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Robust control for Takagi–Sugeno fuzzy systems with time-varying state and input delays
Chaos, Solitons and Fractals 35 (2008) 1003–1008 www.elsevier.com/locate/chaos
Robust control for Takagi–Sugeno fuzzy systems with time-varying state and input delays Chang-Hua Lien b
a,b,*
, Ker-Wei Yu
b
a Department of Electrical Engineering, I-Shou University, Kaohsiung 840, Taiwan, ROC Department of Marine Engineering, National Kaohsiung Marine University, 811, Taiwan, ROC
Accepted 6 June 2006
Abstract The paper investigates the robust control for uncertain Takagi–Sugeno (T–S) fuzzy systems with time-varying state and input delays. Delay-dependent stabilization criterion is proposed to guarantee the asymptotic stabilization of fuzzy systems with parametric uncertainties. The result of [Lee HJ, Park JB, Joo YH. Robust control for uncertain Takagi– Sugeno fuzzy systems with time-varying input delay. ASME J Dyn Syst Meas Control 2005;127:302–6] is extended to uncertain fuzzy systems with time-varying state and input delays. Simulations show that significant improvement over the previous results can be obtained. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Many practical systems suffer time delays frequently. Among them are: chemical engineering systems, manual controls, neural networks, nuclear reactors, rolling mills, inferred grinding models, pollution dynamic models and systems without transmission lines. The impacts of these delays are not only costly but also disastrous, even catastrophic in cases of accidents caused by chemical or nuclear explosions. Takagi–Sugeno (T–S) fuzzy system model concept [2] has received considerable attention in control system analysis and synthesis. The stability and stabilization problems of fuzzy systems have been investigated by using the nonlinear frameworks [1–7]. The stability of fuzzy system is guaranteed if a common positive definite matrix exists [6]. The scientific and engineering communities throughout the world are fascinated by the properties and potentials of T–S fuzzy systems. Thus, a great number of studies and researches had been done on stabilization of T–S fuzzy systems in the past few years [1,3,4]. Among those endeavors, we are indebted to Lyapunov stability theory and LMI approach. Our proofs will be incomplete without them. In this paper, we focus on linear and nonlinear variables and the roles they play in the processes of robust stabilization. In [1], they study the relationship between time-varying input delays and robust stabilization of T–S fuzzy systems. So does the relationship between a group of uncertain parameters and robust stabilization. In [4], robust stabilization of fuzzy systems with time-varying state delay and nonlinear uncertain parameters is considered.
*
Corresponding author. Tel.: +886 7 6577711x6619; fax: +886 7 6577205. E-mail address: [email protected] (C.-H. Lien).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.007
1004
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
Based on the sizes of delays, we classify T–S fuzzy systems either as delay-dependent [1] or delay-independent [4]. Generally speaking, when the values of (time) delays are relatively small, delay-dependent T–S fuzzy systems are less conservative than the delay-independent ones. In [1,4], they enlist Lyapunov–Razumikhin stability theory to explain the stabilization problems of delay-independent T–S fuzzy systems. To determine the outcome of delay-dependent stabilizations, both the Lyapunov–Krasovskii function and the Leibniz–Newton formula are used. LMI (linear matrix inequality) approach [8] finds the upper boundaries of delays in the T–S fuzzy systems. LMI approach also serves to secure robust stabilization of the uncertain T–S fuzzy systems. A numerical example which appears in [1] has shown that the obtained delay-dependent result in this paper is less conservative even in this special case.
