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Original Article

Robust multi-objective H2/HN tracking control based on the Takagi–Sugeno fuzzy model for a class of nonlinear uncertain drive systems

Proc IMechE Part I: J Systems and Control Engineering 226(8) 1107–1118 Ó IMechE 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0959651812443925 pii.sagepub.com

Vahid Azimi1, Mohammad A Nekoui2 and Ahmad Fakharian3

Abstract In this paper a robust H2/HN multi-objective state-feedback controller and tracking design are presented for a class of multiple input/multiple output nonlinear uncertain systems. First, some states (error of tracking) are augmented to the system in order to improve tracking control. Next, uncertain parameters and the quantification of uncertainty on physical 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 linear subsystem with augmented state, an H2/HN multi-objective state-feedback controller is designed by using a linear matrix inequalities approach. Finally, parallel distributed compensation is used to design the controller for the overall system and the total linear system is obtained by use of the weighted sum of the local linear subsystems. Several results show that the proposed method can effectively meet performance requirements such as robustness, good load disturbance rejection, good tracking and fast transient responses for a three-phase interior permanent magnet synchronous motor system.

Keywords Robust control, Takagi–Sugeno fuzzy model, linear matrix inequalities, multi-objective H2/HN controller, parallel distributed 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 synchronous motor (IPMSM) has progressively been replacing direct current and induction motors in a wide range of drives for many industrial applications, such as in robotic actuators, traction and machine tool spindle drives, rolling mills, air conditioning compressors, electrical vehicles, integrated starters/alternators, computer disk drives, domestic applications, automotive and renewable energy conversion systems. The reason why the IPMSM has become so well liked is essentially due to its many pleasant characteristics such as high efficiency, exceptional power density, low inertia, excellent torque generation, admirable speed and acceleration capabilities, wide range of operation speed, strong structure and high torque-to-inertia, torque-to-current and power-to-weight ratios. In many applications, tracking a position and speed trajectory, torque ripple and elimination of reluctance effects are of great importance. However, the existence

of system parameter variations, load torque disturbance and system nonlinearity elements makes this a rather arduous task. Numerous nonlinear and linear controllers have been developed for the IPMSM. Hitherto, much research has been performed on torque ripple reduction and reluctance effects elimination. For example, Colamartino et al.1 proposed a torque ripple minimization with two strategies. The first was based on numerical predetermination of the current waveform 1

Electrical Engineering Department, Islamic Azad University, South Tehran Branch, Islamic Republic of Iran 2 Faculty of Control Engineering, KN Toosi University of Technology, Islamic Republic of Iran 3 Faculty 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]

1108 which is imposed by the control in machine phases and the second was a torque regulation based on its online instantaneous estimation for an IPMSM drive. Chen et al.2 presented the controller-induced parasitic torque ripples for an IPMSM drive. Gulez et al.3 investigated the torque ripple and electromagnetic interference noise minimization using active filter topology and fieldoriented control for an IPMSM drive. For achieving acceptable drive performance and good position and speed tracking, disturbances attenuation and torque control, many issues have been considered in recent years. Lin et al.4 used a nonlinear position controller design with an input–output linearization technique for an IPMSM control system. Yang and Zhong5 implemented a robust speed tracking of permanent magnet synchronous motor (PMSM) servo systems by equivalent disturbance attenuation. Su et al.6 studied the automatic disturbances rejection controller for precise motion control for a PMSM drive. Chou and Liaw7 proposed the development of robust two-degree-offreedom current controllers for a PMSM drive with reaction wheel load. Karabacak and Eskikurt8 proposed a speed and current regulation of a PMSM via nonlinear and adaptive backstepping control. Errouissi and Ouhrouche9 used a nonlinear predictive controller (NPC) for a PMSM. Other papers have studied application of the sliding mode technique to PMSM or IPMSM drive systems.10–13 Most physical dynamical systems are nonlinear in the real world and cannot be represented by linear differential equations. The Takagi–Sugeno (T–S) fuzzy model, which is often used in the literature, can approximate a wide class of highly complex nonlinear systems. The major feature of a T–S fuzzy model is to extract the local dynamics of each fuzzy implication (rule). Published papers have used the T–S fuzzy model technique for different drive systems. For example, Chen and Wu14 presented a robust optimal reference-tracking design method for stochastic synthetic biology systems using a T–S fuzzy approach. Qiu et al.15 proposed a new design of delay-dependent robust HN filtering for discrete-time T–S fuzzy systems with time-varying delay. Wai and Yang16 proposed an adaptive fuzzy neural network control design via a T–S fuzzy model for a robot manipulator including actuator dynamics. The main contribution of the present research is robust tracking H2/HN control based on a T–S fuzzy model for an uncertain nonlinear IPMSM. This paper considers the problem of robust H2/HN control for uncertain nonlinear multiple input/multiple output (MIMO) T–S fuzzy systems which possess not only parameter uncertainties but also external disturbances. Several robust HN schemes based on the use of linear matrix inequality (LMI) theory have been reported previously.17–19 In the method proposed herein, first the nonlinear plant is represented by a T–S fuzzy model. The fuzzy model is described by fuzzy IF–THEN rules which represent local input–output relationships of the nonlinear system. Then, some states are augmented to

