LMI Based Stability Analysis

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 3, JUNE 2011 505

LMI-Based Stability Analysis forFuzzy-Model-Based Control Systems

Using Artificial TS Fuzzy ModelH. K. Lam, Senior Member, IEEE

AbstractThis paper investigates the stability of fuzzy-model-based (FMB) control systems. An alternative stability-analysis ap-proach using an artificial fuzzy system based on the Lyapunovstability theory is proposed. To facilitate the stability analysis,the continuous membership functions of the TakagiSugeno (TS)fuzzy model are represented by the staircase ones. With the niceproperty of the staircase membership functions, it turns the set ofinfinite number of linear-matrix-inequality (LMI) based stabilityconditions into a finite one. Furthermore, the staircase member-ship functions carrying system information can be brought to the

stability conditions to relax the stability conditions. The stability ofthe original FMB control systems is guaranteed by the satisfactionof the LMI-based stability conditions. The proposed stability anal-ysis is applied to the FMB control systems of which the TS fuzzymodel and fuzzy controller do not share the same premise mem-bership functions and, thus, is able to enhance the design flexibilityof the fuzzy controller. A simulation example is given to illustratethe merits of the proposed approach.

Index TermsFuzzy control, linear-matrix inequality (LMI),stability analysis, staircase membership functions, TakagiSugeno(TS) fuzzy model.

I. INTRODUCTION

FUZZY-MODEL-BASED (FMB) control [1] offers a sys-

tematic approach to deal with nonlinear control problems.

With the TakagiSugeno (TS) fuzzy model [2], [3], it offers a

general framework to represent the nonlinear system as an aver-

age weighted sum of linear subsystems. The TS fuzzy model

exhibits a nice property to facilitate the stability analysis and

control synthesis. A fuzzy controller was proposed in [4] and [5]

to close the feedback loop of the nonlinear plant represented by

the TS fuzzy model to form an FMB control system.

Stability is an essential issue for the FMB control systems. It

has drawn the attention of the researchers in the fuzzy-control

community for the past decades. The Lyapunov stability theory

is the most-popular analysis tool to investigate the stability ofthe FMB control systems. Basic stability conditions in the form

of Lyapunov inequalities were reported in [4] and [5]. If there

exists a solution to the stability conditions, the FMB control

Manuscript received November 9, 2009; revised April 27, 2010 and October25,2010; accepted January 5, 2011. Date of publication February 17, 2011; dateof current version June 6, 2011. This work was supported by the Kings CollegeLondon and by the Engineering and Physical Sciences Research Council underProject EP/E05627X/1.

The author is with the Division of Engineering, Kings College London,London, WC2R 2LS, U.K. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TFUZZ.2011.2116027

system is guaranteed to be asymptotically stable. As the mem-

bership functions are not considered, the stability conditions

achieved are very conservative. However, it allows the member-

ship functions to be chosen freely for the fuzzy controller, and

thus, enhances the controller-design flexibility. In [6][8], the

same authors and other researchers proposed some approaches

to bring the boundary information of the membership functions,

such as the lower and upper bounds of the membership func-

tions and/or their multiplications, to the stability analysis forthis class of FMB control systems. As more information of the

system nonlinearity is considered, more relaxed stability con-

ditions can be obtained, compared with the work given in [4]

and [5].

The parallel-distribution-compensation (PDC) technique was

proposed in [9] for the design of fuzzy controller. It is required

that the fuzzy controller share the same premise membership

functions as those of the TS fuzzy model. As some cross terms

of membership functions of the FMB control system are in

common, they can be grouped in the stability analysis, thereby

leading to more relaxed stability conditions. Further relaxed

stability conditions were reported in [10][17] by utilizing the

property of the fuzzy summations at different levels. As the de-sign flexibility of the fuzzy controller under the PDC vanishes

(i.e., the membership functions of the fuzzy controller are not

allowed to be chosen freely but determined by the TS fuzzy

model), the implementation cost (i.e., the cost of implementing

the fuzzy controller physically) will increase if the membership

functions of the TS fuzzy model are complicated. The stabil-

ity conditions and the design of the fuzzy controller were for-

mulated as a linearmatrix-inequality (LMI) problem [4][17].

The solution to the LMI-based stability conditions can be found

numerically using convex-programming techniques [18]. By in-

corporating the boundary information of the membership func-

tions [19], [20], more relaxed stability conditions for the FMBcontrol systems under the PDC can be obtained.

