Robust guaranteed cost output feedback control for uncertain discrete fuzzy systems with state and i

Scientific Journal of Control Engineering June 2013, Volume 3, Issue 3, PP.94-105

Robust Guaranteed Cost Output Feedback Control for Uncertain Discrete Fuzzy Systems with State and Input Delays Xiaona Song 1, 2#, Jinchan Wang 1 1. Electronic and Information Engineering College, Henan University of Science and Technology, Luoyang 471023, China 2. China Airborne Missile Academy, Luoyang 471009, China #

Email: xiaona_97@163.com

Abstract This paper investigates the problem of robust guaranteed cost output feedback control for a class of uncertain discrete fuzzy systems with both discrete and input delays. The system is described by a state-space Takagi-Sugeno (T-S) fuzzy model with input delays and norm-bounded parameter uncertainties. The aim is to design a piecewise output feedback controller which ensures the robust asymptotic stability and minimizes the guaranteed cost of the closed-loop uncertain system. In terms of linear matrix inequalities, a sufficient condition for the solvability of this problem is presented.

Keywords: Robust Guaranteed Cost Control; Output Feedback; Input Delays; Discrete T-S Fuzzy Models

1 INTRODUCTION In recent years, fuzzy systems of the Takagi-Sugeno (T-S) model have attracted considerable attention from scientists [19, 21]. The T-S fuzzy system [20, 26] is one of the most popular fuzzy system models in the model-based fuzzy control. T-S fuzzy models are nonlinear systems described by a set of IF-THEN rules; it has been shown that T-S fuzzy models could approximate any smooth nonlinear function to any specified accuracy within any compact set. Thus it is expected that T-S fuzzy systems can be used to represent a large class of nonlinear systems. Therefore, many stability and control issues related to the T-S fuzzy systems have been studied in the past two decades; see, e.g., [1, 24, 30], and the references cited therein.

On the other hand, time delays are frequently encountered in many practical engineering systems, such as chemical processes, long transmission lines in pneumatic systems [11]. It has been shown that the presence of a time delay in a dynamical system is often a primary source of instability and performance degradation [6, 13]. Therefore, time delay systems have been an attractive research topic in the past years. However, most of the articles are for the state delayed systems and only a few are special for the uncertain systems with both state and input delays. In [5, 12], the robust stabilization of uncertain systems with state and input delays has been attempted in the past by solving the Riccati or Lyapunov-equation. In order to overcome the shortcomings of the Riccati or Lyapunov-equation, robust stabilization methods of uncertain systems with state and input delay are developed based on linear matrix inequalities (LMIs) [18, 31, 32], and the guaranteed cost control problem for uncertain systems with state and input delay has been addressed in [25]. For T-S fuzzy systems with state and input delay, via different approaches, the authors in [14, 15], [3] and [27] have investigated the stabilization, guaranteed cost controller design and robust H∞ controller design problem, respectively.

Recently, guaranteed cost control has attracted lots of attention among control community, because this approach has the advantage of providing an upper bound on a given performance index and thus the system performance degradation is guaranteed to be less than this bound, therefore many authors have researched the guaranteed cost control problem. For example, guaranteed cost control results for uncertain systems with delay has been considered for continuous-time systems in [4, 8, 16, 28] and for discrete time systems in [4, 10, 29]. However, many papers - 94 http://www.sj-ce.org/

dealt with a state feedback control design that requires all state variables are available. In many cases, this condition is too restrictive. So it is meaningful to control a system via output feedback controllers. Very recently, there are many authors investigating the problem of guaranteed cost control for T-S fuzzy systems [3, 9]. It is worth noting that the results in [3] were obtained in the context of continuous fuzzy systems with state and input delays and parameter uncertainties. However, the problem of guaranteed cost output feedback control for uncertain discrete fuzzy systems with state and input delays is still open and remains unsolved, which motivates the present study.

