Exponential stability analysis and controller design of fuzzy systems with time-delay

Journal of the Franklin Institute 348 (2011) 865–883 www.elsevier.com/locate/jfranklin

Exponential stability analysis and controller design of fuzzy systems with time-delay Fei-Peng Dan, Shuai-Tian He Institute of Automation, Southeast University, Nanjing 210096, China Received 6 April 2007; received in revised form 19 February 2011; accepted 23 February 2011 Available online 5 March 2011

Abstract Takagi–Sugeno (T–S) fuzzy models can provide an effective representation of complex nonlinear systems with a series of linear input/output submodels in terms of fuzzy sets and fuzzy reasoning. In this paper, the T–S fuzzy model approach is extended to the stability analysis and controller design for nonlinear systems with time delays. An improved stability condition is proposed by introducing adjustable parameters into the Lyapunov–Krasovskii functional. Stabilization approach for fuzzy state feedback is also presented. Sufficient conditions for the existence of fuzzy feedback gain are derived through the numerical solution of a set of obtained linear matrix inequalities (LMIs). Compared with the existing methods in the literature, the proposed approach has less conservatism and both the sizes of delay and its derivative are involved in the criterion. The dynamical performance of the system can be adjusted by changing the adjustable parameters. Finally, two examples are given to show the effectiveness of the proposed approach. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Since Takagi–Sugeno fuzzy model could provide an effective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning [1], in recent years many researches have been done in the area of modeling systems with T–S models (see, e.g., [2–9] and the references cited therein). Tanaka and Sugeno analyze m2d, the stability of T–S fuzzy model with the Lyapunov second method [10]. They pointed out that to keep the fuzzy systems stable, a common positive definite matrix P, which satisfied all the Lyapunov equations must n

Corresponding author. E-mail address: [email protected] (F.-P. Da).

0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.02.012

866

F.-P. Da, S.-T. He / Journal of the Franklin Institute 348 (2011) 865–883

be found. However, it is very difficult to find the common matrix P. On the other hand, in many real systems, time-delay is frequently encountered and can result in unsatisfactory performance or even instability. Therefore, the research on time-delay systems has attracted much attention in past years (see, e.g., [11–13]). Based on T–S fuzzy models, many researchers have investigated the stability analysis and synthesis problems for time-delay systems, for example, Refs. [14–26]. To weaken the conservatism of the conclusion, some researches based on the Lyapunov second method have been carrying out [5,14–18]. The delay-dependent stability of fuzzy systems with time-delay was studied in Refs. [14,27–31], and the exponential stable conditions of time-delay systems were presented in Refs. [16–18]. The methods of controller design were discussed in Refs. [6,15,18–25,27,32]. In Ref. [15], Cao and Frank analyzed the stability of nonlinear time-delay systems with Takagi–Sugeno fuzzy model, the state feedback gain and the fuzzy observer gain were derived through the numerical solution of a set of obtained linear matrix inequalities, which guaranteed the stability of the closed-loop systems. However, in Ref. [15], the stability analysis was based on delay-independent stability analysis methods and the delay in controller was not discussed. To weaken the conservatism of the results, the state feedback controller was designed to guarantee the stability of the closedloop systems based on the delay-dependent stability analysis [27–31,33,34]. The output feedback controllers were discussed in Refs. [19,20]. Other control methods, such as HN control, adaptive control, variable structure control, sliding mode control and guaranteed cost control, were studied in Refs. [6,21–23,27,32]. In the paper, the Takagi–Sugeno fuzzy model of time-delay systems is given firstly. By introducing adjustable parameters into the Lyapunov–Krasovskii functional, an improved exponential stable condition is discussed. Then the state feedback controller is designed, and the state feedback gain is derived through the numerical solution of a set of obtained linear matrix inequalities. Lastly two illustrative examples are given to show the effectiveness of the obtained results. Notations: Throughout the paper the superscript ‘‘T ’’ stands for matrix transposition, and if not stated, matrices are assumed to have compatible dimensions. The standard notations 4(Z) and o(r) are used to denote the positive (semi-positive) and negative (semi-negative) definite of matrices, respectively. A vector norm 99x99 on Rn is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffi n X 99x99 ¼ x2i : i¼1

