120596= 14311 rad (82∘) and120596
119888119894= 3000 rads
with a margin phase of PM119894= 15707 rad (90∘) The D action
of the speed loop is added after PI is tuned and a small valueis enough to do faster the response but without any overshotwhere 119870
119889= 002 and the PID-PI controller is obtained The
delay time measured in the experiment platform using thisPID-PI speed regulator tuning is 700120583s using the sampletime of 100 120583s (Section 32)
242 SM Controller A sliding-mode speed regulator forinductionmotor is also implemented in order to be comparedwith the proposed GPC regulator This advanced speedregulator was presented by the authors in [7] where theparameters used in its design for the induction motor(Table 1) are obtained considering an uncertainty around 50in the system parameters 119886 = 03947 119887 = 77601 119891 =
17544 lowast119879119871 119896 = 400 120574 = 13 and 120577 = 25 This speed scheme
Table 2 Several stable cases of GPC speed regulator designs
GPC speed regulator designD1 (efficient 1)
Nominal mech parameters120591119898= 38 s
119879119904= 100120583s N = 5 d = 7
N1 = 8 N2 = 12119873119906= 1
120582 = 017
GM120596= 15 dB PM
120596= 8∘
D2 (efficient 2)Nominal mech parameters120591119898= 19 s (119869 = 00285 kgm2)
119879119904= 700120583s N = 5 d= 1
N1 = 2 N2 = 6119873119906= 1 120582 = 1
GM120596= 12 dB PM
120596= 35∘
U1IM 120591119898higher
120591119898= 76 s (J = 0114 kgm2)
GM120596= 25 dB PM
120596= 5∘
U1IM 120591119898higher
120591119898= 38 s (J = 0057 kgm2)
GM120596= 25 dB PM
120596= 25∘
U2IM 120591119898smaller
120591119898= 038 s (119861V = 015Nms)
GM120596= 15 dB PM
120596= 9∘
U2IM 120591119898smaller
120591119898= 019 s (119861V = 015Nms)
GM120596= 12 dB PM
120596= 35∘
PI currents regulators designStator nominal electr parameters 120591
119904= 01487 s (119879 = 20∘C)
119879119904= 100120583s 120596
119888119894= 3000 rads PM
119894= 15707 (90∘)
Us1 120591119904higher 120591
119904= 01612 s (119879 = 0∘C)
PM119894= 90∘
Us2 120591119904smaller 120591
119904= 01040 s (T = 130∘C)
PM119894= 90∘
uses the same two current PI regulators employed inGPC andPI speed schemes and detailed in the previous subsection
243 GPC Controller The tuning of the GPC regulatorrequires choosing the values of two horizons and two weight-ing factors The control horizon 119873
119906 and the output error
weighting factor 120575 both have been fixed to 1 (Section 21)The prediction horizon 119873 determines the size of the 119866
matrix and the f vector and as a consequence the numberof the GPC controllerrsquos coefficients A larger value for 119873
increases the anticipative effect involving a better controland performance of the system However this increases thenumber of the coefficients and the computational cost of thecontrol law which requires increasing the sample time So119873 should be selected to ensure proper dynamic behaviourand low computational cost In this design it has been setto 5 From Section 21 it is known that the minimum andmaximum horizons take into account the delay time of thesystem 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 Considering two GPC
speed regulator designs D1 and D2 using sample times of100 120583s and 700120583s respectively and taking into account thatthe measured experimental delay time is 700 120583s (Section 32)then the delay values are 119889 = 7 for D1 and 119889 = 1 for D2(Table 2)
The control weighting factor 120582 has a direct impact onthe response of the controlled system Hence the higher itsvalue the slower the resulting controlled system On theother hand if its value is too small it can produce thechattering phenomenon that can damage the motor in a realcase Then it is desirable to find a value for 120582 to provide
Mathematical Problems in Engineering 7
+
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
minus180
minus175
minus170
minus165
minus150
minus145
minus140
minus130
minus120
minus100
minus30
minus20
minus10
0
10
20
30
40
6dB
3dB
1dB
05dB
025 dB
GPC D1GPC D1U1GPC D1U2GPC D2
GPC D2U1GPC D2U2PID
Figure 4 Nichols diagram of the GPC and PID speed loops
an effective response of the controlled system Taking intoaccount the recommendation in [15] an easy way to obtain aneffective and stable value for 120582 factor is applying the followingempirical expression
120582opt = 119898 trace (119866119879119866) (32)
where 119898 takes a value inversely proportional to the sampletime For D1 design the 119898 parameter results in a value of 60and for D2119898 equals 2 (tens for 100 120583s units for 700 120583s)
25 Robust Stability Analysis The closed loop stability of themotor 119889-119902 currents with the PI controllers is guaranteed if thePM119894phasemargins are positive and sufficiently highAs to the
speed loop the stability of the controlled system is analysedusing the classic RST poles placement scheme [25] where thecontrol law is obtained as
119894lowast
119904119902(119905) =
1
1199030
[119879 (119911minus1
) 120596lowast
119898(119905) 119879
minus119878 (119911minus1
) 120596119898(119905) minus 119894
lowast
119904119902(119905 minus 1) 119877
1015840
(119911minus1
)]
(33)
minus180 minus135 minus90 minus45 0minus20
minus15
minus10
minus5
0
5 6dB
minus6
minus12
minus20
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
PIPI Us1PI Us2
(dB)
Figure 5 Nichols diagram of the PI current loops
In this sense it is necessary to translate the GPC con-trolled system parameters to its equivalent RST controlledsystem parameters [1] to check the stability of the systemIdentifying the terms between (31) and (33) we can obtainthe general expression of equivalence in both RST and GPCalgorithms for any first order plant with delay time
119877 (119911minus1
) = 1199030+ 1199031119911minus1
+ sdot sdot sdot + 119903119889+1
119911minus(119889+1)
= 1199030+ 119911minus1
1198771015840
(119911minus1
)
119878 (119911minus1
) = 1199040+ 1199041119911minus1
119879 (119911minus1
) = 1199050119911minus1198731 + 119905
1119911minus(1198731+1) + sdot sdot sdot + 119905
119873minus1119911minus1198732
(34)
1199030= 1
1199031= 119870[119892
1198731+1sdot sdot sdot 1198921198732+1
]119879
minus 1
1199032= 119870([119892
1198731+2sdot sdot sdot 1198921198732+2
]119879
minus[1198921198731+1
sdot sdot sdot 1198921198732+1
]119879
)
119903119889= 119870([119892
1198731+119889sdot sdot sdot 1198921198732+119889
]119879