It should be noted that the stability conditions in [9][17]

contain no information about the membership functions of

the TS fuzzy model and the fuzzy controller. Once a fea-

sible solution is found, the stability of the FMB control

system is guaranteed for any shapes of membership functions.

Practically, a fuzzy controller should be designed for a specified

nonlinear plant but not for a family. By considering the nonlin-

ear plant to be controlled, the membership functions are already

known, and it is a waste if we do not consider this information

in the stability analysis. In this paper, we propose an alterna-

tive stability-analysis approach to investigate the stability of the

1063-6706/$26.00 2011 IEEE


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506 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 3, JUNE 2011

FMB control systems (of which the TS fuzzy model and fuzzy

controller not sharing the same premise membership functions)

with the consideration of the information of membership func-

tions. As the membership functions are continuous in values,

the number of stability conditions will become infinite when

the membership functions are included. To circumvent the dif-

ficulty, the staircase membership functions [21] are employedto facilitate the stability analysis. We propose an artificial TS

fuzzy model, with the staircase membership functions replac-

ing the original ones to conduct the stability analysis. It should

be noted that the artificial TS fuzzy model is a mathemati-

cal tool for stability analysis and not necessarily implemented

physically. Stabilization of the artificial FMB control system

(which is formed by the artificial TS fuzzy model and the

fuzzy controller connected in a closed loop) implies that of the

original FMB control system (which is formed by the original

TS fuzzy model and the fuzzy controller). As the staircase

membership functions have finite number of levels, it leads to a

finite number of stability conditions, even though the staircase

membership functions are incorporated. The proposed stabilityconditions are applied to the FMB control system that both TS

fuzzy model and fuzzy controller do not share the same premise

membership functions. It thus allows a greater design flexibil-

ity for the fuzzy controller, and its implementation cost can be

lowered by employing some simple membership functions.

This paper is organizedas follows.In Section II, theTS fuzzy

model and the fuzzy controller are briefly presented. In Section

III, the stability of the FMB control systems is investigated

through the artificial TS fuzzy model. Stability conditions in

terms of LMIs are derived to guarantee the system stability

and synthesize the fuzzy controller. In Section IV, a simulation

example is given to demonstrate the merits of the proposedstability conditions. In Section V, a conclusion is drawn.

II. TAKAGISUGENO FUZZY MODEL AND FUZZY CONTROLLER

WITH STAIRCASE MEMBERSHIP FUNCTIONS

The TS fuzzy model and the proposed fuzzy controller with

staircase membership functions are briefly presented in the fol-

lowing.

A. TakagiSugeno Fuzzy Model

Letting p be the number of fuzzy rules describing the nonlin-

ear plant. The ith rule is of the following format:

Rule i : IF f1 (x(t)) is Mi1 AND AND f (x(t)) is M

i

THEN x(t) = Aix(t) + Bi u(t) (1)

where Mi is a fuzzy term of rule i corresponding to the knownfunction f (x(t)), = 1, 2, . . . , , i = 1, 2, . . . , p, isa positive integer, Ai

nn and Bi n m are the known

constant system and input matrices, respectively, x(t) n isthe system state vector, and u(t) m is the input vector. Thesystem dynamics is described by

x(t) =

p

i=1

wi (x(t)) (Aix(t) + Biu(t)) (2)

wherep

i= 1

wi (x(t)) = 1, wi (x(t)) [0, 1] i (3)

wi (x(t)) =

Mi1

(f1 (x(t))) Mi2

(f2 (x(t))) Mi

(f (x(t)))pk =1

Mk1 (f1 (x(t)))Mk2 (f2 (x(t))) Mk

(f (x(t))

(4)

are the normalized grades of membership, and Mi (f (x(t)))are the grades of membership corresponding to the fuzzy terms

Mi .

B. Fuzzy Controller With Staircase Membership Functions

A fuzzy controller with p fuzzy rules is to be designed for

the nonlinear plant. The jth rule of the fuzzy controller is of the

following format:

Rulej : IF g1 (x(t)) is Nj1 AND . . . AND g (x(t)) is N

j

THENu(t) = Gjx(t) (5)

where Nj is a fuzzy term of rule j corresponding to the knownfunction g(x(t)), = 1, 2, . . . , , j = 1, 2, . . . , p, is apositive integer, and Gj

m n is the feedback gain of rule

j to be designed. The inferred output of the fuzzy controller is

given by

u(t) =

p

j =1mj (x(t))Gjx(t) (6)

wherep

j = 1

mj (x(t)) = 1, mj (x(t)) [0, 1] j (7)

mj (x(t)) =

Nj1(g1 (x(t))) Nj2

(g2 (x(t))) Nj

(g (x(t)))pk =1

Nk1 (g1(x(t)))Nk2 (g2(x(t))) Nk

(g (x(t)))(8)

are the normalized grades of membership, and Nj(g(x(t)))

are the grades of membership corresponding to the fuzzy terms

Nj. To facilitate the stability analysis, staircase membershipfunctions are employed for the fuzzy controller and will be

discussed later on.