which define a matrix norm 99P99 on Rn  n induced by the vector norm on Rn as 99P99 ¼ sup0ax2Rn ð99Px99=99x99Þ, xARn. Rn denotes the n dimensional Euclidean space, Rn  m is the set of all n  m real matrices, lmin ðUÞ; lmax ðUÞ are the minimum and maximum eigenvalues of (U), respectively. And the symmetric terms in a symmetric matrix are     X Y X Y denoted by n, e.g., . ¼  Z YT Z 2. Problem formulation Takagi–Sugeno fuzzy model was brought forward by Takagi and Sugeno in Ref. [1], which provided an effective representation of complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning applied to a series of linear input/output (I/O) submodels. As pointed out in Ref. [1], a Takagi–Sugeno fuzzy time-delay model composed of r rules can

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be described as Rule i:IF a1 ðtÞ is Mi1 ; a2 ðtÞ is Mi2 ;    ; ( _ ¼ Ai xðtÞ þ Adi xðttðtÞÞ þ Bi uðtÞ xðtÞ xðtÞ ¼ jðtÞ;

ap ðtÞ is Mip ;

Then

t 2 ½2t; 0

ð1Þ

where x(t)ARn and u(t)ARm are the state and the output, respectively. Ai, Adi and Bi, i=1, 2, y, r, are real constant matrices with appropriate dimensions; 0rt(t)rt is the bounded delay and satisfies t_ ðtÞrdo1, t and d are real constants; r is the number of fuzzy if-then rules; a1(t), a2(t), y, ap(t) are premise variables, Mij (i=1, 2, y , r; j=1, 2, y, p) is the fuzzy set and j(t), defined in the interval [2t, 0], is the initial condition of the state. It is assumed in this paper that the premise variables do not depend on the input u(t). This assumption is employed to avoid a complicated defuzzification process of fuzzy controllers. Given a pair of (x(t), u(t) ), the final outputs of the fuzzy systems are inferred as follows [15]: r X _ ¼ xðtÞ hi ðaðtÞÞ½Ai xðtÞ þ Adi xðttðtÞÞ þ Bi uðtÞ ð2Þ i¼1

P Q where a(t)=[a1(t), a2(t), y, ap(t)], hi ðaðtÞÞ ¼ wi ðaðtÞÞ= ri¼1 wi ðaðtÞÞ, wi ðtÞ ¼ pj¼1 vij ðaj ðtÞÞ, . It is assumed that wi(a(t))Z0, vP ij(aj(t)) is the grade of membership of aj(t) in MijP r r w ðaðtÞÞ40 for all t. Therefore, h (a(t))Z0 and i i i¼1 i¼1 hi ðaðtÞÞ ¼ 1 for all t. In the following, for simplicity, hi(a(t)) is abbreviated as hi. 3. Stability analysis In the following, we firstly derive the exponential stability condition of unforced systems with time-delay described by 8 r X > < xðtÞ _ ¼ hi ½Ai xðtÞ þ Adi xðttðtÞÞ ð3Þ i¼1 > : xðtÞ ¼ jðtÞ; t 2 ½2t; 0 Definition 1. The unforced system (3) is said to be exponential stable if there exist constants k40 and b40 such that 99xðtÞ99rk sup 99jðBÞ99ebt 2trBr0

where b is called the exponential convergence rate. Lemma 1. [27]. For any real matrices Xi, Yi for 1rirr, and P40 with appropriate dimensions, we have r X r r X X 2 hi hj XiT PYj r hi ðXiT PXi þ YiT PYi Þ; i¼1 j¼1

2

r X r X r X r X

i¼1

hi hj hk hl XijT PYkl r

i¼1 j¼1 k¼1 l¼1

where hi(1rirr) are defined as hiZ0,

r X r X

hi hj ðXijT PXij þ YijT PYij Þ

i¼1 j¼1

Pr

k¼1

hi ¼ 1.