minus[1198921198731+119889minus1
sdot sdot sdot 1198921198732+119889minus1
]119879
)
119903119889+1
= minus119870[1198921198731+119889
sdot sdot sdot 1198921198732+119889
]119879
1199040= 119870[119865
1198731 0sdot sdot sdot 1198651198732 0
]119879
1199041= 119870[119865
1198731 1sdot sdot sdot 1198651198732 1
]119879
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
The prediction of the future output is as follows
119910 (119905 + 119895 + 119889 | 119905) = 119865119895+119889
(119911minus1
) 119910 (119905)
+ 119864119895+119889
(119911minus1
) 119861 (119911minus1
) Δ119906 (119905 + 119895 minus 1)
(20)
where 119865(119911minus1
) and 119864(119911minus1
) polynomials are calculated byoperating 1119860(119911
minus1
)1198732times where 119860(119911
minus1
) = 119860(119911minus1
)Δaccording to the following Diophantine identity
1 = 119864119895(119911minus1
)119860 (119911minus1
) + 119911minus119895
119865119895(119911minus1
) (21)
Taking into account that
119866119895(119911minus1
) = 119864119895(119911minus1
) 119861 (119911minus1
) (22)
the following set of 119895 ahead optimal predictions for the systemexpressed in matrix form (23) can be obtained
119910 = 119866u + 119865 (119911minus1
) 119910 (119905)
+ 1198661015840
(119911minus1
) Δ119906 (119905 + 119895 minus 1) = 119866u + f (23)
where for an119873 prediction horizon one obtains the119866matrixwhich contains values of the step sequence of the plant(induction motor) with119873 times 119873 dimension
119866 =
[[[[[
[
1198921198731
0 sdot sdot sdot 0
1198921198731+1
1198921198731
d
1198921198731+1
d 0
1198921198732
sdot sdot sdot sdot sdot sdot 1198921198731
]]]]]
]
=
[[[[[[
[
1198661198731
(119911minus1
)
1198661198732
(119911minus1
)
]]]]]]
]
(24)
Thus from (22) and (24) we can see that the 119866 matrixdepends only on the process parameters while the f term isthe free response of the system and it is easy to deduce thatit is a vector of119873 elements
f =[[
[
11986511987310
+ 1198651198731 1
119911minus1
11986511987320
+ 1198651198732 1
119911minus1
]]
]
119910 (119905)
+
[[[
[
1198921198731+1
+ 1198921198731+2
119911minus1
+ sdot sdot sdot + 1198921198731+119889
119911minus(119889minus1)
1198921198732+1
+ 1198921198732+2
119911minus1
+ sdot sdot sdot + 1198921198732+119889
119911minus(119889minus1)
]]]
]
Δ119906 (119905 minus 1)
f =
[[[
[
1198651198731
(119911minus1
)
1198651198732
(119911minus1
)
]]]
]
119910 (119905) +
[[[
[
1198661015840
1198731
(119911minus1
)
1198661015840
1198731
(119911minus1
)
]]]
]
Δ119906 (119905 minus 1)
(25)
Assuming that the 120575(119905) control weighting sequence isconstant and equal to 1 then the cost function (19) can bewritten in the following form
119869119888= (119866u + f minus w)
119879
(119866u + f minus w) + 120582u119879u (26)
minus
+K
120596m
GPC120596
Freeresponsecalculation
b01 minus azminus1
zminusdilowastsq120596lowastm
119943
Figure 3 Diagram of the induction motor speed control schemeusing GPC linear regulator
The minimum of the cost function (26) when no con-straints are considered can be calculated by equalling 0 thegradient of 119869
119888 obtaining the following analytical expression
u = (119866119879
119866 + 120582119868)minus1
119866119879
(w minus f) (27)
where u and w are two vectors of119873 elements
u =[[
[
Δ119906 (119905)
Δ119906 (119905 + 119873 minus 1)
]]
]
w =[[
[
119908 (119905 + 1198731)
119908 (119905 + 119873
2)
]]
]
(28)
As the receding horizon strategy is used the control signalapplied to the process is obtained from the first element of uwhere the following expression must be employed
Δ119906 (119905) = 119870 (w minus f) (29)
where 119870 = [11987011198702sdot sdot sdot 119870119873] that is the first row of the
(119866119879
119866 + 120582119868)minus1
119866119879 matrix in (27)
Now replacing the variables 119910(119905) and 119906(119905) by 120596lowast
119898and 119894lowast
119904119902
respectively it is obtained that
Δ119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) (30)
where finally the control law for the induction motor GPCspeed regulation (31) is obtained Figure 3 as follows
119894lowast
119904119902(119905) = 119870 (120596
lowast
119898minus f) + 119894
lowast
119904119902(119905 minus 1) (31)
The analytical solution of the cost function minimizationis possible only if the control signal (31) is not restrictedor limited [1] As induction motors can support statorovercurrents up to 25 times the rated value in short periodsthis GPC design does not need to limit the stator currentsThat is the 119894
lowast
119904119902control signal is not limited allowing the
implementation of the more simple low computational costanalytical solution With regard to the 119894
lowast
119904119889current it is fixed
to a value to produce the nominal rotor flux 119868119904119889 Finally the
calculation of the 119870 119866 119865 and 1198661015840 parameters is carried out
off-line without using any identification algorithm becausethe usual variation of the main parameters of the motorshould be overcome in an effective form by the inherentrobustness of any closed loop controller [22]
6 Mathematical Problems in Engineering
Table 1 Induction motor parameters (manufacturer)
Parameter Symbol ValueStator resistance 119877
119904081Ω
Rotor resistance 119877119903
057ΩMagnetizing inductance 119871
1198980117774H
Stator inductance 119871119904
0120416HRotor inductance 119871
1199030121498H
Nominal rotor flux 120595119903
101WbNumber of poles 119901 4Nominal torque 119879
119873493Nm
Moment of inertia 119869 0057 kgm2
Viscous friction coefficient 119861V 0015Nm(rads)Temperature coeff AlCu 120572 00039Kminus1
24 Regulator Tuning
241 PID and PI Controllers The two current PI regulatorsuse the same tuning valuesThe higher the bandwidth chosenfor these controllers the faster the current loops dynamicsare However in practice any real systemrsquos bandwidth islimited physically In the employed experimental platform(Section 31) which uses the induction motor described inTable 1 this limit is located at 4000 rads up to this value theplatform produces the undesirable chattering phenomenonand dangerous mechanical vibrations that can damage themachine
