proc imeche part i: j systems and control engineering...

Original Article

Proc IMechE Part I:J Systems and Control Engineering226(8) 1107–1118� IMechE 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0959651812443925pii.sagepub.com

Robust multi-objective H2/HN trackingcontrol based on the Takagi–Sugenofuzzy model for a class of nonlinearuncertain drive systems

Vahid Azimi1, Mohammad A Nekoui2 and Ahmad Fakharian3

AbstractIn this paper a robust H2/HN multi-objective state-feedback controller and tracking design are presented for a class ofmultiple input/multiple output nonlinear uncertain systems. First, some states (error of tracking) are augmented to thesystem in order to improve tracking control. Next, uncertain parameters and the quantification of uncertainty on physi-cal parameters are defined by the affine parameter-dependent systems method. Then, to apply the H2/HN controller,the uncertain nonlinear system is approximated by the Takagi–Sugeno fuzzy model. After that, based on each local linearsubsystem with augmented state, an H2/HN multi-objective state-feedback controller is designed by using a linear matrixinequalities approach. Finally, parallel distributed compensation is used to design the controller for the overall systemand the total linear system is obtained by use of the weighted sum of the local linear subsystems. Several results showthat the proposed method can effectively meet performance requirements such as robustness, good load disturbancerejection, good tracking and fast transient responses for a three-phase interior permanent magnet synchronous motorsystem.

KeywordsRobust control, Takagi–Sugeno fuzzy model, linear matrix inequalities, multi-objective H2/HN controller, parallel distrib-uted compensation, three-phase interior permanent magnet synchronous motor

Date received: 19 November 2011; accepted: 24 February 2012

Introduction

In recent years, the interior permanent magnet synchro-nous motor (IPMSM) has progressively been replacingdirect current and induction motors in a wide range ofdrives for many industrial applications, such as inrobotic actuators, traction and machine tool spindledrives, rolling mills, air conditioning compressors, elec-trical vehicles, integrated starters/alternators, computerdisk drives, domestic applications, automotive andrenewable energy conversion systems. The reason whythe IPMSM has become so well liked is essentially dueto its many pleasant characteristics such as high effi-ciency, exceptional power density, low inertia, excellenttorque generation, admirable speed and accelerationcapabilities, wide range of operation speed, strongstructure and high torque-to-inertia, torque-to-currentand power-to-weight ratios.

In many applications, tracking a position and speedtrajectory, torque ripple and elimination of reluctanceeffects are of great importance. However, the existence

of system parameter variations, load torque disturbanceand system nonlinearity elements makes this a ratherarduous task. Numerous nonlinear and linear control-lers have been developed for the IPMSM. Hitherto,much research has been performed on torque ripplereduction and reluctance effects elimination. For exam-ple, Colamartino et al.1 proposed a torque ripple mini-mization with two strategies. The first was based onnumerical predetermination of the current waveform

1Electrical Engineering Department, Islamic Azad University, South

Tehran Branch, Islamic Republic of Iran2Faculty of Control Engineering, KN Toosi University of Technology,

Islamic Republic of Iran3Faculty of Electrical and Computer Engineering, Islamic Azad University,

Qazvin Branch, Islamic Republic of Iran

Corresponding author:

Ahmad Fakharian, Faculty of Electrical and Computer Engineering, Islamic

Azad University, Qazvin Branch, Qazvin 1478735564, Islamic Republic of

Iran.

Email: [email protected]

which is imposed by the control in machine phases andthe second was a torque regulation based on its onlineinstantaneous estimation for an IPMSM drive. Chenet al.2 presented the controller-induced parasitic torqueripples for an IPMSM drive. Gulez et al.3 investigatedthe torque ripple and electromagnetic interference noiseminimization using active filter topology and field-oriented control for an IPMSM drive. For achievingacceptable drive performance and good position andspeed tracking, disturbances attenuation and torquecontrol, many issues have been considered in recentyears. Lin et al.4 used a nonlinear position controllerdesign with an input–output linearization technique foran IPMSM control system. Yang and Zhong5 imple-mented a robust speed tracking of permanent magnetsynchronous motor (PMSM) servo systems by equiva-lent disturbance attenuation. Su et al.6 studied the auto-matic disturbances rejection controller for precisemotion control for a PMSM drive. Chou and Liaw7