In this paper, we consider the robust guaranteed cost output feedback control problem for discrete fuzzy systems with state and input delays. The system to be considered is described by a state-space T-S fuzzy model with input delays and norm-bounded parameter uncertainties. The input delays are assumed to appear in the state equation, and the uncertainties are allowed to be time-varying but norm bounded. The aim is to design a piecewise output feedback controller such that the resulting closed-loop system is robustly asymptotically stable while a desired cost performance can be guaranteed. A sufficient condition for the solvability of this problem is proposed in terms of LMIs, which can be implemented by the cone complementary linearization method in [7]. When these LMIs are feasible, an explicit expression of a desired output feedback controller is also given.

Notation: Throughout this paper, for real symmetric matrices X and Y , the notation X  Y (respectively, X  Y ) means that the matrix X  Y is positive semidefinite (respectively, positive definite). tr (M ) denotes the trace of T matrix M . I is an identity matrix with appropriate dimension. The notation M represents the transpose of the matrix M. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

2 MAIN RESULTS The discrete T-S fuzzy dynamic model is described by fuzzy IF-THEN rules, which locally represent linear inputoutput relations of nonlinear systems. A discrete T-S fuzzy model with state and input delays and parameter uncertainties can be described by Plant Rule i: IF s1 (t ) is  i1 and

and s p (t ) is  ip , then

x(t  1)  [ Ai  Ai (t )]x(t )  [ A1i  A1i (t )]x(t   )  [ Bi  Bi (t )]u (t )  [ B1i  B1i (t )]u (t   ), y (t )  Ci x(t ),

(1)

x( s)   (t ), t  [ , 0], i  1, 2,

, r,

where  ij is the fuzzy set and r is the number of IF-THEN rules; s1 (t ) ,…, s p (t ) are the premise variables. n Throughout this paper, it is assumed that the premise variables do not depend on control variables; x(t )  R is the m s state; u (t )  R is the control input; y(t )  R is the measured output;   0 and   0 are integers representing the time delay of the fuzzy systems;   max( , ) ; Ai , A1i , Bi , B1i , Ci are known real constant matrices; Ai (t ) , A1i (t ) and Bi (t ) , B1i (t ) are real-valued unknown matrices representing time-varying parameter uncertainties, and are assumed to be of the form

 Ai (t )

A1i (t ) Bi (t ) B1i (t )   M i Fi (t )  N i

N1i

N 2i

N 3i  ,

(2)

where M i , Ni , N1i , N 2i , N3i are known real constant matrices, and Fi (t ) is an unknown matrix function with Fi T (t ) Fi (t )  I .

Then the final ourpur of the fuzzy system is inferred as follows: r

x(t  1)   hi ( s(t )) [ Ai  Ai (t )]x(t )  [ A1i  A1i (t )]x(t   ) i 1

 [ Bi  Bi (t )]u (t )  [ B1i  B1i (t )]u (t   ) ,

r

y (t )   hi ( s(t ))Ci x(t ), i 1

where - 95 http://www.sj-ce.org/

(3)

hi ( s(t )) 

s p (t )], in which ij ( s j (t )) is the grade of membership of s j (t ) in ij . Then, it can

s(t )  [ s1 (t ) s2 (t ) be seen that

p i ( s(t )) ,  ( s ( t ))  ij (s j (t )),  i r  ( s ( t )) j  1 i1 i

r

 (s(t ))  0, for all t , h (s(t ))  0,

i (s(t ))  0, i  1, , r ,

i

i

i 1

i  1,

, r,

r

 h (s(t ))  1. i 1

i

Denoting the state space partition as Si iL  R and L as the set of subspace indexes, we can write the dynamic as: n



x(t  1) 

kK ( i )

hk ( s (t )) [ Ak  Ak (t )]x(t )  [ A1k  A1k (t )]x(t   )

 [ Bk  Bk (t )]u (t )  [ B1k  B1k (t )]u (t   ) , y (t ) 



kK ( i )

 Ak (t )

(4)

hk ( s (t ))Ck x(t ), s (t )  Si ,

A1k (t ) Bk (t ) B1k (t )   M k Fk (t )  N k

N1k

N 3k  , k  1, 2,

N 2k

, r,

Fk (t )T Fk (t )  I , t. where 0  hk ( s(t ))  1 and



kK ( i )

hk ( s(t ))  1. For each subspace S i , the set K (i), K (i)  1, 2, , q(i) , contains

the indexes for the system matrices used in the interpolation within that subspace. For operating subspaces, K (i ) contains a single element.