As to the speed regulation in the first case a PID speedcontroller has been used to measure the delay time of thesystem This PID speed regulator has also been designed inthe frequency domain using a bandwidth which is 10 timessmaller than the PI current loop regulators [24] Then theoptimal tuning of the PID speed regulator has been obtainedtaking into account this ratio of bandwidths and the physicallimit of the experiment platform Thus the most efficientPID-PI (speed currents) controller system has been designedwith the following parameter values 120596
119888120596= 300 rads with a
margin phase of PM120596= 14311 rad (82∘) and120596
119888119894= 3000 rads
with a margin phase of PM119894= 15707 rad (90∘) The D action
of the speed loop is added after PI is tuned and a small valueis enough to do faster the response but without any overshotwhere 119870
119889= 002 and the PID-PI controller is obtained The
delay time measured in the experiment platform using thisPID-PI speed regulator tuning is 700120583s using the sampletime of 100 120583s (Section 32)
242 SM Controller A sliding-mode speed regulator forinductionmotor is also implemented in order to be comparedwith the proposed GPC regulator This advanced speedregulator was presented by the authors in [7] where theparameters used in its design for the induction motor(Table 1) are obtained considering an uncertainty around 50in the system parameters 119886 = 03947 119887 = 77601 119891 =
17544 lowast119879119871 119896 = 400 120574 = 13 and 120577 = 25 This speed scheme
Table 2 Several stable cases of GPC speed regulator designs
GPC speed regulator designD1 (efficient 1)
Nominal mech parameters120591119898= 38 s
119879119904= 100120583s N = 5 d = 7
N1 = 8 N2 = 12119873119906= 1
120582 = 017
GM120596= 15 dB PM
120596= 8∘
D2 (efficient 2)Nominal mech parameters120591119898= 19 s (119869 = 00285 kgm2)
119879119904= 700120583s N = 5 d= 1
N1 = 2 N2 = 6119873119906= 1 120582 = 1
GM120596= 12 dB PM
120596= 35∘
U1IM 120591119898higher
120591119898= 76 s (J = 0114 kgm2)
GM120596= 25 dB PM
120596= 5∘
U1IM 120591119898higher
120591119898= 38 s (J = 0057 kgm2)
GM120596= 25 dB PM
120596= 25∘
U2IM 120591119898smaller
120591119898= 038 s (119861V = 015Nms)
GM120596= 15 dB PM
120596= 9∘
U2IM 120591119898smaller
120591119898= 019 s (119861V = 015Nms)
GM120596= 12 dB PM
120596= 35∘
PI currents regulators designStator nominal electr parameters 120591
119904= 01487 s (119879 = 20∘C)
119879119904= 100120583s 120596
119888119894= 3000 rads PM
119894= 15707 (90∘)
Us1 120591119904higher 120591
119904= 01612 s (119879 = 0∘C)
PM119894= 90∘
Us2 120591119904smaller 120591
119904= 01040 s (T = 130∘C)
PM119894= 90∘
uses the same two current PI regulators employed inGPC andPI speed schemes and detailed in the previous subsection
243 GPC Controller The tuning of the GPC regulatorrequires choosing the values of two horizons and two weight-ing factors The control horizon 119873
119906 and the output error
weighting factor 120575 both have been fixed to 1 (Section 21)The prediction horizon 119873 determines the size of the 119866
matrix and the f vector and as a consequence the numberof the GPC controllerrsquos coefficients A larger value for 119873
increases the anticipative effect involving a better controland performance of the system However this increases thenumber of the coefficients and the computational cost of thecontrol law which requires increasing the sample time So119873 should be selected to ensure proper dynamic behaviourand low computational cost In this design it has been setto 5 From Section 21 it is known that the minimum andmaximum horizons take into account the delay time of thesystem 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 Considering two GPC
speed regulator designs D1 and D2 using sample times of100 120583s and 700120583s respectively and taking into account thatthe measured experimental delay time is 700 120583s (Section 32)then the delay values are 119889 = 7 for D1 and 119889 = 1 for D2(Table 2)
The control weighting factor 120582 has a direct impact onthe response of the controlled system Hence the higher itsvalue the slower the resulting controlled system On theother hand if its value is too small it can produce thechattering phenomenon that can damage the motor in a realcase Then it is desirable to find a value for 120582 to provide
Mathematical Problems in Engineering 7
+
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
minus180
minus175
minus170
minus165
minus150
minus145
minus140
minus130
minus120
minus100
minus30
minus20
minus10
0
10
20
30
40
6dB
3dB
1dB
05dB
025 dB
GPC D1GPC D1U1GPC D1U2GPC D2
GPC D2U1GPC D2U2PID
Figure 4 Nichols diagram of the GPC and PID speed loops
an effective response of the controlled system Taking intoaccount the recommendation in [15] an easy way to obtain aneffective and stable value for 120582 factor is applying the followingempirical expression
120582opt = 119898 trace (119866119879119866) (32)
where 119898 takes a value inversely proportional to the sampletime For D1 design the 119898 parameter results in a value of 60and for D2119898 equals 2 (tens for 100 120583s units for 700 120583s)
25 Robust Stability Analysis The closed loop stability of themotor 119889-119902 currents with the PI controllers is guaranteed if thePM119894phasemargins are positive and sufficiently highAs to the
speed loop the stability of the controlled system is analysedusing the classic RST poles placement scheme [25] where thecontrol law is obtained as
119894lowast
119904119902(119905) =
1
1199030
[119879 (119911minus1
) 120596lowast
119898(119905) 119879
minus119878 (119911minus1
) 120596119898(119905) minus 119894
lowast
119904119902(119905 minus 1) 119877
1015840
(119911minus1
)]
(33)
minus180 minus135 minus90 minus45 0minus20
minus15
minus10
minus5
0
5 6dB
minus6
minus12
minus20
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
PIPI Us1PI Us2
(dB)
Figure 5 Nichols diagram of the PI current loops
In this sense it is necessary to translate the GPC con-trolled system parameters to its equivalent RST controlledsystem parameters [1] to check the stability of the systemIdentifying the terms between (31) and (33) we can obtainthe general expression of equivalence in both RST and GPCalgorithms for any first order plant with delay time
119877 (119911minus1
) = 1199030+ 1199031119911minus1
+ sdot sdot sdot + 119903119889+1
119911minus(119889+1)
= 1199030+ 119911minus1
1198771015840
(119911minus1
)
119878 (119911minus1
) = 1199040+ 1199041119911minus1
119879 (119911minus1
