position and current control of an interior...

Vahid AzimiYoung Researchers and Elite Club,

Qazvin Branch,

Islamic Azad University,

Qazvin, Iran 3416616114

e-mail: [email protected]

Ahmad Fakharian1

Department of Electrical and

Computer Engineering,

Islamic Azad University,

Qazvin Branch,

Qazvin, Iran 1478735564


Mohammad Bagher MenhajDepartment of Electrical Engineering,

Amirkabir University of Technology,

Tehran, Iran 3416616114


Position and Current Controlof an Interior Permanent-MagnetSynchronous Motor by UsingLoop-Shaping Methodology:Blending of H‘

Mixed-Sensitivity Problemand T–S Fuzzy Model SchemeThis paper presents a robust mixed-sensitivity H1 controller design via loop-shaping meth-odology for a class of multiple-input multiple-output (MIMO) uncertain nonlinear systems.In order to design this controller, the nonlinear plant is first modeled as several linear subsys-tems by Takagi and Sugeno’s (T–S) fuzzy approach. Both loop-shaping methodology andmixed-sensitivity problem are then introduced to formulate frequency-domain specifications.Afterward for each linear subsystem, a regional pole-placement output-feedback H1 control-ler is employed by using linear matrix inequality (LMI) approach. The parallel distributedcompensation (PDC) is then used to design the controller for the overall system. Several ex-perimental results show that the proposed method can effectively meet the performancerequirements like robustness, good load disturbance rejection, and both tracking and fasttransient responses even in the presence of parameter variations and load disturbance for thethree-phase interior permanent-magnet synchronous motor (IPMSM). Finally, the superior-ity of the proposed control scheme is approved in comparison with the input–output lineariza-tion (I/O linearization) and the H2/H1 controller methods. [DOI: 10.1115/1.4024200]

Keywords: LMI, mixed-sensitivity, loop-shaping, robust control, T–S fuzzy model, three-phase interior permanent-magnet synchronous motor (IPMSM)

1 Introduction

Interior permanent-magnet synchronous motors are extensivelyused in industry, due to their comparatively high efficiency,exceptional power density, and excellent torque generating. Overthe last decade, there have been numerous progresses in the devel-opment of controllers for interior permanent-magnet (IPM) syn-chronous motors as listed below. Lin et al. [1] used a nonlinearposition controller based on input–output feedback linearizationtechnique for an IPMSM control system. Yang and Zhong [2] pro-posed a robust speed-tracking control of permanent-magnet syn-chronous motor (PMSM) servo systems to eliminate speed-tracking error due to time-varying load torque uncertainty and pa-rameter perturbations. Su et al. [3] developed a highly robust auto-matic disturbance rejection controller to implement high-precisionmotion control of permanent-magnet synchronous motors. Chouand Liaw [4] developed robust 2-DOF current and torque controlscheme for a PMSM drive with satellite reaction wheel load. Linet al. [5] designed the adaptive back stepping proportional-integral(PI) sliding-mode controller for interior permanent-magnet syn-chronous motor drive systems. Azimi et al. [6] designed a robustmulti-objective H2/H1 tracking controller based on T–S fuzzymodel for a class of nonlinear uncertain drive systems. The avail-able papers also employed the sliding-mode technique of PMSMor IPMSM drive systems [7,8]. Recent researches show that a T–S

fuzzy model can be utilized to approximate global behavior ofhighly complex nonlinear systems. The large numbers of pub-lished paper have used the T–S fuzzy model technique for differ-ent drive systems [9,10].

The main contribution of this research is position and current con-trol of an IPMSM by using loop-shaping methodology: blending ofrobust mixed-sensitivity H1 problem and T–S fuzzy model scheme.

In this paper, the problem of robust mixed-sensitivity H1 controlof an IPMSM system which possesses not only parameter uncer-tainties but also external disturbances is considered. Several robustH1, loop shaping and mixed-sensitivity problem schemes based onthe use of LMIs theory have also been proposed in Refs. [11–13].

In the proposed method, a Takagi–Sugeno (T–S) fuzzy modelis first designed on behalf of given nonlinear plant. The fuzzymodel is described by fuzzy IF–THEN rules which represent localinput–output relations of a nonlinear system. Both loop-shapingmethodology and mixed-sensitivity problem are then introducedto formulate frequency-domain specifications. Afterward, a regu-lar flowchart is proposed to get optimal weighting functions. Inthis research, tracking of rotor angular position and d-axis currentare selected as the main objectives, in the event that second objec-tive refer to elimination of reluctance effects and torque ripple.Thereafter, a robust mixed-sensitivity H1 output-feedback con-troller with regional pole-placement is designed by LMI-baseddesign techniques for each linear subsystem. The PDC is thenused to design the controller for the overall system. Furthermore,the overall fuzzy model of the system is achieved by fuzzy blend-ing of the local linear system models. Eventually, several resultsshow that the proposed method can effectively meet the per-formance requirements like robustness, good load disturbance

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedApril 21, 2012; final manuscript received March 30, 2013; published online May 27,2013. Assoc. Editor: Eugenio Schuster.

Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2013, Vol. 135 / 051006-1Copyright VC 2013 by ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 05/31/2013 Terms of Use: http://asme.org/terms

rejection, tracking, and also fast transient responses for the three-phase IPMSM system.

The paper is organized as follows: problem statement andIPMSM dynamic model are presented in Sec. 2. In Sec. 3, T–S fuzzymodel of IPMSM and problem formulation are introduced. Section 4describes H1 loop-shaping and the mixed-sensitivity problem. InSec. 5, design of robust tracking controller is proposed. Simulationresults of the closed-loop system with the proposed technique arepresented in Sec. 6 and finally, the paper is concluded in Sec. 7.

2 Problem Statement and IPMSM Dynamic Model

2.1 Problem Statement. The dynamics of IPMSM are non-linear and may also contain uncertain parameters such as viscousdamping coefficient and stator windings resistance. Consequentlythe control performance of IPM synchronous motors in variousapplications is highly touchy to variations of external load andsystem parameters. Noting that these variations are noxious fac-tors in the IPMSM control system, in this paper, an innovative ro-bust position and current control will be presented to reduce theireffects. The major contributions of this study are as follows:

• Successful employment of a proper T–S fuzzy model onbehalf of original nonlinear plant.

• Successful design of feasible robust position and current con-troller based on a suitable T–S fuzzy model in the presence ofparameters uncertainties as well as unknown load disturbance.

• Successful development of transient responses and disturb-ance attenuation of position and current trackings.

• Successful robustness of designed system ensuring that allclosed-loop performance specifications are satisfied in thepresence of unavoidable model uncertainty when the parame-ters in the system dynamic are varied in a wide range.

• Successful superiority of proposed strategy in comparisonwith former design procedures.

• Successful responses of rotor angular position control for var-ious position commands, in order to prove to afford trackingof different position reference inputs.

2.2 IPMSM Dynamic Model. The IPMSM nonlinear dy-namics in the d–q reference frame can be described by the set ofequations [1,6]

dhr

dt¼ xr

dxr

dt¼ 3

2

P0

Jm

Ld � Lq

� �id þ ;f

� �iq �

Bm

Jm

xr �Cl

Jm

diddt¼ �Rs

Ld

id þ P0

Lq

Ld

xriq þ1

Ld

vd

diqdt¼ �P0

;f

Lq

xr � P0

Ld

Lq

xrid �Rs

Lq

iq þ1

Lq

vq

(1)

In Eq. (1), hr is the angular position of the motor shaft, xr is theangular velocity of the motor shaft, id is the direct current and iq isthe quadrature current. Also, ;f is the flux linkage of the permanentmagnet, P0 is the number of pole pairs, Rs is the stator windings re-sistance, Ld and Lq are the direct and quadrature stator inductances,respectively. Moreover, Jm is the rotor moment of inertia, Bm is theviscous damping coefficient and Cl is the load torque. In addition,vd is the direct voltage and vq is the quadrature voltage. The electro-magnetic torque of the motor can be described as follows:

Te ¼3

2P0 Ld � Lq

� �idiq þ ;f iq

� �(2)

The parameters Rs and Bm are supposed to differ from theirnominal values Rs0

and Bm0. The following equation indicates a

state-space representation of the synchronous motor:

_x1 ¼ x2

_x2 ¼ g1x3 þ g2ð Þx4 þ g3x2 �Cl

Jm

_x3 ¼ g4x3 þ g5x2x4 þ g6u1

_x4 ¼ g7x2 þ g8x2x3 þ g9x4 þ g10u2

(3)

where

x ¼ x1 x2 x3 x4½ �T¼ hr xr id iq½ �T (4)

u1 u2½ �T¼ vd vq½ �T (5)

In Eq. (3), the parameters gi can be expressed in the following form:

g1 ¼3

2

P0

Jm

Ld � Lq

� �g2 ¼

3

2

P0

Jm

;f g3 ¼ �Bm

Jm

g4 ¼ �Rs

Ld

g5 ¼ P0

Lq

Ld

g6 ¼1

Ld

g7 ¼ �P0

;f

Lq

g8 ¼ �P0

Ld

Lq

g9 ¼ �Rs

Lq

g10 ¼1

Lq

(6)

3 T–S Fuzzy Model of IPMSM and Problem

Formulation

In this section, the T–S fuzzy dynamic model is described byfuzzy IF–THEN rules, which represent local linear input–outputrelations of nonlinear systems [14,15]. The fuzzy dynamicmodel is proposed by Takagi and Sugeno. Over the last years, largnumber of authors employed T–S fuzzy approach to approximatenonlinear system [16,17]. The ith rule of T–S fuzzy dynamicmodel with parametric uncertainties can be described as follows:

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

_x tð Þ ¼ Ai þ DAi½ �x tð Þ þ B1iþ DB1i

½ �w tð Þ þ B2iþ DB2i

½ �u tð Þ½ �; x 0ð Þ ¼ 0

z tð Þ ¼ C1iþ DC1i

½ �x tð Þ þ D11iþ DD11i

½ �w tð Þ þ D12iþ DD12i

½ �u tð Þ½ �y tð Þ ¼ C2i

þ DC2i½ �x tð Þ þ D21i

þ DD21i½ �w tð Þ þ D22i

þ DD22i½ �u tð Þ½ � i ¼ 1; 2;…; r

(7)

where, Mip is the fuzzy set; r is the number of IF–THEN rules andv1(t) ! vp(t) are the premise variables; x(t) 2 Rn is the state vec-tor; u(t) 2 Rm is the control input vector; w(t) 2 Rq is the disturb-ance input vector; y(t) 2 Rp is the output vector. The matrices:DAi, DB1i, DB2i, DC1i, DC2i, DD12i, DD21i, DD11i, and DD22i rep-resent the uncertainties in the system and satisfy the followingassumption:

DAi¼G x tð Þ;tð ÞH1i; DB1i¼G x tð Þ;tð ÞH2i; DB2i¼G x tð Þ;tð ÞH3i;

DC1i¼G x tð Þ;tð ÞH4i; DD11i¼G x tð Þ;tð ÞH5i; DD12i¼G x tð Þ;tð ÞH6i;

DC2i¼G x tð Þ;tð ÞH7i; DD21i¼G x tð Þ;tð ÞH8i; DD22i¼G x tð Þ;tð ÞH9i

(8)where Hji; j ¼ 1;…; 9 are known matrix functions which charac-terize the structure of the uncertainties. Furthermore, the

051006-2 / Vol. 135, SEPTEMBER 2013 Transactions of the ASME


following inequality holds: G x tð Þ; tð Þ � @; @ > 0. Considering thenonlinear state space model (1), the parameters Rs and Bm andload torque disturbance input Cl are supposed to vary. Accordingto the prementioned local linearization approach, the local linearmodel matrices for the system (3) with mentioned variations atthe ith selected operating point are obtained as follows:

Ai ¼ A0i þ BmABmþ RsARs

(9)

B1¼

0

� 1

Jm

0

0

2666664

3777775 B2¼

0 0

0 0

g6 0

0 g10

266664

377775 A0i¼

0 1 0 0

0 0 g1x4 g2

0 g5x4 0 0

0 g7 g8x2 0

266664

377775(10)

ABm¼

0 0 0 0

0 � 1

Jm

0 0

0 0 0 0

0 0 0 0

266664

377775 ARs

¼

0 0 0 0

0 0 0 0

0 0 � 1

Ld

0

0 0 0 � 1

Lq

266666664

377777775(11)

where A0i, ABm, ARs

, B1, B2 are known real matrices with appropri-ate dimensions in nonlinear model (3). In view of the set of matri-ces (9)–(11), �2 and �4 appear only in matrix A0i. since thenonlinear system (1) is an affine model accordingly, xi cannot bepresent at Bi. In other words, matrix A0i is only influenced by thenonlinearity term �2 and �4. The overall fuzzy model can bedefined as following form:

_x tð Þ ¼Xr

i¼1

li v tð Þð Þ Ai þ DAi½ �x tð Þ þ B1iþ DB1i

½ �w tð Þ þ B2iþ DB2i

½ �u tð Þ½ �; x 0ð Þ ¼ 0

z tð Þ ¼Xr

i¼1

li v tð Þð Þ C1iþ DC1i


½ �w tð Þ þ D12iþ DD12i

½ �u tð Þ½ �

y tð Þ ¼Xr

i¼1

li v tð Þð Þ C2iþ DC2i


½ �w tð Þ D22iDD22i

½ �u tð Þ½ �

(12)

where

v tð Þ ¼ v1 tð Þ … vp tð Þ½ � (13)

And weighting function is

li v tð Þð Þ ¼ -i v tð Þð ÞXr

i¼1

-i v tð Þð Þ(14)

-i v tð Þð Þ ¼Yp

k¼1

Mik vk tð Þð Þ (15)

And it should be noted that

-i v tð Þð Þ � 0; i ¼ 1; 2;…; r;Xr

i¼1

-i v tð Þð Þ > 0

li v tð Þð Þ � 0; i ¼ 1; 2;…; r;Xr

i¼1

li v tð Þð Þ ¼ 1

(16)

4 The H‘ Loop-Shaping and the Mixed-Sensitivity

Problem

Loop shaping is a design procedure to formulate frequency-domain specifications as H1 constraints problems [18–21]. To geta feeling for the loop-shaping methodology, consider the generalpattern loop of Fig. 1.

In this figure, P(s) is the generalized plant, K(s) is the control-ler, u are the control signals, y are the measured variables, ware the exogenous signals and z are the main objectives. Inthis frequency-domain method, the design specifications arereflected as gain constraints on the various closed-loop transferfunctions. Where the main closed-loop transfer functions are

sensitivity function and complementary sensitivity function, aswell as the gain constraints are shaping filters. These shapingobjectives are then turned into uniform H1 bounds by means ofshaping filters, and H1 synthesis algorithms are applied to com-pute an adequate controller. As a matter of fact, the H1 norm of atransfer function F(s) is corresponds to the peak gain of the fre-quency response F(jx) that is denoted as

F1 ¼ sup rmaxðFðjwÞÞ (17)

The optimal H1 control problem can be interpreted as minimiz-ing the effect of the worst-case disturbance w on the output z. Theclosed-loop transfer function from w to z is given by Tzw sð Þ.Hence, the optimal H1 control seeks to minimize ||F(P,K)||1 overall stabilizing controllers K(s). Alternatively, we can specify somemaximum value c for the closed-loop RMS gain as