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Theorem 1. The unforced system with time-delay described by (3) is exponential stable if there exist matrices P40, Q40, R40, X40 and Y such that 2 T 3 aAi P þ aPAi þ Q þ Y þ Y T þ tX aPAdi Y T gtATi R 6  ðd1ÞQ gtATdi R 7 4 5o0;   gtR i ¼ 1; 2; . . .; r;   gR Y Z0  X

ð4Þ ð5Þ

where a40 and gZ0 are the adjustable parameters. Proof. Construct the following Lyapunov–Krasovskii (L–K) functional: V ðtÞ ¼ aV1 þ V2 þ gV3

ð6Þ

where Z

T

V1 ¼ x ðtÞPxðtÞ;

Z0 Z

t T

V2 ¼

x ðsÞQxðsÞds; ttðtÞ

T

T

t

V3 ¼ t

_ x_ T ðsÞRxðsÞdsdu

tþu

T

and P=P 40, Q=Q 40 and R=R 40 are to be determined. The derivative of the L–K functional V(t) along the solution of Eq. (3) is V_ ðtÞ ¼ aV_ 1 þ V_ 2 þ gV_ 3 :

ð7Þ

Then V_ 1 ¼

r X r X

hi hj f½Ai xðtÞ þ Adi xðttðtÞÞT PxðtÞ þ xT ðtÞP½Aj x þ Adj xðttðtÞÞg:

i¼1 j¼1

By Lemma 1, we get V_ 1 r

r X

T

hi x ðtÞ

"

ATi P þ PAi

PAdi

ATdi P

0

i¼1

# xðtÞ

ð8Þ

where xðtÞ ¼ ½xT ðtÞxT ðttðtÞÞT .

"

V_ 2 ¼ x ðtÞQxðtÞð1_t ðtÞÞx ðttðtÞÞQxðttðtÞÞrx ðtÞ T

T

T

Q

0

0

ðd1ÞQ

# xðtÞ:

ð9Þ

By Lemma 1, we obtain _ V_ 3 ¼ tx_ T ðtÞRxðtÞ

Z

t

_ x_ T ðsÞRxðsÞds tt

# Z t ATi _ ¼t hi hj x ðtÞ T R½Aj Adj xðtÞ x_ T ðsÞRxðsÞds A tt di i¼1 j¼1 " T# Z t r X Ai T _ rt hi x ðtÞ T R½Ai Adi xðtÞ x_ T ðsÞRxðsÞds: Adi ttðtÞ i¼1 r X r X

"

T

ð10Þ

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Substituting Eqs. (8)–(10) into (7) yields (" T ) " T# # r X aAi P þ aPAi þ Q aPAdi Ai T V_ ðtÞr hi x ðtÞ þ gt T R½Ai Adi  aATdi P ðd1ÞQ Adi i¼1 ( " # Z t T r X P þ aPA þ Q aPA aA i di i _ xðtÞg hi xT ðtÞ x_ T ðsÞRxðsÞdsr aATdi P ðd1ÞQ ttðtÞ i¼1 " T# ) Z t Ai _ x_ T ðsÞRxðsÞds þgt T R½Ai Adi  xðtÞg Adi ttðtÞ #T  # " Z t " xðsÞ r X _ _ xðsÞ gR Y þ hi xT ðtÞGi xðtÞ dsr T xðtÞ xðtÞ Y X ttðtÞ i¼1 where Gi ¼

"

aATi P þ aPAi þ Q þ Y þ Y T þ tX "

þgt

ATi ATdi

#

aATdi PY  gR R½Ai Adi ; YT

aPAdi Y T

#

ðd1ÞQ Y

 Z0:

X

By Schur complement, from Eq. (4), we have Gio0. Let c ¼ minflmin ðGi Þ; 1rirrg40, then 2 V_ ðxðtÞÞrc99xðtÞ99 :

Since 99xðtÞ99Z99xðtÞ99 so V_ ðxðtÞÞrc

n X

2

x2i ðtÞ ¼ c99xðtÞ99 :

ð11Þ

i¼1

Let f ðzÞ ¼ ayz þ tyzetz þ 2gdyt2 zetz þ 2gdyt2 z

e2tz 1d

where y ¼ maxflmax ðPÞ; lmax ðQÞ; lmax ðRÞg, d ¼ maxflmax ðATi Ai Þ; lmax ðATdi Adi Þg. It is clear that f(z) is a continuous function for zA(N,þN) and f ð0Þ ¼ 0;

f ðþ1Þ ¼ þ1:

For c40, by using mean value theorem, there exists a constant b40 such that f ðbÞ ¼ c that is abyc þ btyebt þ 2gdbyt2 ebt þ 2gdbyt2

e2bt ¼ 0: 1d

ð12Þ

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From Eqs. (6) and (11), we can get 2 Z0 Z t    d  bt 2 bt bt 4 _ _ e V ðtÞ ¼ e bV ðtÞ þ V ðtÞ re gby 99xðsÞ99 dsdu dt tþu t