) = 1199050119911minus1198731 + 119905
1119911minus(1198731+1) + sdot sdot sdot + 119905
119873minus1119911minus1198732
(34)
1199030= 1
1199031= 119870[119892
1198731+1sdot sdot sdot 1198921198732+1
]119879
minus 1
1199032= 119870([119892
1198731+2sdot sdot sdot 1198921198732+2
]119879
minus[1198921198731+1
sdot sdot sdot 1198921198732+1
]119879
)
119903119889= 119870([119892
1198731+119889sdot sdot sdot 1198921198732+119889
]119879
minus[1198921198731+119889minus1
sdot sdot sdot 1198921198732+119889minus1
]119879
)
119903119889+1
= minus119870[1198921198731+119889
sdot sdot sdot 1198921198732+119889
]119879
1199040= 119870[119865
1198731 0sdot sdot sdot 1198651198732 0
]119879
1199041= 119870[119865
1198731 1sdot sdot sdot 1198651198732 1
]119879
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 1 Induction motor parameters (manufacturer)
Parameter Symbol ValueStator resistance 119877
119904081Ω
Rotor resistance 119877119903
057ΩMagnetizing inductance 119871
1198980117774H
Stator inductance 119871119904
0120416HRotor inductance 119871
1199030121498H
Nominal rotor flux 120595119903
101WbNumber of poles 119901 4Nominal torque 119879
119873493Nm
Moment of inertia 119869 0057 kgm2
Viscous friction coefficient 119861V 0015Nm(rads)Temperature coeff AlCu 120572 00039Kminus1
24 Regulator Tuning
241 PID and PI Controllers The two current PI regulatorsuse the same tuning valuesThe higher the bandwidth chosenfor these controllers the faster the current loops dynamicsare However in practice any real systemrsquos bandwidth islimited physically In the employed experimental platform(Section 31) which uses the induction motor described inTable 1 this limit is located at 4000 rads up to this value theplatform produces the undesirable chattering phenomenonand dangerous mechanical vibrations that can damage themachine
As to the speed regulation in the first case a PID speedcontroller has been used to measure the delay time of thesystem This PID speed regulator has also been designed inthe frequency domain using a bandwidth which is 10 timessmaller than the PI current loop regulators [24] Then theoptimal tuning of the PID speed regulator has been obtainedtaking into account this ratio of bandwidths and the physicallimit of the experiment platform Thus the most efficientPID-PI (speed currents) controller system has been designedwith the following parameter values 120596
119888120596= 300 rads with a
margin phase of PM120596= 14311 rad (82∘) and120596
119888119894= 3000 rads
with a margin phase of PM119894= 15707 rad (90∘) The D action
of the speed loop is added after PI is tuned and a small valueis enough to do faster the response but without any overshotwhere 119870
119889= 002 and the PID-PI controller is obtained The
delay time measured in the experiment platform using thisPID-PI speed regulator tuning is 700120583s using the sampletime of 100 120583s (Section 32)
242 SM Controller A sliding-mode speed regulator forinductionmotor is also implemented in order to be comparedwith the proposed GPC regulator This advanced speedregulator was presented by the authors in [7] where theparameters used in its design for the induction motor(Table 1) are obtained considering an uncertainty around 50in the system parameters 119886 = 03947 119887 = 77601 119891 =
17544 lowast119879119871 119896 = 400 120574 = 13 and 120577 = 25 This speed scheme
Table 2 Several stable cases of GPC speed regulator designs
GPC speed regulator designD1 (efficient 1)
Nominal mech parameters120591119898= 38 s
119879119904= 100120583s N = 5 d = 7
N1 = 8 N2 = 12119873119906= 1
120582 = 017
GM120596= 15 dB PM
120596= 8∘
D2 (efficient 2)Nominal mech parameters120591119898= 19 s (119869 = 00285 kgm2)
119879119904= 700120583s N = 5 d= 1
N1 = 2 N2 = 6119873119906= 1 120582 = 1
GM120596= 12 dB PM
120596= 35∘
U1IM 120591119898higher
120591119898= 76 s (J = 0114 kgm2)
GM120596= 25 dB PM
120596= 5∘
U1IM 120591119898higher
120591119898= 38 s (J = 0057 kgm2)
GM120596= 25 dB PM
120596= 25∘
U2IM 120591119898smaller
120591119898= 038 s (119861V = 015Nms)
GM120596= 15 dB PM
120596= 9∘
U2IM 120591119898smaller
120591119898= 019 s (119861V = 015Nms)
GM120596= 12 dB PM
120596= 35∘
PI currents regulators designStator nominal electr parameters 120591
119904= 01487 s (119879 = 20∘C)
119879119904= 100120583s 120596
119888119894= 3000 rads PM
119894= 15707 (90∘)
Us1 120591119904higher 120591
119904= 01612 s (119879 = 0∘C)
PM119894= 90∘
Us2 120591119904smaller 120591
119904= 01040 s (T = 130∘C)
PM119894= 90∘
uses the same two current PI regulators employed inGPC andPI speed schemes and detailed in the previous subsection
243 GPC Controller The tuning of the GPC regulatorrequires choosing the values of two horizons and two weight-ing factors The control horizon 119873
119906 and the output error
weighting factor 120575 both have been fixed to 1 (Section 21)The prediction horizon 119873 determines the size of the 119866
matrix and the f vector and as a consequence the numberof the GPC controllerrsquos coefficients A larger value for 119873
increases the anticipative effect involving a better controland performance of the system However this increases thenumber of the coefficients and the computational cost of thecontrol law which requires increasing the sample time So119873 should be selected to ensure proper dynamic behaviourand low computational cost In this design it has been setto 5 From Section 21 it is known that the minimum andmaximum horizons take into account the delay time of thesystem 119873
1= 1 + 119889 and 119873
2= 119873 + 119889 Considering two GPC
speed regulator designs D1 and D2 using sample times of100 120583s and 700120583s respectively and taking into account thatthe measured experimental delay time is 700 120583s (Section 32)then the delay values are 119889 = 7 for D1 and 119889 = 1 for D2(Table 2)
The control weighting factor 120582 has a direct impact onthe response of the controlled system Hence the higher itsvalue the slower the resulting controlled system On theother hand if its value is too small it can produce thechattering phenomenon that can damage the motor in a realcase Then it is desirable to find a value for 120582 to provide
Mathematical Problems in Engineering 7
+
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
minus180
minus175
minus170
minus165
minus150
minus145
minus140
minus130
minus120
minus100
minus30
minus20