2. Problem formulation and main result Consider a continuous-time uncertain fuzzy system with time-varying state and input delays, which is represented by a Takagi–Sugeno (T–S) fuzzy model [2] composed of a set of fuzzy implications. Each implication is expressed by a linear uncertain time-delay system and the ith rule of the T–S model is written in the following form: Rule i: If z1(t) is about Mi1, . . . , zr(t) is about Mir, then x_ ðtÞ ¼ ½Ai þ DAi ðtÞxðtÞ þ ½Bi þ DBi ðtÞxðt hðtÞÞ þ ½C i þ DC i ðtÞuðt sðtÞÞ; xðtÞ ¼ /ðtÞ; t 2 ½H ; 0; i ¼ 1; 2; . . . ; m;
ð1aÞ ð1bÞ
t P 0;
where z1(t), z2(t), . . . , zr(t) are premise variables, Mij, i = 1, 2, . . . , m, j = 1, . . . , r are fuzzy sets, m is the rules number. _ 6 hD ; sðtÞ 6 sM ; s_ ðtÞ 6 sD ; hM > 0; sM > 0, and xðtÞ 2 Rn is the state, uðtÞ 2 Rp is the input, hðtÞ 6 hM ; hðtÞ H = max{hM, sM}. The matrices Ai ; Bi 2 Rnn , and C i 2 Rnp , are known, and the initial vector /i belongs to the set of continuously functions. The perturbed matrices DAi(t), DBi(t), DCi(t), are some time-varying functions satisfying ½ DAi ðtÞ DBi ðtÞ DC i ðtÞ ¼ Di F i ðtÞ ½ E1i
E2i
E3i ;
ð2aÞ
where Di, Eji, i 2 {1, . . . , m}, j 2 {1, 2, 3}, are some given constant matrices, Fi(t) is an unknown real time-varying function with appropriate dimension and bounded as follows: F i ðtÞT F i ðtÞ 6 I;
i 2 f1; . . . ; mg 8t P 0;
ð2bÞ
where the notation A 6 B stands that the matrix B A is symmetric positive semi-definite. The final state output of fuzzy system is inferred as follows: x_ ðtÞ ¼
m X
wi ðzðtÞÞ f½Ai þ DAi ðtÞxðtÞ þ ½Bi þ DBi ðtÞxðt hðtÞÞ
i¼1
,
þ ½C i þ DC i ðtÞuðt sðtÞÞg
m X
wi ðzðtÞÞ
i¼1
¼ xðtÞ ¼
m X i¼1 m X
gi ðzðtÞÞ f½Ai þ DAi ðtÞxðtÞ þ ½Bi þ DBi ðtÞxðt hðtÞÞ þ ½C i þ DC i ðtÞuðt sðtÞÞg; gi ðzðtÞÞ /i ðtÞ;
t P 0;
t 2 ½H ; 0;
ð3aÞ ð3bÞ
i¼1
Pm Q where wi ðzðtÞÞ ¼ rj¼1 Xij ðzj ðtÞÞ; gi ðzðtÞÞ ¼ wi ðzðtÞÞ i¼1 wi ðzðtÞÞ; Xij ðzj ðtÞÞ isPthe grade of membership of zj(t) in m fuzzy set M . In this paper, we assume w (z(t)) P 0, i 2 {1, . . . , m}, and ij i i¼1 wi ðzðtÞÞ > 0. Hence gi(z(t)) P 0 and Pm g ðzðtÞÞ ¼ 1, for all t P 0. i¼1 i In this paper, we use the following fuzzy rule for fuzzy-model-based control: Rule i: If z1(t) is Mi1, . . . , zr(t) is Mir, then uðtÞ ¼ K i xðtÞ; pn
where K i 2 R
uðtÞ ¼
t P 0;
; i 2 f1; . . . ; mg, are control gains. The final feedback control is inferred as follows:
m X i¼1
gi ðzðtÞÞ K i xðtÞ;
t P 0:
ð4Þ
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
1005
Lemma 1 [9]. Let Z, M, N, and F(t) be the matrices of appropriate dimensions. Assume that Z is symmetric and FT(t)F(t) 6 I, then Z þ MF ðtÞN þ N T F T ðtÞM T < 0; if and only if there exists a scalar e > 0 satisfying Z þ e1 ðeMÞðeMÞT þ e1 N T N < 0: Now we present a delay-dependent condition for the asymptotic stability of system (1) with (2). Theorem 1. If for a given l > 0, there exist some positive definite symmetric matrices P 0 ; P 1 ; P 2 ; R1 ; R2 , matrices K i 2 Rpn ; i 2 f1; . . . ; mg, and some positive constants eij, i, j 2 {1, . . . , m}, such that the following LMI conditions hold for all i, j 2 {1, . . . , m}, 2 3 N11ij N12ij N13ij N14ij N15ij N16ij 6 N22ij 0 N24ij 0 N26ij 7 6 7 6 7 6 N33ij N34ij 0 N36ij 7 7 < 0; Nij ¼ 6 ð5Þ 6 N44ij N45ij 0 7 6 7 6 7 4 N55ij 0 5
N66ij
where * represents the symmetric form in the matrix and N11ij ¼ Ai P 0 þ P 0 ATi þ P 1 þ P 2 R1 R2 ; N14ij ¼ N26ij ¼
l P 0 ATi ; N15ij P 0 ET2i ; N33ij ¼
N44ij ¼ 2l P 0 þ
h2M
¼ eij Di ;
N16ij ¼
N12ij ¼ Bi P 0 þ R1 ; P 0 ET1i ;
