Generalised H2 controller synthesis for uncertain discrete-time fuzzy

Generalised H2 controller synthesis for uncertain discrete-time fuzzy systems via basis-dependent Lyapunov functions S. Zhou and G. Feng Abstract: The generalised H2 control problem is investigated for a class of discrete-time fuzzy systems with uncertainties. The uncertain fuzzy dynamic model is used to represent a class of uncertain discrete-time complex nonlinear systems which include both linguistic information and system uncertainties. Using basis-dependent Lyapunov functions an H2 control design approach is developed. It is shown that the controller can be obtained by solving a set of linear matrix inequalities. It is also shown that the basis-dependent results are less conservative than the basis-independent ones. Numerical examples including a discrete chaotic Lorenz system are also given to demonstrate the applicability of the proposed approach.

1

Introduction

Fuzzy models of the Takagi – Sugeno (T –S) type have been widely studied recently. This model consists of local dynamics which are represented by linear state-space models and the fuzzy blending of these local models. It has been shown [1] that the more general T – S fuzzy systems with affine terms are universal function approximators in the sense that they can 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. The control design can be carried out on the basis of the fuzzy model via the parallel distributed compensation scheme [2] or the local control approach. The main advantage of the T– S fuzzy modelling and control approach is that model-based analysis and control techniques can be applied. Over the past few years there have been significant research efforts on the analysis and control design for T– S fuzzy systems [3 – 6]. On the other hand, since uncertainties often degrade system performance and may even lead to instability, a number of results have appeared on stability analysis and control synthesis for uncertain fuzzy systems [7, 8]. The stability with guaranteed H2 [9] performance or H1 performance [4 – 6] is most determined by checking a set of algebraic Riccati equations (AREs) or linear matrix inequalities (LMIs). In most cases it is required that a common positive-definite matrix can be found to satisfy all the AREs or LMIs. However, this is a difficult problem to solve since such a matrix might not exist in many cases, especially for highly nonlinear complex systems. More recently, there # IEE, 2005 IEE Proceedings online no. 20045164 doi:10.1049/ip-cta:20045164 Paper received 20th April 2004 S. Zhou is with Institute of Automation, Qufu Normal University, Qufu 273165, Shandong, P.R. China G. Feng is with Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong E-mail: [email protected]

74

appeared a number of results on stability analysis and control synthesis of dynamic systems based on parameterdependent Lyapunov functions [10, 11], piecewise Lyapunov functions [3, 7], or basis-dependent Lyapunov function [12, 13]. It is shown that with the use of a parameter-dependent or basis-dependent Lyapunov function less conservative results can be obtained than those with the use of a single Lyapunov quadratic function [10, 11, 13]. This paper focuses on generalised H2 control problems for uncertain discrete-time T– S fuzzy models. The uncertainty is assumed to have a linear fractional form, which includes the norm-bounded uncertainty as a special case and can describe a class of rational nonlinearities [14]. With the introduction of some matrix variables, a new control design approach is proposed based on basisdependent Lyapunov functions. It is shown that the solution of the control design problem can be obtained by solving a set of linear matrix inequalities, which can be facilitated by commercially available software. Notation. Throughout this paper a real symmetric matrix P . 0 (
0) denotes P being a positive definite (or positive semidefinite) matrix, and A . (
)B means A 2 B . (
)0. A real matrix M t denotes a matrix M’s transpose; I is used to denote an identity matrix with proper dimensions. Matrices, if not explicitly stated, are assumed to have compatible dimensions, and l2[0, 1) refers to the space of square summable infinite vector sequences, whereas k.k2 stands for the usual l2[0, 1) norm. 2