proposed the development of robust two-degree-of-freedom current controllers for a PMSM drive withreaction wheel load. Karabacak and Eskikurt8 pro-posed a speed and current regulation of a PMSM vianonlinear and adaptive backstepping control. Errouissiand Ouhrouche9 used a nonlinear predictive controller(NPC) for a PMSM. Other papers have studied applica-tion of the sliding mode technique to PMSM orIPMSM drive systems.10–13 Most physical dynamicalsystems are nonlinear in the real world and cannot berepresented by linear differential equations. TheTakagi–Sugeno (T–S) fuzzy model, which is often usedin the literature, can approximate a wide class of highlycomplex nonlinear systems. The major feature of a T–Sfuzzy model is to extract the local dynamics of eachfuzzy implication (rule). Published papers have used theT–S fuzzy model technique for different drive systems.For example, Chen and Wu14 presented a robust opti-mal reference-tracking design method for stochasticsynthetic biology systems using a T–S fuzzy approach.Qiu et al.15 proposed a new design of delay-dependentrobust HN filtering for discrete-time T–S fuzzy systemswith time-varying delay. Wai and Yang16 proposed anadaptive fuzzy neural network control design via a T–Sfuzzy model for a robot manipulator including actuatordynamics.

The main contribution of the present research isrobust tracking H2/HN control based on a T–S fuzzymodel for an uncertain nonlinear IPMSM. This paperconsiders the problem of robust H2/HN control foruncertain nonlinear multiple input/multiple output(MIMO) T–S fuzzy systems which possess not onlyparameter uncertainties but also external disturbances.Several robust HN schemes based on the use of linearmatrix inequality (LMI) theory have been reported pre-viously.17–19 In the method proposed herein, first thenonlinear plant is represented by a T–S fuzzy model.The fuzzy model is described by fuzzy IF–THEN ruleswhich represent local input–output relationships of thenonlinear system. Then, some states are augmented to

the system in order to improve tracking control andtracking error reduction. After that, uncertain para-meters and the quantification of uncertainty on physi-cal parameters are defined by the affine parameter-dependent systems method. Next, based on each linearsubsystem with augmented state, an H2/HN multi-objective state-feedback controller is designed by usingthe LMI approach. Parallel distributed compensation(PDC) is then utilized to design the controller for theoverall system. Thus the overall fuzzy model of the sys-tem is achieved by fuzzy ‘‘blending’’ of the local linearsubsystem models. Several results show that the pro-posed method can effectively meet performancerequirements like robustness, good load disturbancerejection responses, good tracking responses and fasttransient responses for the three-phase IPMSM system.

The paper is organized as follows. Preliminary con-cepts are presented first and then the IPMSM dynamicmodel is introduced. The problem statement and designof the robust tracking controller are presented next.Simulation results of the closed-loop system with theproposed technique follow, and finally conclusions aredrawn.

Preliminary concepts

Multi-objective H2/HN framework

In this subsection, multi-objective state-feedback synth-esis by the LMI framework is described.17,19–21 Themain design objectives of the multi-objective controllerare:

(a) HN performance (for tracking, disturbance rejec-tion or robustness aspects);

(b) H2 performance (for linear quadratic Gaussian(LQG) aspects);

(c) robust pole placement specifications (to ensurefast and well-damped transient responses, reason-able feedback gain, etc.).

Denoting by TN(s) and T2(s) the closed-loop transferfunctions from w to zN and z2, respectively, our goal isto design a state-feedback law u=Kx that:

(a) maintains the root-mean-square (RMS) gain (HN

norm) of TN below some prescribed value g0 . 0;(b) maintains the H2 norm of T2 (LQG cost) below

some prescribed value n0 . 0;(c) minimizes an H2/HN trade-off criterion of the

form

ajjT‘jj‘2 +bjjT2jj2

2 ð1Þ

(d) places the closed-loop poles in a prescribed LMIregion.

To get a feeling for the multi-objective H2/HN

methodology, consider the general pattern loop ofFigure 1.

1108 Proc IMechE Part I: J Systems and Control Engineering 226(8)

LMI formulation given a state-space realization is

_x=Ax +B1w+B2uz‘ =C1x+D11w+D12uz2 =C2x+D21w+D22u ð2Þ

Of the plant P, the closed-loop system is given in state-space form by

_x= A+B2Kð Þx+B1wz‘ =(C1 +D12K)x+D11wz2 = C2 +D22Kð Þx+D21w ð3Þ

Taken separately, our three design objectives have thefollowing LMI formulation.