ð1 sD Þ P 2 R2 ;
R1 þ
s2M
R2 ;
N13ij ¼ C i K j þ R2 ;
N22ij ¼ ð1 hD Þ P 1 R1 ;
N34ij ¼ l
N45ij ¼ eij l Di ;
K Tj C Ti ;
N36ij ¼
N24ij ¼ l P 0 BTi ;
K Tj ET3i ;
N55ij ¼ N66ij ¼ eij I:
for all Then uncertain fuzzy time-delay system (1) with (2) is asymptotically stabilizable by (4) with K j ¼ K j P 1 0 _ 6 hD ; sðtÞ 6 sM ; s_ ðtÞ 6 sD . hðtÞ 6 hM ; hðtÞ Proof. Define the Lyapunov functional Z t Z xT ðsÞP 1 xðsÞds þ V ðxt Þ ¼ xT ðtÞP 0 xðtÞ þ þ sM
Z
thðtÞ
t
xT ðsÞP 2 xðsÞds þ hM
tsðtÞ
Z
t
ðs ðt hM ÞÞ_xT ðsÞR1 x_ ðsÞds thM
t
ðs ðt sM ÞÞ_xT ðsÞR2 x_ ðsÞds;
ð6Þ
tsM
P P where P0, P1, P2, R1, R2 > 0. By system (3) with mi¼1 gi ðzðtÞÞ ¼ 1 and mj¼1 gj ðzðt sðtÞÞÞ ¼ 1, the time derivatives of V(xt), along the trajectories of system (1) with (2) and (4) satisfy V_ ðxt Þ ¼
m X
gi ðzðtÞÞ fxT ðtÞðP 0 Ai þ ATi P 0 ÞxðtÞ þ xT ðtÞðP 0 DAi ðtÞ þ DATi ðtÞP 0 ÞxðtÞ
i¼1
þ 2xT ðtÞP 0 ðBi þ DBi ðtÞÞxðt hðtÞÞg m X m X gi ðzðtÞÞ gj ðzðt sðtÞÞÞ f2xT ðtÞP 0 ðC i þ DC i ðtÞÞK j xðt sðtÞÞg i¼1
j¼1
T _ ðt hðtÞÞP 1 xðt hðtÞÞ þ x ðtÞP 1 xðtÞ ð1 hðtÞÞx T
þ xT ðtÞP 2 xðtÞ ð1 s_ ðtÞÞxT ðt sðtÞÞP 2 xðt sðtÞÞ Z t þ h2M x_ T ðtÞR1 x_ ðtÞ hM x_ T ðsÞR1 x_ ðsÞds þ s2M x_ T ðtÞR2 x_ ðtÞ sM
Z
thM t
x_ T ðsÞR2 x_ ðsÞds: tsM
1006
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
By the inequality in [10], we have Z Z t T hM x_ ðsÞR1 x_ ðsÞds 6 hðtÞ thM
sM
Z
t T
x_ ðsÞR1 x_ ðsÞds 6
Z
thðtÞ
t T
x_ ðsÞR2 x_ ðsÞds 6 sðtÞ
tsM
Z
!T
t
x_ ðsÞds
Z
T
x_ ðsÞR2 x_ ðsÞds 6
tsðtÞ
Z
x_ ðsÞds :
R1
thðtÞ
t
!
t
ð7aÞ
thðtÞ
!T
t
x_ ðsÞds
tsðtÞ
Z
!
t
x_ ðsÞds :
R2
ð7bÞ
tsðtÞ
From Leibniz–Newton formula, we have Z t x_ ðsÞds ¼ xðtÞ xðt hðtÞÞ; thðtÞ t
Z
ð7cÞ x_ ðsÞds ¼ xðtÞ xðt sðtÞÞ:
tsðtÞ
From (7c), (7a) and (7b) can be rewritten as Z t x_ T ðsÞR1 x_ ðsÞds 6 ðxðtÞ xðt hðtÞÞÞT R1 ðxðtÞ xðt hðtÞÞÞ; hM sM
Z
ð7dÞ
thM t
x_ T ðsÞR2 x_ ðsÞds 6 ðxðtÞ xðt sðtÞÞÞT R2 ðxðtÞ xðt sðtÞÞÞ:
ð7eÞ
tsM
From (7d), (7e), and the following result with a given constant l > 0: ( m X m X T T 2l x_ ðtÞðP 0 Þ_xðtÞ þ l x_ ðtÞP 0 gi ðzðtÞÞ gj ðzðt sðtÞÞÞ ½ðAi þ DAi ðtÞÞxðtÞ þ ðBi þ DBi ðtÞÞxðt hðtÞÞ i¼1
) ðC i þ DC i ðtÞÞK j xðt sðtÞÞ
j¼1
(
þl
m X m X i¼1
gi ðzðtÞÞ gj ðzðt sðtÞÞÞ ½ðAi þ DAi ðtÞÞxðtÞ
j¼1
)T P 0 x_ ðtÞ ¼ 0;
þ ðBi þ DBi ðtÞÞxðt hðtÞÞ ðC i þ DC i ðtÞÞK j xðt sðtÞÞ one has V_ ðxt Þ 6
m X m X i¼1
gi ðzðtÞÞ gj ðzðt sðtÞÞÞ ½UT Rij U;
j¼1
where UT ¼ xT ðtÞ xT ðt hðtÞÞ 2 R11ij R12ij R13ij 6 R22ij 0 6 Rij ¼ 6 4 R33ij
xT ðt sðtÞÞ 3 R14ij R24ij 7 7 7; R34ij 5
ð8Þ
x_ T ðtÞ ,
R44ij
R11ij ¼ P 0 Ai þ ATi P 0 þ P 0 DAi ðtÞ þ DATi ðtÞP 0 þ P 1 þ P 2 R1 R2 ; R13ij ¼ P 0 ðC i þ DC i ðtÞÞK j þ R2 ; T
R24ij ¼ l ðBi þ DBi ðtÞÞ P 0 ;
T
R14ij ¼ l ðAi þ DAi ðtÞÞ P 0 ;
R33ij ¼ ð1 sD Þ P 2 R2 ;
P 0 ¼ P 1 0 ;
1 P 1 ¼ P 1 0 P 1P 0 ;
1 P 2 ¼ P 1 0 P 2P 0 ;
P 1 0
R22ij ¼ ð1 hD Þ P 1 R1 ;
R34ij ¼ l K Tj ðC i þ DC i ðtÞÞT P 0 ;
R44ij ¼ 2l P 0 þ h2M R1 þ s2M R2 :
Pre- and post-multiplying the matrix Rij in (8) by diag P 1 0
R12ij ¼ P 0 ðBi þ DBi ðtÞÞ þ R1 ;
P 1 0
1 R1 ¼ P 1 0 R1 P 0 ;
> 0 with P 1 0 1 R2 ¼ P 1 0 R2 P 0 ;
K j ¼ K j P 1 0 ;
we have 2
N11ij 6 b ij ¼ 6 R 6 4
N12ij N22ij
N13ij N23ij
N33ij
3 N14ij N24ij 7 7 T 7 þ wij F i ðtÞxij þ xTij F Ti ðtÞwij ; 0 5 N44ij
ð9Þ
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
0 , Nklij, k, l 2 {1, 2, . . . , 6} are defined in (5). By the b ij < 0 in (9). The condition R b ij < 0 in (9) will also Lemma 1 and Schur complement of [8], LMI (5) is equivalent to R equivalent to Rij < 0 in (8) for all i, j 2 {1, . . . , m}. By Eq. (8) with the matrix Rij < 0, there exists a constant m > 0 such that where wij ¼ DTi
0 0 l DTi , xij ¼ E1i P 0
1007
E2i P 0
E3i K j
V_ ðxt Þ 6 m kxðtÞk2 :
ð10Þ
From (6) and (10), we can conclude that the fuzzy system (1) with (2) is asymptotically stabilizable by (4) [11].
