Page 1

Stability Analysis and Control Design for 2-D Fuzzy Systems via

Basis-dependent Lyapunov Functions§

Xiaoming Chen∗

James Lam∗

Huijun Gao†

Shaosheng Zhou‡

Abstract

This paper investigates the problem of stability analysis and stabilization for two-dimensional (2-D) discrete

fuzzy systems. The 2-D fuzzy system model is established based on the Fornasini-Marchesini local state-space

(FMLSS) model, and a control design procedure is proposed based on a relaxed approach in which basis-dependent

Lyapunov functions are used. First, nonquadratic stability conditions are derived by means of linear matrix inequal-

ity (LMI) technique. Then, by introducing an additional instrumental matrix variable, the stabilization problem for

2-D fuzzy systems is addressed, with LMI conditions obtained for the existence of stabilizing controllers. Finally,

the effectiveness and advantages of the proposed design methods based on basis-dependent Lyapunov functions are

shown via two examples.

Keywords: 2-D System; Basis-dependent Lyapunov function; Control Design; Fuzzy System; Stability Analysis.

1 Introduction

As is well known, many practical systems can be modeled as two-dimensional (2-D) systems [?, ?], such as those in

image data processing and transmission, thermal processes, gas absorption and water stream heating. During the last

few decades, the investigation of 2-D systems in the control and signal processing fields has attracted considerable

attention and many important results have been reported to the literature. Among these results, the stability problem

of 2-D systems has been investigated in [?, ?, ?, ?]. The controller and filter design problems have been addressed in

[?, ?, ?, ?, ?, ?, ?, ?, ?]. In addition, the model reduction of 2-D systems has also been solved in [?, ?].

Despite the success, it has been made aware that many basic issues of 2-D systems remain to be addressed. 2-D

nonlinear system is one of the challenging research areas because until now, there has been no systematic and effective

approach to handle the problem of 2-D nonlinear systems. The difficulty in modeling the nonlinearity is one of the

reasons. It is noticeable that, in the one-dimensional (1-D) case, the Takagi-Sugeno (T-S) fuzzy model [?, ?, ?, ?] has

shed some light on this difficult problem, based on the fact that the T-S fuzzy model can approximate the smoothly

nonlinear system on a compact set. This model formulates the 1-D nonlinear systems into a framework consisting of

a set of local models which are smoothly connected by some membership functions. Based on the local linearities,

the stability and performance analysis approaches for 1-D linear systems can be fully developed for 1-D nonlinear

∗Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong (Emails: james.lam@hku.hk;

h1095051@hku.hk).

†Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin, Heilongjiang Province, 150001, China

(Email: hjgao@hit.edu.cn).

‡Department of Automation, Hangzhou Dianzi University, Hangzhou 310018, Zhejiang, P.R. China (Email: sszhouyah@yahoo.com.cn).

§This work was partially supported by HKU RGC Grant 7137/09E, the National Natural Scinece Foundation of China (60825303,

60834003), the 973 project (2009CB320600), the Foundation for the Author of National Excellent Doctoral Dissertation of China (2007B4),

and the Key Laboratory of Integrated Automation for the Process Industry (Northeastern University), Ministry of Education.

1

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systems in this framework. In virtue of this advantage, a number of important issues in 1-D nonlinear fuzzy control

systems have been well studied. Among these results, stability analysis has been studied in [?, ?, ?], systematic design

procedures have been proposed in [?], robustness and optimality have been investigated in [?, ?, ?, ?].

On the other hand, it is noted that the aforementioned research efforts have been focused on the use of a single

quadratic Lyapunov function [?, ?], which tends to yield more conservative conditions. More recently, there appeared

a number of results on stability analysis and control synthesis of 1-D dynamic systems based on basis-dependent Lya-

punov functions [?, ?, ?, ?]. It is shown that, with the use of a basis-dependent Lyapunov function, less conservative

results can be obtained than those with the use of a single Lyapunov quadratic function. Examples of reduced con-

servative conditions based on basis-dependent Lyapunov functions can be found in [?, ?, ?]. Motivated by the above

observations, one may naturally ask: Can we represent the 2-D nonlinear systems using the T-S fuzzy model and thus

solve the problems of 2-D nonlinear fuzzy control systems based on the approach towards the use of basis-dependent

Lyapunov functions?