Problem formulation and preliminaries

The following T –S fuzzy dynamic model can be used to represent a complex discrete-time system with both fuzzy inference rules and local analytic-linear models: Plant rule i: If j1(k) is F i1, and . . . , jr(k) is F ir, then xkþ1 ¼ Ai xk þ Bi vk þ Gi uk zk ¼ Ci xk þ F i uk ; i ¼ 1; 2; . . . ; s where j1 , . . . , jr are the premise variables, F 1i , . . . , F ri are fuzzy sets, xk [ R n is the state, vk [ R p is the disturbance IEE Proc.-Control Theory Appl., Vol. 153, No. 1, January 2006

input and vk [ l2[0, 1), uk [ R m is the control input, zk [ R q is the regulated output, and Ai, Bi , Ci , Gi and Fi are system matrices with parametric uncertainties. The system matrices with parametric uncertainties are assumed to satisfy

 Bi Gi Ai Ai Bi Gi ¼ Ci H2i DNBi Fi Ci 0 F i

 H1i i i þ D NAC ð1Þ NBi NGF H2i D ¼ ½I  FðkÞJ 1 FðkÞ I  JJ t . 0

ð2Þ ð3Þ

where Ai , Bi , Ci , Gi , Fi , H1i , H2i , N iAC, N iB, N iGF and J are known real constant matrices of appropriate dimensions for all uncertain matrices F [ R ij satisfying FðkÞF t ðkÞ  I

To the purpose of development, the following lemmas are needed. Lemma 1: Suppose D is given by (2) – (4). With matrices M ¼ M t, S and N of appropriate dimensions, the inequality M þ SDN þ N t Dt S t , 0

holds for all F(k) such that F(k)F t(k)  I, if and only if, for some d . 0 2 3 dM S dN t 4 S t I J t 5 , 0 ð9Þ dN J I Lemma 2: Suppose Pi . 0, Pj . 0, and Pl . 0. If Ati Pj Ai  Pi , 0;

Proof: Note that

Remark 1: The uncertainties are modelled as a structured linear fractional form. It is shown that every rational nonlinear system possesses a linear fractional representation [14]. The linear fractional parametric uncertainties have been investigated in the robust control setting as in [15, 16]. It is easy to see that when J ¼ 0 the linear fractional uncertainty reduces to norm-bounded one. Notice also that condition (3) guarantees that I 2 FJ is invertible.

which gives

SD :

xkþ1 ¼

s P

hi ½jðkÞðAi xk þ Bi vk þ Gi uk Þ

ð5Þ

i¼1

zk ¼

s P

hi ½jðkÞðCi xk þ F i uk Þ

ð6Þ

i¼1

By definition the fuzzy basis functions satisfy hi ½jðkÞ
0;

i ¼ 1; . . . ; s and

s P

hi ½jðkÞ ¼ 1

ð7Þ

i¼1

The objective of this paper is to design a feedback controller for system SD with a guaranteed performance in the generalised H2 sense [7, 17], that is, given a g . 0, find a controller such that the resulting closed-loop system is globally stable when vk ; 0 and the generalised H2 norm of the closed-loop system defined as   NP 1 H2 :¼ sup kzN k : x0 ¼ 0; ðvtk vk Þ  1 N [f1;2;...g

k¼0

is less than g. In this case the closed-loop system is said to be globally stable with generalised H2-norm bound g. In what follows we drop the argument of hi(jk) for clarity. IEE Proc.-Control Theory Appl., Vol. 153, No. 1, January 2006

ð10Þ

Ati Pj Al þ Atl Pj Ai  Pi  Pl , 0

The class of parametric uncertainties D satisfying (1) –(4) is said to be admissible.

where mij[jj(k)] is the grade of membership function of jj(k) in F ij. By using a standard singleton fuzzifier, product inference, and centre average defuzzifier, a more compact presentation of the Takagi –Sugeno discrete-time fuzzy model is given by

Atl Pj Al  Pl , 0

then

ð4Þ

It is assumed that the premise variables do not depend on the control variables and the disturbance. The fuzzy basis functions are then given by Qr j¼1 mij ½jj ðkÞ ; i ¼ 1; . . . ; s hi ½jðkÞ ¼ Ps Qr l¼1 j¼1 mlj ½jj ðkÞ