h
Remark 1. In general, two conditions for the time-varying delays h(t) and s(t) are usually considered: _ 6 hD ; sðtÞ 6 sM ; s_ ðtÞ 6 sD . (A1) hðtÞ 6 hM ; hðtÞ (A2) h(t) 6 hM, s(t) 6 sM. In (A1), the slow variation condition h_ i ðtÞ 6 hD < 1 is necessary in this delay-dependent result [7]. In Theorem 1, the slow variation conditions hD < 1 and sD < 1 are not imposed in this paper. In (A2) (i.e. unknown hD and sD), Theorem 1 is also valid with P 1 ¼ 0 and P 2 ¼ 0. In [1], delay-dependent stabilization criterion for fuzzy systems with time-varying input delay is proposed by using Lyapunov–Razumikhin stability theory. In this paper, new Lyapunov–Krasovskii functional and Leibniz–Newton formula are used to obtain delay-dependent stabilization result and the result of [1] is extend to the stabilization of fuzzy systems with time-varying state and input delays.
3. Numerical example Consider the nonlinear mass–spring–damper mechanical system in [1]. The T–S fuzzy models can be constructed as follows: Plant rules [1]: Rule 1: If x1(t) is about C1, then x_ ðtÞ ¼ ½A1 þ DA1 ðtÞxðtÞ þ ½C 1 þ DC 1 ðtÞuðt sðtÞÞ:
ð11aÞ
Rule 2: If x1(t) is about C2, then x_ ðtÞ ¼ ½A2 þ DA2 ðtÞxðtÞ þ ½C 2 þ DC 2 ðtÞuðt sðtÞÞ; 0:5 0:1 1 0:1 0 0:1nðtÞ ; A2 ¼ ; DA1 ðtÞ ¼ DA2 ðtÞ ¼ ; where A1 ¼ 1 0 1 0 0 0 0 0:1nðtÞ DA2 ðtÞ ¼ ; jnðtÞj 6 1; C1 and C2 are defined in [1]. 0 0 From (1), (2) and (11), we have " pffiffiffiffiffiffiffi # 0 0 0:1 ; D1 ¼ D2 ¼ ; B1 ¼ B2 ¼ 0 0 0
pffiffiffiffiffiffiffi E11 ¼ E12 ¼ 0 0:1 ;
ð11bÞ 1 C1 ¼ C2 ¼ ; 0
E21 ¼ E22 ¼ ½ 0
0 ;
DA1 ðtÞ ¼
E31 ¼ E32 ¼ 0:
In order to show the improvement of this paper, a comparison for the upper bound of input delay to guarantee the asymptotic stabilization of fuzzy system is shown in Table 1. The obtained result in this paper improves the result of [1] in this example.
Table 1 A comparison for the upper bounds of input delay for fuzzy system (11) Results
Upper bound of input delay and fuzzy control gains in (4)
[1]
sM = 0.4223 (sD unknown) K 1 ¼ ½ 0:7623 0:5212 ; K 2 ¼ ½ 0:3690 0:5362 sM = 1.75 (sD unknown) K 1 ¼ ½ 0:3622 0:299 ; K 2 ¼ ½ 0:3625 0:299
This paper (l = 2)
1008
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
4. Conclusion In this paper, the robust control problem for a class of uncertain fuzzy system with time-varying state and input delays has been investigated. Based on the LMI approach, delay-dependent criterion has been proposed to guarantee the asymptotic stabilization of fuzzy systems. An example has been provided to demonstrate the reduced conservativeness of the proposed result.
Acknowledgement The research reported here was supported by the National Science Council of Taiwan, ROC under grant no. NSC 93-2213-E-214-020.