In this paper, we provide an affirmative answer to the above question. The 2-D fuzzy system model is established

based on the Fornasini-Marchesini local state-space (FMLSS) model [?, ?], and the controller design procedure is

presented based on a relaxed approach in which basis-dependent Lyapunov functions are used. First, nonquadratic

stability is derived by means of linear matrix inequality (LMI) technique [?]. Then, by introducing an additional

instrumental matrix variable, the stabilization problem for 2-D fuzzy systems is addressed, with LMI conditions

obtained for the existence of stabilizing controllers. Finally, two illustrative examples are provided to show the

effectiveness and it is shown that the results based on basis-dependent Lyapunov functions are less conservative than

those based on basis-independent Lyapunov functions.

The rest of the paper is organized as follows. The problem under consideration is formulated in Section 2. Stability

analysis is given in Section 3, based on which controller designs are presented in Section 4. Illustrative examples

are given in Section 5 to demonstrate the effectiveness of the results. Finally, the paper is concluded in Section

6.

Notation: Thenotationusedthroughoutthepaperisstandard. ThesuperscriptT standsformatrixtransposition; Rn

denotes the n-dimensional Euclidean space and the notation P>0 means that P is real symmetric and positive definite;

The notation |·| refers to the Euclidean vector norm; and λmin(·), λmax(·) denote the minimum and the maximum

eigenvalues of the corresponding matrix respectively. In symmetric block matrices or long matrix expressions, we

use an asterisk (∗) to represent a term that is induced by symmetry and diag{···} stands for a block-diagonal matrix.

Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2Problem Formulation

Let us first recall the well known 2-D discrete FMLSS model [?] given by:

F : xi+1,j+1= A1xi,j+1+A2xi+1,j+B1ui,j+1+B2ui+1,j,

(1)

where xi,j ∈ Rnis the local state vector and ui,j ∈ Rmis the input; A1,A2,B1,B2are system matrices. Similar

to the well-established fuzzy model of 1-D system [9], we consider 2-D discrete fuzzy model based on a suitable

choice of a set of linear subsystems, according to rules associated with some physical knowledge and some linguistic

characterization of the properties of the system. The linear subsystems describe, at least locally, the behavior of the

nonlinear system for a pre-defined region of the state space. The T-S model for the nonlinear system is given by the

following IF-THEN rules:

2

Page 3

Model Rule k: IF θ1(i,j)is µk1and θ2(i,j)is µk2and ... and θp(i,j)is µkp, THEN

xi+1,j+1= A1kxi,j+1+A2kxi+1,j+B1kui,j+1+B2kui+1,j,

where µk1, ..., µkpare fuzzy sets; A1k, B1k, A2k, B2kare constant matrices; r is the number of IF-THEN rules and

θi,j=

θ1(i,j),

θ2(i,j),...,

θp(i,j)

is the premise variable vector. Throughout the paper, it is assumed that the

premise variables do not depend on the input variables explicitly. Then, the final output of the fuzzy system is inferred

as

S : xi+1,j+1=

∑

k=1

where

[]

r

hk(θi,j)

{

A1kxi,j+1+A2kxi+1,j+B1kui,j+1+B2kui+1,j

}

,

(2)

hk(θi,j)=

ωk(θi,j)/

r

∑

k=1

ωk(θi,j),

ωk(θi,j)=

p

∏

l=1

µkl(θl(i,j)),

with µkl(θl(i, j)) ∈ [0,1] representing the grade of membership of θl(i, j) in µkl. We have

r

∑

k=1

ωk(θi,j)

ωk(θi,j)>

0,

≥

0,

k = 1,2,...,r,

for all i, j. Therefore, for all i, j we have

r

∑

k=1

hk(θi,j)=

1,

hk(θi,j)

≥

0,

k = 1,2,...,r.

The boundary conditions are defined by

Xh

0

=

[

[

xT

0,1

xT

0,2

...

xT

0,M

]T

]T

,

Xv

0

=

xT

1,0

xT

2,0

...

xT

N,0

.

Denote

Xr= sup{??xi,j

{

η=1

??: i+ j = r, i, j ∈ Z}.

|x0,η|2+|xη,0|2)}

Assumption 1 The boundary condition is assumed to satisfy

lim

N→∞

N

∑

(

< ∞.

Then we give the following stability definition which will be used in the paper.