ð8Þ

Pj . 0 ¼) ðAi  Al Þt Pj ðAi  Al Þ
0 Ati Pj Al þ Atl Pj Ai  Ati Pj Ai þ Atl Pj Al

ð11Þ

The conclusion follows from (10) and (11). 3

Robust stability and H2 performance analysis

We analyse the stability and H2 performance for the nominal unforced system discussed in the last Section. Consider the nominal unforced system (5) and (6) given by S : xkþ1 ¼

s X

hi ðAi xk þ Bi vk Þ

ð12Þ

hi Ci xk

ð13Þ

i¼1

zk ¼

s X i¼1

Then we have the following results. Theorem 1: The system S is stable with generalised H2norm bound g, if there exist matrices fXi . 0gsi¼1 for all i, j [ f1, . . . , sg, such that " # Xi Xi Cit ,0 ð14Þ Ci Xi g2 I 2 3 Xi 0 Xi ATi 6 7 6 0 ð15Þ I Bti 7 4 5,0 Ai Xi Bi Xj Proof: Let Vk ¼

xtk

s X

! h i Pi x k

ð16Þ

i¼1

with Pi ¼ Xi1 When vk ; 0, system (12) becomes ! s X h i Ai x k xkþ1 ¼

ð17Þ

ð18Þ

i¼1

75

By some algebraic manipulation, with hþ j U hj(jkþ1), the difference of Vk given by DVk U Vkþ1 2 Vk along the solution of system (18) is " !# s s X s X X t þ t hj hi hl ðAi Pj Al  Pi Þ xk DVk jð18Þ ¼ xk j¼1

" ¼ xtk

s X

i¼1 l¼1 s X

hþ j

j¼1

þ

h2i ðAti Pj Ai  Pi Þ

i¼1

s X s X

!# hi hl ðAti Pj Al

þ

Atl Pj Ai

 Pi  Pl Þ

xk

l¼1 i.1

Premultiplying diag(X 21 i, , I, I ) and postmultiplying diag(X21 i , I, I ) to (15) gives 2 3 Pi 0 Ati 4 0 I Bti 5 , 0 Ai Bi P1 j

Inequality (23) can be rewritten as ( 
t 
N 1 s X s X X   Ai Pi t 2 þ jk hi hj Pj Ai Bi  JN  t B 0 i i¼1 j¼1 k¼0 
t 
s X s X s X   Ai Pi þ þ hi h j hl Pj Al Bl  t B 0 i j¼1 l¼1 i.l
t
   A Pl 0 jk þ it Pj Ai Bi  Bi 0 I


 Pi Bi  0

  Ati t P j Ai Bi

 0 ,0 I

VN ,

Ati Pj Ai  Pi Bti Pj Ai

Ati Pj Bi t B i Pj B i  I

" ztN zN  g2 VN ¼ xtN

N 1 X

,0

¼

ð20Þ

þ ð21Þ

where DVkj(12) defines the difference of Vk along the solution of system (12). It is noted that ( s X s X s X t DVk jð12Þ ¼ jk hi hþ j hl 


 Pi Bl  0

0



0

jk

where

jk :¼



xtk

t vtk

ð22Þ

Then JN ¼

N1 P k¼0


 76

(

jtk

s X s X s X i¼1 j¼1 l¼1

Pi

0

0

I

hi hþ j hl




Ati Bti



 Pj A l

Bl





jk

ð25Þ

xtN

# hi hl ðCit Cl  g2 Pi Þ xN

i¼1 l¼1 s X

h2i ðCit Ci  g2 Pi Þ

s X s X

hi hl ½ðCit Cl

) 2

 g Pi Þ þ

ðClt Ci

2

 g Pl Þ xN

i¼1 l.i

It follows from (25) and Lemma 2 that for any N and vk [ l2[0, 1) ztN zN  g2 VN , 0

ð26Þ

Inequalities (24) and (26) imply that the system S is of generalised H2-norm bound g and thus the proof is completed.