References [1] Lee HJ, Park JB, Joo YH. Robust control for uncertain Takagi–Sugeno fuzzy systems with time-varying input delay. ASME J Dyn Syst Meas Control 2005;127:302–6. [2] Takagi T, Sugeno M. Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybernet 1985;15:116–32. [3] Cao YY, Frank PM. Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach. IEEE Trans Fuzzy Syst 2000;8:200–11. [4] Wang RJ, Lin WW, Wang WJ. Stabilizability of linear quadratic state feedback for uncertain fuzzy time-delay systems. IEEE Trans Syst Man Cybernet B 2004;34:1288–92. [5] Lien CH. Further results on delay-dependent robust stability of uncertain fuzzy systems with time-varying delay. Chaos, Solitons & Fractals 2006;28:422–7. [6] Tanaka K, Sugeno M. Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst 1992;45:135–56. [7] Li C, Wang H, Liao X. Delay-dependent robust stability of uncertain fuzzy systems with time-varying delays. IEE Proc Control Theory Appl 2004;151:417–21. [8] Boyd SP, Ghaoui LE, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Philadelphia: SIAM; 1994. [9] Singh V. Robust stability of cellular neural networks with delay: linear matrix inequality approach. IEE Proc Control Theory Appl 2004;151:125–9. [10] Gu K, Kharitonov VL, Chen J. Stability of time-delay systems. Boston: Birkhauser; 2003. [11] Kolmanovskii VB, Myshkis A. Applied theory of functional differential equations. Netherlands: Kluwer Academic Publishers; 1992.
Robust control for Takagi–Sugeno fuzzy systems with time-varying state and input delays Chang-Hua Lien b
a,b,*
, Ker-Wei Yu
b
a Department of Electrical Engineering, I-Shou University, Kaohsiung 840, Taiwan, ROC Department of Marine Engineering, National Kaohsiung Marine University, 811, Taiwan, ROC
Accepted 6 June 2006
Abstract The paper investigates the robust control for uncertain Takagi–Sugeno (T–S) fuzzy systems with time-varying state and input delays. Delay-dependent stabilization criterion is proposed to guarantee the asymptotic stabilization of fuzzy systems with parametric uncertainties. The result of [Lee HJ, Park JB, Joo YH. Robust control for uncertain Takagi– Sugeno fuzzy systems with time-varying input delay. ASME J Dyn Syst Meas Control 2005;127:302–6] is extended to uncertain fuzzy systems with time-varying state and input delays. Simulations show that significant improvement over the previous results can be obtained. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Many practical systems suffer time delays frequently. Among them are: chemical engineering systems, manual controls, neural networks, nuclear reactors, rolling mills, inferred grinding models, pollution dynamic models and systems without transmission lines. The impacts of these delays are not only costly but also disastrous, even catastrophic in cases of accidents caused by chemical or nuclear explosions. Takagi–Sugeno (T–S) fuzzy system model concept [2] has received considerable attention in control system analysis and synthesis. The stability and stabilization problems of fuzzy systems have been investigated by using the nonlinear frameworks [1–7]. The stability of fuzzy system is guaranteed if a common positive definite matrix exists [6]. The scientific and engineering communities throughout the world are fascinated by the properties and potentials of T–S fuzzy systems. Thus, a great number of studies and researches had been done on stabilization of T–S fuzzy systems in the past few years [1,3,4]. Among those endeavors, we are indebted to Lyapunov stability theory and LMI approach. Our proofs will be incomplete without them. In this paper, we focus on linear and nonlinear variables and the roles they play in the processes of robust stabilization. In [1], they study the relationship between time-varying input delays and robust stabilization of T–S fuzzy systems. So does the relationship between a group of uncertain parameters and robust stabilization. In [4], robust stabilization of fuzzy systems with time-varying state delay and nonlinear uncertain parameters is considered.
*
Corresponding author. Tel.: +886 7 6577711x6619; fax: +886 7 6577205. E-mail address: [email protected] (C.-H. Lien).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.007
1004
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
Based on the sizes of delays, we classify T–S fuzzy systems either as delay-dependent [1] or delay-independent [4]. Generally speaking, when the values of (time) delays are relatively small, delay-dependent T–S fuzzy systems are less conservative than the delay-independent ones. In [1,4], they enlist Lyapunov–Razumikhin stability theory to explain the stabilization problems of delay-independent T–S fuzzy systems. To determine the outcome of delay-dependent stabilizations, both the Lyapunov–Krasovskii function and the Leibniz–Newton formula are used. LMI (linear matrix inequality) approach [8] finds the upper boundaries of delays in the T–S fuzzy systems. LMI approach also serves to secure robust stabilization of the uncertain T–S fuzzy systems. A numerical example which appears in [1] has shown that the obtained delay-dependent result in this paper is less conservative even in this special case.