FðP;KÞ1 < c (18)

where c is guaranteed H1 ratio between z and w. In this case, theclosed-loop transfer function Tzw(s) in frequency domain can berepresented as follows:

Fig. 1 The control structure

Journal of Dynamic Systems, Measurement, and Control SEPTEMBER 2013, Vol. 135 / 051006-3


Tzw sð Þ ¼ F P;Kð Þ ¼ WT sð ÞTðsÞWs sð ÞSðsÞ

� �(19)

where S(s) is the sensitivity transfer matrix (the transfer functionfrom r to e) and T(s) is the complementary sensitivity transfer ma-trix (the transfer function from r to y) that they can be expressedas

S sð Þ ¼ ðI þ G sð ÞKðsÞÞ�1

T sð Þ ¼ G sð ÞKðsÞðI þ G sð ÞKðsÞÞ�1(20)

The WS(s) and WT(s) are two frequency dependent weightingfunctions (shaping filters), the sensitivity and the complementarysensitivity weighting functions, respectively. As it is known, shap-ing T(s) is desirable for tracking problems, noise attenuation andfor robust stability with respect to multiplicative output uncertain-ties, On the other hand, since S(s) relates the error signals withreferences and disturbances, shaping the sensitivity function willpermit the performance (in terms of command tracking and dis-turbance attenuation) of the system to be controlled. Accordingly,in order to realize above requirement, following close-loop trans-fer functions can be normalized as following forms:

Ws sð ÞS sð Þ1< 1 WT sð ÞT sð Þ1< 1 (21)

Noting that the rotor angular position and the d-axis current areselected as the main objectives, as a result it can conclude thatnumbers of both system outputs and tracking errors are 2. There-fore, the size of weighting function matrices WS(s) and WT(s) arechosen 2� 2 that can be expressed as follows:

T sð Þ ¼ Tyr ¼T1 ¼ Ty1r1

T2 ¼ Ty2r2

� �S sð Þ ¼ Ter ¼

S1 ¼ Te1r1

S2 ¼ Te2r2

� �(22)

where y1 and y2 are angular position and d-axis current; r1 and r2

are position command and direct current reference input; e1 and e2

are the tracking errors. The sensitivity and the complementarysensitivity weighting functions WT(s), WS(s) are designed as two2� 2 square diagonal matrices. Where WTii

sð Þ and WSiisð Þ are the

diagonal elements of WT(s), WS(s), respectively, that they can berewritten as follows:

WT sð Þ ¼ WTiisð ÞI2�2 Ws sð Þ ¼ WSii

sð ÞI2�2 (23)

The transfer functions WSiisð Þ and WTii

sð Þ must be stable, mini-mum phase and additionally should be proper. As well as WSii

sð Þand WTii

sð Þ must be low-pass and high-pass filters, respectively. Inorder to determine the performance and robustness weights, prac-tical formulas of WSii

sð Þ and WTiisð Þ are suggested as follows:

WTiisð Þ ¼ dsþ 1

dsþ 2WSii

sð Þ ¼ asþ wc

sþ wcb(24)

where “a” high frequency disturbances gain, “b” is the gain forlow frequency control signal, “d” is a constant and “wc” is thecrossover frequency. wc is the frequency which differentiates thehigh frequency disturbance signal and the low frequency controlsignal. Also, wc indicates the minimum bandwidth of the transferfunctions which are weighted. In order to design the optimalweighting functions, the regular weights design method (RWDM)can be expressed as a flowchart in Table 1.

Consequently, RWDM selects optimal weighting functions inorder to formulate performance and robustness specifications ofclose-loop system. In the RWDM, the coefficients a, b, c, wc

should be decremented or incremented as far as whole of clausesin each step are realized and finally controller K is returned.

5 Design of Robust Tracking Controller

In this section, focus to design a local pole-placement output-feedback controller for each linear subsystem

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

u tð Þ ¼ Kiy tð Þ; i ¼ 1; 2;…; r(25)

where Ki (i ¼ 1; 2;…; r) are the local controller gains. In order todesign whole controller, the concept of PDC is employed for thesystem (7) [14,17]. According to PDC approach, the control lawof the whole system is the weighted sum of the local feedbackcontrol of the various subsystems that can be expressed asfollows:

u tð Þ ¼Xr

j¼1

ljKjy tð Þ (26)

where the local pole-placement output-feedback gains Kj aredetermined by LMI-based design techniques in order to achievethe design requirements [22,23]. The LMI formulation is applica-ble to design local controllers that are introduced as follows.