As a performance measure for fuzzy system (5), the cost function is written as 

J  [ xT (t )Q1 x(t )  uT (t ) R1u(t )] t 0

where Q1 and R1 are given positive-definite symmetric matrices.

Now, we consider the following piecewise discrete-time output feedback controller

xc (t  1)  Aci xc (t )  Bci y(t ), xc ( )  xc ( )  xc (0)  0, u(t )  Cci xc (t ), i  L.

(5)

where xc (t )  R is the controller state; Aci , Bci and Cci are matrices to be determined later. n

Form (4)-(5), the closed-loop system can be obtained as

x (t  1) 



kK ( i )

hk ( s(t ))[ Aki x (t )  A1k x (t   )  Bki x (t   )], (6)

x   , x   , x0  0 , where

x (t )   x(t )T

xc (t )T  ,    x T T

T

0 ,    x T

T

0 , 0   x0T

T

0  ,

and

Ak  Ak  Ak (t ), A1k  A1k  A1k (t ), Bk  Bk  Bk (t ), B1k  B1k  B1k (t ),

 A Aki   k  Bci Ck

 A1k Bk Cci   , A1k   Aci   0

 B1k Cci 0  , Bki   0  0

0 .

0

Then, the performance measure for fuzzy system (6) can be written as 



t 0

t 0

J  [ xT (t )Qx(t )  uT (t ) Ru(t )]  x T (t )CciT QCci x (t ), - 96 http://www.sj-ce.org/

(7)

where

Q I 0  Cci   , Q   0 Cci  0

0 .

R

In this section, an LMI approach will be developed to solve the problem of robust output feedback guaranteed cost control of uncertain fuzzy systems with state and input delays formulated in the previous section. We first give the following results which will be used in the proof of our main results.

Lemma 1 [22]. Let A, D, S ,W and F be real matrices of appropriate dimensions with W  0 and F satisfying F T F  I . Then we have the following: 1) For any scalar   0 and vectors X , Y  R , n

2 X T DFSY   1 X T DDT X  Y T S T SY .

2) For any scalar   0 such that W   DD  0, T

( A  DFS )T W 1 ( A  DFS )  AT (W   DDT )1 A   1S T S .

Lemma 2 [2]. If P is a positive definite matrix and matrices A and B are of appropriate dimensions such that AT PA  P  0 and BT PB  P  0 , then we have

AT PA  BT PB  2P  0 .

Lemma 3 [23]. Given any matrices X, Y and Z with appropriate dimensions such that Y > 0. Then, we have

X T Z  Z T X  X T YX  Z TY 1Z. Then, we come to the stability result.

Theorem 1. Consider the discrete time fuzzy system (6) If there exist matrix G  0 , and scalar  such that the following matrix inequalities are satisfied

 G  Q  Z  0   0   Aki G  NTG k   Cci G  0 

*

*

*

*

*

Q 0 A1k G N1Tk G 0

* Z Bki G N 3Tk G 0

* * G 0 0

* * *  0

* * * * Q 1

0

0

 M kT

0

0

*  * *  *   0, *  *  

where

Q  GT QG, Z  GT ZG,  A Aki   k  Bci Ck

Bk Cci   A1k , A1k    Aci   0

0  B1k Cci B  , ki  0 0 

0 , 0 

and

M  M k   k  , Nk   Nk  0 

N 2 k Cci  , N1k   N1k

0 , N 3k   N 3k Cci

0 .

Then, the closed-loop system (6) is asymptotically stable and the cost function (7) satisfies the following bound: 1

1

l 

l 

J  x T (0)G 1 x (0)   x T (l )Qx (l )   x T (l )Zx (l ).