Definition 1 The 2-D discrete fuzzy system S in (2) is said to be asymptotically stable if lim

input and any boundary conditions such that X0< ∞.

r→∞Xr= 0 under the zero

3

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3Stability Analysis

In this subsection, we are concerned with the stability analysis of the 2-D discrete fuzzy systems and we will give

some stability conditions obtained with the use of a basis-dependent Lyapunov function. Before presenting Theorem

1, we first introduce the following lemma which will be used in the proof of Theorem 1.

Lemma 1 For matrices P ≥ 0, A, B with appropriate dimensions, the following matrix inequality holds.

ATPB+BTPA ≤ ATPA+BTPB.

(3)

Then, the following theorem shows that the stability of 2-D discrete fuzzy systems can be guaranteed if there exist

some matrices satisfying certain LMIs.

Theorem 1 Consider the 2-D fuzzy system S in (2) under Assumption 1. The 2-D discrete fuzzy system S in (2) is

asymptotically stable if there exist matrices Xk> 0,Yk≥ 0 and Zk> 0 satisfying

where k = 1,2,...,r; 1 ≤ m < n ≤ r and Qk:= Xk+2Yk+Zk.

−Xm

∗

∗

−Ym

−Zm

∗

AT

AT

−Qk

∗

AT

AT

−Qk

AT

AT

0

−Qk

1mQk

2mQk

<

0,

(4)

−Xm−Xn

∗

∗

∗

−Ym−Yn

−Zm−Zn

∗

∗

1mQk

2nQk

1nQk

2mQk

<

0,

(5)

Proof. To establish the stability of system S, assume ui,j= 0. Then the system S in (2) can be represented by

xi+1,j+1=

r

∑

k=1

hk(θi,j){A1kxi,j+1+A2kxi+1,j

}.

First, by Schur complement equivalence [?], LMIs (4) and (5) are equivalent to

[

AT

1mQkA1m−Xm

∗

AT

AT

AT

AT

1mQkA2m−Ym

2mQkA2m−Zm

1nQkA2m−Ym−Yn

2nQkA2n−Zm−Zn

]

]

<

0,

(6)

[

AT

1mQkA1m+AT

1nQkA1n−Xm−Xn

∗

1mQkA2n+AT

2mQkA2m+AT

<

0.

(7)

Consider the following index

J :=W1−W2,

(8)

with

W1

=

[

? xT

xT

i+1,j+1

(

k=1

xT

i+1,j+1

)

](

r

∑

k=1

h+

kPk

)[

xi+1,j+1

xi+1,j+1

]

,

W2

=

r

∑

hkPk

? x,

4

Page 5

where h+

k= hk(θi+1,j+1), ? x =

[

xi,j+1

xi+1,j

]

, and Pk:=

[

Xk

∗

Yk

Zk

]

> 0. By some algebraic manipulations, we have

J

=

xT

i+1,j+1

[

(

II

](

r

∑

k=1

)

h+

kPk

)[

I

I

]

(

xi+1,j+1−? xT

∑

k=1

)}

(

r

∑

k=1

hkPk

)

? x

=

xT

i+1,j+1

r

∑

k=1

(

h+

kQk

xi+1,j+1−? xT

hm(θi,j)hn(θi,j)M1

r

hkPk

)

? x

=

? xT

? xT

{

r

∑

k=1

h+

k

r

∑

m=1

r

∑

n=1

? x

=

r

∑

k=1

h+

k

r

∑

m=1h2

∑

m(θi,j)M2

+

r

∑

m=1

m<nhm(θi,j)hn(θi,j)M3

r

∑

? x

≤

? xT

? xTΨ? x,

AT

AT

r

∑

k=1

h+

k

m=1h2

∑

m(θi,j)M2

+

r

∑

m=1

m<nhm(θi,j)hn(θi,j)M4

? x

=

(9)

where M1, M2, M3, M4satisfying

M1

=

[

[

[

[

1mQkA1n−Xm

2mQkA1n−Ym

AT

∗

AT

AT

AT

1mQkA2n−Ym

2mQkA2n−Zm

AT

AT

]

,

M2

=

1mQkA1m−Xm

1mQkA2m−Ym

2mQkA2m−Zm

1nQkA1m−Xm−Xn

∗

1nQkA1n−Xm−Xn

∗

]

AT

AT

,

M3

=

1mQkA1n+AT

1mQkA2n+AT

2mQkA2n+AT

AT

AT

1nQkA2m−Ym−Yn

2nQkA2m−Zm−Zn

1nQkA2m−Ym−Yn

2nQkA2n−Zm−Zn

]

]

,

M4

=

AT

1mQkA1m+AT

1mQkA2n+AT

2mQkA2m+AT

.