Table 1 RWDM

Answer: NO Step number Step description Answer: YES

— Start1 Get ith local linear subsystem, Gi and new

input values: a, b, d, wc

Go to next step

2 Design weighting matrices, Build the plant Pand merge(Gi, WT(s), WS(s))

Go to next step

3 Specify LMI pole-placement region, D andFind local controller by LMI approach, K

Go to next step

4 Find the close-loop responses: S(s): {S1(s),S2(s)} and T(s):{T1(s), T2(s)}

Go to next step

Go back to: Step 2 and/or Step 5 5 Are true these two constraints?rmax S jwð Þð Þ < c rmin W�1

s ðjwÞ� �

rmax T jwð Þð Þ < crmin W�1T ðjwÞ

� � Go to next step

Go back to: Step 2 and/or Step 5 6 Is the guaranteed H1 less than 1? Go to next stepGo back to: Step 2 and/or Step 5 7 Is close-loop system steady state and tran-

sient responses expected for tracking anddisturbance attenuation

Go to next step

— K is optimal and WT(s), WS(s) are desirable Go to next step— Stop



Main objective is to design an output-feedback controller u¼Ky as follows:

• It maintains the H1 norm of T1(s) (RMS gain) below someprescribed value c0> 0

• It maintains the H2 norm of T2(s) (LQG cost) below someprescribed value v0> 0

• It places the closed-loop poles in some prescribed LMI regionD

• It minimizes a trade-off criterion of the formajjT1jj

21 þ bjjT2jj

22.

where T1(s) and T2(s) are the closed-loop transfer functionsfrom w to z1 and from w to z2, respectively. The linear fuzzy sub-plant P(s) for the control structure Fig. 1 is given in state-spaceform by

_x ¼ Aixþ B1iwþ B2i

u

z1 ¼ C1ixþ D11i

wþ D12iu

z2 ¼ C2ixþ D21i

wþ D22iu

y ¼ Cyixþ Dy1i

w

8>>><>>>:

(27)

And related controller K(s) is introduced as

_n ¼ Aknþ Bky

_u ¼ Cknþ Dky

((28)

By merging P(s), K(s), and u¼K y, the closed-loop system is con-cluded as following form:

_xcl ¼ Aclixcl þ Bcli

w

z1 ¼ Ccl1ixcl þ Dcl1i

w

z2 ¼ Ccl2ixcl þ Dcl2i

w

8><>: (29)

Our three design objectives can be expressed as follows:

• H‘ performance: The closed-loop RMS gain from w to z1does not exceed c> 0 if and only if there exists a symmetricmatrix X1 such that

AclX1 þ X1ATcl Bcl X1CT

cl1

BTcl �I DT

cl1

Ccl1X1 Dcl1 �c2I

24

35 < 0;X1 > 0 (30)

• H2 performance: The H2 norm of the closed-loop transferfunction from w to z2 does not exceed � > 0 if and only ifDcl2¼ 0 and there exist two symmetric matrices X2 and Qsuch that

Q Ccl2X2

X2CTcl2 X2

� �>0

AclX2þX2ATcl Bcl

BTcl �I

� �<0 Trace Qð Þ<m2

(31)

• Pole placement: The closed-loop poles lie in the LMI region

D ¼ z 2 C : LþMzþMTz < 0�

L ¼ LT ¼ kij

� 1�i;j�m

M ¼ lij

n o1�i;j�m

(32)

If and only if there exists a symmetric matrix Xpol satisfying

kijXpolþlij AþB2Kð ÞXpolþlijXpolþljiXpol AþB2Kð ÞTh i

1�i;j�m<0

Xpol>0 (33)

For tractability in the LMI framework, we seek a single Lyapunovmatrix: X:¼X1¼X2¼Xpol that enforces all three sets of con-straints. Factorizing X as

X ¼ X1XT2 X1 :¼ R I

MT 0

� �X2 :¼ 0 S

I NT

� �

And introducing the change of controller variables

Ak :¼ NAkMT þ NBkCyRþ SB2CkMT þ S Aþ B2DkCy

� �R

Bk :¼ NBk þ SB2Dk

Ck :¼ CkMT þ DkCyR (34)

The inequality constraints on v are readily turned into LMI con-straints in the variables R, S, Q, AK, BK, CK, and DK. This leads tothe following suboptimal LMI formulation of our multi-objectivesynthesis problem:

Minimize ac2 þ b Trace (Q) over R, S, Q, AK, BK, CK, DK, andc2 satisfying:

ARþ RAT þ B2CK þ CTKBT

2 ATK þ Aþ B2DkCy B1 þ B2DkDy1 H

H ATSþ SAþ BKCy þ CTy BT

K SB1 þ BkDy1 H

H H �I H

C1Rþ D12CK C1 þ D12DKCy D11 þ D12DKDy1 �c2I

266664

377775 < 0

Q C2Rþ D22CK C2 þ D22DKCy

H R I

H I S

264

375 > 0

"kij

R I

I S

!þ lij

ARþ B2CK Aþ B2DkCy

AK SAþ BKCy

!þ lji

RAT þ CTKBT

2 ATK

Aþ B2DkCy

� �TATSþ CT

y BTK

!#1�i;j�m

< 0

Trace Qð Þ < m20 c2 < c2

0 D21 þ D22DKDy1 ¼ 0

(35)

where Q is a symmetric matrix as the sum of the elements on themain diagonal of Q must be less than v0; c0 and v0 are prescribedvalues, as H1 norm of T1(s) (Tz1w) and H2 norm of T2(s) (Tz2w)

lie below of them, respectively. Given optimal solutions c*, Q* ofthis LMI problem, the closed-loop H1 and H2 performances arebounded by



jjT1jj1 < c� jjT2jj2 <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTraceðQ�Þ

p(36)

The main goal of this research is design of a proper controlwhich guarantees robust performance in presence of parametersvariation as well as load torque disturbance. In this case, two con-trol objectives are defined. First, rotor angular position and sec-ond, the d-axis current of motor that both of them must trackreference trajectories r1 and r2, respectively. In fact, d-axis currentmust be converged to zero, due to elimination of reluctanceeffects and torque ripple. Therefore, in order to achieve theseobjectives, both of tracking errors should be minimized. Men-tioned goals are realized through constructing of the objectives zin an appropriate control loop. Under the above considerations,the structure of the mixed-sensitivity control loop is shown sche-matically in Fig. 2.