1. HN performance. The closed-loop RMS gain fromw to zN does not exceed g if and only if there existsa symmetric matrix XN such that

A+B2Kð ÞX‘ +X‘ A+B2Kð ÞT B1 X‘ C1 +D12Kð ÞTB1

T �I D11T

C1 +D12Kð ÞX‘ D11 �g2I

24

35\ 0

X‘ . 0 ð4Þ

2. H2 performance. The closed-loop H2 norm of T2

does not exceed n if there exist two symmetricmatrices X2 and Q such that

Q C2 +D22Kð ÞX2

X2 C2 +D22Kð ÞT X2

� �. 0

A+B2Kð ÞX2 +X2 A+B2Kð ÞT B1

B1T �I

� �\ 0

Trace Qð Þ\ n2 ð5Þ

3. Pole placement. The closed-loop poles lie in theLMI region D

D= z 2 C : L+Mz+MT�z\ 0� �

L=LT = lij

� �14i, j4m

M= mij

� �14i, j4m

ð6Þ

if and only if there exists a symmetric matrix Xpol

satisfying

lijXpol +mij A+B2Kð ÞXpol +mijXpol +mjiXpol A+B2Kð ÞTh i

14i, j4m

\ 0 Xpol . 0 ð7Þ

These three sets of conditions add up to a no-convexoptimization problem with variables Q, K, XN, X2 andXpol. For tractability in the LMI framework, we seek asingle Lyapunov matrix X:=XN=X2=Xpol thatenforces all three objectives. With the change of vari-able Y:=KX, this leads to the following suboptimalLMI formulation of our multi-objective state-feedbacksynthesis problem.

Minimize ag2 +b Trace(Q) over Y, X, Q and g2,satisfying

AX+XAT +B2Y+YTB2T B1 XC1

T +YTD12T

B1T �I D11

T

C1X+D12Y D11 �g2I

24

35\ 0

Q C2X+D22YXC2

T +YTD22T X

� �. 0

lij +mij AX+B2Yð ÞXpol +mji XAT +YTB2T

� �� 14i, j4m

\ 0

Trace Qð Þ\ n02

g2 \ g20 ð8Þ

Denoting the optimal solution by (X*, Y*, Q*, g*), thecorresponding state-feedback gain is given byK*=Y*(X*)21. This gain guarantees the worst-caseperformance

jjT‘jj‘ \ g�

jjT2jj2 \ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTrace(Q�)

pð9Þ

T–S fuzzy dynamic model

The T–S fuzzy dynamic model is described by fuzzy IF–THEN rules, which represent local linear input–outputrelationships of nonlinear systems.22,23 The fuzzydynamic model is proposed by Takagi and Sugeno. Theith rule of the T–S fuzzy dynamic model with para-metric uncertainties in the H2/HN multi-objectiveframework can be described as follows.

Plant rule i

IF v1 tð Þ is Mi1 and . . . and vp tð Þ is Mip THEN

_x tð Þ= Ai +DAi½ �x tð Þ+ B1i +DB1i½ �w tð Þ½+ B2i +DB2i½ �u tð Þ�, x 0ð Þ=0

z‘ tð Þ= C1i +DC1i½ �x tð Þ+ D11i +DD11i½ �w tð Þ½+ D12i +DD12i½ �u tð Þ�

z2 tð Þ= C2i +DC2i½ �x tð Þ+ D21i +DD21i½ �w tð Þ½+ D22i +DD22i½ �u tð Þ�

i=1, 2, . . . , r ð10Þ

where Mip is the fuzzy set, r is the number of IF–THENrules and v1(t)!vp(t) are the premise variables. ThePDC technique is used to design the controller for theoverall system and the total linear system is obtained byuse of the weighted sum of the local linear subsystems.The concept of PDC and the overall fuzzy model areshown in Figure 2. The overall fuzzy model is accord-ingly of the following form (Figure 2(a))

Figure 1. The control structure.

Azimi et al. 1109

_x tð Þ=Xr

i=1mi v tð Þð Þ Ai +DAi½ �x tð Þ½

+ B1i +DB1i½ �w tð Þ+ B2i +DB2i½ �u tð Þ�, x 0ð Þ=0

z‘ tð Þ=Xr

i=1mi v tð Þð Þ C1i +DC1i½ �x tð Þ½

+ D11i +DD11i½ �w tð Þ+ D12i +DD12i½ �u tð Þ�z2 tð Þ=

Xr

i=1mi v tð Þð Þ C2i +DC2i½ �x tð Þ½

+ D21i +DD21i½ �w tð Þ D22iDD22i½ �u tð Þ� ð11Þ

where v tð Þ= v1 tð Þ . . . vp tð Þ½ � and the weightingfunction is

mi v tð Þð Þ= î v tð Þð ÞPri=1 î v tð Þð Þ

î v tð Þð Þ=Ypk=1

Mik vk tð Þð Þ ð12Þ

The matrices DAi, DB1i, DB2i, DC1i, DC2i, DD11i, DD12i

and DD22i represent the uncertainties in the system andsatisfy the following assumptions