Proof: As for discrete time Lyapunov function candidate, we consider the function of the form - 97 http://www.sj-ce.org/

(8)

t 1

V (t )  x T (t ) Px (t ) 



x T ( j )Qx ( j ) 

j t 

t 1

 x

T

( j ) Zx ( j ).

j t 

Then along trajectories of the system (6), we have

V (t )  x T (t  1) Px (t  1)  x T (t ) Px (t )  x T (t )Qx (t )  x T (t )Zx (t )  x T (t   )Qx (t   )  x T (t   ) Zx (t   ) 

 

kK ( i ) mK ( i )

hk ( s (t ))hm ( s (t ))[ Aki x (t )  A1k x (t   )  Bki x (t   )]T

 P[ Aki x (t )  A1k x (t   )  Bki x (t   )]  x T (t ) Px (t )  x T (t )Qx (t )  x T (t ) Zx (t )  x T (t   )Qx (t   )  x T (t   )Zx (t   ) q (i )

  hk2 ( s (t ))e(t )T {[ Abki  M k Fk N k ]T P[ Abki  M k Fk N k ]  Z c }e(t ) k 1

q (i )

  hk ( s (t ))hm ( s (t ))e(t )T {[ Abki  M k Fk N k ]T P[ Abmi  M m Fm N m ] k m

 [ Abmi  M m Fm N m ]T P[ Abki  M k Fk N k ]  2Z c }e(t ), where T

x (t   )T  , Abki   Aki A1k Bki  , 0   P  Q  Z 0  0 Q 0  .

N3k  , Z c    0 0  Z 

x (t   )T

e(t )   x (t )T

Nk   Nk

N1k

Then,

V  x T (t )CciT QCci x (t ) q (i )

  hk2 ( s (t ))e(t )T {[ Abki  M k Fk N k ]T P[ Abki  M k Fk N k ]  Z c }e(t ) k 1

q (i )

  hk ( s (t ))hm ( s (t ))e(t )T {[ Abki  M k Fk N k ]T P[ Abmi  M m Fm N m ] k m

 [ Abmi  M m Fm N m ]T P[ Abki  M k Fk N k ]  2Z c }e(t ), where

  P  Q  Z  CciT QCci  Zc   0  0 

0 0   Q 0  , 0  Z 

with G  P1 , it follows from the Schur complement that (8) is equivalent to T Abki ( P 1   M k M kT )1 Abki   1 N kT N k ]  Z c  0.

Now, by Lemma 1, it can be shown that

[ Abki  M k Fk N k ]T P[ Abki  M k Fk N k ]  Z c  0.

According to Lemma 2, we have

[ Abki  M k Fk N k ]T P[ Abmi  M m Fm N m ]  [ Abmi  M m Fm N m ]T P[ Abki  M k Fk N k ]  2Z c  0.

Thus, we can get that V   x (t )Cci QCci x (t )  0 , for all e(t )  0.

Therefore, the closed-loop system (6) is asymptotically stable. Furthermore, we obtain T

T







t 0

t 0

t 0

J  [ xT (t )Qx(t )  uT (t ) Ru(t )]  x T (t )CciT QCci x (t )   V , - 98 http://www.sj-ce.org/

for any nonzero initial state x0  Sr 0. This completes the proof.

Now, we are in a position to present a solution to the robust output feedback guaranteed cost controllers by means of a system of matrix inequalities.

Theorem 2: Consider the uncertain discrete fuzzy delay system (6) and cost function (7), if there exist symmetric matrices X , Y , matrices M , N , Aci , Bci , Cci and scalar   0 , such that the following matrix inequalities hold:

 Y  I 

I   0,  X 

(9)

and

 1  0   0   J1ki  J 4 ki   J 7i  0   J9  J  9  J10 k   0

* * Q * 0  2 J 2 k J 3ki J 5 k J 6 ki 0 0 0 0 0 0 0 0 0 0 0 J12i

* * * 1 0 0 J 8k 0 0 0 J13k

* * * * 3 0 0 0 0 J11i 0

* * * * *  4 0 0 0 0 0

* * * * * *  I 0 0 0 0

* * * * * * * Q 1 0 0 0

* * * * * * * *  Z 1 0 0

* * * * * * * * * I 0

* *  *  * *  *   0, *  * *  *  I 

(10)