It should be noted that the fuzzy controller with continuous or

staircase membership functions is in the same form of (6). When

mj (x(t)) is governed by a staircase function, the fuzzy controller

becomes a staircase-membership-function fuzzy controller. By

sampling each state variable, we denote the sample state vec-

tor as x(t). The grades of membership of the fuzzy controllercorresponding to x(t), which is denoted as mj (x(t)), can beobtained. The value ofmj (x(t)) is equal to mj (x(t)), depend-

ing on the domain in which x(t) is lying. For example, let us


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LAM: LMI-BASED STABILITY ANALYSIS FOR FUZZY-MODEL-BASED CONTROL SYSTEMS USING ARTIFICIAL TS FUZZY MODEL 507

consider mj (x1 (t)) depending on x1 (t) only and x1 (t) beingsampled at 5, 4, . . . , 4, 5. For 4.5 < x1 (t) 5.5, we havex1 (t) = 5, and mj (x1 (t)) = mj (x1 (t)) = mj (5). By this way,the membership grades of the staircase membership functions

can be defined.

III. STABILITY ANALYSISIn this section, the stability of the FMB control system, which

is formed by the TS fuzzy model (2) and the proposed fuzzy

controller (6) connected in a closed loop, is investigated. From

(2) and (6), the FMB control system is obtained as follows:

x(t) =

pi=1

wi(x(t))

Aix(t)+Bi

p

j = 1

mj (x(t))Gjx(t)

=

pi=1

pj =1

wi (x(t))mj (x(t)) (Ai +BiGj )x(t). (9)

Remark 1: It was reported in [5] and [9] that the FMB con-trol system (9) is guaranteed to be asymptotically stable if

there exists a symmetric positive-definite matrix P such that

(Ai +BiGj )TP+P (Ai +BiGj ) < 0 for all i and j. As the

stability conditions are independent of the membership func-

tions wi (x(t)) and mj (x(t)), the membership functions mj (x(t))are allowed to be chosen flexibly and lead to lower implemen-

tation cost of the fuzzy controller when simple membership

functions are employed. However, the stability conditions are

very conservative. When mi (x(t)) = wi (x(t)) for all i are con-sidered (under the PDC design), various relaxed LMI-based

stability conditions can be found in [9][17]. If the membership

functions of the TS fuzzy model wi (x(t)) are very complicatedin structure, it will lead to a high implementation cost for the

fuzzy controller.

To facilitate the stability analysis of (9), the following artifi-

cial FMB control system is proposed:

x(t) =

pi=1

pj =1

( wi (x(t)) + wi (x(t), x(t)))

mj (x(t)) (Ai +BiGj )x(t) Ce(t)

=

pi=1

pj =1

( wi (x(t)) + wi (x(t), x(t)))

mj (x(t)) (Ai +BiGj ) x(t)

+

pi=1

pj = 1

( wi (x(t)) + wi (x(t), x(t)))

mj (x(t)) (Ai +BiGj ) e(t) Ce(t) (10)

where x(t) n is the system state vector,e(t) = x(t) x(t),wi (x(t)) is a staircase nonlinear function to be designed,wi (x(t), x(t)) takes the scalar value either i or i at anyinstant, and C = CT n n is a matrix to be determined.In the following, the matrix C is determined in a way that

e(t) 0 as time t . As a result, x(t) x(t) as time t

can be achieved, and thus, the stability of the artificial FMB

control system implies that of the original FMB control system

(9).

Remark 2: Similar to the approach in Remark 1, we denote

the sample grades of membership of the TS fuzzy model as

wi (x(t)) corresponding to the sample state vector x(t). Thevalue of wi (x(t)) is equal to wi (x(t)) depending on the do-

main ofx

(t) lying in. For example, considering wi (x1 (t))depending on x1 (t) only and x1 (t) being sampled at 5, 4,. . . , 4, 5. For 4.5 < x1 (t) 5.5, we have x1 (t) = 5. Thus,wi (x1 (t)) = wi (x1 (t)) = wi (5) is determined.