DAi =F x tð Þ, tð ÞH1i, DB1i =F x tð Þ, tð ÞH2i

DB2i =F x tð Þ, tð ÞH3i; DC1i =F x tð Þ, tð ÞH4i

DD11i =F x tð Þ, tð ÞH5i, DD12i =F x tð Þ, tð ÞH6i

DC2i =F x tð Þ, tð ÞH7i, DD21i =F x tð Þ, tð ÞH8i

DD22i =F x tð Þ, tð ÞH9i ð13Þ

where Hji; j=1, . . . , 9 are known matrix functionswhich characterize the structure of the uncertainties.Furthermore, the following inequality holds:F x tð Þ, tð Þ4∂, ∂. 0. For a fuzzy controller design, it issupposed that the fuzzy system is locally controllable.Then, the local state-feedback multi-objective H2/HN

controller is designed as follows

IF v1 tð Þ is Mi1 and . . . and vp tð Þ is Mip THEN

u tð Þ=Kix tð Þ, i=1, 2, . . . , r ð14Þ

where Ki is the multi-objective H2/HN controller gain.Then, the final T–S fuzzy controller is (Figure 2(b))

u tð Þ=Xrj=1

mjKjx tð Þ ð15Þ

where Kj is the H2/HN controller for every linear sub-system and u(t) is the final controller for the overall sys-tem. The closed-loop state-space system from the fuzzysystem model, equation (11), with the fuzzy controller,equation (15), is given by

_x tð Þ=Xr

i=1

Xr

j=1mimj Ai +B2iKj

� ��+ DAi +DB2iKj

� ��x tð Þ

+ B1i +DB1i½ �w tð Þ�, x 0ð Þ=0

z‘ tð Þ=Xr

i=1

Xr

j=1mimj½ C1i +D12iKj

� ��+ DC1i +DD12iKj

� ��x(t)

+ D11i +DD11i½ �w tð Þ�

z2 tð Þ=Xr

i=1

Xr

j=1mimj C2i +D22iKj

� ��+ DC2i +DD22iKj

� ��x tð Þ

+ D21i +DD21i½ �w tð Þ� ð16Þ

PMSM model dynamics

The nonlinear electrical and mechanical equations forthe three-phase IPMSM in the d–q reference frame canbe written as follows4

dur

dt=vr

dvr

dt=

3

2

P0

JmLd � Lq

� �id + ;f

� iq �

Bm

Jmvr �

Cl

Jm

diddt

= � Rs

Ldid +P0

Lq

Ldvriq +

1

Ldvd

diqdt

= � P0;fLq

vr � P0Ld

Lqvrid �

Rs

Lqiq +

1

Lqvq ð17Þ

In equation (17) ur is the angular position of the motorshaft, vr is the angular velocity of the motor shaft, id isthe direct current and iq is the quadrature current. ;f isthe flux linkage of the permanent magnet, P0 is thenumber of pole pairs, Rs is the stator windings resis-tance, Ld and Lq are the direct and quadrature statorinductances, respectively. Jm is the rotor moment ofinertia, Bm the viscous damping coefficient and Cl isthe load torque. vd is the direct voltage and vq is thequadrature voltage. The electromagnetic torque of themotor can be described as

Te =3

2P0 Ld � Lq

� �idiq + ;fiq

� ð18Þ

The parameters Rs and Bm are supposed to differ fromtheir nominal values Rs0 and Bm0

.4 The following

Figure 2. (a) The T–S fuzzy system; (b) the T–S fuzzycontroller.


equation indicates a state-space representation of thesynchronous motor

_x1 =x2

_x2 = h1x3 +h2ð Þx4 +h3x2 �Cl

Jm

_x3 =h4x3 +h5x2x4 +h6u1

_x4 =h7x2 +h8x2x3 +h9x4 +h10u2 ð19Þ

In equation (19)

x= x1 x2 x3 x4½ �T = ur vr id iq½ �T

u1 u2½ �T = vd vq½ �T ð20Þ

and

h1 =3

2

P0

JmLd � Lq

� �, h2 =

3

2

P0

Jm;f

h3 = � Bm

Jm, h4 = � Rs

Ld

h5 =P0Lq

Ld, h6 =

1

Ld

h7 = � P0;fLq

, h8 = � P0Ld

Lq

h9 = � Rs

Lq, h10 =

1

Lqð21Þ

Problem statement and design of therobust tracking controller

Proposed structure

In this paper the purpose is to design a suitable controlwhich guarantees robust performance in the presenceof parameter variations and load disturbance. In thiscase the control objectives are tracking of the rotorangular position and elimination of reluctance effectsand torque ripple (the direct current has to follow aconstant reference, zero). Therefore to achieve accuratetracking, the tracking errors (e1, e2) should be mini-mized. Our goal is to design a state-feedback controllerthat maintains the RMS gain (HN norm) of theseerrors below some prescribed value g0 . 0. In order toobtain this minimization two extra states are augmen-ted to the IPMSM system, as described in the followingsubsection.