(10)

Hence, from the conditions in (6) and (7), we have Ψ < 0. Then, for ? x ̸= 0, we have

W2

W1−W2

=

−

? xT(−Ψ)? x

α −1,

λmin(−Ψ)

λmax

∑

? xT

)> 0, we have α < 1. Obviously,

α ≥W1

W2

(

r

∑

k=1hkPk

)

? x

≤ −

λmin(−Ψ)

λmax

(

r

∑

k=1hkPk

)

=

where α := 1−

λmin(−Ψ)

λmax

(

r

∑

k=1hkPk

). Since

(

r

k=1hkPk

≥ 0,

that is, α belongs to [0,1) and is independent of ? x. Therefore, we have

W1≤ αW2,

that is,

[

xT

i+1,j+1

xT

i+1,j+1

](

r

∑

k=1

h+

kPk

)[

xi+1,j+1

xi+1,j+1

]

≤

α

[

xT

i,j+1

xT

i+1,j

](

r

∑

k=1

hkPk

)[

xi,j+1

xi+1,j

]

.

(11)

5

Page 6

Then, it can be established that

[

xT

η,1

xT

η,1

](

r

∑

k=1

h+

kPk

)[

xη,1

xη,1

]

≤

α

[

∑

k=1

r

∑

k=1

xT

η−1,1

xT

η,0

](

r

∑

k=1

hkPk

)[

xη−1,1

xη,0

]

≤

α

r

hk

{xT

{xT

η,0(Yk+Zk)xη,0+xT

η−1,1(Xk+Yk)xη−1,1

}

≤

...

α

hk

η,0Qkxη,0+xT

η−1,1(Xk+Yk)xη−1,1

},

[

xT

1,η

xT

1,η

](

r

∑

k=1

h+

kPk

)[

x1,η

x1,η

]

≤

α

[

∑

k=1

xT

0,η

xT

1,η−1

](

r

∑

k=1

hkPk

)[

x0,η

x1,η−1

]

≤

α

r

hk

{xT

1,η−1(Yk+Zk)x1,η−1+xT

0,ηQkx0,η

}.

Adding both sides of the above inequality system yields

η+1

∑

j=0

xT

η+1−j,j

(

(

r

∑

k=1

h+

kQk

)

xη+1−j,j

≤

α

η

∑

j=0

xT

η−j,j

r

∑

k=1

hkQk

)

xη−j,j+xT

η+1,0

(

r

∑

k=1

hkQk

)

xη+1,0+xT

0,η+1

(

r

∑

k=1

hkQk

)

x0,η+1.

Using this relationship iteratively, we can obtain

(

k=1

η+1

∑

j=0

xT

η+1−j,j

r

∑

h+

kQk

)

xη+1−j,j

≤

αη+1xT

0,0

(

[

r

∑

k=1

hkQk

)

(

x0,0

+

η

∑

j=0

αj

xT

η+1−j,0

r

∑

k=1

hkQk

)

xη+1−j,0+xT

0,η+1−j

(

r

∑

k=1

hkQk

)

x0,η+1−j

]

,

≤

η+1

∑

j=0

αj

[

xT

η+1−j,0

(

r

∑

k=1

hkQk

)

xη+1−j,0+xT

0,η+1−j

(

r

∑

k=1

hkQk

)

x0,η+1−j

]

.

Therefore, we have

η+1

∑

j=0

??xη+1−j,j

??2

≤

µ

η+1

∑

j=0

αj[??xη+1−j,0

(

λmin

∑

??2+??x0,η+1−j

??2]

,

(12)

where

µ :=

λmax

r

∑

k=1hkQk

r

k=1h+

)

).

(

kQk

We note that

µ ≤ τ :=

r

∑

i=1λmax(Qi)

εmini(λmin(Qi))

6

Page 7

with 0 < ε < 1. Now denote χκ:=

κ

∑

j=0

??xκ−j,j

??2, and then using the above inequality, we have

}

χ0

χ1

≤

≤

...