Proc IMechE Part I: J Systems and Control Engineering 226(8) the system in order to improve tracking control and tracking error reduction. After that, uncertain parameters and the quantification of uncertainty on physical parameters are defined by the affine parameterdependent systems method. Next, based on each linear subsystem with augmented state, an H2/HN multiobjective state-feedback controller is designed by using the LMI approach. Parallel distributed compensation (PDC) is then utilized to design the controller for the overall system. Thus the overall fuzzy model of the system is achieved by fuzzy ‘‘blending’’ of the local linear subsystem models. Several results show that the proposed method can effectively meet performance requirements like robustness, good load disturbance rejection responses, good tracking responses and fast transient responses for the three-phase IPMSM system. The paper is organized as follows. Preliminary concepts are presented first and then the IPMSM dynamic model is introduced. The problem statement and design of the robust tracking controller are presented next. Simulation results of the closed-loop system with the proposed technique follow, and finally conclusions are drawn.

Preliminary concepts Multi-objective H2/HN framework In this subsection, multi-objective state-feedback synthesis by the LMI framework is described.17,19–21 The main design objectives of the multi-objective controller are: HN performance (for tracking, disturbance rejection or robustness aspects); (b) H2 performance (for linear quadratic Gaussian (LQG) aspects); (c) robust pole placement specifications (to ensure fast and well-damped transient responses, reasonable feedback gain, etc.).

(a)

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

maintains the root-mean-square (RMS) gain (HN norm) of TN below some prescribed value g 0 . 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 + bjjT2 jj2 2

ð1Þ

(d) places the closed-loop poles in a prescribed LMI region. To get a feeling for the multi-objective H2/HN methodology, consider the general pattern loop of Figure 1.

Azimi et al.

1109 These three sets of conditions add up to a no-convex optimization problem with variables Q, K, XN, X2 and Xpol. For tractability in the LMI framework, we seek a single Lyapunov matrix X:=XN=X2=Xpol that enforces all three objectives. With the change of variable Y:=KX, this leads to the following suboptimal LMI formulation of our multi-objective state-feedback synthesis problem. Minimize ag2 + b Trace(Q) over Y, X, Q and g 2, satisfying

Figure 1. The control structure.

2

LMI formulation given a state-space realization is x_ = Ax + B1 w + B2 u z‘ = C1 x + D11 w + D12 u z2 = C2 x + D21 w + D22 u

ð2Þ

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

ð3Þ

Taken separately, our three design objectives have the following LMI formulation. HN performance. The closed-loop RMS gain from w to zN does not exceed g if and only if there exists a symmetric matrix XN such that 2

ðA + B2 KÞX‘ + X‘ ðA + B2 KÞT 4 B1 T ðC1 + D12 KÞX‘

B1 I D11

3 X‘ ðC1 + D12 KÞT 5\0 D11 T g2 I

ð4Þ

X‘ . 0

2.

H2 performance. The closed-loop H2 norm of T2 does not exceed n if there exist two symmetric matrices X2 and Q such that   Q ðC2 + D22 KÞX2 .0 X2 ðC2 + D22 KÞT X2   ðA + B2 KÞX2 + X2 ðA + B2 KÞT B1 \0 B1 T I TraceðQÞ \ n

3.

2

ð5Þ

Pole placement. The closed-loop poles lie in the LMI region D   D = z 2 C : L + Mz + MTz \ 0   L = LT = lij 14i, j4m   M = mij 14i, j4m ð6Þ if and only if there exists a symmetric matrix Xpol satisfying h

lij Xpol + mij ðA + B2 KÞXpol + mij Xpol + mji Xpol ðA + B2 KÞT \0

Xpol . 0

TraceðQÞ \ n0 2 g2 \ g20

x_ = ðA + B2 KÞx + B1 w z‘ = (C1 + D12 K)x + D11 w z2 = ðC2 + D22 KÞx + D21 w

1.