Z

t

2

99xðsÞ99 ds þ ðabycÞ99xðtÞ99

þby

2



ttðtÞ

2

rebt 4gby 

Z0 Z Zt t

t

2

_ 99xðsÞ99 dsdu þ by

Z

tþu

3

t

2

2

99xðsÞ99 ds þ ðabycÞ99xðtÞ99 5

tt

Z t 2 2 _ ðs þ ttÞ99xðsÞ99 ds þ by 99xðsÞ99 ds þ ðabycÞ tt tt   Z t Z t i 2 2 2 2 bt _ 99xðtÞ99 re gtby 99xðsÞ99 ds þ by 99xðsÞ99 ds þ ðabycÞ99xðtÞ99 : ¼ ebt gby

tt

ð13Þ

tt

For any T40, integrating both sides of Eq. (13) in the interval [0 T], we have Z TZ t 2 bT _ ebt 99xðsÞ99 dsdt e V ðxðTÞÞV ðxð0ÞÞrgtby Z0 T Ztt Z T t 2 2 bt þ by e 99xðsÞ99 dsdt þ ðabycÞ ebt 99xðtÞ99 dt: tt

0

0

Since Z

T 0

Z

t

2

e 99xðsÞ99 dsdt ¼ bt

Z

tt

Z

T bt

t

2

99xðsÞ99 ds ¼

e dt tt

0

Z

Tt

þ

2

99xðsÞ99 ds

Z0

0

_ e 99xðsÞ99 dsdtrte

tt

bt

ebt dt

Z

T

2

99xðsÞ99 ds

Z

Z

Tt

2

tebðsþtÞ 99xðsÞ99 ds

2

tebðsþtÞ 99xðsÞ99 ds ¼ tebt 

Z

Z

T

t

T

2

_ ebs 99xðsÞ99 ds:

hi ½Ai xðtÞ þ Adi xðttðtÞÞ:

i¼1

r

r X i¼1

2

ebs 99xðsÞ99 ds

Considering that r X

ebt dt s

t

_ 99xðtÞ99 ¼:

T

0 T

Tt

2

sþt

Tt

2

þ

bt

Z 0

ebt dt þ

tebðsþtÞ 99xðsÞ99 ds þ

Z

t

99xðsÞ99 ds

sþt

t

we obtain Z TZ

2

t

Z s

0

r

Z0

pffiffiffi hi ½99Ai xðtÞ99 þ 99Adi xðttðtÞÞ99r d½99xðtÞ99 þ 99xðttðtÞÞ99

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it is not difficult to check that Z T Z TZ t 2 2 2 _ ebt 99xðsÞ99 dsdtr2dtebt ebs ½99xðsÞ99 þ 99xðstðsÞÞ99 ds 0 tt t Z T  Z T 2 2 bt bs bs ¼ 2dte e 99xðsÞ99 ds þ e 99xðstðsÞÞ99 ds t ZtT  Z T ebt 2 2 bt bs bs ¼ 2dte e 99xðsÞ99 ds þ e 99xðsÞ99 ds 1d 2t ZtT  Z Z t T ebt ebt 2 2 2 bt bs bs bs ¼ 2dte e 99xðsÞ99 ds þ e 99xðsÞ99 ds þ e 99xðsÞ99 ds : 1d t 1d 2t t Then, taking Eq. (12) into account, we have   e2bt ebT V ðxðTÞÞrV ðxð0ÞÞ þ abyc þ btyebt þ 2gdbyt2 ebt þ 2gdbyt2 1d   Z T e2bt ebt 99xðtÞ99dt þ btyebt þ 2gdbyt2 ebt þ 2gdbyt2  1d 0 Z0 Z e2bt t bt 2 2  ebt 99xðtÞ99 dt þ 2gdbyt2 e 99xðtÞ99 dt 1d 2t t   bt 2 2 e ¼ V ðxð0ÞÞ þ bty þ 2gdbyt þ 2gdbyt ebt 1d Z Z0 e2bt t bt 2 2 2  ebt 99xðtÞ99 dt þ 2gdbyt2 e 99xðtÞ99 dtrk sup 99jðBÞ99 1d 2t 2trBr0 t

ð14Þ bt

bt

2

2

where k ¼ ay þ tye þ 2gdyt ðe 1Þ þ 4gdyt ðe Noticing that

2bt

=1dÞ40.