minus10
0
10
20
30
40
6dB
3dB
1dB
05dB
025 dB
GPC D1GPC D1U1GPC D1U2GPC D2
GPC D2U1GPC D2U2PID
Figure 4 Nichols diagram of the GPC and PID speed loops
an effective response of the controlled system Taking intoaccount the recommendation in [15] an easy way to obtain aneffective and stable value for 120582 factor is applying the followingempirical expression
120582opt = 119898 trace (119866119879119866) (32)
where 119898 takes a value inversely proportional to the sampletime For D1 design the 119898 parameter results in a value of 60and for D2119898 equals 2 (tens for 100 120583s units for 700 120583s)
25 Robust Stability Analysis The closed loop stability of themotor 119889-119902 currents with the PI controllers is guaranteed if thePM119894phasemargins are positive and sufficiently highAs to the
speed loop the stability of the controlled system is analysedusing the classic RST poles placement scheme [25] where thecontrol law is obtained as
119894lowast
119904119902(119905) =
1
1199030
[119879 (119911minus1
) 120596lowast
119898(119905) 119879
minus119878 (119911minus1
) 120596119898(119905) minus 119894
lowast
119904119902(119905 minus 1) 119877
1015840
(119911minus1
)]
(33)
minus180 minus135 minus90 minus45 0minus20
minus15
minus10
minus5
0
5 6dB
minus6
minus12
minus20
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
PIPI Us1PI Us2
(dB)
Figure 5 Nichols diagram of the PI current loops
In this sense it is necessary to translate the GPC con-trolled system parameters to its equivalent RST controlledsystem parameters [1] to check the stability of the systemIdentifying the terms between (31) and (33) we can obtainthe general expression of equivalence in both RST and GPCalgorithms for any first order plant with delay time
119877 (119911minus1
) = 1199030+ 1199031119911minus1
+ sdot sdot sdot + 119903119889+1
119911minus(119889+1)
= 1199030+ 119911minus1
1198771015840
(119911minus1
)
119878 (119911minus1
) = 1199040+ 1199041119911minus1
119879 (119911minus1
) = 1199050119911minus1198731 + 119905
1119911minus(1198731+1) + sdot sdot sdot + 119905
119873minus1119911minus1198732
(34)
1199030= 1
1199031= 119870[119892
1198731+1sdot sdot sdot 1198921198732+1
]119879
minus 1
1199032= 119870([119892
1198731+2sdot sdot sdot 1198921198732+2
]119879
minus[1198921198731+1
sdot sdot sdot 1198921198732+1
]119879
)
119903119889= 119870([119892
1198731+119889sdot sdot sdot 1198921198732+119889
]119879
minus[1198921198731+119889minus1
sdot sdot sdot 1198921198732+119889minus1
]119879
)
119903119889+1
= minus119870[1198921198731+119889
sdot sdot sdot 1198921198732+119889
]119879
1199040= 119870[119865
1198731 0sdot sdot sdot 1198651198732 0
]119879
1199041= 119870[119865
1198731 1sdot sdot sdot 1198651198732 1
]119879
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
+
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
minus180
minus175
minus170
minus165
minus150
minus145
minus140
minus130
minus120
minus100
minus30
minus20
minus10
0
10
20
30
40
6dB
3dB
1dB
05dB
025 dB
GPC D1GPC D1U1GPC D1U2GPC D2
GPC D2U1GPC D2U2PID
Figure 4 Nichols diagram of the GPC and PID speed loops
an effective response of the controlled system Taking intoaccount the recommendation in [15] an easy way to obtain aneffective and stable value for 120582 factor is applying the followingempirical expression
120582opt = 119898 trace (119866119879119866) (32)
where 119898 takes a value inversely proportional to the sampletime For D1 design the 119898 parameter results in a value of 60and for D2119898 equals 2 (tens for 100 120583s units for 700 120583s)
25 Robust Stability Analysis The closed loop stability of themotor 119889-119902 currents with the PI controllers is guaranteed if thePM119894phasemargins are positive and sufficiently highAs to the
speed loop the stability of the controlled system is analysedusing the classic RST poles placement scheme [25] where thecontrol law is obtained as
119894lowast
119904119902(119905) =
1
1199030
[119879 (119911minus1
) 120596lowast
119898(119905) 119879
minus119878 (119911minus1
) 120596119898(119905) minus 119894
lowast
119904119902(119905 minus 1) 119877
1015840
(119911minus1
)]
(33)
minus180 minus135 minus90 minus45 0minus20
minus15
minus10
minus5
0
5 6dB
minus6
minus12
minus20
Open-loop phase (deg)
Ope
n-lo
op g
ain
(dB)
PIPI Us1PI Us2
(dB)
Figure 5 Nichols diagram of the PI current loops
In this sense it is necessary to translate the GPC con-trolled system parameters to its equivalent RST controlledsystem parameters [1] to check the stability of the systemIdentifying the terms between (31) and (33) we can obtainthe general expression of equivalence in both RST and GPCalgorithms for any first order plant with delay time
119877 (119911minus1
) = 1199030+ 1199031119911minus1
+ sdot sdot sdot + 119903119889+1
119911minus(119889+1)
= 1199030+ 119911minus1
1198771015840
(119911minus1
)
119878 (119911minus1
) = 1199040+ 1199041119911minus1
119879 (119911minus1
) = 1199050119911minus1198731 + 119905
1119911minus(1198731+1) + sdot sdot sdot + 119905
119873minus1119911minus1198732
(34)
1199030= 1
1199031= 119870[119892
1198731+1sdot sdot sdot 1198921198732+1
]119879
minus 1
1199032= 119870([119892
1198731+2sdot sdot sdot 1198921198732+2
]119879
minus[1198921198731+1
sdot sdot sdot 1198921198732+1
]119879
)
119903119889= 119870([119892
1198731+119889sdot sdot sdot 1198921198732+119889
]119879
minus[1198921198731+119889minus1
sdot sdot sdot 1198921198732+119889minus1
]119879
)
119903119889+1
= minus119870[1198921198731+119889
sdot sdot sdot 1198921198732+119889
]119879
1199040= 119870[119865
1198731 0sdot sdot sdot 1198651198732 0
]119879
1199041= 119870[119865
1198731 1sdot sdot sdot 1198651198732 1
]119879
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
1199050= 1198701
1199051= 1198702
119905119873minus1
= 119870119873
(35)
where it should be noted that the polynomial119879multiplies thefuture references vector (33) Thus the controller should bedesigned so that the controlled system presents all the polesinside the unit circle In order to achieve this a graphicalrepresentation (Nichols chart) of the open loop system ismore appropriate as GM
120596gain margin and PM
120596phase
margin can be observedThe robust stability is analysed taking into account the