Remark 3: The artificial FMB control system (10) and the

staircase membership functions are mathematical tools to facil-

itate the stability analysis of the original FMB control system

(9). In order to show their advantages, two cases of FMB control

systems with the continuous and staircase membershipfunctions

are discussed in the following.

Case 1: Considering mj (x(t)) as ordinary membership func-

tions (continuous membership functions), the FMB control

system (9) has continuous membership grades, i.e., wi (x(t))

and mj (x(t)). By taking the Lyapunov functions as V(t) =x(t)TPx(t), where P is a symmetric positive-definite matrix,along the line of derivation in [5] and [9], the stability condition

governing the system stability of (9) is achieved as

=

pi= 1

pj = 1

wi (x(t))mj (x(t))

(Ai +BiGj )TP

+P (Ai +BiGj )

< 0.

With the consideration of the membership functions in the

stability analysis, it has to make sure that < 0 for all val-ues of wi (x(t)) and mj (x(t)). Thus, the number of LMIs be-comes infinite, and it is impractical to find the solution nu-

merically. A workaround is given in [5] and [9]; from Remark

1, the system stability is guaranteed by (Ai +BiGj )TP +

P (Ai +BiGj ) < 0 for all i and j. When these stability condi-tions are satisfied, < 0 holds. However, the stability condi-tions are very conservative.

Case 2: We consider the FMB control systems with the stair-

case membership functions. For the FMB control system (10),

wi (x(t)) + wi (x(t), x(t)) and mj (x(t)) are both staircase func-tions and take discrete values in a finite set. When e(t) 0 (asshown later on), the FMB control system (10) becomes

x(t) =

pi=1

pj =1

( wi (x(t)) + wi (x(t), x(t)))mj (x(t))

(Ai +BiGj )x(t).

As the membership functions are staircase and take discrete

values in a finite set, the stability of the FMB control system

(10) is guaranteed by

pi= 1

pj =1

( wi (x(t)) + wi (x(t), x(t)))mj (x(t))mj (x(t))

(Ai +BiGj )TP+P(Ai +BiGj )

< 0


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for all i, j, and all possible values ofwi (x(t)) + wi (x(t), x(t))and mj (x(t)). Compared with the two sets of stability conditions

considered in the two cases, in order to incorporate the mem-

bership function to the stability conditions, the one in Case 1

has to consider all continuous values of wi (x(t)) and mj (x(t))(i.e., infinite number of values). It is thus impractical to find

the solution using convex-programming techniques. However,the proposed one with staircase membership functions in Case

2 only needs to consider the discrete membership grades in a

finite set. Thus, a finite number of LMIs with the staircase mem-

bership functions carrying the nonlinearity information of the

plant can be obtained.

The stability analysis of the FMB control system (9) is in-

vestigated through the artificial FMB control system (10) in the

following section. For brevity, wi (x(t)), mj (x(t)), wi (x(t)), andwi (x(t), x(t)) are denoted as wi , mj , wi , and wi , respectively.The equality

pi=1 wi =

pj =1 mj =

pi= 1

pj = 1 wi mj = 1

given by the property of the membership functions is utilized in

the following stability analysis. From (9) and (10), we consider

the error system as follows:

e(t)= x(t) x(t)

=

pi= 1

pj =1

(wi wi wi ) mj (Ai +BiGj )x(t) +Ce(t).

(11)

Let us consider the following quadratic Lyapunov function can-

didate:

V(t) = x(t)TP1 x(t) + e(t)TP2e(t) (12)

where 0 < P1 = PT1

nn , and 0 < P2 = PT2

n n . In

the following analysis, it will be shown that V(t) < 0 for x(t)and e(t) (excluding x(t) = e(t) = 0), which implies the asymp-totic stability of the systems (10) and (11), i.e., x(t) 0, ande(t) 0 as time t . Once e(t) 0 is achieved, we havex(t) x(t), which implies the asymptotic stability of the orig-

inal FMB control system (9), i.e., x(t) 0. The artificial FMBcontrol system (10) is a mathematical tool to facilitate the sta-bility analysis of the original FMB control system (9) using the

property of the staircase membership functions. From (10) to

(12), we have (13), shown at the bottom of the page, where wi ,i = 1, 2, . . . , p are the staircase functions designed such that|wi wi | i , where i is a scalar. We choose

wi = i sgn

p

j = 1

mje(t)TP2 (Ai +BiGj )x(t)

(14)

where the sign function is defined as

sgn(z) =

1, z > 01, otherwise.