2. Problem formulation and main result Consider a continuous-time uncertain fuzzy system with time-varying state and input delays, which is represented by a Takagi–Sugeno (T–S) fuzzy model [2] composed of a set of fuzzy implications. Each implication is expressed by a linear uncertain time-delay system and the ith rule of the T–S model is written in the following form: Rule i: If z1(t) is about Mi1, . . . , zr(t) is about Mir, then x_ ðtÞ ¼ ½Ai þ DAi ðtÞxðtÞ þ ½Bi þ DBi ðtÞxðt hðtÞÞ þ ½C i þ DC i ðtÞuðt sðtÞÞ; xðtÞ ¼ /ðtÞ; t 2 ½H ; 0; i ¼ 1; 2; . . . ; m;
ð1aÞ ð1bÞ
t P 0;
where z1(t), z2(t), . . . , zr(t) are premise variables, Mij, i = 1, 2, . . . , m, j = 1, . . . , r are fuzzy sets, m is the rules number. _ 6 hD ; sðtÞ 6 sM ; s_ ðtÞ 6 sD ; hM > 0; sM > 0, and xðtÞ 2 Rn is the state, uðtÞ 2 Rp is the input, hðtÞ 6 hM ; hðtÞ H = max{hM, sM}. The matrices Ai ; Bi 2 Rnn , and C i 2 Rnp , are known, and the initial vector /i belongs to the set of continuously functions. The perturbed matrices DAi(t), DBi(t), DCi(t), are some time-varying functions satisfying ½ DAi ðtÞ DBi ðtÞ DC i ðtÞ ¼ Di F i ðtÞ ½ E1i
E2i
E3i ;
ð2aÞ
where Di, Eji, i 2 {1, . . . , m}, j 2 {1, 2, 3}, are some given constant matrices, Fi(t) is an unknown real time-varying function with appropriate dimension and bounded as follows: F i ðtÞT F i ðtÞ 6 I;
i 2 f1; . . . ; mg 8t P 0;
ð2bÞ
where the notation A 6 B stands that the matrix B A is symmetric positive semi-definite. The final state output of fuzzy system is inferred as follows: x_ ðtÞ ¼
m X
wi ðzðtÞÞ f½Ai þ DAi ðtÞxðtÞ þ ½Bi þ DBi ðtÞxðt hðtÞÞ
i¼1
,
þ ½C i þ DC i ðtÞuðt sðtÞÞg
m X
wi ðzðtÞÞ
i¼1
¼ xðtÞ ¼
m X i¼1 m X
gi ðzðtÞÞ f½Ai þ DAi ðtÞxðtÞ þ ½Bi þ DBi ðtÞxðt hðtÞÞ þ ½C i þ DC i ðtÞuðt sðtÞÞg; gi ðzðtÞÞ /i ðtÞ;
t P 0;
t 2 ½H ; 0;
ð3aÞ ð3bÞ
i¼1
Pm Q where wi ðzðtÞÞ ¼ rj¼1 Xij ðzj ðtÞÞ; gi ðzðtÞÞ ¼ wi ðzðtÞÞ i¼1 wi ðzðtÞÞ; Xij ðzj ðtÞÞ isPthe grade of membership of zj(t) in m fuzzy set M . In this paper, we assume w (z(t)) P 0, i 2 {1, . . . , m}, and ij i i¼1 wi ðzðtÞÞ > 0. Hence gi(z(t)) P 0 and Pm g ðzðtÞÞ ¼ 1, for all t P 0. i¼1 i In this paper, we use the following fuzzy rule for fuzzy-model-based control: Rule i: If z1(t) is Mi1, . . . , zr(t) is Mir, then uðtÞ ¼ K i xðtÞ; pn
where K i 2 R
uðtÞ ¼
t P 0;
; i 2 f1; . . . ; mg, are control gains. The final feedback control is inferred as follows:
m X i¼1
gi ðzðtÞÞ K i xðtÞ;
t P 0:
ð4Þ
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
1005
Lemma 1 [9]. Let Z, M, N, and F(t) be the matrices of appropriate dimensions. Assume that Z is symmetric and FT(t)F(t) 6 I, then Z þ MF ðtÞN þ N T F T ðtÞM T < 0; if and only if there exists a scalar e > 0 satisfying Z þ e1 ðeMÞðeMÞT þ e1 N T N < 0: Now we present a delay-dependent condition for the asymptotic stability of system (1) with (2). Theorem 1. If for a given l > 0, there exist some positive definite symmetric matrices P 0 ; P 1 ; P 2 ; R1 ; R2 , matrices K i 2 Rpn ; i 2 f1; . . . ; mg, and some positive constants eij, i, j 2 {1, . . . , m}, such that the following LMI conditions hold for all i, j 2 {1, . . . , m}, 2 3 N11ij N12ij N13ij N14ij N15ij N16ij 6 N22ij 0 N24ij 0 N26ij 7 6 7 6 7 6 N33ij N34ij 0 N36ij 7 7 < 0; Nij ¼ 6 ð5Þ 6 N44ij N45ij 0 7 6 7 6 7 4 N55ij 0 5
N66ij
where * represents the symmetric form in the matrix and N11ij ¼ Ai P 0 þ P 0 ATi þ P 1 þ P 2 R1 R2 ; N14ij ¼ N26ij ¼
l P 0 ATi ; N15ij P 0 ET2i ; N33ij ¼
N44ij ¼ 2l P 0 þ
h2M
¼ eij Di ;
N16ij ¼
N12ij ¼ Bi P 0 þ R1 ; P 0 ET1i ;
ð1 sD Þ P 2 R2 ;
R1 þ
s2M
R2 ;
N13ij ¼ C i K j þ R2 ;
N22ij ¼ ð1 hD Þ P 1 R1 ;
N34ij ¼ l
N45ij ¼ eij l Di ;