parametric uncertainties of the induction motor in two limitreal cases of each control loop detailed in Table 2These limitsare used to determine the robustness of the speed and currentregulators as these limits are both critical in real casesHenceas the controller is lineal any uncertainty case located into thedefined limits has the stability guaranteed
First the following two uncertainty cases for the limitsof robustness related with the mechanical parameters of theinductionmotor are considered in the proposedGPCTheU1takes 119869 a 100 higher and the U2 that considers 119861V 10 timeshigher increasing 2 times and reducing 10 times respectivelythe 120591119898mechanical time constant of the machine The D1 and
D2 designs are presented for the speed GPC regulatorTheD1uses a sample time of 100120583s andnominal 119869of 0057 kgm2 andD2 a sample time of 700 120583s and a nominal 119869 of 00285 kgm2On the other hand another two limit cases are considered forthe PI current controllers which are related with the statorwindings resistances (119877
119904) modified by their temperature Us1
determines the uncertainty when the temperature of thewindings is 0∘C (before starting in a cold place) and Us2considers the heating case of the stator windings when theinduction motor is working at rating power for half an houror more where the stator windings are around 130∘C As it isknown the temperature variation produces the proportionalincrease of the stator windings resistance
119877119891= 1198770(1 + 120572 (119879
119891minus 1198790)) (36)
and the proportional decrease of the 120591119904stator electrical time
constant
120591119904=
119871119904
119877119904
(37)
Table 2 shows the two efficient designs for the GPC speedregulator D1 and D2 taking into account different nominalmoments of inertia of the induction motor and sample timesin each case The two uncertainties cases U1 and U2 havebeen based in the D1 andD2 designs but even so the stabilityof the controlled system is guaranteed observing the gainand phase margins obtained from Figure 4 In Table 2 theefficient design for the two PI current controllers is detailed
with Us1 and Us2 uncertainties cases Figure 5 shows that thetemperature variations in the stator windings practically havenot any effect
3 Simulation and Experimental Results
31 Experiment Platform The employed platform is com-posed by a PC with MatLab7Simulink R2007a dsControl321 and the DS1103 Controller Board real time interfaceof dSpace with a floating point PowerPC processor to1 GHz and a set of electric machines that includes a M2AA132M4ABB commercial inductionmotor of 75 kWof die-castaluminium squirrel-cage type (1445 rpm) Table 1 connectedto a DC bus of 540V by VSI inverter and a 190U2 Unimotorsynchronous AC servo in motor of 106 kW to generate theload torque (controlled in torque) presented in [15] Anincremental encoder of 4096 pulses is employed to measurethe mechanical speed of the induction motor The rotor fluxof the induction motor has been set to its nominal value of101Wb keeping the flux current command 119894
lowast
119904119902 that is 119868
119904119889 to
a constant value of 861 A Finally as the SVPWMmodulatorfrequency is fixed at 10 kHz then the sample time employedfor the PI current controllers is 100 120583s
32 Speed Tracking Using D1 design for the GPC controllersimulation and experimental tests are carried out with atrapezoidal speed reference of 1445 rpm and 033Hz addingan square form load torque of 30Nm (starting from thesecond period of the speed reference)
Results are shown in Figure 6 where very satisfactoryspeed tracking can be observed obtaining a stationary speederror of about 2 rpm (0138) in the experimental caseboth without and with load torque In addition the electro-magnetic torque does not present any aggressiveness andconsequently it will not generate the undesirable chatteringphenomenon as seen in the third graph Moreover around119905 = 4 s and 119905 = 55 s the motor is working at nominal speedand torque (Table 2) obtaining an excellent speed trackingConsidering the electromagnetic torque reference current119894lowast
119904119902 it should be noted that it is not limited which justifies the
employed analytical Moreover it can be seen that the rotorflux current 119894
119904119889 takes the value imposed by its reference in
accordance with the previous assumption of the theoreticalsection (see Section 23) and consequently that the rotor fluxremains constant at its nominal value for a short time after itis requested verifying the decoupling of its components Thegreat similarity between the simulation and the experimentaltests validates the presented GPC speed regulator and theyshow that the choice of the first ordermodel for the inductionmotor is correct
Figure 7 shows the speed response and the electromag-netic torque reference current for D1 of GPC speed controllerand PID speed regulator cases It can be observed that thePI regulator has a delay of 700120583s (119889 = 7) and that the GPCcontroller generates the control signal 12 sample times (119873
2)
before the speed reference compensating for the delay timeof the system and anticipating 5 steps (119873) the reaction of thespeed response
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp real
(a)
0 1 2 3 4 5 6minus4minus20 2 4
Time (s)
Spee
d er
ror
SimExp
(b)
0 1 2 3 4 5 6
minus500
50
Time (s)
Te
(Nm
)
Te simTe exp
TLoad
(c)
0 1 2 3 4 5 6minus50minus20
02050
Time (s)
Sim refExp ref
i sq
ref (
A)
(d)
0 1 2 3 45 60
5
10
Time (s)
ReferenceSim real
Exp real
i sd
(A)
(e)
0 1 2 3 4 5 60
05
1
Time (s)
Phir
(Wb)
Sim realExp real
(f)
0 1 2 3 4 5 6minus40minus20
02040
Time (s)
Stat
or cu
rren
ti A
(A)
iA simiA exp
(g)
Figure 6 Simulation and real tests of the GPC-PI model with D1 design with unknown load torque
In Figure 8 the performance comparison between theregulator proposed in this paper GPC1 and a previouslyproposed GPC one [15] based on a second order transferfunction (15) GPC2 can be observed The tests are based ona 1200 rpm and 075Hz a saw-toothed speed reference andan unknown load torque of 25Nm starting from 165 s Com-paring the two speed responses it is possible to appreciatethat the proposed GPC speed regulator (GPC1) is better than
the previous version (GPC2) because the speed response issimilar but is less oscillatory