τ(|x0,0|2+|x0,0|2),

τα(|x0,0|2+|x0,0|2)+(|x1,0|2+|x0,1|2)

{

{

,

χN

≤

ταN(|x0,0|2+|x0,0|2)+αN−1(|x1,0|2+|x0,1|2)+...+(|xN,0|2+|x0,N|2)

Adding both sides of the above inequality system yields

}

.

N

∑

η=0

χη

≤

τ1−αN+1

1−α

N

∑

k=0

{

|xk,0|2+|x0,k|2}

.

Then from Assumption 1, the right side of the above inequality is bounded, which means: lim

0 as i+ j → ∞, so, lim

Remark 1 Theorem 1 provides the LMI based conditions for the asymptotic stability of 2-D fuzzy systems, which can

be solved efficiently by employing standard numerical software [?]. Actually from the proof of Theorem 1, we see that

the conditions M2< 0 and M3< 0 can be used for the stability analysis of system S. However, it is noticed that

the product terms between the system matrices and the matrix Qkcan not be eliminated in this case. Therefore, the

conditions M2< 0 and M3< 0 is not powerful for controller synthesis.

η→∞χη=0, that is,??xi,j

??2→

r→∞Xr= 0 and then the proof is completed.

?

If the basis-dependent Lyapunov functions reduce to a common quadratic Lyapunov function, by following similar

lines as in the proof of Theorem 1, we obtain the following corollary.

Corollary 1 Consider the fuzzy system S in (2) with Assumption 1. The 2-D discrete fuzzy system S in (2) is

asymptotically stable if there exist matrices X > 0,Y ≥ 0 and Z > 0 satisfying

where m,n = 1,2,...,r; m < n ≤ r and Q := X +2Y +Z.

Remark 2 From Corollary 1, we can find that the basis-independent result is a special case of basis-dependent result.

Thus Theorem 1 is less conservative than that based on Corollary 1.

−X

∗

∗

−Y

−Z

∗

1mQ

AT

−Q

∗

AT

AT

−Q

AT

AT

0

−Q

1mQ

2mQ

<

0,

−X

∗

∗

∗

−Y

−Z

∗

∗

AT

1nQ

2mQ

2nQ

<

0.

(13)

Remark 3 From the proof of Theorem 1, we see that when the systems are linear time-invariant and the basis-

dependentLyapunovfunctionsbecomebasis-independentLyapunovfunctions, M1≡M2andM3, M4disappear. There-

fore, LMIs (4) and (5) become

which has been obtained in [?]. From this point of view, Theorem 1 and Corollary 1 can be seen as an extension of

[?] to 2-D fuzzy systems.

−X

∗

∗

−Y

−Z

∗

AT

AT

−Q

1Q

2Q

< 0,

(14)

7

Page 8

Since Theorem 1 is derived from Tuan’s results [?], in the following we will show that we can also establish the

asymptotic stability on the basis of another elegant stability result for 2-D systems proposed in [?]. As the proof is

analogous to that of Theorem 2 in [?], it is omitted for brevity.

Theorem 2 LMIs (4) and (5) in Theorem 1 hold if and only if there exist matrices Rk> 0 and Tk> 0 satisfying

where k,m,n = 1,2,...,r; m < n ≤ r.

Remark 4 Similar to Remark 3, when the systems are linear time-invariant and the basis-dependent Lyapunov func-

tions become common quadratic Lyapunov functions, LMIs (15) and (16) will reduce to

which has been obtained in [?].

Tm−Rn

∗

∗

0

−Tm−Tn

∗

∗

0AT

AT

−Rk

AT

AT

0

−Rk

1mRk

2mRk

−Tm

∗

AT

AT

−Rk

∗

<

0,

(15)

Tm−Rm+Tn−Rn

∗

∗

∗

1mRk

2nRk

1nRk

2mRk

<

0,

(16)

T −R

∗

∗

0AT

AT

−R

1R

2R

−T

∗

< 0,

(17)

Remark 5 Theorem 2 is in fact equivalent to Theorem 1 (please refer to Theorem 3 in [?]). In the following, we will

only present the stabilization results based on Theorem 1, and equivalent results based on Theorem 2 can be readily

obtained by employing similar arguments.