The mentioned goals are realized via constructingthe objectives z and extra states in an appropriate con-trol loop. Under the above considerations, the fuzzyrobust control loop proposed is based on the structuredepicted in Figure 3.

Augmented states

To guarantee robust performance in the presence ofparameter and load torque variations, a suitable controlhas to be designed. There are two control objectives.First, the rotor angular position x1=ur must track a

reference trajectory r1. Second, the direct current x3=idhas to track a constant reference r2=0. This objective isequivalent to the nonlinear electromagnetic torquebeing linearized to avoid reluctance effects and torqueripple.

Therefore, to achieve these objectives, the outputsof the integrator are considered as extra statevariables

x5 =

ðt

0

e1 dð Þdd, e1 = r1 � x1

x6 =

ðt

0

e2 dð Þdd, e2 = r2 � x3 ð22Þ

Consequently the four states in equation (19) areincreased to six states with addition of the following

_x5 = r1 � x1, _x6 = r2 � x3

xaug = x1 x2 x3 x4 x5 x6½ �T ð23Þ

Affine parameter-dependent system

Affine parameter-dependent systems are defined as

E rð Þ _x=A rð Þx+B1 rð Þw+B2 rð Þuz‘ =C1 rð Þx+d11 rð Þw+d12 rð Þuz2 =C2 rð Þx+d12 rð Þw+d22 rð ÞuA rð Þ+jE rð Þ B rð Þ

C rð Þ D rð Þ

� �=

A0 + jE0 B0

C0 D0

� �

+ r1

Ar1+ jEr1

Br1

Cr1Dr1

� �+ . . .

+ rn

Arn+ jErn

Brn

CrnDrn

� �

Figure 3. Block diagram of the robust controller andaugmented states control loop.

Azimi et al. 1111

B(r)= B1 rð Þ B2 rð Þ½ �

C rð Þ= C1 rð Þ C2 rð Þ½ �T

D rð Þ= d11 rð Þ d12 rð Þd12 rð Þ d22 rð Þ

� �ð24Þ

In this case the exogenous inputs are

w=dr1r2

24

35=

Cl

x1refx3ref

24

35 ð25Þ

where r and w are the uncertain parameter and externalinput, respectively. The parameters Rs and Bm and loadtorque disturbance Cl are supposed to vary (for instance,Rs has high variations due to temperature). The formali-zation of these variations is declared as follows

B1 rð Þ=B10 +ClB1Cl

B10 =

0 0 00 0 00 0 00 0 00 1 00 0 1

26666664

37777775

B1Cl=

0 0 0� 1

Jm0 0

0 0 00 0 00 0 00 0 0

26666664

37777775

B2 =

0 00 0

h6 00 h10

0 00 0

26666664

37777775

ð26Þ

and

A pð Þ=A0i +BmABm+RsARs

A0i =

0 1 0 0 0 00 0 h1x4 h2 0 00 h5x4 0 0 0 00 h7 h8x2 0 0 0�1 0 0 0 0 00 0 �1 0 0 0

26666664

37777775

ABm=

0 0 0 0 0 00 � 1

J 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

26666664

37777775

ARs=

0 0 0 0 0 00 0 0 0 0 00 0 � 1

Ld0 0 0

0 0 0 � 1Lq

0 0

0 0 0 0 0 00 0 0 0 0 0

26666664

37777775

ð27Þ

According to equation (27), which emerges from equa-tions (19) and (23), just A0i contains nonlinear para-meters. Therefore to linearize the system, the T–S fuzzymodel method is utilized.

PMSM T–S fuzzy model

First, to linearize the system, the T–S fuzzy modelmethod is utilized. Because two nonlinear elementsexist in A0i, four plant rules are assigned

v1 tð Þ=x2 tð Þ, v2 tð Þ=x4 tð Þ ð28Þ

in which v1 and v2 are fuzzy variables. To attain mem-bership functions, we should calculate the minimumand maximum values of v1(t) and v2(t)

4

x2 2 02000½ �, x4 2 012½ �minv1 tð Þ=0, maxv1 tð Þ=2000

minv2 tð Þ=0, maxv2 tð Þ=12 ð29Þ

Therefore x2 and x4 can be represented by four mem-bership functions M1, M2, M3 and M4 as follows

v1 tð Þ=x2 tð Þ=M1 v1 tð Þð Þ:2000+M2 v1 tð Þð Þ:0v2 tð Þ=x4 tð Þ=M3 v2 tð Þð Þ:12+M4 v2 tð Þð Þ:0 ð30Þ

Also, because M1, M2, M3 and M4 are essentially fuzzysets, according to fuzzy mathematics

M1 v1 tð Þð Þ+M2 v1 tð Þð Þ=1

M3 v2 tð Þð Þ+M4 v2 tð Þð Þ=1 ð31Þ

As a result, in accordance with equations (30) and(31) the membership functions can be calculatedas

M1 =v1

2000, M2 =1� v1

2000

M3 =v212

, M4 =1� v212

ð32Þ

The membership functions according to equation (32)are shown in Figure 4.