Remark 4: As the artificial FMB control system (10) is not

necessarily implemented, and to facilitate stability analysis of

FMB control system (9) only, the complexity of wi and wi de-fined in (14) will not complicate the physical implementation of

the proposed fuzzy controller (6). Although the values of some

matrices are unknown in (14) at the moment, it is straightfor-

ward to see that wi takes the value of either i or I, whichoffers enough information for the subsequent stability analysis.

V(t) = x(t)TP1 x(t) + x(t)TP1 x(t) + e(t)

TP2e(t) + e(t)TP2 e(t)

=

pi=1

pj =1

( wi + wi )mj (Ai +BiGj ) x(t) +

pi= 1

pj = 1

( wi + wi )mj (Ai +BiGj ) e(t) Ce(t)

TP1 x(t)

+ x(t)TP1

pi=1

pj =1

( wi + wi )mj (Ai +BiGj ) x(t) +

pi=1

pj = 1

( wi + wi )mj (Ai +BiGj ) e(t) Ce(t)

+

p

i=1p

j =1(wi wi wi ) mj (Ai +BiGj )x(t) + Ce(t)

T

P2e(t)

+ e(t)TP2

p

i=1

pj =1

(wi wi wi ) mj (Ai +BiGj )x(t) + Ce(t)

=

pi= 1

pj =1

( wi + wi )mj x(t)T

(Ai +BiGj )

TP1

+P1 (Ai +BiGj )

x(t)+e(t)T

p

i=1

pj = 1

( wi + wi )mj (Ai +BiGj )T CT

P1 x(t)

+ x(t)TP1

p

i=1

pj = 1

( wi + wi )mj (Ai +BiGj ) C

e(t) + 2 p

i= 1

pj = 1

(wi wi wi ) mje(t)TP2 (Ai +BiGj )x(t)

+ e(t)T CTP2 +P2C e(t). (13)


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From (14), the second-last term on the right-hand side of (13)

becomes

p

i= 1

p

j =1

(wi wi wi ) mje(t)TP2 (Ai +BiGj )x(t)

pi= 1

i sgn

p

k =1

mke(t)TP2 (Ai +BiGk )x(t)

pj = 1


+

pi=1

|wi wi |

pj = 1


pi= 1

i

pj = 1


+

pi=1

|wi wi |

p

j = 1


=

pi= 1

(i + |wi wi |)

p

j = 1


0. (15)

From (13) and (15), let us denote X = P11 , z(t) = X1 x(t)

and z(t) = X1

e(t) and choose C = X1

, P2 = X1

> 0and Gj = NjX1 , where is a nonzero positive scalar, and

Nj m n , j = 1, 2, . . . , p; we have (16), shown at the bottom

of this page, where

( wi , wi , mj )

=

pi= 1

pj = 1

( wi + wi )mj

XATi +AiX

+BiNj +NT

j BTi

pi=1

pj = 1

( wi + wi )mjXATi +N

Tj B

Ti

+ I 2I

and the symbol denotes the transposed-matrix element at

the corresponding position.Remark 5: It can be seen from (16) that V(t) 0 (i.e.,

equality holds when x(t) = z(t) = e(t) = e(t) = 0) is truewhen ( wi , wi , mj ) < 0, which implies the asymptotic sta-bility of the artificial FMB control system (10) and the error

system (11), i.e., x(t) 0, and e(t) 0 as time t . As

V(t)

pi=1

pj = 1

( wi + wi )mj x(t)T ( (Ai +BiGj )

TP1 +P1 (Ai +BiGj ) ) x(t)

+ e(t)T p

i=1

p

j =1

( wi + wi )mj (Ai +BiGj )T CTP1 x(t)

+ x(t)TP1

p

i=1

pj = 1

( wi + wi )mj (Ai +BiGj ) C

e(t) + e(t)T CTP2 +P2C e(t)

=

pi=1

pj = 1

( wi + wi )mj x(t)TX1X

(Ai +BiGj )

TP1

+P1 (Ai +BiGj )

XX1 x(t)

+ e(t)TX1X

p

i= 1

pj = 1

( wi + wi )mj (Ai +BiGj )T +X1

X1 x(t)