where

 Q 1 0   NN T 0   N 0  N T 0 X I    Z     , , , , (11) 1       2 3 4 0 I   1   NN T     0 I  I Y  0  0 R  0  XAk  Bci Ck   NN kT NN kT Y   XA1k 0   0 Aci J1ki    , J 2k    , (12)  , J 3ki   B C 0  , J 4 ki   T T 0  Ak AkY  Bk Cci   A1k 0   1k ci  Cci N 2 k  CciT N 3Tk N T 0   NN T 0  I Y  I Y  T T J 5 k   1k J  , J 8k   M k X M k  , J 9   , (13)  , J 7i    , 6 ki   T 0 0 0  0 Cci  0 N   0  0  0 Cci   B1Tk X 0  0  0 0 J10 k   J  J  J  , , , (14)  12i  ,  11i   13k  T 0 0 N 2 k Y  Cci 0   0 0 0  Aci  XAk Y  Bci Ck Y  XBk Cci  MAci N T , Bci  MBci , Cci  Cci N T , MN T  I  XY .

(15)

Then, (5) is a guaranteed cost control law and

J *  xT q1 x  xT z1 x  x0T Xx0 ,

(16)

where

I  I  0 Q   , z1   I 0 z   , 0  0 for any nonzero initial state x  Sr 0 , x  Sr 0 , x0  Sr 0 , is a guaranteed cost for the uncertain system. q1   I

Proof: Applying the Schur complement formula to (10) and by Lemma 3 results in a new inequality, then pre- and 1 1 1 post- multiplying the obtained matrix inequalities by diag{I , I , I , I , N , I , I , I , N , N , I , , I } and diag{I , I , I , I , N T , I , I , I , N T , N T , I , , I } , respectively, then we have - 99 http://www.sj-ce.org/

 1  0   0   J1ki  J 4 ki   J 7i  0   J9  J  9

* Q 0 J 2k

* * Z J 3ki

* * * 1

* * * *

* * * *

* * * *

* * * *

J 5k 0 0 0 0

J 6 ki 0 0 0 0

0 0 J 8k 0 0

 I 0 0 0 0

*  4 0 0 0

* *  I 0 0

* * * Q 1 0

      *   0,  *  *   *   Z 1  * * * *

(17)

where

0  N kT  N1Tk 0  CciT N 3Tk 0  N kT Y  J 3ki , J  , J   , J  .

0  4 ki CciT N 2Tk CciT N 2Tk Y  5 k  0 0  6 ki  0 0 Now, from (9), it is easy to see I  XY is nonsingular. Therefore, there always exits nonsingular matrices M and N such that (15) holds. Now we introduce the following nonsingular matrices  XB C   1k ci  B1k Cci

 X 1   T M

1

I I Y  , 2   .  T 0 0 N 

Let G   21 , then by some calculation, we have

1

1

where   M X (Y  X ) XM

T

Y G T N

 0.

N ,  

Thus we have G  0. The matrix inequalities in (17) can be rewritten, then by the Schur complement fornula, we have

 1T G1   T2 Q 2   T2 Z  2  0   0  T 1 Aki  2   N kT  2  Cci  2   0 

* Q 0 T 1 A1k N1Tk

     (18)  0,   0 0 0 0 Q 1 *  0 0  M kT 1 0 0  I  T T T T Now, pre- and post- multiplying the matrix inequalities in the above by diag{1 , G , G , 1 , I , I , I } and 1 1 diag{1 , G, G, 1 , I , I , I } , respectively, then we have  G  Q  Z  0   0   Aki G  NTG k   Cci G  0 

*

*

*

*

*

Q 0 A1k G N1Tk G 0

* Z Bki G N 3Tk G 0

* * G 0 0

* * *  I 0

* * * * Q 1

0

0

 M kT

0

0

* * Z 1T Bki N 3Tk

* * * * * * T 1 G1 * 0  I

*   *  *   * 0. *   *   I 

Finally, by Theorem 1, the desired result follows immediately. This completes the proof.