According to this structure, the nonlinear dynamic is firstapproximated with some local linear models in four rules which

are represented by T–S fuzzy approach. Both shaping filters WS(s)and WT(s) are then designed and augmented plant P is created.Thereafter, local controllers are designed for each linear subplantpi based on LMI approach. After that the total augmented linearsystem P is obtained by using the weighted sum of the local linearsubsystems and is utilized rather than original nonlinear system.Moreover, according to PDC approach, the control law of thewhole system K is designed. On the other hand, fuzzy weights (li)that are shown in Fig. 2 are updated by using a fuzzy weightsonline computation component that is shown in Fig. 3.

• In the first block of this component, angular velocity of themotor shaft (x2¼xr) and the quadrature current (x4¼ iq) aremeasured in real time from IPMSM and nonlinear terms (fuzzyvariables: v1¼ x2, v2¼ x4) are then built by these measurements.

• In the second block of FWOC component, the values ofmembership functions (Mi) in current values of x2, x4

“nonlinearities” are calculated.

Fig. 2 The fuzzy robust control loop structure

Fig. 3 FWOC component



• In the third block, new fuzzy weights (14)–(16) are calculatedand they are sent to main control structure (Fig. 2). Finally,by using whole system and global controller, a tracking loop(Fig. 2) is applied to the system in order to achieve desirablespecifications such as tracking performance, bandwidth, dis-turbance rejection, and robustness for the close-loop system.

6 Simulation Results

In this section, we show the effectiveness of the proposedmethod by performing some simulation studies over an IPMSMmodel. The motor used in the paper is made by Hsin-Ting Com-pany, Taiwan. The type of motor is the 130-750MS-ZK-L2. ThisIPMSM is a three-phase, four poles, rated 0.75 HP, with a2000 rpm rated speed. In the experimentation, the maximum volt-age and the continuous rated armature current are set to 230 V and12 A [1,6]. The parameters of the IPMSM are shown in Table 2.In this case, the stator windings resistance Rs and the viscousdamping coefficient Bm are varied between 650% and load torquedisturbance is unknown [1,6,7].

According to Eq. (11), the x2 and x4 are nonlinear terms andreferring to the IPMSM characteristic and the territory of the sys-tem operating points, we should calculate the minimum and maxi-mum values of x2 and x4 as: v1 tð Þ ¼ x2 2 0; 2000½ � and

v2 tð Þ ¼ x4 2 012½ �. Where v1 and v2 are fuzzy variables. In viewof above limitations, membership functions can be calculated as

M1 ¼v1

2000M2 ¼ 1� v1

2000M3 ¼

v2

12M4 ¼ 1� v2

12

In view of the above equations, the membership functions Mi isdemonstrated in Fig. 4.

• In the first step, the system (3) is represented by a T–S fuzzymodel using the fuzzy rules given in Eq. (7). For this designproblem, the rules r1–r4 are constructed for the T–S fuzzymodel and referring to Eqs. (4)–(6) the four linear subsystemsare then calculated. Referring to Eqs. (9)–(11), all systemmatrices are constant except A0i

that varies with respect toeach rule as given below

A01¼

0 1 0 0

0 0 �1145 1860

0 49 0 0

0 �20 �82119 0

26664

37775

A02¼

0 1 0 0

0 0 0 1860

0 0 0 0

0 �20 �82119 0

26664

37775

A03¼

0 1 0 0

0 0 �1145 1860

0 49 0 0

0 �20 0 0

26664

37775

A04¼

0 1 0 0

0 0 0 1860

0 0 0 0

0 �20 0 0

26664

37775

• In the second step, referring to Sec. 4, the weighting matricesWT(s) and WS(s) are designed as follows:

WT sð Þ ¼ WTiisð ÞI2�2 ¼

0:001sþ 1

0:001sþ 2I2�2

Ws sð Þ ¼ WSiisð ÞI2�2 ¼

0:5sþ 10

sþ 0:01I2�2

Fig. 4 The membership functions for (a) M1(v1) and M2(v1) and(b) M3(v2) and M4(v2)

Table 2 The parameters of IPMSM

Rs 1.9 XBm0 (without load) 0.03 N m s/radBm0 (with load) 0.0341 N m s/radLd 0.0151 HLq 0.031 H;f 0.31 V s/radJm0 (without load) 0.0005 kg m2

Jm0 (with load) 0.0227 kg m2

P0 2

Fig. 5 Pole-placement region



• In the third step, by using the LMI formulations fully pre-sented in Sec. 5, we can calculate the local controllers foreach linear subsystem based on purposed control loop (Fig.2), above weighting matrices and LMI technique (27)–(36).