+ x(t)TX1 p

i=1

pj =1

( wi + wi )mj (Ai +BiGj ) + X11

XX1e(t) e(t)T X1X1 + X1X1 e(t)

=

pi=1

pj = 1

( wi + wi )mjz(t)T

XATi +AiX

+BiNj +NT

j BTi

z(t) + z(t)T

p

i= 1

pj = 1


Tj B

Ti

+ I

z(t)

+ z(t)T

p

i= 1

pj =1

( wi + wi )mj (AiX +BiNj ) + I

z(t) 2z(t)T z(t)

= z(t)z(t)

T

( wi , wi , mj ) z(t)z(t) (16)


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e(t) = x(t) x(t), e(t) 0 leads to x(t) x(t). As a result,considering both conditions x(t) 0, and x(t) x(t), thisimplies the asymptotic stability of the FMB control system of

(9), i.e., x(t) 0 as time t .The stability analysis results are summarized in the following

theorem.

Theorem 1: The FMB control system (9), which is formed bya nonlinear plant represented by the TS fuzzy model in the form

of (2) and the fuzzy controller in the form of (6) connected in

a closed loop, is guaranteed to be asymptotically stable if there

exist a nonzero positive scalar and matricesX = XT nn

and Nj m n , j = 1, 2, . . . , p, such that the following LMIs

hold.

X > 0

> 0.

( wi (x(t)), wi (x(t), x(t)), mj (x(t))) < 0 for all finite num-ber of discrete values ofwi (x(t)), wi (x(t), x(t)),and mj (x(t)),

where wi (x(t), x(t)) takes the value of either i or iat any instant, and i is a positive scalar. The functionswi (x(t)) and the membership functions of the fuzzy controller(6), mj (x(t)), are designed as staircase functions satisfying|wi (x(t)) wi (x(t))| i . The feedback gains are defined asGj = NjX

1 for all j.

The proof of Theorem 1 follows immediately from

(12) and (16). From (12), it is required that P1 > 0and P2 > 0, leading to X > 0 with the considera-tion of X = P11 , P2 = X

1 > 0, and > 0. From (16), ( wi (x(t)), wi (x(t), x(t)), mj (x(t))) < 0 is required to make

sure that V(t) < 0 (excluding x(t) = z(t) = e(t) = e(t) = 0).

Remark 6: It can be seen from the fuzzy controller (6) thatthe staircase membership functions mj introduce discontinuity

to the control signal. Similar to the switching/sliding-mode con-

trol [22], the system states are continuous but nonsmooth due

to the integral action of the nonlinear plant (2). As a result, con-

sidering the FMB control systems (9) and (10), although there

exist switching components (i.e., sign function and staircase

membership functions), it will not cause discontinuity/jumping

in the system states x(t) and x(t) but will cause nonsmoothness.Let us denote 1 , 2 , . . . , q, where q is an integer, and 0 0 and V(t) < 0, it can be concluded that V(t) is monotonicdecreasing. As a result, the stability of the FMB control system

with staircase membership functions is guaranteed in the sense

of Lyapunov.

Remark 7: Unlike some published work, the proposed sta-

bility analysis has a nice property for relaxing the stability

conditions by considering the staircase membership functions

of both TS fuzzy model and fuzzy controller. It can be seen

that wi (x(t)), wi (x(t), x(t)), and mj (x(t)), which carry the

nonlinearities of the plant and fuzzy controller, are considered

and incorporated to the stability conditions in Theorem 1. The

proposed stability conditions, however, suffer from the follow-

ing two drawbacks. 1) The membership functions of the fuzzy

controller are required to be staircase. However, it may not be

a problem as the staircase membership functions can be con-

structed based on their continuous ones. By rounding off the

inputs of the fuzzy controller to discrete level, the continu-ous membership functions will become the staircase ones. 2)

The number of LMIs will become larger when the number of

discrete grades of membership is increasing, and thus, the com-

putational demand required to find the solution to the stability

conditions will be increased. As the nature of the TS fuzzy

model and fuzzy controller, this problem is difficult to over-

come for the moment in the sense of theoretical development,

but can be alleviated by the advanced computing technology. It

should be noted that the existing stability conditions [12][17]

share the same drawback of high computational demand as

well.

Remark 8: The staircase function wi has to be chosen care-

fully to avoid wi + wi < 0. Referring to (16), it may lead tono solution for the stability conditions in Theorem 1 due to the

negative value of wi + wi in ( wi , wi , mj ) at a few points,while the rest are positive.