2

V ðxðTÞÞZalmin ðPÞ99xðtÞ99

ð15Þ

from Eqs. (14) and (15), we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 99xðTÞ99r sup 99jðBÞ99eðb=2ÞT almin ðPÞ2trBr0 for all T40. So, the unforced system (3) is exponential stable. This completes the proof.

&

Remark 1. In the existing delay-dependent stability conditions in the literature, the L–K functional is often chosen as V ðtÞ ¼ xT ðtÞPxðtÞ þ

Z

t

xT ðsÞQxðsÞds þ ttðtÞ

Z0 Z t

t

_ x_ T ðsÞRxðsÞdsdu:

tþu

It is easy to find that this is a special case of the proposed L–K functional (6) when a=g=1. The reason for introducing adjustable parameters in Eq. (6) is that firstly when

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the adjustable parameters g have taken different values, the influences of the integral term V3(t) would be changed in the delay-dependent stability analysis. Thus compared with the existing ones, the obtained delay-dependent criterion could have lower conservatism by choosing proper adjustable parameters. Secondly, in solving LMIs, the matrices P, Q and R should be self-regulated to obtain the solutions of LMIs, but this regulated range is very small such that the dynamic performance of closed-loop system can be little optimized. With the proposed method, the regulated range of ~ and R~ (where P~ ¼ aP; Q ~ ¼ Q and R ~ Q ~ ¼ gR) can be enlarged by choosing proper P, adjustable parameters, then feedback gains can be tuned. Thus the dynamic performance of closed-loop system can be optimized. To the best of our knowledge, there is little work dealing with such adjustable parameters in literature at present stage. Remark 2. In the current researches on the delay-dependent stability conditions for fuzzy systems [14,19,21,22,27–31], some of them only involve the size of delay-derivative [19,22], and some only involve the size of delay [14,21,27–29]. In this paper and in Refs. [30,31], both the sizes of delay and delay-derivative are involved, which weaken the conservatism of the results. Remark 3. In the proof of exponential stability conditions in Refs. [16,18], the term ebt, for any b40, was introduced into the Lyapunov functional, and ebt occurred in the stability conditions, therefore the constant b40 must be chosen firstly when solving the stability conditions. In this paper and in Ref. [17], the term ebt is not introduced into the Lyapunov functional, and ebt does not occur in the stability conditions, so the constant b40 is not considered when solving the stability conditions. From the proofs in the paper and in Ref. [17], it is easy to find that the exponential convergence rate b is determined by the system matrices and the stability conditions. When the system matrices are fixed, comparing the stability conditions in this paper with that in Ref. [17], it can be seen that the convergence rate b is fixed in Ref. [17] and is adjustable by choosing adjustable parameters in this paper. Another difference is that the stability conditions in Ref. [17] are delay-independent, but in this paper it is delay-dependent. In Theorem 1, if X ¼ g1 Y T R1 Y , the condition (5) can be removed, this is because #   " gR Y gR Y ¼ ¼ G T GZ0  g1 Y T R1 Y  X " # ðgX Þ1=2 ðgX Þ1=2 Y where G ¼ : 0 0 By the Schur complement, the following corollary is obtained. Corollary 1. The unforced system with time-delay described by Eq. (3) is exponential stable if there exist matrices P40, Q40, R40 and Y such that 3 2 T aAi P þ aPAi þ Q þ Y þ Y T aPAdi Y T gtATi R YT 7 6  ðd1ÞQ gtATdi R 0 7 6 7o0; i ¼ 1; 2; . . .; r 6 7 6   gtR 0 5 4    gt1 R where a40 and gZ0 are the adjustable parameters.