In order to design state-space feedback gains (Ki) for each sub-system, the following steps are done:

• Specify the LMI region D (32), in order to place the closed-loop poles in this region (pole placement) and also to guaran-tee some minimum decay rate and closed-loop damping. Theaforementioned region is shown in Fig. 5, as: the intersectionof the half-plane is x<�8 and center of sector is at the originwith inner angle p/2. D1 : x < �8 and h ¼ p=2

• Choose a four-entry vector specifying the H2/H1 trade-offcriterion: [c0 �0 a b]¼ [0 0 1 0]. As a matter of fact, in thiscase, constraint and criterion of H2 is not used (b¼ 0) andtwo design objectives, H1 performance and pole placementare only employed.

• Minimize H2/H1 cost function subject to the above-mentioned pole-placement constraint using Eqs. (30), (31),and (34)–(36).

Finally, the overall fuzzy system is obtained by using aweighted average defuzzifer and also the control law of the wholesystem is designed by using the PDC approach. Global proposedT–S fuzzy model can represents the original nonlinear system inthe prespecified domains [0, 2000] rpm� [0, 12] A on the x2-x4

space for various operating point.Figure 6 compares states of the proposed T–S fuzzy model in

comparison with the original nonlinear system [1], where the solidlines denote the state variables by the T–S fuzzy model x̂ tð Þ andthe dashed lines indicate states of original nonlinear system xðtÞ in

Fig. 6 Time responses of the proposed model (solid) and the original nonlinear model (dashed): (a)Angular position of motor shaft, (b) angular speed, (c) d-axis current, and (d) q-axis current

Fig. 7 Load torque

Fig. 8 Disturbance rejection on angular position (a) with stepload torque (1 Nm) and (b) with benchmark load torque (Fig. 5)



Eq. (1). Accordingly, design of T–S fuzzy model is required tosatisfy x̂ tð Þ ! xðtÞ and this condition guarantees that the steady-state errors between x̂ tð Þ and xðtÞ converges to 0. As it is evidentin this figure, the time responses of the proposed T–S fuzzy modelcan follow the responses of the nonlinear system (1) withoutsteady-state errors for the range x2 2 ½0; 2000� and x4 2 ½0; 12�,

which means that the fuzzy model can represents the original sys-tem in the prespecified domains with suitable approximation. Onthe other word, the proposed fuzzy model can be replaced onbehalf of given original system (1) in the prespecified domains.

In actuality, the motor is used to convert the electrical energyinto mechanical energy. Accordingly, an external load is added tothe drive system. The external load can be selected one of twotypes of following disturbances: First, another synchronous motoris coupled to the shaft of the main IPMSM motor in order torequest a load torque [6,7]. The manners of the load torque Cl

applied to the synchronous motor are represented in Fig. 7. Sec-ond, a 1 kg weight is located at a certain location of motor. As aresult, the weight can provide the external load 1 Nm that inter-preted with step input.

Figures 8(a) and 8(b) demonstrate disturbance rejection ofangular position over three different methods, when step load tor-que and benchmark load torque (Fig. 7) act on system, respec-tively, although the IPMSM is first controlled to reach a fixedposition “90 deg” by proposed controller.

These figures show disturbance attenuation performance of theproposed method in comparison with both the H2/H1 controller(represented in Ref. [6]) and the I/O linearization technique (rep-resented in Ref. [1]). The notations on Figs. 8(a) and 8(b) aredescribed as follows: Eip is peak error; Eip-p is peak to peak error.

• E1p and E1p-p are peak and peak to peak errors, when step andbenchmark disturbances (Fig. 7) act on IPMSM, respectively,as well as proposed controller is used.

• E2p and E2p-p are peak and peak to peak errors, when IPMSMis influenced by step and benchmark disturbances, respec-tively, while H2/H1 controller (represented in Ref. [6]) isemployed.

• E3p and E3p-p are peak and peak to peak errors, when IPMSMis affected by step and benchmark disturbances, respectively;moreover, I/O linearization technique (represented in Ref.[1]) is utilized.

As you can view, the proposed method has better disturbanceattenuation in both types of external loads.

Table 3 abridges disturbance rejection performances when sys-tem in influenced by both kind of disturbances. According to thistable, the proposed method has the smallest peak and peak-to-peak error values (E1p and E1p-p) in comparison with other meth-ods. Indeed, in view of the comparisons between three methods, it

Table 3 The position control performance

Method ProposedH2/H1

(represented in Ref. [6])I/O linearization

(represented in Ref. [1])

Tsi (s) 0.15 0.28 0.4Eip (deg) 0.06 0.2 1.1Eip-p (deg) 0.3 0.75 4.2

Fig. 9 d-axis current

Fig. 10 position tracking responses at different positionreference

Fig. 11 Comparison of transient responses (a) certain stepposition and (b) rectangular position command

Table 4 Performance of proposed method for various LMIregions

LMI regionD

D1 D2 D3 D4Item x<�8 h¼p/2 x<�8 h¼ 2p/3 x< 0 h¼p/2 x< 0 h¼ 2p/3

Tsi (s) 0.15 0.15 0.3 0.3Eip (deg) 0.06 0.1 0.06 0.1Eip-p (deg) 0.3 0.7 0.3 0.7

Fig. 12 Angular position responses with varying Rs and Bm



can deduced that E1p<E2p and E3p as well as on the other handE1p-p<E2p-p and E3p-p.