4Stabilization of 2-D Fuzzy Systems

In this section, we shall deal with the problem of stabilization for systems via a parallel distributed compensation

(PDC) fuzzy controller. More specifically, we are interested in finding a PDC fuzzy controller such that the closed-

loop system with this controller is asymptotically stable.

In the PDC design, each control rule is designed from the corresponding rule of a T-S fuzzy model. The designed

fuzzy controller shares the same fuzzy sets with the fuzzy model in the premise parts. For the fuzzy models in (2), we

construct the following fuzzy controller via the PDC:

Control Rule k: IF θ1(i,j)is µk1and θ2(i,j)is µk2and ... and θp(i,j)is µkp, THEN

ui,j= −Fkxi,j.

The overall fuzzy controller is represented by

ui,j= −

r

∑

k=1

hk(θi,j)Fkxi,j.

First, the closed-loop system with the PDC fuzzy controller can be given by

r

∑

p=1q=1

Before stating the main result of this section, we present the following proposition first, which is useful in establishing

our results.

xi+1,j+1

=

r

∑

hp(θi,j)hq(θi,j)

{

(A1p−B1pFq)xi,j+1+(A2p−B2pFq)xi+1,j

}

.

8

Page 9

Proposition 1 Consider the 2-D fuzzy system S in (2) with given boundary condition. The 2-D discrete fuzzy system

S in (2) is asymptotically stable if there exist matrices Xk> 0, Yk≥ 0, Zk> 0 and Vksatisfying

where k = 1,2,...,r; 1 ≤ m < n ≤ r and Qk:= Xk+2Yk+Zk.

Proof. If LMIs (18) and (19) hold, we have Vk+VT

have Qk> 0, so that Vkis nonsingular. In addition, we have(Qk−VT

−VT

Therefore, we can conclude from (18) and (19) that

Performing a congruence transformation to (21) and (22) by diag{I,I,V−1

−Xm

∗

∗

AT

AT

Qk−Vk−VT

∗

−Ym

−Zm

∗

AT

AT

1mVk

2mVk

Qk−Vk−VT

AT

AT

0

Qk−Vk−VT

k

<

0,

(18)

−Xm−Xn

∗

∗

∗

−Ym−Yn

−Zm−Zn

∗

∗

1mVk

2nVk

1nVk

2mVk

k

k

<

0,

(19)

k−Qk> 0. From the conditions Xk> 0, Yk≥ 0, Zk> 0, we

k

)Q−1

k.

k(Qk−Vk) ≥ 0, which implies

kQ−1

kVk≤ Qk−Vk−VT

(20)

−Xm

∗

∗

−Ym

−Zm

∗

AT

AT

kQ−1

AT

AT

0

−VT

1mVk

2mVk

−VT

kVk

<

0,

(21)

−Xm−Xn

∗

∗

∗

−Ym−Yn

−Zm−Zn

∗

∗

AT

AT

kQ−1

∗

1mVk

2nVk

1nVk

2mVk

−VT

kVk

kQ−1

kVk

<

0.

(22)

kQk

}and diag{I,I,V−1

kQk,V−1

kQk

}yields

(4) and (5), and then the proof is completed.

?

Based on Proposition 1, we are in a position to establish conditions to the stabilization problem for system S in

(2).

Theorem 3 The 2-D fuzzy system S in (2) can be stabilized via a PDC fuzzy controller if there exist matrices Xk> 0,

Yk≥ 0, Zk> 0, Fland Glsatisfying

lAT

GT

Qk−GT

−Xm

∗

∗

1m−Fl

lAT

−Ym

−Zm

∗

TBT

TBT

GT

GT

lAT

lAT

Qk−GT

GT

GT

1m−Fl

2m−Fl

TBT

TBT

1m

2m

l−Gl

TBT

TBT

0

l−Gl

<

0,

(23)

−Xm−Xp

∗

∗

∗

−Ym−Yp

−Zm−Zp

∗

∗

GT

1m

lAT

lAT

1p−Fl

2m−Fl

1p

2p−Fl

2p2m

l−Gl

∗

Qk−GT

<

0.

(24)

Moreover, if the above conditions have feasible solutions, the controller gain matrices are given by

Fl= FlG−1

l,

(25)

where Qk? Xk+2Yk+Zk; k, l = 1,2,...,r; 1 ≤ m ≤ p ≤ r.