ðvtk vk Þ

k¼0

i¼1 j¼1 l¼1   Ati Pj A l t Bi

ð24Þ

i¼1

ðDVk jð12Þ  vtk vk Þ




ðvtk vk Þ

s X s X

(



For any nonzero vk [ l2[0, 1) and zero initial condition x0 ¼ 0, one has JN ¼



Note that

It follows from (21) and by using lemma 2 that DVkj(18) , 0 which P establishes the stability of the system xkþ1 ¼ ( si¼1 hiAi)xk. Now consider the H2 performance. For any N [ f1, 2, . . .g, let

k¼0

N 1 X

Cit Ci  g2 Pi , 0

ð19Þ

Ati Pj Ai  Pi . 0

NP 1

0 I

By using Schur complement equivalence, one has by (14) and (17) that

which implies

J N ¼ VN 

I

k¼0

or




It follows from (19) and lemma 2 that for any N, JN , 0. In turn it implies that for any nonzero vk [ l2[0, 1)

yielding


0

ð23Þ

Remark 2: If one uses Lyapunov function Vk ¼ xtkPxk the corresponding conditions can be obtained as follows  8
X XCit > > ,0 > 2 > < Ci X g I 2 3 ð27Þ X 0 XAti >4 > t 5 > 0 I B , 0 > i : Ai X Bi X The common quadratic Lyapunov function-based result is a special case of basis-dependent results. Thus results based on basis-dependent Lyapunov functions are less conservative than those based on single Lyapunov quadratic functions. Next, we present an alternative result. Theorem 2: The nominal unforced system S is stable with generalised H2-norm bound g if there exist matrices fXi . 0gsi¼1 and fVigsi¼1 for all i, j [ f1, . . . , sg such that " # Xi  ðVj þ Vtj Þ Vtj Cit ,0 ð28Þ Ci Vj g2 I 2 3 Xi  ðVj þ Vtj Þ 0 Vtj Ati 6 0 I Bti 7 ð29Þ 4 5,0 Ai Vj

Bi

Xj

IEE Proc.-Control Theory Appl., Vol. 153, No. 1, January 2006

Proof: Assume that the inequalities (28) and (29) holds, which gives 0 , Xi , Vj þ Vtj . The inequality t 21 Vj
Vj þ (Xi 2 Vj)tX21 i (Xi 2 Vj)
0 implies that Vj Xi t Vj 2 Xi which yields " # Vtj Xi1 Vj Vtj Cit ,0 ð30Þ Ci Vj g2 I 2 3 Vtj Xi1 Vj 0 Vtj Ati 6 7 ð31Þ 0 I Bti 5 , 0 4 Ai V j

Bi

Bi

6 4

Xi  ðVj  Vtj Þ

0

0 Ai Vj

I Bi

Vtj Ati

3

Bti 7 5,0 Xj

ð37Þ

Then the proof can be concluded by invoking theorem 2. 4

Control design

Consider the generalised H2 control design problem of (SD). We choose the following fuzzy control law:

Xj

t Note that Vj is invertible. Premultiplying diag(V2 j , I, I ) 21 and postmultiplying diag(Vj , I, I ) to (31) and premultiplyt 21 ing diag(V2 j , I ) and postmultiplying diag(Vj , I ) to (30), yield, respectively,
 Xi1 Cit ,0 ð32Þ Ci g2 I 2 3 Xi1 0 Ati 6 7 ð33Þ I Bti 5 , 0 4 0