3 AX + XAT + B2 Y + YT B2 T B1 XC1 T + YT D12 T T T 4 5\0 B1 I D11 C1 X + D12 Y D11 g2 I   Q C2 X + D22 Y .0 T T T X XC2 + Y D22    lij + mij ðAX + B2 YÞXpol + mji XAT + YT B2 T 14i, j4m \ 0

i 14i, j4m

ð7Þ

ð8Þ

Denoting the optimal solution by (X*, Y*, Q*, g*), the corresponding state-feedback gain is given by K*=Y*(X*)21. This gain guarantees the worst-case performance jjT‘ jj‘ \ g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjT2 jj2 \ Trace(Q )

ð9Þ

T–S fuzzy dynamic model The T–S fuzzy dynamic model is described by fuzzy IF– THEN rules, which represent local linear input–output relationships of nonlinear systems.22,23 The fuzzy dynamic model is proposed by Takagi and Sugeno. The ith rule of the T–S fuzzy dynamic model with parametric uncertainties in the H2/HN multi-objective framework can be described as follows. Plant rule i IF v1 ðtÞ is Mi1 and . . . and vp ðtÞ is Mip THEN x_ ðtÞ = ½½Ai + DAi xðtÞ + ½B1i + DB1i wðtÞ + ½B2i + DB2i uðtÞ, xð0Þ = 0 z‘ ðtÞ = ½½C1i + DC1i xðtÞ + ½D11i + DD11i wðtÞ + ½D12i + DD12i uðtÞ z2 ðtÞ = ½½C2i + DC2i xðtÞ + ½D21i + DD21i wðtÞ + ½D22i + DD22i uðtÞ i = 1, 2, . . . , r

ð10Þ

where Mip is the fuzzy set, r is the number of IF–THEN rules and v1(t)!vp(t) are the premise variables. The PDC technique is used to design the controller for the overall system and the total linear system is obtained by use of the weighted sum of the local linear subsystems. The concept of PDC and the overall fuzzy model are shown in Figure 2. The overall fuzzy model is accordingly of the following form (Figure 2(a))

1110

Proc IMechE Part I: J Systems and Control Engineering 226(8) where Ki is the multi-objective H2/HN controller gain. Then, the final T–S fuzzy controller is (Figure 2(b)) uðtÞ =

r X

mj Kj xðtÞ

ð15Þ

j=1

where Kj is the H2/HN controller for every linear subsystem and u(t) is the final controller for the overall system. The closed-loop state-space system from the fuzzy system model, equation (11), with the fuzzy controller, equation (15), is given by Xr Xr   mm Ai + B2i Kj x_ ðtÞ = i=1 j=1 i j   + DAi + DB2i Kj xðtÞ + ½B1i + DB1i wðtÞ, xð0Þ = 0 Xr Xr   z‘ ðtÞ = m m ½ C1i + D12i Kj i=1 j=1 i j   + DC1i + DD12i Kj x(t)

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

x_ ðtÞ =

Xr i=1

mi ðvðtÞÞ½½Ai + DAi xðtÞ

+ ½B1i + DB1i wðtÞ+½B2i + DB2i uðtÞ, xð0Þ = 0 Xr m ðvðtÞÞ½½C1i + DC1i xðtÞ z‘ ðtÞ = i=1 i + ½D11i + DD11i wðtÞ + ½D12i + DD12i uðtÞ Xr m ðvðtÞÞ½½C2i + DC2i xðtÞ z2 ð t Þ = i=1 i + ½D21i + DD21i wðtÞ½D22i DD22i uðtÞ

where vðtÞ = ½ v1 ðtÞ function is

ð11Þ

. . . vp ðtÞ  and the weighting

ˆi ðvðtÞÞ i = 1 ˆi ðvðtÞÞ p Y ˆi ðvðtÞÞ = Mik ðvk ðtÞÞ

mi ðvðtÞÞ = Pr

ð12Þ

k=1

The matrices DAi, DB1i, DB2i, DC1i, DC2i, DD11i, DD12i and DD22i represent the uncertainties in the system and satisfy the following assumptions DAi = FðxðtÞ, tÞH1i ,

DB1i = FðxðtÞ, tÞH2i

DB2i = FðxðtÞ, tÞH3i ;

DC1i = FðxðtÞ, tÞH4i

DD11i = FðxðtÞ, tÞH5i ,

DD12i = FðxðtÞ, tÞH6i

DC2i = FðxðtÞ, tÞH7i ,

DD21i = FðxðtÞ, tÞH8i

DD22i = FðxðtÞ, tÞH9i

ð13Þ

where Hji ; j = 1, . . . , 9 are known matrix functions which characterize the structure of the uncertainties. Furthermore, the following inequality holds: FðxðtÞ, tÞ4∂, ∂ . 0. For a fuzzy controller design, it is supposed that the fuzzy system is locally controllable. Then, the local state-feedback multi-objective H2/HN controller is designed as follows IF v1 ðtÞ is Mi1 and . . . and vp ðtÞ is Mip THEN uðtÞ = Ki xðtÞ, i = 1, 2, . . . , r