Four plant rules are considered to cover the fourlocal linear subsystems near specific operation pointsdespite the fact that, in the system dynamics, just A0i isvaried in each rule.

Rule 1

IF v1 tð Þ is M1 and v2 tð Þ is M3

THEN A1 pð Þ=A01 +BmABm+RsARs

Rule 2



Rule 3

IF v1 tð Þis M2 and v2 tð Þ is M3



Rule 4



ð33Þ

H2/HN controller design and closed-loop system

The concept of PDC is utilized with equation (33) todesign a fuzzy state-feedback controller. The fuzzy con-troller shares the same fuzzy sets with the fuzzy system,so the ith control rule is as follows.

Rule 1


THEN u tð Þ=K1x tð Þ

Rule 2



Rule 3



Rule 4


THEN u tð Þ=K4x tð Þ ð34Þ

where Ki, i=1, 2, . . . , r are the local H2/HN controllergains to be determined.

According to equations (26) and (27) just A0i variesand ABm

, ARs, B, C and D are constant in each rule. B,

C and D are obtained based on objectives z2 and zN

and external input w. B is defined as in equation (26)and Ai is as represented in equation (27)

z‘ =x5x6

� �=

e1e2

� �ð35Þ

z2 =

x1x3u1u2

2664

3775=

u

idvdvq

2664

3775 ð36Þ

w=dr1r2

24

35=

Cl

x1refx3ref

24

35 ð37Þ

This section focuses on the design of the H2/HN track-ing controller for each linear subsystem of the PMSMrepresented by equation (33). The overall fuzzy modeland final fuzzy H2/HN controller based on equations(11) and (15) are achieved by the T–S fuzzy defuzzifica-tion process and defined by equations (38) and (39),respectively. Finally, the closed-loop state-space systemfrom the fuzzy system model, equation (38), withthe controller, equation (39), and by the use of theproposed control structure (Figure 3), is given byequation (40)

Afuzzy =X4

i=1mi½A0i +BmABm

+RsARs� ð38Þ

Kfuzzy =X4

i=1miKi ð39Þ

and

Bfuzzy = B1fuzzyB2fuzzy

� B1fuzzy =

X4

i=1mi½B10 +ClB1Cl

� B2fuzzy =X4

i=1miB2

Aclose loop =Afuzzy +B2fuzzyKfuzzy ð40Þ

In the control structure, which is shown in Figure 3,first the original system, equation (17), is approximatedwith some local linear models so that each rule is repre-sented by the T–S fuzzy approach. Then, in order toobtain accurate tracking of tool position and direct cur-rent, two extra states are augmented to the T–S fuzzymodel and build up the augmented plant. Separate con-trollers for each linear subplant based on the LMIapproach are designed. After that the total system (P)is obtained by using the weighted sum of the local lin-ear subsystems and it is utilized instead of the originalnonlinear system. So according to the PDC approach,the control law of the whole system (K) is the weightedsum of the local feedback control of all subsystems.Total system (P) and whole controller (K) have beenshown in both Figures 2 and 3.

As can be observed in Figure 2, in order to constructP and K based on equations (11) and (15), a way isneeded to calculate and update the fuzzy weights givenin equation (12). In the present work the fuzzy weightsmi are updated by using a fuzzy weights online compu-tation (FWOC) component that has been designedbefore. The FWOC component includes three blocksthat may be explained as follows.

Figure 4. (a) The membership functions for M1(v1(t)) andM2(v1(t)); (b) the membership functions for M3(v2(t)) andM4(v2(t)).

Azimi et al. 1113

1. In the first block, values of the angular velocity ofthe motor shaft x2=vr and the quadrature currentx4=iq are measured from the IPMSM system inreal time and nonlinear terms (fuzzy variables:v1=x2, v2=x4) are built, equation (28).

2. In the second block, the values of membershipfunctions Mi mentioned in fact of the nonlineari-ties are calculated, equation (32).

3. In the third block, new fuzzy weights, equation(12), are calculated and they are sent to the maincontrol structure (Figures 2 and 3).

Finally by using the whole system and global con-troller, a tracking loop (Figure 3) is applied to the sys-tem in order to achieve desired specifications such astracking performance, bandwidth, disturbance rejectionand robustness for the closed-loop system.