Ai

2

Regulator rule i: If j1(k) is F i1, and . . . jr(k) is F ir, then

The overall state-feedback fuzzy controller is represented by uk ¼

Remark 3: Theorem 2 is motivated by [10]. With the introduction of some new additional matrices Vj we obtain a linear matrix inequality in which the Lyapunov matrix Xi is not involved in any product with the state matrices Ai and output matrices Ci. This feature enables one to derive a condition with less conservativeness due to the extra degrees of freedom. It is noted that the introduced matrices Vj are not even constrained to be symmetric. This result will play a crucial role for control design later on. Now consider the unforced system with uncertainty. Theorem 3: The unforced system SD is robustly stable with generalised H2-norm bound g, if there exist scalars e 1 . 0, e 2 . 0 and matrices fXi . 0gsi¼1 and fVjgsi¼1 for all i, j [ f1, . . . , sg such that 2 it 3 0 Vtj NAC Xi  ðVj þ Vtj Þ Vtj Cit 6 Ci Vj e1 g2 I H2i 0 7 6 7 6 7 , 0 ð34Þ t t 4 5 0 H2i I J i NAC Vj 0 J I 2 3 it Xi  ðVj þ Vtj Þ 0 Vtj Ati 0 Vtj NAC 6 7 0 e2 I e2 Bti 0 e2 NBit 7 6 6 7 6 Ai V j e2 Bi Xj H1i 0 7 6 7,0 6 7 t t 0 0 H1i I J 4 5 i i NAC Vj e2 NB 0 J I ð35Þ Proof: By lemma 1 and inequalities (34) and (35) one has that there exist matrices fXi . 0gsi¼1 and fVjgsi¼1 for all i, j [ f1, . . . , sg such that " # Xi  ðVj þ Vtj Þ Vtj Cit ,0 ð36Þ Ci Vj g2 I IEE Proc.-Control Theory Appl., Vol. 153, No. 1, January 2006

s P

hi K i x k

ð38Þ

i¼1

Under control law (38), the closed-loop system of SD is given by

Xj

Premultiplying diag(Xi, I ) and postmultiplying diag(Xi, I ) to (32), and premultiplying diag(Xi, I, I ) and postmultiplying diag(Xi, I, I ) to (33) yield, respectively, (14) and (15). The result then follows from theorem 1.

i ¼ 1; . . . ; s

uk ¼ K i x k ;

ScD :

xkþ1 ¼

s X s X

hi hj ½ðAi þ Gi Kj Þxk þ Bi vk 

ð39Þ

hi hj ½ðCi þ F i Kj Þxk 

ð40Þ

i¼1 j¼1

zk ¼

s X s X i¼1 j¼1

When D ¼ 0 the nominal closed-loop system becomes Sc :

xkþ1 ¼

s X s X

hi hj ðAij xk þ Bi vk Þ

ð41Þ

hi hj ðCij xk Þ

ð42Þ

Cij :¼ Ci þ Fi Kj

ð43Þ

i¼1 j¼1

zk ¼

s X s X i¼1 j¼1

where Aij :¼ Ai þ Gi Kj ;

Then the following result is easy to obtain. Lemma 3: Closed-loop system Sc is stable with generalised H2-norm bound g if there exist matrices fXi . 0gsi¼1 for all i, j, l [ f1, . . . , sg such that
 Xi Xi Cijt ,0 ð44Þ Cij Xi g2 I 2 3 Xi 0 Xi Atij 6 0 I Bti 7 ð45Þ 4 5,0 Aij Xi Bi Xl Proof : The proof of this lemma is very similar to the proof of theorem 1. With minor modification, one can follow the same line as given by theorem 1 to complete the proof. Theorem 4: If there exist matrices fXi . 0gsi¼1, fVigsi¼1 and fYigsi¼1 for all i, j, l [ f1, . . . , sg such that " # Xi  ðVj þ Vtj Þ Vtj Cit þ Yjt Fit ,0 ð46Þ Ci Vj þ Fi Yj g2 I 2 3 Xi  ðVj þ Vtj Þ 0 Vtj Ati þ Yjt Git 6 7 0 I Bti 4 5 , 0 ð47Þ Ai Vj þ Gi Yj

Bi

Xl 77

2

then the nominal closed-loop system Sc is stable with generalised H2-norm bound g, and the controller gains are given by Yj V1 j