ð14Þ

+ ½D11i + DD11i wðtÞ Xr Xr   z2 ð t Þ = mm C2i + D22i Kj i=1 j=1 i j   + DC2i + DD22i Kj xðtÞ + ½D21i + DD21i wðtÞ

ð16Þ

PMSM model dynamics The nonlinear electrical and mechanical equations for the three-phase IPMSM in the d–q reference frame can be written as follows4 dur = vr dt  dvr 3 P0  Bm Cl = Ld  Lq id + ;f iq  vr  2 Jm dt Jm Jm did Rs Lq 1 =  id + P0 vr iq + vd Ld dt Ld Ld diq ;f Ld Rs 1 =  P0 vr  P0 vr id  iq + vq Lq dt Lq Lq Lq

ð17Þ

In equation (17) ur is the angular position of the motor shaft, vr is the angular velocity of the motor shaft, id is the direct current and iq is the quadrature current. ;f is the flux linkage of the permanent magnet, P0 is the number of pole pairs, Rs is the stator windings resistance, Ld and Lq are the direct and quadrature stator inductances, respectively. Jm is the rotor moment of inertia, Bm the viscous damping coefficient and Cl is the load torque. vd is the direct voltage and vq is the quadrature voltage. The electromagnetic torque of the motor can be described as Te =

 3  P0 Ld  Lq id iq + ;f iq 2

ð18Þ

The parameters Rs and Bm are supposed to differ from their nominal values Rs0 and Bm0 .4 The following

Azimi et al.

1111

equation indicates a state-space representation of the synchronous motor x_ 1 = x2 x_ 2 = ðh1 x3 + h2 Þx4 + h3 x2 

Cl Jm

x_ 3 = h4 x3 + h5 x2 x4 + h6 u1 x_ 4 = h7 x2 + h8 x2 x3 + h9 x4 + h10 u2

ð19Þ

In equation (19) x = ½ x1 ½ u1

x2

x3

u 2  T = ½ vd

x4  T = ½ ur

vr

id

iq T

vq  T

ð20Þ

and  3 P0  3 P0 L d  L q , h2 = ;f 2 Jm 2 Jm Bm Rs h3 =  , h4 =  Jm Ld Lq 1 h5 = P 0 , h6 = Ld Ld ;f Ld h7 =  P0 , h8 =  P0 Lq Lq Rs 1 h9 =  , h10 = Lq Lq h1 =

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

ð21Þ

Problem statement and design of the robust tracking controller Proposed structure In this paper the purpose is to design a suitable control which guarantees robust performance in the presence of parameter variations and load disturbance. In this case the control objectives are tracking of the rotor angular position and elimination of reluctance effects and torque ripple (the direct current has to follow a constant reference, zero). Therefore to achieve accurate tracking, the tracking errors (e1, e2) should be minimized. Our goal is to design a state-feedback controller that maintains the RMS gain (HN norm) of these errors below some prescribed value g 0 . 0. In order to obtain this minimization two extra states are augmented to the IPMSM system, as described in the following subsection. The mentioned goals are realized via constructing the objectives z and extra states in an appropriate control loop. Under the above considerations, the fuzzy robust control loop proposed is based on the structure depicted in Figure 3.

Augmented states To guarantee robust performance in the presence of parameter and load torque variations, a suitable control has 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=id has to track a constant reference r2=0. This objective is equivalent to the nonlinear electromagnetic torque being linearized to avoid reluctance effects and torque ripple. Therefore, to achieve these objectives, the outputs of the integrator are considered as extra state variables ðt x5 =

e1 ðdÞdd, e1 = r1  x1

0

ðt x6 =

e2 ðdÞdd, e2 = r2  x3

ð22Þ

0

Consequently the four states in equation (19) are increased to six states with addition of the following x_ 5 = r1  x1 , x_ 6 = r2  x3 xaug = ½ x1

x2

x3

x4

x5

x6 T

ð23Þ

Affine parameter-dependent system Affine parameter-dependent systems are defined as EðrÞx_ = AðrÞx + B1 ðrÞw + B2 ðrÞu z‘ = C1 ðrÞx + d11 ðrÞw + d12 ðrÞu 

z2 = C2 ðrÞx + d12 ðrÞw + d22 ðrÞu   AðrÞ + jEðrÞ BðrÞ A0 + jE0 = C0 CðrÞ D ð rÞ   Ar1 + jEr1 Br1 + r1 + ... Cr1 Dr1   Arn + jErn Brn + rn Crn Drn