Figure 9 compares the GPC1 SM PI and PID speedregulators in the same test conditions as in the previous caseComparing the responses of the GPC1 and SM regulators itcan be observed that the predictive response is a little fasterthan sliding modesrsquo response Moreover the effect of the loadtorque in the GPC1 case is minor than the SM case which
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
29988 29990 29992 29994 29996498
499
500
501
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC
Exp PID
N = 5 d = 7
(a)
29988 29990 29992 29994 29996
minus40
minus20
0
Time (s)
Exp GPCExp PID
d + N = 7 + 5 = 12
Torq
ue cu
rren
t ref
ilowast sq
(A)
(b)
Figure 7 Comparative experimental tests speed responses (a) and torque reference currents (b) of GPC D1 controller versus PI regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp GPC2
Figure 8 Comparative experimental tests speed responses of GPC1D1 controller versus GPC2 D1 regulator
145 15 155 16 165 17 175 18500550600650700750800
Time (s)
Roro
r spe
ed (r
pm)
ReferenceExp GPC1Exp SM
Exp PIDExp PI
Figure 9 Comparative experimental tests speed responses of GPC1D1 SM PI and PID speed regulators
demonstrates the robustness of the approach when varyingload torques are applied In addition it can be seen that thePID is faster than PI due to derivative effect Also it can beobserved that the GPC1 regulatorrsquos response is considerablyfaster than PIDrsquos response
Figures 10 and 11 show the comparative experimental testsof the GPC1 SM and PID speed regulators using constantacceleration and variable acceleration cases respectivelyFigure 10 shows that the speed tracking without any loadtorque (between 0 and 225 s) is very similar for the threeregulators though it can be observed that the speed erroris about 2 rpm for GPC while its magnitude is increasedto 4 and 5 rpm in the SM and PID cases When the loadtorque is considered the speed tracking is considerably worsefor SM and PID cases while the GPC maintains a goodspeed tracking (fourth graph (a)) Moreover GPC regulatorsupports better than the SM and PID controllers the loadtorque changes (fourth graph on the right (b)) Figure 11shows that the speed tracking without any load torque(between 0 and 225 s) is similar for the three regulatorsalthough the speed error is 2 rpm for GPC while it increasesto 4 an 5 rpm in the SM cases and about 8 rpm in the PIDcase Moreover the GPC regulator provides a faster responsein the initial half period (fourth graph on the left (a)) Whenthe load torque appears the speed tracking is considerablyworse for PID case while the GPC and SM maintain a goodspeed tracking (fourth graph on the right (b)) though theGPC response is a little faster than sliding modersquos response
Therefore the use of the first order transfer functionmodel is justified because the first order model simplifies thecomputational cost and the controlled performance containsless oscillations than the second order case
33 Load Disturbance Uncertainties and Measurement NoiseRejection One of the issues that usually exists in real appli-cations is load disturbance and in the previous tests theproposed GPC speed controller has demonstrated its perfor-mance even in presence of this effect Moreover parametricuncertainties that is the change of values in the inductionmotor parameters can arise These have been considered asU1 and U2 in Table 2 Finally the measurement noise in thetwo loop signals in the rotor speed and in the stator currenthas a negative impact on the controllers
In this sense the graphs of Figure 12 show the simulationand real tests using a 1200 rpm and 033Hz trapezoidal speedreference and an unknown square form load disturbance
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
0 1 2 3 4 5 6minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5
minus202
5
Time (s)
Spee
d er
ror
Exp GPC1Exp SMExp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
29 3 31 315 32minus1445minus1000
minus5000
50010001445
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
(a)
Time (s)
ReferenceExp GPC1
Exp SMExp PID
(b)
375 38 385 39 395
minus1445
minus1360
minus1330
minus1300
Roto
r spe
ed (r
pm)
285 295 305
TLo
ad(N
m)
TLoad
Figure 10 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for constant acceleration case
of 30Nm The GPC speed regulator is designed taking intoaccount the nominal parameters of the motor D2 design(Table 2) In the simulation case the induction motor hasthe nominal values in its parameters and the unknown loaddisturbance in its shaftonly is considered In the experimentalcase an additional software based measurement white noise
is induced in the two feedback signals 120596119898and 119894119860(usingMat-
LabdSControl) In this way robustness againstmeasurementnoises is demonstrated Additionally in this case parameteruncertainties and load disturbances are introduced this isa greater stator (Cu) resistance at 130∘C and a 100 greatermoment of inertia Us2 andU2 cases for D2 design in Table 2
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
37 375 38 385minus500
0
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
ReferenceExp GPC1
Exp SMExp PID
(b)
01 012 014 016 018
500
Time (s)
Roto
r spe
ed (r
pm)
(a)
0 1 2 3 4 5 6minus500
0
500
Time (s)
Roto
r spe
ed (r
pm)
ReferenceExp GPC1
Exp SMExp PID
0 1 2 3 4 5 6
minus5minus2
025
Time (s)
Spee
d er
ror
Exp GPC1Exp SM
Exp PID
0 1 2 3 4 5 6
minus50
0
50
Time (s)
TL
TLo
ad(N
m)
Figure 11 Comparative experimental tests speed responses of GPC1 D1 SM PI and PID speed regulators for variable acceleration case
Observing the graphs of Figure 10 it can be seen that themotor response is very good in spite of all the adversities anda satisfactory speed tracking is achievedThe electromagnetictorque manifests some activity due to the sensor noises butnot due to the controller action as observed in the simulationtest in the second graph As to real computational cost of theGPC-PI controller as the calculation of the parameters119870 119866
119865 and 1198661015840 is realized offline its value is the same as the PI-PI
controller 10120583s employing a PowerPC processor at 1 GHz
4 Conclusions
The contribution of this work consists of the combination ofthe GPC algorithm in the speed loop with a PI based control
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
07 071
11941197120012031206
076 078 08 082 084
1140
11801200
Te
(Nm
)
Te simTe exp
iA simiA expiA exp and noise
0 05 1 15 2 25 3minus1300minus1000
minus500
0
500
10001300
Time (s)
0 05 1 15 2 25 3Time (s)
Roto
r spe
ed (r
pm)
ReferenceSim real