Remark 9: In this paper, the stability of the FMB control

systems (9) with the TS fuzzy model and the fuzzy controller

not sharing the same membership functions, i.e., wi (x(t)) =mi (x(t)) for all i, is investigated. The stability conditions in [5]

and [9] can be applied to this class of FMB control systems,

while those in [9][17] cannot be applied. It has been shown

in [5] and [9] that the FMB control system in the form of (9)

is guaranteed to be asymptotically stable if X = XT > 0 and

Qij = XAT

i +AiX +BiNj +NT

j BTi < 0 for all i and j. It

is shown below that if there exists a solution for the stability

conditions in [5] and [9], i.e., X > 0 and Qij < 0 for all i and j,it is also the solution for the stability conditions in Theorem 1.

Rewriting ( wi , wi , mj ) as

pi= 1

pj =1

wi mjQij +

pi=1

pj =1

wi mjQij

pi= 1

pj =1


Tj B

Ti

+ I 2I

and considering the solution of the stability conditions in [5]

and [9] such that Qij < 0 for all i and j, it is obvious thatpi= 1

pj = 1 wi mjQij < 0 as wi can be regarded as the sample

point of wi . In this case, ( wi , wi , mj ) < 0 can be achievedby choosing a sufficiently small value of wi and a sufficientlylarge nonzero positive value of . However, the solution of thestability conditions in Theorem 1 may not be the solution for

those in [5] and [9].

Let us consider the stability conditions in [9][17], which

require that the fuzzy controller and the TS fuzzy model

share the same premise membership functions, i.e., mi (x(t))

= wi (x(t)) for all i. If there exists a solution to the sta-bility conditions in [9][17],

pi=1

pj =1 wi wjQij < 0,

andp

i= 1p

j = 1 wi wjQij < 0 are both true. Hence,


7/9


( wi , wi , mj ) can be written as (*), shown at the bottom of thepage. ( wi , wi , mj ) < 0 can be achieved with the solution ofthe stability conditions in [9][17]. If the value of wi is chosento be sufficiently small, the magnitude ofmj wj is sufficientlysmall by properly choosing the membership functions mj and

the value of > 0 is sufficiently large. Hence, it can be shown

that the solution of the stability conditions in [9][17] is alsoa solution of the proposed stability conditions by choosing the

membership functions with a sufficiently small discrete level.

However, the solution of the proposed stability conditions may

not be the solution for those in [9][17]. It will be shown later on

through a simulation example that the proposed stability con-

ditions are able to offer more relaxed stability analysis results.

IV. SIMULATION EXAMPLE

An example is given in this section to illustrate the merits

of the proposed fuzzy-control approach. Let us consider the

following TS fuzzy model with three rules [16]:

Rule i : IF x1 (t) is Mi1

THEN x(t) = Aix(t) + Bi u(t), i = 1, 2, and 3

(17)

where

A1 =

1.59 7.290.01 0

A2 =

0.02 4.640.35 0.21

A3 =

a 4.330 0.05

, B1 =

10

, B2 =

80

B3 =

b + 61

, 2 a 12, and 2 < b < 50.

The membership functions of the TS fuzzy model are shown

in Fig. 1.

A three-rule fuzzy controller with staircase membership func-

tions is employed to close the feedback loop. The continuous

membership functions are defined as m1 (x1 (t)) = 1 (1/1 +e(x 1 (t)+4) ), m2 (x1 (t)) = 1 m1 (x1 (t)) m3 (x1 (t)), andm3 (x3 (t)) = 1/1 + e

(x1 (t)4) , which are graphically shown

in Fig. 2. To obtain the staircase membership function, x1 (t) is

sampled at interval of = 1. The staircase membership func-tions are defined as mi (x1 (t)) = mi (h) for (h 0.5) < x1 (t)

Fig. 1. Membership functions of the fuzzy model. (Left triangle in solid line)w1 (x1 (t)), (trapezoid in solid line) w2 (x1 (t)), and (right triangle in solid line)w3 (x1 (t)). Staircase functions. (Left dotted line) w1 (x1 (t)), (middle dashedline) w2 (x1 (t)), and (right dashed-dotted line) w3 (x1 (t)).