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873

4. Controller design Based on the idea of parallel distributed compensation (PDC) [35], we consider the following control law for the Eq. (2): Rule i: IF a1 ðtÞ is Mi1 ; a2 ðtÞ is Mi2 ; . . .; ap ðtÞ is Mip ; uðtÞ ¼ Fi xðtÞ; i ¼ 1; 2; . . .; r

Then

where FiARm  n are the state feedback gains. The overall state feedback fuzzy control law is represented by r X uðtÞ ¼  hi Fi xðtÞ:

ð16Þ

i¼1

From Eqs. (2) and (16), we get the closed-loop system as r X r r X X _ ¼ xðtÞ hi hj ½ðAi Bi Fj ÞxðtÞ þ Adi xðttðtÞÞ ¼ h2i ½Mii xðtÞ þ Nii xðttðtÞÞ i¼1 j¼1 r X

þ

i¼1

hi hj ½ðMij þ Mji ÞxðtÞ þ ðNij þ Nij ÞxðttðtÞÞ

ð17Þ

i;j¼1;ioj

where Mij=AiBiFj, Nij=Adi. The design of the state feedback fuzzy controller is to determine the local feedback gains Fi such that the closed-loop system with time-delay described by Eq. (17) is exponential stable. Analogous to the analysis in Section 3, we have the following result. Theorem 2. There exists a state feedback fuzzy control law (16) such that the closed-loop fuzzy system with time-delay described by Eq. (17) is exponential stable if there exist matrices P40, Q40, R40, X40 and Y satisfying the following matrix inequalities for all i, j=1, 2, y, r, which satisfy hihja0 2 3 aMiiT P þ aPMii þ Q þ Y þ Y T þ tX aPNii Y T gtMiiT R 6  ðd1ÞQ gtNiiT R 7 4 5o0; i ¼ 1; 2; . . .; r;   gtR ð18Þ 2

Yij

aPðNij þ Nji Þ2Y T

6 4   

2ðd1ÞQ 

gR

Y



X

gtðMijT þ MjiT ÞR

3

7 gtðNijT þ NjiT ÞR 5o0; 2gtR

1riojrr;

ð19Þ

 Z0;

ð20Þ

where Yij ¼ aðMijT þ MjiT ÞP þ aPðMij þ Mji Þ þ 2Q þ 2Y þ 2Y T þ 2tX and a40 and gZ0 are the adjustable parameters. Proof. Similar to the proof of Theorems 1, Theorem 2 can be proved.

&

In Eqs. (18) and (19), since the feedback gains Fi ði ¼ 1; 2; . . .; rÞ are involved in the matrices Mij and Nij, the feedback gains cannot be obtained by solving Eqs. (18)–(20) with

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874

LMIs tools in MATLAB directly. In order to get the feedback gains, some transformations are needed. Pre- and post-multiplying Eqs. (18) and (19) by diagfP1 ; P1 ; R1 g and its transpose, respectively, and denoting P ¼ P1 , Q ¼ P1 QP1 , Y ¼ P1 YP1 , X ¼ P1 XP1 , R ¼ R1 , we obtain 2 3 T T aPMiiT þ aMii P þ Q þ Y þ Y þ tX aNii PY gtPMiiT 6 7 6  ðd1ÞQ gtPNiiT 7 4 5o0; i ¼ 1; 2; . . .; r; 



gtR

ð21Þ 2

aðNij þ Nji ÞP2Y

Yij

6 6  4 

T

gtPðMijT þ MjiT Þ

3

7 gtPðNijT þ NjiT Þ 7 5o0; 2gtR

2ðd1ÞQ 

1riojrr

ð22Þ

T

where Yij ¼aPðMijT þ MjiT Þ þ aðMij þ Mji ÞP þ 2Q þ 2Y þ 2Y þ 2tX . Pre- and post-multiplying Eq. (20) by diagfP1 ; P1 g and its transpose, respectively, we have " # 1 gPR P Y Z0:  X Since 1

1

ðPRÞT R ðPRÞT ¼ PR P2P þ RZ0; we have " 1 gPR P

# "

2gPgR Z  X Y



# Y : X

By inserting Mij=AiBiFj, Nij=Adi into Eqs. (21) and (22) and let F i ¼ Fi P, we obtain the following result. Theorem 3. There exists a state feedback fuzzy control law (16) such that the closed-loop fuzzy system with time-delay described by Eq. (17) is exponential stable if there exist matrices P40 , Q40 , R40 , X 40 and Y ; F i satisfying the following LMIs for all i, j=1, 2, y, r, which satisfy hihja0 2 ii 3 D11 Dii12 Dii13 6  ðd1ÞQ Dii 7 i ¼ 1; 2; . . .; r; ð23Þ 4 23 5o0;  2 6 6 4



gtR

Dij11 þ Dji11

Dij12 þ Dji12

 