9

Page 10

Proof. According to Proposition 1, the closed-loop system is asymptotically stable if there exist matrices Xk, Yk,

ZkandVlsatisfying

The congruence transformations to (26) and (27) by diag{V−1

−Xm

∗

∗

−Ym

−Zm

∗

(A1m−B1mFl)TVl

(A2m−B2mFl)TVl

Qk−Vl−VT

(A1p−B1pFl)TVl

(A2m−B2mFl)TVl

0

Qk−Vl−VT

,V−1

ll

l

<

0,

(26)

−Xm−Xp

∗

∗

∗

−Ym−Yp

−Zm−Zp

∗

∗

(A1m−B1mFl)TVl

(A2p−B2pFl)TVl

Qk−Vl−VT

∗

l

l

<

0.

(27)

l

,V−1

}and diag{V−1

l

,V−1

l

,V−1

l

,V−1

l

}together

with a change of variables by

Xm

Zm

? V−T

? V−T

l

XmV−1

ZmV−1

l

, Ym?V−T

, Gl?V−1

l

YmV−1

, Fl? FlV−1

l

,

llll

(28)

yield LMIs (23) and (24). In addition, we know that if LMIs (23) and (24) are feasible, the control law can be given

by (25) and the proof is completed.

?

Remark 6 Theorem 3 solves the stabilization problem on the basis of Proposition 1. It should be pointed out that

the LMI conditions in Theorem 1 contain product terms between the system matrices and the matrix Qk(which is a

substitute for Xk+2Yk+Zk). Therefore, it is not an easy task to solve the stabilization problem based on Theorem

1. On the other hand, by introducing the slack variable V, Proposition 1 eliminates the product terms involving the

matrix Qk. In such a way, the dilated LMI conditions in Proposition 1 are not only preferable for stability analysis of

the systems, but also powerful for controller synthesis.

5Illustrative Examples

In this section, we will use two examples to illustrate the applicability of the approach proposed in this paper.

Example 1 We consider a 2-D system with the following system matrices:

A1

=

[

[

−0.15+γsin(x1(i,j+1))

−0.15

0.06

−0.5

−0.3

0.5

0.0025

]

,

A2

=

0.005

]

,

(29)

where xi,jis the local state of coordinates (i, j) and

xi,j∈

(

−π

2,π

2

)

.

We have the fuzzy model for the nonlinear system

xi+1,j+1=

2

∑

k=1

hk(θi,j)

{

A1kxi,j+1+A2kxi+1,j+B1kui,j+1+B2kui+1,j

}

,

10

Page 11

with

A11

=

[

[

γ −0.15

−0.15

0.06

−0.3

0.5

0.0025

]

, A12=

[

−γ −0.15

−0.15

0.06

−0.3

0.5

0.0025

]

,

A21

=

−0.5

0.005

]

, A22=

[

−0.5

0.005

]

and

h1(θi,j)=

1+sin(x1(i,j+1))

2

1−sin(x1(i,j+1))

2

,

h2(θi,j)=

.

Whenγ =0.5, usingLMIControlToolboxtosolveLMIs(13)inCorollary1, weobtainthefollowingfeasiblesolutions:

[

−26.1721

By solving LMIs (4) and (5) in Theorem 1, our results give the following feasible solutions:

[

−0.0484

[

−0.0262

[

−0.0257

[

−0.1265

Figure 1 and Figure 2 show the state variables of the above system. It shows that both the basis-dependent and basis-

independent results can guarantee stability for the system S in (2).

When γ = 0.8, using LMI Control Toolbox to solve LMIs (13) in Corollary 1, the LMIs are infeasible. However, by

solving LMIs (4) and (5) in Theorem 1, our results give the following feasible solutions:

[

−0.4595

[

−0.1041

[

−0.0713

[

−0.7390

Figure 3 and Figure 4 show the state variables of the above system. It shows that the basis-dependent results

can guarantee stability for the system S in (2), while basis-independent results cannot. Therefore results based on

basis-dependent Lyapunov functions are less conservative than those based on single quadratic Lyapunov functions.

X =

66.8992

−26.1721

70.1356

]

, Y =

[

28.8264

−9.1054

−9.1054

38.7482

]

, Z =

[

48.2373

−5.7484

−5.7484

73.1166

]

.