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* * * * *

* * * * *

Remark 1. Note that bound (16) obtained in Theorem 2 depends on the initial condition of system (1). To remove this dependence on the initial condition, we will assume that the initial state of system (1) is arbitrary but belongs to n T the set   {x(i )  R : x(i )  U i , i i  1, i   , ,0}, where U is a given matrix. The cost bound (16) then leads to

J  max {U T XU }  max {U T q1U }  max {U T z1U }.

Remark 2 . Theorem 2 provides a sufficient condition for the solvability of the guaranteed cost output feedback control problem for uncertain fuzzy systems with input delays. It is worth pointing out that the matrix inequality in 1 1 Theorem 2 is not an LMI because of the terms Q and Z . In order to solve this non-convex problem, we propose the following non-linear minimization problem involving LMI conditions minimise tr (QQ  ZZ ) subject to

 1  0   0   J1ki  J 4 ki   J 7i  0   J9   J9 J  10 k  0

*

*

*

*

*

*

*

*

*

Q

*

*

*

*

*

*

*

*

0 J 2k

 2 J 3ki

* 1

* *

* *

* *

* *

* *

* *

J 5k

J 6 ki

0

3

*

*

*

*

*

0 0

0 0

0 J 8k

0 0

 4 0

*  I

* *

* *

* *

0

0

0

0

0

0

Q

*

*

0

0

0

0

0

0

0

Z

*

0 0

0 J12i

0

J11i 0

0 0

0 0

0 0

0 0

I 0

J13k

* *  *  * *  *   0, *  *  * *   I 

(19)

and

where  j , j  1,

Z I  Q I   Y  I  (20)   0, Q  0, Z  0, Q  0, Z  0,   0,    I  X   0,  I Z I Q       , 4 and J1ki , J 2k , J 3ki , J 4ki , J 5k , J 6ki , J 7i , J8k , J 9 , J10k , J11i , J12i , J13k are given in (12)-(14).

If the solution of the above minimization problem is 4n, then, by Theorem 2, it can be seen that the guaranteed cost control problem is solvable and a desired guaranteed cost piecewise discrete-time output feedback controller can be obtained as in (5). Then the proposed non-linear minimization problem can be solved by the cone complementary linearization method in [7].

Remark 3. Based on Theorem 2, the following algorithm can be developed to get the output feedback controllers.

Algorithm 1: Step 1. Fixing the matrices Ai , A1i , Bi , B1i , Ci and M i , Ni , N1i , N 2i , N3i , i  1, 2 , then solving the LMI (19)-(20). If QQ  I and ZZ  I are satisfied, then go to step 2. Otherwise, go on step 1.

Step 2. If the solutions to X , Y , Aki , Bci , Cci are found in step 1, then by the given N, we can determine M to satisfy (15), then go to step 3. Otherwise, go to step 1.

Step 3. Using matrices X, Y, M, N obtained in step 2, then solving the LMI (19)-(20) again. If QQ  I and ZZ  I are satisfied, then go to step 4. Otherwise, go to step 1.

Step 4. In step 3, if the solutions to Aci , Bci , Cci are found, then guaranteed cost output feedback controller for each subspace can be obtained. Otherwise, go to step 1.

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3 SIMULATION EXAMPLE The uncertain discrete fuzzy system with input delays considered in this example is with two rules: Plant Rule i: IF s1 (t ) is  i1 , then

x(t  1)  [ A1  A1 (t )]x(t )  [ A11  A11 (t )]x(t   )  [ B1  B1 (t )]u (t )  [ B11  B11 (t )]u (t   ), y(t )  C1 x(t ), i  1, 2,

where

0 0.5 0 0   1  0.2 0.3     A1   0.05 0.8 0  , A11   0 0.1 0.1  , B1   0.1   0  0  0 0.3 0.1  0 0.2  0  0.2 0 0.5  0.8  2 0.2 0       0  , A12   0 0.1 C1   0.5 5 0  , A2  0.05 0.8  0 0 0.3 0.1   0 0.2 0 0  0 0.1  1 0.2 0   0   B12   0.3 0.1 0.2  , C2   0.5 5 0  , 0.2 0 0   0 0.2 0.1

0 0.6  0 0.1  0.1   0.2 0  , B11   0.3 0.1 0.2  ,  0 0.2 0.1 0.1 0.3 0   0.3 0 0.6   0.1  , B2   0.1 0.2 0  ,   0.2  0.1 0.3   0

and Ai (t ) , A1i (t ) , Bi (t ) , B1i (t ) (i=1,2) can be represented in the form of (2) with

M1  0.1 0 0.2 , N1  0.2 , N11  0 , N21  0.4 , N31  0.2 , M 2  M1 , N 2  N1 , N12  N11 , T

N 22  N 21 , N32  N31 .