Kj ¼

60 6 6 6 Ai Vj þ Gi Yj 6 6 40 i i NAC Vj þ NGF Yj

ð48Þ

Proof: With (43) and (48), inequalities (46) and (47) become " # Xi  ðVj þ Vtj Þ Vtj Cijt ,0 ð49Þ Cij Vj g2 I 2 3 Xi  ðVj þ Vtj Þ 0 Vtj Atij 6 0 I Bti 7 ð50Þ 4 5,0 Aij Vj

Bi

Bi

Xl

t Note that Vj is invertible. Premultiplying diag(V2 j , I, I, I ) 21 and postmultiplying diag(Vj , I, I, I ) to (52), yield respectively, " # Cijt Xi1 ,0 ð53Þ Cij g2 I 2 3 Atij Xi1 0 6 7 ð54Þ I Bti 5 , 0 4 0

Aij

Bi

Remark 4: We use different methods in choosing the slack variables fVigsi¼1 in theorem 2 and theorem 4. The choice of slack variables fVigsi¼1 in theorem 2 is based on the index j which is generated by hþ j U hj(jkþ1). The choice of slack variables fVigsi¼1 in theorem 4 is not based on this type of index, but instead on the index generated by hj U hj(jk). The reason for doing so is that in theorem 4 all the slack variables fVigsi¼1 are used to obtain the control gain matrices fKjgsj¼1. Thus if one chooses them in the ways of theorem 2, the control gain matrices fKjgsj¼1 cannot be determined. Theorem 5: If there exist scalars e 1 . 0, e 2 . 0 and matrices fXi . 0gsi¼1, fVigsi¼1,fYigsi¼1for all i, j, l [ f1, . . . , sg such that 2 Xi  ðVj þ Vtj Þ ðCi Vj þ Fi Yj Þt 6 CV þFY e1 g2 I i j i j 6 6 4 0 H2it i i NAC Vj þ NGF Yj 0 3 i i 0 ðNAC Vj þ NGF Y j Þt 7 H2i 0 7 ð55Þ 7,0 5 I Jt

78

ðAi Vj þ Gi Yj Þt

 e2 I

e2 Bti Xl H1it 0

0

i i ðNAC Vj þ NGF Yj Þt

0 H1i

e2 NBit 0 Jt I

I J

I

3 7 7 7 7 7 7 5

ð56Þ

then the uncertain closed-loop system ScD is robustly stable with generalised H2-norm bound g, and the controller gains are given by Kj ¼ Yj V1 j

ð57Þ

Proof: By (57), inequalities (55) and (56) become 2 3 Xi  ðVj þ Vtj Þ Vtj Cijt 0 Vtj Nijt 6 Cij Vj e1 g2 I H2i 0 7 6 7 6 7 , 0 ð58Þ 4 0 H2it I Jt 5 2

Nij Vj Xi  ðVj þ Vtj Þ 0

60 6 6 6 Aij Vj 6 6 40

0

 e2 I e 2 Bi 0

e2 NBi

Nij Vj

J Vtj Atij

Xl

Premultiplying diag(Xi, I ) and postmultiplying diag(Xi, I ) to (53), and premultiplying diag(Xi, I, I ) and postmultiplying diag(Xi, I, I ) to (54) yield (44) and (45), respectively. The result then follows from lemma 3.

J

0

e2 B i 0 e2 NBi

Xl

which gives 0 , Xi , Vj þ Vtj . The inequalities t 21 (Xi 2 Vj)tX21 i (Xi 2 Vj)
0 imply that Vj Xi Vj
Vj þ t Vj 2 Xi yielding " # Vtj Xi1 Vj Vtj Cijt ,0 ð51Þ Cij Vj g2 I 2 3 Vtj Xi1 Vj 0 Vtj Atij 6 7 ð52Þ 0 I Bti 5 , 0 4 Aij Vj