B0 D0



1112 B(r) = ½ B1 ðrÞ

Proc IMechE Part I: J Systems and Control Engineering 226(8) B2 ðrÞ  T

CðrÞ = ½ C1 ðrÞ C2 ðrÞ    d ðrÞ d12 ðrÞ DðrÞ = 11 d12 ðrÞ d22 ðrÞ

In this case the exogenous inputs are 2 3 2 3 Cl d w = 4 r1 5 = 4 x1ref 5 x3ref r2

ð24Þ

PMSM T–S fuzzy model ð25Þ

where r and w are the uncertain parameter and external input, respectively. The parameters Rs and Bm and load torque disturbance Cl are supposed to vary (for instance, Rs has high variations due to temperature). The formalization of these variations is declared as follows B1 ðrÞ = B10 + Cl B1Cl 3 2 0 0 0 60 0 07 7 6 60 0 07 7 B10 = 6 60 0 07 7 6 40 1 05 0 0 1 3 2 0 0 0 6 1 0 07 7 6 Jm 6 0 0 07 7 B1Cl = 6 6 0 0 07 7 6 4 0 0 05 0 0 0 3 2 0 0 6 0 0 7 7 6 6 h6 0 7 7 6 B2 = 6 7 6 0 h10 7 4 0 0 5 0 0

First, to linearize the system, the T–S fuzzy model method is utilized. Because two nonlinear elements exist in A0i, four plant rules are assigned v1 ðtÞ = x2 ðtÞ,

v2 ðtÞ = x4 ðtÞ

ð28Þ

in which v1 and v2 are fuzzy variables. To attain membership functions, we should calculate the minimum and maximum values of v1(t) and v2(t)4 x2 2 ½02000,

x4 2 ½012

minv1 ðtÞ = 0,

maxv1 ðtÞ = 2000

minv2 ðtÞ = 0,

maxv2 ðtÞ = 12

ð29Þ

Therefore x2 and x4 can be represented by four membership functions M1, M2, M3 and M4 as follows v1 ðtÞ = x2 ðtÞ = M1 ðv1 ðtÞÞ:2000 + M2 ðv1 ðtÞÞ:0 v2 ðtÞ = x4 ðtÞ = M3 ðv2 ðtÞÞ:12 + M4 ðv2 ðtÞÞ:0

ð30Þ

Also, because M1, M2, M3 and M4 are essentially fuzzy sets, according to fuzzy mathematics M1 ðv1 ðtÞÞ + M2 ðv1 ðtÞÞ = 1 M3 ðv2 ðtÞÞ + M4 ðv2 ðtÞÞ = 1

ð26Þ

ð31Þ

As a result, in accordance with equations (30) and (31) the membership functions can be calculated as v1 v1 , M2 = 1  2000 2000 v2 v2 M3 = , M4 = 1  12 12

M1 =

and AðpÞ = A0i + Bm ABm + Rs ARs 3 2 0 1 0 0 0 0 6 0 0 h1 x 4 h2 0 0 7 7 6 6 0 h 5 x4 0 0 0 07 7 6 A0i = 6 h7 h8 x 2 0 0 0 7 7 6 0 4 1 0 0 0 0 05 0 0 1 0 0 0 3 2 0 0 0 0 0 0 60 1 0 0 0 07 J 7 6 60 0 0 0 0 07 7 6 A Bm = 6 7 60 0 0 0 0 07 40 0 0 0 0 05 0 0 0 0 0 0 3 2 0 0 0 0 0 0 60 0 0 0 0 07 7 6 6 0 0  L1 0 0 07 d 7 A Rs = 6 60 0 0  L1q 0 0 7 7 6 40 0 0 0 0 05 0 0 0 0 0 0

According to equation (27), which emerges from equations (19) and (23), just A0i contains nonlinear parameters. Therefore to linearize the system, the T–S fuzzy model method is utilized.

ð32Þ

The membership functions according to equation (32) are shown in Figure 4. Four plant rules are considered to cover the four local linear subsystems near specific operation points despite the fact that, in the system dynamics, just A0i is varied in each rule. Rule 1 IF v1 ðtÞ is M1 and v2 ðtÞ is M3 THEN A1 ðpÞ = A01 + Bm ABm + Rs ARs

Rule 2 IF v1 ðtÞ is M1 and v2 ðtÞ is M4 THEN A2 ðpÞ = A02 + Bm ABm + Rs ARs ð27Þ

Rule 3 IF v1 ðtÞis M2 and v2 ðtÞ is M3 THEN A3 ðpÞ = A03 + Bm ABm + Rs ARs

Azimi et al.