2ðd1ÞQ 

Dij13 þ Dji13

3

7 Dij23 þ Dji23 7 5o0; 2gtR

1riojrr;

ð24Þ

F.-P. Da, S.-T. He / Journal of the Franklin Institute 348 (2011) 865–883

"

2gPgR 

# Y Z0 X

875

ð25Þ

where T

T

Dij11 ¼ aPATi þ aAi PaF j BTi aBi F j þ Q þ Y þ Y þ tX ; T

T

Dij12 ¼ aAdi PY ; Dij13 ¼ gtðPATi F j BTi Þ; Dij23 ¼ gtPATdi and a40, gZ0 are the adjustable parameters. The state feedback gains can be constructed as Fi ¼ F i P

1

, i=1, 2, y., r.

Remark 4. When the LMIs (23)–(25) have solutions, it is obvious that the solution matrices P; Q; R; X ; Y and F i are all influenced by the adjustable parameters. Therefore, when the adjustable parameters take different values, the different feedback gains are obtained. Since different feedback gains lead to different dynamic performance of closed-loop systems, the dynamic performance of closed-loop systems can be optimized by choosing proper adjustable parameters. 5. Simulation examples

Example 1. In this example, we consider system (3) with system matrices as follows:         2 0 1 0 1:5 1 1 0 A1 ¼ ; Ad1 ¼ ; A2 ¼ ; Ad2 ¼ : 0 0:9 0 1 0 0:75 1 0:85 Let a=0.1 and g=1. When time-varying delay t(t)t, that is d=0, according to Corollary 1 in Ref. [31] and Theorem 1 in this paper, the upper boundary of t is 2.3937 and 2.6835, respectively. When d=0.4, by using Corollary 1 in Ref. [31] and Theorem 1 in this paper, the upper boundary of t(t) is 1.4431 and 1.5041, respectively. Hence, it can be concluded that the result obtained in Theorem 1 is less conservative than the one in Ref. [31] when d=0 and 0.4. Example 2. To illustrate the proposed results, we apply the above analysis technique to the example in Ref. [15]. The fuzzy model is described as Rule 1: v v x1 ðtÞ þ ð1eÞ x1 ðttÞ is about 0ðradÞ; 2L 2L _ ¼ A1 xðtÞ þ Ad1 xðttÞ þ B1 uðtÞ: then xðtÞ

If fðtÞ ¼ x2 ðtÞ þ e

Rule 2: v v x1 ðtÞ þ ð1eÞ x1 ðttÞ 2L 2L is about pðradÞ orpðradÞ; then _ ¼ A2 xðtÞ þ Ad2 xðttÞ þ B2 uðtÞ: xðtÞ

If fðtÞ ¼ x2 ðtÞ þ e

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where 3 3 2 v v 0 0 0 0 ð1eÞ 2v3 L L 7 7 6 6 7 7 6 v 6 v 6l7 7 7 6 6 7 A1 ¼ 6 e L 0 0 7; Ad1 ¼ 6 ð1eÞ L 0 0 7; B1 ¼ 6 4 0 5; 7 7 6 6 2 2 5 5 4 v 4 v 0 e ð1eÞ v 0 0 0 2L 2L 3 3 2 2 v v 0 0 0 0 e ð1eÞ 2v3 L L 7 7 6 6 7 7 6 6 v v 6l7 7 6 6 0 07 7 A2 ¼ 6 e L 7; Ad2 ¼ 6 ð1eÞ L 0 0 7; B2 ¼ 6 4 0 5; 7 7 6 6 2 2 5 5 4 4 ov ov 0 e ð1eÞ ov2 0 0 0 2L 2L 2

e

and l=2.8, L=5.5 and v=1.0, the constant e defined in [0,1] is the retarded coefficient where the limits 0 and 1 corresponding to full delay term and no delay term, respectively. In this example, let e=0.7, o=10/p and the membership function as    1 1 h1 ðfÞ ¼ 1 ; h2 ðfÞ ¼ 1h1 ðfÞ: 1 þ e3ðfðtÞ0:5pÞ 1 þ e3ðfðtÞþ0:5pÞ Using LMI tools in MATLAB, we get the maximum values of t according to different adjustable parameters a and g, which are shown in Table 1. From Table 1, we can conclude that the maximum values of t are influenced by a and g: When a=g ¼ 1, the maximum value of t changes little; when a=go1, the maximum values of t become smaller and when a=g41, the maximum values of t become larger. To illustrate the effectiveness of the proposed approach in the paper, we take state feedback control approach to control the above system. The fuzzy control laws are Table 1 Maximum values of t with different a and g. t

a=1

a=0.5

a=5

g=1 g=0.5 g=5

36.33 72.70 10.13

25.60 36.36 6.79

182.03 365.03 36.38

Table 2 Feedback gains with different a and g. Gains

a=1

a=0.5

g=1

F1=[4.6928 F2=[4.6161

13.5111 13.5816

1.0920] 0.8635]