Exp realExp real and noise
minus400
40
80
0 05 1 15 2 25 3
minus30
minus20
0
20
30
Time (s)
145minus15minus10minus5 0 5 10 15
Stat
or cu
rren
ti A
(A)
TLoad
Figure 12 Simulation and real tests of the proposed GPC (D2 design) speed regulator with high-level noise in the two feedback loops andunknown load torque
in the current loops using an easy and effective design wherethe robust stability is demonstrated for typical inductionmotorsrsquo parametric uncertainties The GPC speed controllerdesign is based on the first order model of the inductionmotor with a delay time which is compensated This regu-lator design is simpler to implement than other predictiveproposed schemes as neither constraints nor robustnessterms have been taken into account
The proposed controller has been tested using varioussimulation and experimental tests in the presence of theparametric uncertainties unknown load disturbance andmeasurement noise in the loop signals the rotor speedand the stator current The experiment demonstrates theeffectiveness of the approach Moreover the presented resultsalso show that the GPC speed regulator is considerably fasterthan the classic PID and slightly faster than the advanced
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
SM speed controller using the same computational costThiswork demonstrates that the GPC-PI controller is an effectivespeed control algorithm in both adverse and acceptableconditions its robustness is clearly shown the proposedcontrol scheme is also easy to tune and to implement in a realsystem and therefore it can be used in industrial applications
Acknowledgments
The authors are very grateful to the UPVEHU for itssupport throughout the project GUI0708 and the BasqueGovernment for the support to this work throughout theproject S-PE09UN12
References
[1] E F Camacho and C Bordons Model Predictive ControlSpringer London UK 2nd edition 2007
[2] Y Xia G-P Liu M Fu and D Rees ldquoPredictive controlof networked systems with random delay and data dropoutrdquoControl Theory and Applications vol 3 no 11 pp 1476ndash14862009
[3] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart I The basic algorithmrdquo Automatica vol 23no 2 pp 137ndash148 1987
[4] D W Clarke C Mohtadi and P S Tuffs ldquoGeneralized predic-tive controlmdashpart II Extensions and interpretationsrdquoAutomat-ica vol 23 no 2 pp 149ndash160 1987
[5] J Yu J Gao Y Ma and H Yu ldquoAdaptive fuzzy tracking controlfor a permanent magnet synchronous motor via backsteppingapproachrdquo Mathematical Problems in Engineering vol 2010Article ID 391846 13 pages 2010
[6] C-F Huang J-S Lin T-L Liao C-Y Chen and J-J YanldquoQuasi-sliding mode control of chaos in permanent magnetsynchronous motorrdquo Mathematical Problems in Engineeringvol 2011 Article ID 964240 10 pages 2011
[7] O Barambones and P Alkorta ldquoA robust vector control forinduction motor drives with an adaptive sliding-mode controllawrdquo Journal of the Franklin Institute vol 348 no 2 pp 300ndash3142011
[8] S Zeng ldquoAdaptive speed control design for brushed permanentmagnet DC motor based on worst-case analysis approachrdquoMathematical Problems in Engineering vol 2013 Article ID698050 15 pages 2013
[9] R Kennel A Linder and M Linke ldquoGeneralized PredictiveControl (GPC)-ready for use in drive applicationsrdquo in Pro-ceedings of the IEEE 32nd Annual Power Electronics SpecialistsConference pp 1839ndash1844 Vancouver Canada June 2001
[10] R Hedjar R Toumi P Boucher and D Dumur ldquoCascadednonlinear predictive control of induction motorrdquo EuropeanJournal of Control vol 10 no 1 pp 65ndash80 2004
[11] K Barra and K Benmahammed ldquoNew extended cascaded pre-dictive control with multiple reference models ECGPCMRMof an induction motor drive with efficiency optimizationrdquoJournal of Electrical Engineering vol 58 no 2 pp 71ndash78 2007
[12] M Tonnes and H Rasmussen ldquoRobust self-tuning control ofAC-servo driverdquo in Proceedings of the European Conference onPower Electronics and Application pp 3049ndash3053 November1991
[13] P Boucher D Dumur and A Ehrlinger ldquoUnified approach ofequality and inequality constraints in GPCrdquo in Proceedings of
the IEEE International Conference on Control Applications pp894ndash899 Dearborn Mich USA September 1996
[14] MAGama andMMahfouf ldquoSpeed control of an experimentalhot-rolling mill using generalized predictive controlrdquo in Pro-ceedings of the International Control Conference (CCA rsquo06) pp1ndash6 September 2006
[15] P Alkorta O Barambones A J Garrido and I GarridoldquoSVPWM linear Generalized Predictive Control of induc-tion motor drivesrdquo in Proceedings of the IEEE InternationalSymposium on Industrial Electronics (ISIE rsquo08) pp 588ndash593Cambridge UK July 2008
[16] P Rodrıguez and D Dumur ldquoGeneralized predictive controlrobustification under frequency and time-domain constraintsrdquoIEEE Transactions on Control Systems Technology vol 13 no 4pp 577ndash587 2005
[17] E S de Santana E Bim and W C do Amaral ldquoA predictivealgorithm for controlling speed and rotor flux of inductionmotorrdquo IEEE Transactions on Industrial Electronics vol 55 no12 pp 4398ndash4407 2008
[18] P Cortes L Vattuone and J Rodrıguez ldquoPredictive currentcontrol with reduction of switching frequency for three phasevoltage source invertersrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo11) pp 1817ndash1822 Gdansk Poland June 2011
[19] J Guzinski andH Abu-Rub ldquoSpeed sensorless inductionmotordrive with predictive current controllerrdquo IEEE Transactions onIndustrial Electronics vol 60 no 2 pp 699ndash709 2013
[20] E F Camacho ldquoConstrained generalized predictive controlrdquoIEEE Transactions on Automatic Control vol 38 no 2 pp 327ndash332 1993
[21] C-W Park and W-H Kwon ldquoTime-delay compensation forinduction motor vector control systemrdquo Electric Power SystemsResearch vol 68 no 3 pp 238ndash247 2004
[22] K Ogata Modern Control Engineering Prentice Hall UpperSaddle River NJ USA 3rd edition 1997
[23] N Mohan Advanced Electric Drives University of MinnesotaMineapolis Minn USA 1st edition 2001
[24] B K Bose Modern Power Electronics and AC Drives PrenticeHall Upper Saddle River NJ USA 2002
[25] K J Astrom and B Wittenmark Computer Controller SystemsTheory andDesigns PrenticeHall Upper Saddle River NJ USA3rd edition 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of