Xi  ðVj þ Vtj Þ

e2 Bti Xl H1it 0

I 3 Vtj Nijt 0 e2 NBit 7 7 7 7,0 H1i 0 7 7 I J t 5 J I ð59Þ 0

where i i Nij :¼ NAC þ NGF Kj

ð60Þ

By lemma 1 and inequalities (58) and (59), there exist matrices fXi . 0gsi¼1 and fVjgsi¼1 for all i, j, l [ f1, . . . , sg such that " # Xi  ðVj þ Vtj Þ Vtj Cijt ,0 Cij Vj g2 I 2 3 Xi  ðVj þ Vtj Þ 0 Vtj Atij 6 0 I Bti 7 4 5,0 Aij Vj Bi Xl where Aij :¼ Ai þ Gi Kj ;

Cij :¼ Ci þ F i Kj

Following a similar development as in the proof of theorem 4, the desired result can be established. 5 5.1

Numerical examples Example 1

Consider the nominal unforced system (5) and (6) given by S : xkþ1 ¼

s X

hi ðAi xk þ Bi vk Þ

ð61Þ

i¼1

IEE Proc.-Control Theory Appl., Vol. 153, No. 1, January 2006

zk ¼

s X

If xk(1) is about M2, then

hi ðCi xk þ Di vk Þ

ð62Þ 2 Þxk þ ðB2 þ H12 DNB2 Þvk xkþ1 ¼ ðA2 þ H12 DNAC

i¼1

2 þ ðG2 þ H12 DNGF Þuk

where
A1 ¼
B1 ¼

0:291

1

0 

0:95

0:1 0:1


;

 C1 ¼ 0:1

 ;

B2 ¼  0:1 ;


A2 ¼ 0:1

0:1

0

1

0:2



2 2 zk ¼ ðC2 þ H22 DNAC Þxk þ ðF2 þ H22 DNGF Þuk

; with



2

0:1  C2 ¼ 0:09

 0:05 ;

2

The purpose here is to verify that system S is stable with generalised H2-norm bound g . 0. If one uses Lyapunov function Vk ¼ x tkPxk, the corresponding conditions can be obtained as follows, for all i [ f1, 2g:  8
X XCit > > ,0 > 2 > < C i X g I 2 3 ð63Þ X 0 XAti >4 > t 5 > 0 I B , 0 > i : Ai X Bi X

5.2

1  Ts

7 M1 Ts 5;

0

M1 Ts

1  bTs

1  sTs

sTs

0

hTs

1  Ts

0

M2 T s

2

0:1 2 3 1 6 7 G1 ¼ 4 0 5; 

0

3

7 M2 Ts 5

1  bTs 3 0:1 6 7 B2 ¼ 4 0:1 5;

3

0:1 6 7 B1 ¼ 4 0:1 5;

2

0:1 2 3 1 6 7 G2 ¼ 4 0 5

0

0

 C1 ¼ 0:1 0:1 0:3 ;   C2 ¼ 0:09 0:05 0:03

one has that (64) is infeasible, which implies that (63) are infeasible for any g . 0. However, by using MATLAB LMI Control Toolbox to solve LMIs (20) or (15) with g ¼ 0.6, our results give the following feasible solutions:

 3:8324 1:0149 1:5825 1:1538 X1 ¼ ; X2 ¼ 1:0149 1:6882 1:1538 3:9073

This example shows that theorems 1 and 2 (basis-dependent results) can guarantee stability with generalised H2-norm bound g ¼ 0.6 for the system S, while basis-independent results cannot. As expected, results based on basisdependent Lyapunov functions are less conservative than those based on single Lyapunov quadratic function.

hT s

6 A2 ¼ 4

Using MATLAB LMI Control Toolbox to solve LMIs

 X XAt1 X XAt2 , 0; ,0 ð64Þ A1 X X A2 X X

Furthermore, to solve LMIs (28) and (29) with g ¼ 0.6, the feasible solutions are