Figure 9 shows d-axis current tracking of proposed and H2/H1(represented in Ref. [6]) methods to zero, in order to eliminationof reluctance effects and torque ripple.

In view of this figure, magnitude band of distortion over d-axiscurrent tracking in proposed method is less than H2/H1 approach.In fact, proposed controller can traces reference current, 0 A, moresmooth than another method and consequently in our method, re-luctance effects and torque ripple can be eliminated better thanH2/H1 method. The notations on Fig. 9 are described as: DBi: dis-tortion band

• DB1 and DB2 are distortion bands on current trackingresponses, when proposed and H2/H1 controllers are used,respectively. Referring to Fig. 9, the magnitude bands of dis-tortion are DB1¼ 0.2 and DB2¼ 0.4 and as can be easilyobserved that DB1<DB2. As a result, it can conclude that inproposed method d-axis current is tracked to 0 A, with lessdistortion band DB1 in comparison with H2/H1 method withDB2.

Figure 10 illustrates the position tracking responses for differ-ent position commands, when the proposed controller is used.According to this figure, the proposed system has satisfactory per-formance for various position commands.

Figures 11(a) and 11(b) show the position tracking responses ofthe proposed, H2/H1 (represented in Ref. [6]) and I/O lineariza-tion (represented in Ref. [1]) controllers for a certain position(180 deg) and rectangular position commands, respectively. Thenotations on Figs. 11(a) and 11(b) are described as: TSi is settlingtime.

• TS1, TS2, and TS3 are settling times of tracking responses,when proposed, H2/H1 and I/O linearization controllers areused, respectively. According to these figures, the proposedcontroller has the best design traits with regard to settlingtime and raise time in comparison with other methods. Table3 also summarizes the numerical results of transientresponses for aforementioned methods. Referring to this ta-ble, the proposed method has the smallest settling time value(Ts1) in comparison with other methods. Indeed, in view ofthe comparison between three methods, it can conclude thatTs1< Ts2 and Ts3.

In fact, numerical results of Table 3 are fascinated by the afore-mentioned LMI region D1 : x < �8 and h ¼ p=2. Table 4 presentsobtained values of settling time Tsi, peak Eip, and peak-to-peak Eip-p

error for various LMI regions (pole-placement regions) over theposition tracking response of the proposed method.

Figure 12 illustrates the position responses, when both the statorwindings resistance Rs and the viscous damping coefficient Bm arevaried between 650%. As you can see, the system has goodrobustness when the parameters in the systems dynamic are variedin a wide range.

Table 4 proves that D1 is the optimal region among regions ofthis table, because all of settling time, peak and peak-to-peakerrors based on this region have the smallest values in comparisonwith other regions. In fact, values of errors are increased when theinner angle h is expanded, as well as settling time is raised whenthe intersection of the half-plane x is reduced. Therefore, we haveused design results based on the optimal LMI Region D1.

Figures 13(a) and 13(b) demonstrate W�1s and W�1

T , that aregreater than S1, S2 and T1, T2 over frequency domain, respectively.Furthermore, the weights W�1

s and W�1T also are greater than var-

iations of sensitivity functions and complementary sensitivityfunctions. Indeed, these variations are occurred over S and T dueto parameters uncertainties.

7 Conclusions

In this paper, a robust mixed-sensitivity H1 controller has beendesigned to tracking and disturbance attenuation of position andcurrent for an MIMO nonlinear uncertain T–S fuzzy model ofIPMSM systems. An LMI approach has been developed and amixed-sensitivity problem has been given to formulate frequency-domain specifications and achieve the robustness against nonlin-ear system uncertainties. The numerical results on IPMSMshowed that the robust control system has small position and cur-rent tracking errors and has desired robustness against load torquedisturbance and parameter variations. In addition, the superiorityof the proposed control scheme was approved through simulationsin compared with the input–output linearization and the H2/H1methods (represented in Refs. [1,6], respectively). Referring toTable 3, the proposed method had smallest settling time on posi-tion tracking response and the lowest amount of undershoot ondisturbance rejection response. The major achievements of this

Fig. 13 (a) W�1s matrix as an upper bound of the S1 and S2 uncertainty and (b) W�1

T matrix as an upper bound of the T1 and T2

uncertainty



research are delivered as follows: (i) The performance require-ments like good load disturbance rejection, tracking and fast tran-sient responses in the proposed method were better than those ofthe other methods,(ii) The proposed method had satisfactorytracking for various angular position commands, (iii) The pro-posed controller had good robustness when parameter trajectoriesof Rs(t) and Bm(t) were changed in real time, and (iv) direct cur-rent of system can be tracked to 0 A for elimination of reluctanceeffects and torque ripple.

Acknowledgment

The authors wish to thank the Referees and the Associate Editorfor their constructive comments and helpful suggestions that havehelped to improve the quality of this paper.

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