1113 and external input w. B is defined as in equation (26) and Ai is as represented in equation (27)     x e z‘ = 5 = 1 ð35Þ x6 e2 2 3 2 3 u x1 6 x3 7 6 i d 7 7 6 7 ð36Þ z2 = 6 4 u1 5 = 4 vd 5 vq u2 2 3 2 3 Cl d w = 4 r1 5 = 4 x1ref 5 ð37Þ x3ref r2

Figure 4. (a) The membership functions for M1(v1(t)) and M2(v1(t)); (b) the membership functions for M3(v2(t)) and M4(v2(t)).

Rule 4 IF v1 ðtÞ is M2 and v2 ðtÞ is M4 THEN A4 ðpÞ = A04 + Bm ABm + Rs ARs

ð33Þ

This section focuses on the design of the H2/HN tracking controller for each linear subsystem of the PMSM represented by equation (33). The overall fuzzy model and final fuzzy H2/HN controller based on equations (11) and (15) are achieved by the T–S fuzzy defuzzification process and defined by equations (38) and (39), respectively. Finally, the closed-loop state-space system from the fuzzy system model, equation (38), with the controller, equation (39), and by the use of the proposed control structure (Figure 3), is given by equation (40) X4 Afuzzy = m ½A0i + Bm ABm + Rs ARs  ð38Þ i=1 i X4 mK ð39Þ Kfuzzy = i=1 i i and

H2/HN controller design and closed-loop system The concept of PDC is utilized with equation (33) to design a fuzzy state-feedback controller. The fuzzy controller shares the same fuzzy sets with the fuzzy system, so the ith control rule is as follows. Rule 1 IF v1 ðtÞ is M1 and v2 ðtÞ is M3 THEN uðtÞ = K1 xðtÞ

Rule 2 IF v1 ðtÞ is M1 and v2 ðtÞ is M4 THEN uðtÞ = K2 xðtÞ

Rule 3 IF v1 ðtÞ is M2 and v2 ðtÞ is M3 THEN uðtÞ = K3 xðtÞ

Rule 4 IF v1 ðtÞ is M2 and v2 ðtÞ is M4 THEN uðtÞ = K4 xðtÞ

ð34Þ

where Ki, i = 1, 2, . . . , r are the local H2/HN controller gains to be determined. According to equations (26) and (27) just A0i varies and ABm , ARs , B, C and D are constant in each rule. B, C and D are obtained based on objectives z2 and zN

 Bfuzzy = B1fuzzy B2fuzzy X4 X4 B1fuzzy = m ½B + C B  B = mB 1 l 1 2 i 0 C fuzzy l i=1 i=1 i 2 Aclose loop = Afuzzy + B2fuzzy Kfuzzy

ð40Þ

In the control structure, which is shown in Figure 3, first the original system, equation (17), is approximated with some local linear models so that each rule is represented by the T–S fuzzy approach. Then, in order to obtain accurate tracking of tool position and direct current, two extra states are augmented to the T–S fuzzy model and build up the augmented plant. Separate controllers for each linear subplant based on the LMI approach are designed. After that the total system (P) is obtained by using the weighted sum of the local linear subsystems and it is utilized instead of the original nonlinear system. So according to the PDC approach, the control law of the whole system (K) is the weighted sum of the local feedback control of all subsystems. Total system (P) and whole controller (K) have been shown in both Figures 2 and 3. As can be observed in Figure 2, in order to construct P and K based on equations (11) and (15), a way is needed to calculate and update the fuzzy weights given in equation (12). In the present work the fuzzy weights mi are updated by using a fuzzy weights online computation (FWOC) component that has been designed before. The FWOC component includes three blocks that may be explained as follows.

1114 1.

2.

3.

Proc IMechE Part I: J Systems and Control Engineering 226(8) 2

0 6 0 6 6 0 A02 = 6 6 0 6 4 1 0 2 0 6 0 6 6 0 A03 = 6 6 0 6 4 1 0 2 0 6 0 6 6 0 A04 = 6 6 0 6 4 1 0 2 0 0 60 0 6 61 0 C=6 60 0 6 40 0 0 0 2 0 0 60 0 6 60 0 D=6 60 0 6 40 0 0 0

In the first block, values of the angular velocity of the motor shaft x2=vr and the quadrature current x4=iq are measured from the IPMSM system in real time and nonlinear terms (fuzzy variables: v1=x2, v2=x4) are built, equation (28). In the second block, the values of membership functions Mi mentioned in fact of the nonlinearities are calculated, equation (32). In the third block, new fuzzy weights, equation (12), are calculated and they are sent to the main control structure (Figures 2 and 3).

Finally by using the whole system and global controller, a tracking loop (Figure 3) is applied to the system in order to achieve desired specifications such as tracking performance, bandwidth, disturbance rejection and robustness for the closed-loop system.