K Tj C Ti ;
N36ij ¼
N24ij ¼ l P 0 BTi ;
K Tj ET3i ;
N55ij ¼ N66ij ¼ eij I:
for all Then uncertain fuzzy time-delay system (1) with (2) is asymptotically stabilizable by (4) with K j ¼ K j P 1 0 _ 6 hD ; sðtÞ 6 sM ; s_ ðtÞ 6 sD . hðtÞ 6 hM ; hðtÞ Proof. Define the Lyapunov functional Z t Z xT ðsÞP 1 xðsÞds þ V ðxt Þ ¼ xT ðtÞP 0 xðtÞ þ þ sM
Z
thðtÞ
t
xT ðsÞP 2 xðsÞds þ hM
tsðtÞ
Z
t
ðs ðt hM ÞÞ_xT ðsÞR1 x_ ðsÞds thM
t
ðs ðt sM ÞÞ_xT ðsÞR2 x_ ðsÞds;
ð6Þ
tsM
P P where P0, P1, P2, R1, R2 > 0. By system (3) with mi¼1 gi ðzðtÞÞ ¼ 1 and mj¼1 gj ðzðt sðtÞÞÞ ¼ 1, the time derivatives of V(xt), along the trajectories of system (1) with (2) and (4) satisfy V_ ðxt Þ ¼
m X
gi ðzðtÞÞ fxT ðtÞðP 0 Ai þ ATi P 0 ÞxðtÞ þ xT ðtÞðP 0 DAi ðtÞ þ DATi ðtÞP 0 ÞxðtÞ
i¼1
þ 2xT ðtÞP 0 ðBi þ DBi ðtÞÞxðt hðtÞÞg m X m X gi ðzðtÞÞ gj ðzðt sðtÞÞÞ f2xT ðtÞP 0 ðC i þ DC i ðtÞÞK j xðt sðtÞÞg i¼1
j¼1
T _ ðt hðtÞÞP 1 xðt hðtÞÞ þ x ðtÞP 1 xðtÞ ð1 hðtÞÞx T
þ xT ðtÞP 2 xðtÞ ð1 s_ ðtÞÞxT ðt sðtÞÞP 2 xðt sðtÞÞ Z t þ h2M x_ T ðtÞR1 x_ ðtÞ hM x_ T ðsÞR1 x_ ðsÞds þ s2M x_ T ðtÞR2 x_ ðtÞ sM
Z
thM t
x_ T ðsÞR2 x_ ðsÞds: tsM
1006
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
By the inequality in [10], we have Z Z t T hM x_ ðsÞR1 x_ ðsÞds 6 hðtÞ thM
sM
Z
t T
x_ ðsÞR1 x_ ðsÞds 6
Z
thðtÞ
t T
x_ ðsÞR2 x_ ðsÞds 6 sðtÞ
tsM
Z
!T
t
x_ ðsÞds
Z
T
x_ ðsÞR2 x_ ðsÞds 6
tsðtÞ
Z
x_ ðsÞds :
R1
thðtÞ
t
!
t
ð7aÞ
thðtÞ
!T
t
x_ ðsÞds
tsðtÞ
Z
!
t
x_ ðsÞds :
R2
ð7bÞ
tsðtÞ
From Leibniz–Newton formula, we have Z t x_ ðsÞds ¼ xðtÞ xðt hðtÞÞ; thðtÞ t
Z
ð7cÞ x_ ðsÞds ¼ xðtÞ xðt sðtÞÞ:
tsðtÞ
From (7c), (7a) and (7b) can be rewritten as Z t x_ T ðsÞR1 x_ ðsÞds 6 ðxðtÞ xðt hðtÞÞÞT R1 ðxðtÞ xðt hðtÞÞÞ; hM sM
Z
ð7dÞ
thM t
x_ T ðsÞR2 x_ ðsÞds 6 ðxðtÞ xðt sðtÞÞÞT R2 ðxðtÞ xðt sðtÞÞÞ:
ð7eÞ
tsM
From (7d), (7e), and the following result with a given constant l > 0: ( m X m X T T 2l x_ ðtÞðP 0 Þ_xðtÞ þ l x_ ðtÞP 0 gi ðzðtÞÞ gj ðzðt sðtÞÞÞ ½ðAi þ DAi ðtÞÞxðtÞ þ ðBi þ DBi ðtÞÞxðt hðtÞÞ i¼1
) ðC i þ DC i ðtÞÞK j xðt sðtÞÞ
j¼1
(
þl
m X m X i¼1
gi ðzðtÞÞ gj ðzðt sðtÞÞÞ ½ðAi þ DAi ðtÞÞxðtÞ
j¼1
)T P 0 x_ ðtÞ ¼ 0;
þ ðBi þ DBi ðtÞÞxðt hðtÞÞ ðC i þ DC i ðtÞÞK j xðt sðtÞÞ one has V_ ðxt Þ 6
m X m X i¼1
gi ðzðtÞÞ gj ðzðt sðtÞÞÞ ½UT Rij U;
j¼1
where UT ¼ xT ðtÞ xT ðt hðtÞÞ 2 R11ij R12ij R13ij 6 R22ij 0 6 Rij ¼ 6 4 R33ij
xT ðt sðtÞÞ 3 R14ij R24ij 7 7 7; R34ij 5
ð8Þ
x_ T ðtÞ ,
R44ij
R11ij ¼ P 0 Ai þ ATi P 0 þ P 0 DAi ðtÞ þ DATi ðtÞP 0 þ P 1 þ P 2 R1 R2 ; R13ij ¼ P 0 ðC i þ DC i ðtÞÞK j þ R2 ; T
R24ij ¼ l ðBi þ DBi ðtÞÞ P 0 ;
T
R14ij ¼ l ðAi þ DAi ðtÞÞ P 0 ;
R33ij ¼ ð1 sD Þ P 2 R2 ;
P 0 ¼ P 1 0 ;
1 P 1 ¼ P 1 0 P 1P 0 ;
1 P 2 ¼ P 1 0 P 2P 0 ;
P 1 0
R22ij ¼ ð1 hD Þ P 1 R1 ;
R34ij ¼ l K Tj ðC i þ DC i ðtÞÞT P 0 ;
R44ij ¼ 2l P 0 þ h2M R1 þ s2M R2 :
Pre- and post-multiplying the matrix Rij in (8) by diag P 1 0
R12ij ¼ P 0 ðBi þ DBi ðtÞÞ þ R1 ;
P 1 0
1 R1 ¼ P 1 0 R1 P 0 ;
> 0 with P 1 0 1 R2 ¼ P 1 0 R2 P 0 ;
K j ¼ K j P 1 0 ;
we have 2
N11ij 6 b ij ¼ 6 R 6 4
N12ij N22ij
N13ij N23ij
N33ij
3 N14ij N24ij 7 7 T 7 þ wij F i ðtÞxij þ xTij F Ti ðtÞwij ; 0 5 N44ij
ð9Þ
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
0 , Nklij, k, l 2 {1, 2, . . . , 6} are defined in (5). By the b ij < 0 in (9). The condition R b ij < 0 in (9) will also Lemma 1 and Schur complement of [8], LMI (5) is equivalent to R equivalent to Rij < 0 in (8) for all i, j 2 {1, . . . , m}. By Eq. (8) with the matrix Rij < 0, there exists a constant m > 0 such that where wij ¼ DTi
0 0 l DTi , xij ¼ E1i P 0
1007
E2i P 0
E3i K j
V_ ðxt Þ 6 m kxðtÞk2 :
ð10Þ
From (6) and (10), we can conclude that the fuzzy system (1) with (2) is asymptotically stabilizable by (4) [11].