The membership function and partition of subspace are defined as in Fig. 1.

FIG. 1 MEMBERSHIP FUNCTION AND PARTITION OF SUBSPACES

Si , i  1, 2,3

Then the final output of the fuzzy system is as inferred as follows: 2

x(t  1)   hk ( s(t )) [ Ak  Ak (t )]x(t )  [ A1k  A1k (t )]x(t   )  [ Bk  Bk (t )]u (t )  [ B1k  B1k (t )]u (t   ) , k 1

2

y (t )   hk ( s(t ))Ck x(t ), s (t )  Si , k 1

where

 1  3  2 1 h1 ( x1 (t ))    x1 3 3  1  

x1  1, x1  1, x1  1,

 2  3  1 1 h2 ( x1 (t ))    x1 3 3  0  

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x1  1, x1  1, x1  1.

In this example, the cost function is given in (7) with Q1  R1  10I .

Now, using the cone complementary linearization method in [7], we can find that one solution to the non-linear minimization problem in Remark 2 is as follows

 6.7912 0.0057 0.0219   0.4863 0.0775 0.1453   X   0.0057 3.4364 1.6278  , Y  0.0775 0.6536 0.1873 ,    0.0219 1.6278 6.8351   0.1453 0.1873 0.5968  And by Remark 1, we choose U  diag (1.5,1.5,1.5) , it is easy to show that the corresponding closed-loop cost function satisfied J *  21.1518.

Now, we choose

0.2 0  1  N  0.3 0.5 0.1 .  0.1 0.2 0.5 Then, M can be obtained by (15). Using X , Y , M , N obtained in step 2 and 4, we can get Aci , Bci , Cci (i=1,2,3).

For the initial condition x0  [0.5, 0.2, 0.1]T , we apply the piecewise output feedback controller to the fuzzy system and simulate the behaviors of the closed-loop systems, the simulation results of the state response of the nonlinear system are given in Fig. 2, 3 and 4.

FIG. 2 STATE RESPONSE

x1 (t )

FIG. 3 STATE RESPONSE

x2 (t )

FIG. 4 STATE RESPONSE x3 (t ) From these simulation results, it can be seen the designed fuzzy output feedback controller ensures the robust asymptotic stability of the delay fuzzy system and minimizes the guaranteed cost of the closed-loop uncertain system.

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4 CONCLUSIONS The problem of robust output feedback guaranteed cost control for uncertain discrete T-S fuzzy systems with parameter uncertainties and both state and input delays has been studied. In terms of LMIs, a sufficient condition for the existence of piecewise output feedback controller, which robustly stabilizes the uncertain delay systems and minimizes the guaranteed cost of the closed-loop uncertain system, has been obtained. It is shown that the piecewise guaranteed cost output feedback controller is simple and practical. Example has been provided to show the effectiveness of the proposed method.

ACKNOWLEDGMENT This work is supported by the National Natural Science Foundation of China under Grant 61203047.

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AUTHORS 1

Xiaona Song received her Ph. D. degree

Jinchan Wang received her Ph. D. degree

in control theory and control engineering

in Physical Electronics from Southeast

from Nanjing University of Science and

University, China in 2009. Now, she is an

Technology, China in 2011. Now, she is

associate professor in Henan University of

an associate professor in Henan University

Science and Technology, China. Her

of Science and Technology, China. Her

research

interests include fuzzy system and robust

reliability and failure.

control. Email: xiaona_97@163.com

- 105 http://www.sj-ce.org/

mainly

focuses

on

device