Simulation results The motor type used in this paper is the 130-750MSZK-L2. This IPMSM is three-phase, four-pole, with 0.75 HP (horse power) and with 2000 r/min rated speed. The maximum voltage and the continuous rated armature current are set to 230 V and 12 A.4 The parameters of the IPMSM are shown in Table 1. The stator windings resistance Rs and the viscous damping coefficient Bm are varied between 650% and the load torque disturbance is unknown.4,10 According to the approach outlined above in subsections ‘‘Affine parameter-dependent system’’ and ‘‘PMSM T–S fuzzy model’’, the local linear model matrices for the nonlinear augmented states of the IPMSM at the ith selected operating point are obtained as follows 2

0 6 0 6 6 0 A01 = 6 6 0 6 4 1 0

1 0 0 1145 49 0 20 82119 0 0 0 1

0 1860 0 0 0 0

0 0 0 0 0 0

1.

2. 3.

Table 1. Parameters of the IPMSM. Parameter

Value

Rs Bm0 (without load) Bm0 (with load) Ld Lq ;f Jm0 (without load) Jm0 (with load) P0

1.9 O 0.03 Nms/rad 0.0341 Nms/rad 0.0151 H 0.031 H 0.31 Vs/rad 0.0005 kg.m2 0.0227 kg.m2 2

0 1860 0 0 0 0

0 0 0 0 0 0

3 0 07 7 07 7 07 7 05 0 3

1 0 0 0 0 0 1145 1860 0 0 7 7 49 0 0 0 07 7 20 0 0 0 07 7 0 0 0 0 05 0 1 0 0 0 3 1 0 0 0 0 0 0 1860 0 0 7 7 0 0 0 0 07 7 20 0 0 0 07 7 0 0 0 0 05 0 1 0 0 0 3 0 0 1 0 0 0 0 17 7 0 0 0 07 7 1 0 0 07 7 0 0 0 05 0 0 0 0 3 0 0 0 0 0 07 7 0 0 07 7 0 0 07 7 0 1 05 0 0 1

Next, in order to design state-space feedback gains Ki for each subsystem, the following steps are undertaken.

3

0 07 7 07 7 07 7 05 0

1 0 0 0 0 0 20 82119 0 0 0 1

Specify the LMI region, equation (6), in order to place the closed-loop poles in this region (pole placement) and also to guarantee some minimum decay rate and closed-loop damping. The mentioned region is shown in Figure 5, as the intersection of the half-plane x \ 25 and of the sector centered at the origin and with inner angle 2p/3. Choose a four-entry vector specifying the H2/HN cost function, equation (1): [g0 n0 a b]=[0 0 1 1]. Minimize the H2/HN cost function, equation (1), subject to the mentioned pole placement constraint by using equations (4), (5), (8) and (9).

The local H2/HN control gain matrices Ki, the final fuzzy H2/HN controller matrix Kfuzzy and the overall fuzzy model matrix Afuzzy are obtained as follows 2

0:0033 6 0 6 6 4 6 0:0097 K1 = 10 6 6 0:0002 4 0:0386 0:1467

3T 0:0011 7 0 7 0:0002 7 7 0:0001 7 7 0:0009 5 3:8313

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Figure 5. The pole placement region.

3T 0:0004 0:0012 7 6 0 0 7 6 7 6 0:0049 0:0001 46 7 K2 = 10 6 7 0:0001 0:0001 7 6 4 0:0051 0:0002 5 0:0177 4:3223 3T 2 0:3078 0:1441 6 0:0009 0:0025 7 7 6 6 0:0157 0:0104 7 7 K3 = 103 6 6 0:0252 0:0210 7 7 6 4 3:3985 2:5003 5 0:0336 0:0940 3T 2 0 65:1757 6 0 0:5657 7 7 6 7 6 0:8028 0 7 K4 = 100 6 6 0 14:7752 7 7 6 4 0 774:3000 5 59:1158 0 2 0 1 0 0 6 0 60 410 1858 6 6 0 54 126 0 Afuzzy = 6 6 0 20 68405 61 6 4 1 0 0 0 0 0 1 0 3T 2 0:0037 0:0027 7 6 0 0 7 6 6 0:0055 0:0002 7 7 Kfuzzy = 104 6 6 0:0003 0:0003 7 7 6 4 0:0417 0:0257 5 0:0312 3:4662 2

0 0 0 0 0 0

3 0 07 7 07 7 07 7 05 0

To calculate the overall controller use is made of the PDC technique and the whole system is obtained by using a weighted average defuzzifer. The above proposed T–S fuzzy model and controller can exactly represent the nonlinear system in the region

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

½0, 12A3½0, 2000 r=min on the x2–x4 space for various operating points. Figure 6 compares system states for both the proposed H2/HN model and the original nonlinear system, which is presented in Lin et al.4 As is evident from Figure 4, the time responses of the proposed T–S fuzzy model exactly follow the responses of the nonlinear differential equations (17), which means that the fuzzy model can exactly represent the original system in the pre-specified domains. These time responses of states are influenced by step inputs which are shown in Figure 6. In actuality, the motor is used to convert electrical energy into mechanical energy. Accordingly, an external load is added to the drive system. The external load

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

Figure 8. d-Axis current.