h
Remark 1. In general, two conditions for the time-varying delays h(t) and s(t) are usually considered: _ 6 hD ; sðtÞ 6 sM ; s_ ðtÞ 6 sD . (A1) hðtÞ 6 hM ; hðtÞ (A2) h(t) 6 hM, s(t) 6 sM. In (A1), the slow variation condition h_ i ðtÞ 6 hD < 1 is necessary in this delay-dependent result [7]. In Theorem 1, the slow variation conditions hD < 1 and sD < 1 are not imposed in this paper. In (A2) (i.e. unknown hD and sD), Theorem 1 is also valid with P 1 ¼ 0 and P 2 ¼ 0. In [1], delay-dependent stabilization criterion for fuzzy systems with time-varying input delay is proposed by using Lyapunov–Razumikhin stability theory. In this paper, new Lyapunov–Krasovskii functional and Leibniz–Newton formula are used to obtain delay-dependent stabilization result and the result of [1] is extend to the stabilization of fuzzy systems with time-varying state and input delays.
3. Numerical example Consider the nonlinear mass–spring–damper mechanical system in [1]. The T–S fuzzy models can be constructed as follows: Plant rules [1]: Rule 1: If x1(t) is about C1, then x_ ðtÞ ¼ ½A1 þ DA1 ðtÞxðtÞ þ ½C 1 þ DC 1 ðtÞuðt sðtÞÞ:
ð11aÞ
Rule 2: If x1(t) is about C2, then x_ ðtÞ ¼ ½A2 þ DA2 ðtÞxðtÞ þ ½C 2 þ DC 2 ðtÞuðt sðtÞÞ; 0:5 0:1 1 0:1 0 0:1nðtÞ ; A2 ¼ ; DA1 ðtÞ ¼ DA2 ðtÞ ¼ ; where A1 ¼ 1 0 1 0 0 0 0 0:1nðtÞ DA2 ðtÞ ¼ ; jnðtÞj 6 1; C1 and C2 are defined in [1]. 0 0 From (1), (2) and (11), we have " pffiffiffiffiffiffiffi # 0 0 0:1 ; D1 ¼ D2 ¼ ; B1 ¼ B2 ¼ 0 0 0
pffiffiffiffiffiffiffi E11 ¼ E12 ¼ 0 0:1 ;
ð11bÞ 1 C1 ¼ C2 ¼ ; 0
E21 ¼ E22 ¼ ½ 0
0 ;
DA1 ðtÞ ¼
E31 ¼ E32 ¼ 0:
In order to show the improvement of this paper, a comparison for the upper bound of input delay to guarantee the asymptotic stabilization of fuzzy system is shown in Table 1. The obtained result in this paper improves the result of [1] in this example.
Table 1 A comparison for the upper bounds of input delay for fuzzy system (11) Results
Upper bound of input delay and fuzzy control gains in (4)
[1]
sM = 0.4223 (sD unknown) K 1 ¼ ½ 0:7623 0:5212 ; K 2 ¼ ½ 0:3690 0:5362 sM = 1.75 (sD unknown) K 1 ¼ ½ 0:3622 0:299 ; K 2 ¼ ½ 0:3625 0:299
This paper (l = 2)
1008
C.-H. Lien, K.-W. Yu / Chaos, Solitons and Fractals 35 (2008) 1003–1008
4. Conclusion In this paper, the robust control problem for a class of uncertain fuzzy system with time-varying state and input delays has been investigated. Based on the LMI approach, delay-dependent criterion has been proposed to guarantee the asymptotic stabilization of fuzzy systems. An example has been provided to demonstrate the reduced conservativeness of the proposed result.
Acknowledgement The research reported here was supported by the National Science Council of Taiwan, ROC under grant no. NSC 93-2213-E-214-020.
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