Fig. 2. Membership functions of the fuzzy controller. (Left solid line)m1 (x1 (t)), (middle solid line) m2 (x1 (t)), and (right solid line) m3 (x1 (t)).Staircase membership functions. (Left dotted line) m1 (x1 (t)), (middle dashedline) m2 (x1 (t)), and (right dashed-dotted line) m3 (x1 (t)).

(h + 0.5), where i = 1, 2, and 3, and h = ,. . ., 10, 9,. . . , 10, . . . , .

Based on this setting, the staircase functions wi (x1 (t)) (asshown in Fig. 1) and wi are chosen according to Remark 8. It canbe checked that i = 0.0417 for all i satisfying the conditions

pi=1

pj = 1

wi (mj + wj wj )Qij +

pi= 1

pj =1

wi mjQij

pi= 1

pj =1


Tj B

Ti

+ I 2I

=

pi= 1

pj = 1

wi wjQij +

pi= 1

pj =1

wi (mj wj )Qij +

pi= 1

pj = 1

wi mjQij

p

i=1p

j =1( wi + wi )mj

XATi +N

Tj B

Ti

+ I 2I

. ()


8/9


Fig. 3. Stabilization region of the stability conditions in Theorem 1 with =1 (which is denoted by o) for the system parameters 2 a 12 and 2 b 50.

Fig. 4. Stabilization regions of the stability conditions in Theorem 1 with = 0.5 (denoted by ) and = 0.25 (denoted by o), respectively, for thesystem parameters 2 a 12 and 2 b 50.

of |wi (x1 (t)) wi (x1 (t))| i . The stability conditions inTheorem 1 are employed to determine the stability of the FMB

control system with the help of MATLAB LMI toolbox, and

the stabilization region, as indicated by open circles (o), is

shown in Fig. 3. Referring to Fig. 3, each circle indicates that

there exists a feasible solution to the stability conditions in

Theorem 1.

For comparison purposes, we employ the stability condi-

tions in [5] and [9], which can be applied to the case that both

TS fuzzy model and fuzzy controller have different member-

ship functions, to determine the stabilization region. However,

no. stabilization region can be found. Furthermore, the stability

conditions in [6][8] for unmatched premise membership func-

tions, which consider the boundary information of the mem-

bership functions of the TS fuzzy model and fuzzy controller,

are employed to determine the stabilization region; however, no

stabilization region can be found. It should be noted that since

the membership functions of both TS fuzzy model and fuzzy

controller considered in this simulation example are different,

the stability conditions in [9][17] cannot be applied.

To show how the value of influences the stabilization re-gion given by the proposed stability conditions, we choose as 0.5 and 0.25, such that we have i for all i as 0.0208 and0.0104, respectively. Fig. 4 shows the stabilization regions for

both cases, which are indicated by for = 0.5 and o for= 0.25. It can be seen that a larger stabilization region can beobtained for a smaller value of .

V. CONCLUSION

The stability of the FMB control systems has been investi-

gated based on an artificial fuzzy system. A set of finite number

of LMI-based stability conditions has been derived based on the

Lyapunov stability theoryto replace theoriginal onewith infinite

number of stability conditions. The proposed approach is able

to bring the membership functions of both TS fuzzy model and

fuzzy controller to the stability conditions. The proposed stabil-

ity conditions are applied to the FMB control systems with TS

fuzzy model and fuzzy controller not sharing the same premisemembership functions, thereby widening the applicability of the

FMB control approach. A simulation example has been given

to illustrate the merits of the proposed approach.

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H. K. Lam (M98SM10) received the B.Eng.(Hons.) and Ph.D. degrees from the Department ofElectronic and Information Engineering, The HongKong Polytechnic University, Hong Kong, in 1995and 2000, respectively.

In 2000 and 2005, he was with the Department ofElectronic and Information Engineering, The HongKong Polytechnic University, as a Postdoctoral Fel-low and a Research Fellow, respectively. In 2005, hebecame a Lecturer with the Kings College London,London, U.K. He is the coeditor for two edited vol-

umes: Control of Chaotic Nonlinear Circuits (World Scientific, 2009) and theComputational Intelligence and its Applications (World Scientific, 2011). He isthe coauthor of the bookStability Analysis of Fuzzy-Model-Based Control Sys-tems (Springer, 2011). His current research interests include intelligent controlsystems and computational intelligence.

Dr. Lam is currently an Associate Editor of the IEEE TRANSACTIONS ONFUZZY SYSTEMS and the International Journal of Fuzzy Systems.

Documents

LMI Based Stability Analysis