Simulation results

The motor type used in this paper is the 130-750MS-ZK-L2. This IPMSM is three-phase, four-pole, with0.75HP (horse power) and with 2000 r/min rated speed.The maximum voltage and the continuous rated arma-ture current are set to 230 V and 12 A.4The parametersof the IPMSM are shown in Table 1. The stator wind-ings resistance Rs and the viscous damping coefficientBm are varied between 650% and the load torque dis-turbance is unknown.4,10

According to the approach outlined above in subsec-tions ‘‘Affine parameter-dependent system’’ and‘‘PMSM T–S fuzzy model’’, the local linear modelmatrices for the nonlinear augmented states of theIPMSM at the ith selected operating point are obtainedas follows

A01 =

0 1 0 0 0 00 0 �1145 1860 0 00 49 0 0 0 00 �20 �82119 0 0 0�1 0 0 0 0 00 0 �1 0 0 0

26666664

37777775

A02 =

0 1 0 0 0 00 0 0 1860 0 00 0 0 0 0 00 �20 �82119 0 0 0�1 0 0 0 0 00 0 �1 0 0 0

26666664

37777775

A03 =

0 1 0 0 0 00 0 �1145 1860 0 00 49 0 0 0 00 �20 0 0 0 0�1 0 0 0 0 00 0 �1 0 0 0

26666664

37777775

A04 =

0 1 0 0 0 00 0 0 1860 0 00 0 0 0 0 00 �20 0 0 0 0�1 0 0 0 0 00 0 �1 0 0 0

26666664

37777775

C=

0 0 0 0 1 00 0 0 0 0 11 0 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 0

26666664

37777775

D=

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 1 00 0 0 0 1

26666664

37777775

Next, in order to design state-space feedback gains Ki

for each subsystem, the following steps are undertaken.

1. Specify the LMI region, equation (6), in order toplace the closed-loop poles in this region (pole pla-cement) and also to guarantee some minimumdecay rate and closed-loop damping. The men-tioned region is shown in Figure 5, as the intersec-tion of the half-plane x \ 25 and of the sectorcentered at the origin and with inner angle 2p/3.

2. Choose a four-entry vector specifying the H2/HN

cost function, equation (1): [g0 n0 a b]=[0 0 1 1].3. Minimize the H2/HN cost function, equation (1),

subject to the mentioned pole placement constraintby using equations (4), (5), (8) and (9).

The local H2/HN control gain matrices Ki, the finalfuzzy H2/HN controller matrix Kfuzzy and the overallfuzzy model matrix Afuzzy are obtained as follows

K1 =104

0:0033 �0:00110 0

�0:0097 0:00020:0002 �0:0001�0:0386 0:00090:1467 3:8313

26666664

37777775

T

Table 1. Parameters of the IPMSM.

Parameter Value

Rs 1.9 OBm0 (without load) 0.03 Nms/radBm0 (with load) 0.0341 Nms/radLd 0.0151 HLq 0.031 H;f 0.31 Vs/radJm0 (without load) 0.0005 kg.m2Jm0 (with load) 0.0227 kg.m2P0 2


K2 =104

0:0004 �0:00120 0

�0:0049 0:00010:0001 �0:0001�0:0051 0:0002�0:0177 4:3223

26666664

37777775

T

K3 =103

0:3078 �0:14410:0009 �0:0025�0:0157 0:01040:0252 �0:0210�3:3985 2:50030:0336 0:0940

26666664

37777775

T

K4 =100

0 �65:17570 �0:5657

�0:8028 00 �14:77520 774:3000

59:1158 0

26666664

37777775

T

Afuzzy =

0 1 0 0 0 00 �60 �410 1858 0 00 54 �126 0 0 00 �20 �68405 �61 0 0�1 0 0 0 0 00 0 �1 0 0 0

26666664

37777775

Kfuzzy =104

0:0037 �0:00270 0

�0:0055 0:00020:0003 �0:0003�0:0417 0:02570:0312 3:4662

26666664

37777775

T

To calculate the overall controller use is made of thePDC technique and the whole system is obtained byusing a weighted average defuzzifer. The above pro-posed T–S fuzzy model and controller can exactly rep-resent the nonlinear system in the region

½0, 12�A3½0, 2000� r=min on the x2–x4 space for variousoperating points.

Figure 6 compares system states for both the pro-posed H2/HN model and the original nonlinear system,which is presented in Lin et al.4 As is evident fromFigure 4, the time responses of the proposed T–S fuzzymodel exactly follow the responses of the nonlinear dif-ferential equations (17), which means that the fuzzymodel can exactly represent the original system in thepre-specified domains. These time responses ofstates are influenced by step inputs which are shown inFigure 6.