 1:0415 0:1202 1:1178 0:1291 ; X2 ¼ X1 ¼ 0:1202 1:2156 0:1291 1:0698

 1:0487 0:0831 1:0393 0:0816 V1 ¼ ; V2 ¼ 0:0828 0:9994 0:0834 0:9970

sTs

6 A1 ¼ 4

s¼2

3

1  sTs

H11

F1 ¼ F2 ¼ 0:5 2 0:06 6 ¼ H12 ¼ 4 0 0

1 2 NAC ¼ NAC

0

0

2 3 0 6 7 ¼ 4 0 5;

0

NB1 ¼ 0;

3

7 0 5;

0:06

0:06  0

H21 ¼ H22 ¼ 0 0 2 3 s s 0 6 7 ¼ 103 4 b 0 0 5; 0

2 NGF



0

h

1 NGF

2 3 0 6 7 ¼ 4 0 5; 0

NB2 ¼ 0

0 2

0

6 J ¼ 40 0

0 0

3

7 0 05 0 0

Example 2

In this example we demonstrate the performance of the proposed new generalised H2 fuzzy control method with the problem of chaotic Lorenz system. Consider the following discrete-time chaotic Lorenz system, which is based on the example given in [8] with sampling time Ts ¼ 0.002 s: Plant rules: If xk(1) is about M1, then 1 Þxk þ ðB1 þ H11 DNB1 Þvk xkþ1 ¼ ðA1 þ H11 DNAC 1 þ ðG1 þ H11 DNGF Þuk 1 1 zk ¼ ðC1 þ H21 DNAC Þxk þ ðF1 þ H21 DNGF Þuk

IEE Proc.-Control Theory Appl., Vol. 153, No. 1, January 2006

Fig. 1 State trajectories of open-loop system 79

6

Conclusions

The generalised H2 control for a class of uncertain discretetime fuzzy systems has been studied. Linear matrix inequality conditions of stability with H2-norm bound for discrete-time fuzzy systems were given via fuzzy basis-dependent Lyapunov function which is less conservative than those using a single quadratic Lyapunov function. Sufficient conditions for robust H2 performance analysis and control were obtained in the form of LMIs. Numerical examples illustrated the results.

Fig. 2 State trajectories of closed-loop system

7

As in [8], choose [M1 , M2] as [220, 30] and the same membership function, and (s, h, b ) ¼ (10, 28, 8/3) for chaos to emerge. The purpose here is to control the discrete-time chaotic Lorenz system such that the closed-loop system is stable with a given generalised H2-norm bound g. The performance level is chosen as g ¼ 0.9. Using MATLAB LMI Control Toolbox to solve LMIs (55) and (56), we obtain a feasible set of solutions as follows: 2 3 14:7785 3:6903 0:0331 6 7 X1 ¼ 4 3:6903 6:0206 0:0646 5;

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0:0331 2

14:3307 6 X2 ¼ 4 3:7020 0:0144 2

12:8153 6 V1 ¼ 4 0:9118 0:0131 2

0:0646 3:7020 6:0102 0:0868 3:5436 6:0791 0:0683

13:2841 6 V2 ¼ 4 0:6281

3:5714 6:0679

0:0061

0:0687



5:5817 3 0:0144 7 0:0868 5 5:5830 3 0:0139 7 0:0739 5; 5:5889 3 0:0143 7 0:0738 5 5:5890

 Y1 ¼ 10:6372 0:0299 0:0027 ;   Y2 ¼ 12:4747 0:0547 0:0024 e1 ¼ 17:9255; e2 ¼ 0:3400 By theorem 5, we have the local state feedback gains given by   K1 ¼ 0:8663 0:5100 0:0094 ;  K2 ¼ 0:9655

0:5594

0:0103



To illustrate the behaviour of the control action, Fig. 1 depicts the behaviour of the open-loop state trajectories with starting point x0 ¼ [10 210 210]t which subsequently generated a chaotic motion. With the fuzzy control applied, the state trajectories of the closed-loop system are shown in Fig. 2. It can be observed that the chaotic system is stabilised without oscillatory behaviour.

80

References

IEE Proc.-Control Theory Appl., Vol. 153, No. 1, January 2006