Figure 9. Comparison of transient responses of the proposed H2/HN controller and the feedback linearization controller: (a) with step load torque; (b) with benchmark load torque.

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

Figure 10. Comparison of responses of the proposed H2/HN controller and the feedback linearization controller with rectangular position command.

1 kg weight is put at a certain location from the motor; as a result, the weight can provide the external load of 1 Nm that is proposed with step input. Figure 8 shows the real direct current response that tracks 0 A in order to eliminate reluctance effects and torque ripple. Figures 9 and 10 show the responses of the certain step position (180°) and the rectangular position commands, respectively, for the proposed H2/HN controller and the feedback linearization controller (presented in Lin et al.4). Although Figure 9(a) is influenced by step load torque and measurement noise, Figure 9(b) shows this comparison at 180°, noise and benchmark external load (Figure 7). In Figures 9 and 10, the notation used is as follows: Ts is settling time, where Ts1 and Ts2 are the settling times of tracking responses when the proposed H2/HN and feedback linearization controller is used, respectively. According to Figures 9 and 10, the settling time and rise time for the proposed H2/HN controller are better than for the feedback linearization controller. Table 2 also summarizes the results of transient responses for the two mentioned methods. Referring to Table 2, the proposed H2/HN method has smaller settling time value than the feedback linearization method on position tracking responses. Figure 11(a) and (b) demonstrates the disturbance rejection on angular position with the two different methods, with step load torque and benchmark load torque (Figure 7), respectively, although first the IPMSM is controlled to reach a fixed position, 180°. As a matter of fact, Figure 11(a) and (b) is a zoom of

Table 2. Comparison of disturbance rejection and tracking responses in the two methods. Method

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

Emax (°) Ep-p (°) Ts (s)

Proposed H2/HN control

Feedback linearization control4

0.20 0.75 0.28

1.1 4.2 0.4

Azimi et al.

1117 methods, from which it is observed that the peak and peak-to-peak values in the proposed H2/HN method are smaller than those in the feedback linearization technique on disturbance rejection responses. Figure 12 illustrates the position responses when the parameters of the stator windings resistance Rs and the viscous damping coefficient Bm are varied between 650%. As can be seen, the system has good robustness when the parameters in the dynamic systems are varied over a wide range.

Conclusions

Figure 11. Comparison of the proposed H2/HN controller and the feedback linearization controller for disturbance rejection on angular position: (a) with step load torque (1 Nm); (b) with benchmark load torque (Figure 7).

In this paper, a robust H2/HN position and direct current tracking controller has been designed for a nonlinear, MIMO and uncertain PMSM system. First, in order to improve tracking control, some states (error of tracking) were augmented to the system. Then, to approximate uncertain nonlinear systems, the T–S fuzzy linear model was employed. After that, based on each linear model with augmented state, an H2/HN multi-objective state feedback controller was developed to achieve the robustness design of nonlinear uncertain systems. Finally by using the PDC approach the overall fuzzy controller was calculated. Simulation results on a three-phase IPMSM showed that the robust proposed position and current control system has small position and current tracking error, desired robustness against load torque disturbance and parameter variations and good transient responses, load disturbance responses and tracking responses. Funding

Figure 12. Angular position responses of the proposed H2/HN controller with varying J and B.

Figure 9(a) and (b), between t=0.6 s and t = 1.2 s and t = 0.5 s and t = 1.8 s, respectively. The notation used in Figure 11(a) and (b) is as follows: Emax is the peak error and Ep-p is the peak-to-peak error, where E1max and E1p-p are the peak and peak-topeak errors when step and benchmark disturbances (Figure 7) respectively act on the IPMSM and proposed H2/HN controller is used; and E2max and E2pp are the peak and peak-to-peak errors, when step and benchmark disturbances (Fig. 7) respectively act on the IPMSM and the feedback linearization controller (presented in Lin et al.4) is used. Figure 11(a) and (b) shows the comparison of the proposed H2/HN method and the feedback linearization technique in the presence of disturbances. As can be seen, the proposed H2/HN method has better disturbance attenuation under both types of external load. Table 2 summarizes the disturbance rejection differences (values of Emax, Ep-p and Ts (sec)) in the two

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