In actuality, the motor is used to convert electricalenergy into mechanical energy. Accordingly, an exter-nal load is added to the drive system. The external load

Figure 6. Time responses of the proposed H2/HN model(solid) and the original nonlinear model (dashed): (a) angularposition of the motor shaft; (b) angular speed; (c) d-axis current;(d) q-axis current.

Figure 5. The pole placement region.

Azimi et al. 1115

is obtained by using one of two types of disturbance.First, another synchronous motor is coupled to theshaft of the main PMSM in order to request a load tor-que.10 The manner of the load torque Cl applied to thesynchronous motor is presented in Figure 7. Second, a

1 kg weight is put at a certain location from the motor;as a result, the weight can provide the external load of1Nm that is proposed with step input.

Figure 8 shows the real direct current response thattracks 0 A in order to eliminate reluctance effects andtorque ripple.

Figures 9 and 10 show the responses of the certainstep position (180�) and the rectangular position com-mands, respectively, for the proposed H2/HN controllerand the feedback linearization controller (presented inLin et al.4). Although Figure 9(a) is influenced by stepload torque and measurement noise, Figure 9(b) showsthis comparison at 180�, noise and benchmark externalload (Figure 7). In Figures 9 and 10, the notation usedis as follows: Ts is settling time, where Ts1 and Ts2 arethe settling times of tracking responses when the pro-posed H2/HN and feedback linearization controller isused, respectively.

According to Figures 9 and 10, the settling time andrise time for the proposed H2/HN controller are betterthan for the feedback linearization controller. Table 2also summarizes the results of transient responses forthe two mentioned methods. Referring to Table 2, theproposed H2/HN method has smaller settling time valuethan the feedback linearization method on positiontracking responses.

Figure 11(a) and (b) demonstrates the disturbancerejection on angular position with the two differentmethods, with step load torque and benchmark loadtorque (Figure 7), respectively, although first theIPMSM is controlled to reach a fixed position, 180�.As a matter of fact, Figure 11(a) and (b) is a zoom of

Figure 9. Comparison of transient responses of the proposedH2/HN controller and the feedback linearization controller: (a)with step load torque; (b) with benchmark load torque.

Figure 10. Comparison of responses of the proposed H2/HN

controller and the feedback linearization controller withrectangular position command.

Figure 7. Benchmark load torque (Nm versus time).

Figure 8. d-Axis current.

Table 2. Comparison of disturbance rejection and trackingresponses in the two methods.

Method

ProposedH2/HN control

Feedback linearizationcontrol4

Emax (�) 0.20 1.1Ep-p (�) 0.75 4.2Ts (s) 0.28 0.4


Figure 9(a) and (b), between t=0.6 s and t=1.2 sand t=0.5 s and t=1.8 s, respectively. The notationused in Figure 11(a) and (b) is as follows: Emax isthe peak error and Ep-p is the peak-to-peak error,where E1max and E1p-p are the peak and peak-to-peak errors when step and benchmark disturbances(Figure 7) respectively act on the IPMSM and pro-posed H2/HN controller is used; and E2max and E2p-p are the peak and peak-to-peak errors, when stepand benchmark disturbances (Fig. 7) respectively acton the IPMSM and the feedback linearization con-troller (presented in Lin et al.4) is used.

Figure 11(a) and (b) shows the comparison of theproposed H2/HN method and the feedback lineariza-tion technique in the presence of disturbances. As canbe seen, the proposed H2/HN method has better distur-bance attenuation under both types of external load.Table 2 summarizes the disturbance rejection differ-ences (values of Emax, Ep-p and Ts (sec)) in the two

methods, from which it is observed that the peak andpeak-to-peak values in the proposed H2/HN methodare smaller than those in the feedback linearizationtechnique on disturbance rejection responses.

Figure 12 illustrates the position responses when theparameters of the stator windings resistance Rs and theviscous damping coefficient Bm are varied between650%. As can be seen, the system has good robustnesswhen the parameters in the dynamic systems are variedover a wide range.

Conclusions

In this paper, a robust H2/HN position and direct cur-rent tracking controller has been designed for a non-linear, MIMO and uncertain PMSM system. First, inorder to improve tracking control, some states (error oftracking) were augmented to the system. Then, toapproximate uncertain nonlinear systems, the T–Sfuzzy linear model was employed. After that, based oneach linear model with augmented state, an H2/HN

multi-objective state feedback controller was developedto achieve the robustness design of nonlinear uncertainsystems. Finally by using the PDC approach the overallfuzzy controller was calculated. Simulation results on athree-phase IPMSM showed that the robust proposedposition and current control system has small positionand current tracking error, desired robustness againstload torque disturbance and parameter variations andgood transient responses, load disturbance responsesand tracking responses.

Funding

This research received no specific grant from any fund-ing agency in the public, commercial, or not-for-profitsectors.

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