X1

=

0.7461

−0.0484

0.8671

]

]

]

]

, X2=

[

[

[

[

0.4919

−0.0227

0.1607

−0.0070

0.4298

−0.0059

1.2431

−0.0427

−0.0227

0.5640

]

]

]

]

,

Y1

=

0.2556

−0.0262

0.2920

, Y2=

−0.0070

0.1291

,

Z1

=

0.6641

−0.0257

0.8673

, Z2=

−0.0059

0.5641

,

Q1

=

1.9214

−0.1265

2.3185

, Q2=

−0.0427

1.3862

.

X1

=

1.2577

−0.4595

1.0741

]

]

]

]

, X2=

[

[

[

[

0.7389

−0.2237

0.2872

−0.0292

0.5048

−0.0417

1.8181

−0.3237

−0.2237

0.7193

]

]

]

]

,

Y1

=

0.5106

−0.1041

0.4019

, Y2=

−0.0292

0.2010

,

Z1

=

0.7286

−0.0713

1.1435

, Z2=

−0.0417

0.7484

,

Q1

=

3.0074

−0.7390

3.0215

, Q2=

−0.3237

1.8696

.

11

Page 12

0

5

10

15

0

5

10

15

−1

−0.5

0

0.5

1

Fig. 1. State variable x1of open-loop system in Example 1 (γ = 0.5)

0

5

10

15

0

5

10

15

−0.5

0

0.5

1

Fig. 2. State variable x2of open-loop system in Example 1 (γ = 0.5)

12

Page 13

0

5

10

15

0

5

10

15

−0.5

0

0.5

1

1.5

Fig. 3. State variable x1of open-loop system in Example 1 (γ = 0.8)

0

5

10

15

0

5

10

15

−0.5

0

0.5

1

Fig. 4. State variable x2of open-loop system in Example 1 (γ = 0.8)

13

Page 14

Example 2 Consider a 2-D system with two variables xi,j+1, xi+1,jand the following system matrices

[

[

[

For simplicity, we assume that

x(i,j)∈

We have the fuzzy model for the nonlinear system

{

with

[

00.5

[

[

where

A1

=

0.5sin(x1(i,j+1))

0

]

0.5

0.5

0

0.5

]

,

A2

=

0.5

0

1

1

,

B1

=

]

,B2=

[

0.5

1

]

.

(30)

(

−π

2,π

2

)

.

xi+1,j+1=

2

∑

k=1

hk(θi,j)

A1kxi,j+1+A2kxi+1,j+B1kui,j+1+B2kui+1,j

}

,

(31)

A11

=

0.50

]

, A12=

[

−0.5

0

0

0.5

[

]

0.5

0.5

,

A21

=

A22=

0.5

0

1

1

]

, B11= B12=

]

,

B21

=

B22=

0.5

1

]

h1(θi,j)=

1+sin(x1(i,j+1))

2

1−sin(x1(i,j+1))

2

,

h2(θi,j)=

.

Figure 5 and Figure 6 show the state variables of the above system. It can be seen that the open-loop system is not

asymptotically stable. Our purpose now is to design a controller such that the closed-loop system is asymptotically

stable. By solving LMIs (23) and (24) in Theorem 3, we can obtain a feasible solution with

[

−0.0277

[

0.01090.0389

[

0.01560.0677

[

0.00980.2436

Then, from (25), the corresponding controller gain matrices are given by

X1

=

0.1349

−0.0277

0.0980

]

, X2=

[

0.0634

0.0172

0.0172

0.0689

]

,

Y1

=

0.03630.0109

]

]

]

, Y2=

[

[

[

0.0545

0.0145

0.0145

0.0451

]

]

]

,

Z1

=

0.05300.0156

, Z2=

0.0590

0.0094

0.0094

0.0683

,

Q1

=

0.26050.0098

, Q2=

0.2314

0.0557

0.0557

0.2275

.

F1

F2

=[ 0.01921.0082 ],

=[ 0.16421.0791 ].

Figure 7 and Figure 8 show that the state variables of the closed-loop system converge to zero. This shows that the

PDC fuzzy controller designed in the paper can stabilize the originally unstable system.

14

Page 15

0

2

4

6

8

10

0

5

10

0

50

100

150

200

Fig. 5. State variable x1of open-loop system in Example 2

0

2

4

6

8

10

0

5

10

0

50

100

150

200

Fig. 6. State variable x2of open-loop system in Example 2

15