F1=[4.5181 F2=[4.3022

8.5908 1.8476

g=0.5

F1=[5.1341 F2=[4.7223

17.7082 15.9239

1.7081] 1.2028]

F1=[4.8867 F2=[4.8598

14.1080 14.5916

g=5

F1=[2.5786 F2=[2.6402

2.4099 3.3653

F1=[1.5840 F2=[1.6010

0.6212 0.8752

0.0747] 0.0749]

a=5 0.5399] 0.5020] 1.1873] 0.9718] 0.0098] 0.0098]

F1=[4.7632 F2=[4.2348

19.4182 17.4088

1.6547] 1.0221]

F1=[5.4979 F2=[4.9319

24.7483 22.6172

1.8625] 1.1811]

F1=[4.5122 F2=[4.4064

13.1043 12.9318

1.0215] 0.7882]

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described as Rule i: If a1 ðtÞ is Mi1 ; a2 ðtÞ is Mi2 ; . . .; ap ðtÞ is Mip then uðtÞ ¼ Fi xðtÞ; i ¼ 1; 2:

Fig. 1. Response curves of x1 (a=0.5).

Fig. 2. Response curves of x2 (a=0.5).

877

878

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The initial condition of the state is [0.15p, 0.25, p5]T and let t=1. With different adjustable parameters, the state feedback gains are shown in Table 2 and the state response curves of closed-loop systems are shown in Figs. 1–9.

Fig. 3. Response curves of x3 (a=0.5).

Fig. 4. Response curves of x1 (a=1).

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Fig. 5. Response curves of x2 (a=1).

Fig. 6. Response curves of x3 (a=1).

From Tables 1 and 2 we can find that when the adjustable parameters take different values, the maximum value of tand the feedback gains are also different, which are illustrated in Remark 4. Different feedback gains lead to different dynamic performance of closed-loop systems, which are illustrated by Figs. 1–9.

880

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Fig. 7. Response curves of x1 (a=5).

Fig. 8. Response curves of x2 (a=5).

From Ref. [15] we know that the open-loop system is unstable. But with the method proposed in this paper, the closed-loop system is stable, which illustrates the effectiveness of the proposed method. It is obvious that when the adjustable parameters take different

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Fig. 9. Response curves of x3 (a=5) (Note 1. In Figs. 1–9, ‘‘—’’ stands for g=0.5, ‘‘——’’stands for g=1, ‘‘——’’ stands for g=5.).

values, the dynamic performance of closed-loop systems is also different. Therefore in the practical applications, the better dynamic performance can be got by choosing proper adjustable parameters. 6. Conclusions In this paper, an improved exponential stability condition is presented for time-delay systems with Takagi–Sugeno fuzzy model. Based on the proposed exponential stability conditions, the state feedback controller is designed, and the state feedback gains are derived through the numerical solution of a set of obtained linear matrix inequalities. The dynamical performance of closed-loop system can be adjusted by changing the adjustable parameters. Lastly, two illustrative examples are given to show the effectiveness of the proposed method. Although the better dynamic performance can be gotten by choosing proper adjustable parameters, how to choose the proper adjustable parameters is an interesting and meaningful work, which will be done next. References [1] T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control, IEEE Transactions on Systems, Man and Cybernetics 15 (1) (1985) 116–132. [2] C.C. Ku, P.H. Huang, W.J. Chang, Passive fuzzy controller design for nonlinear systems with multiplicative noises, Journal of the Franklin Institute 347 (5) (2010) 732–750. [3] M. Ababneh, A.M. Almanasreh, H. Amasha, Design of digital controllers for uncertain chaotic systems using fuzzy logic, Journal of the Franklin Institute 346 (6) (2009) 543–556.

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