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Augmented Lyapunov–Krasovskii functional approaches to robust stability criteria for uncertain Takagi–Sugeno fuzzy systems with time-varying delays
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Fuzzy Sets and Systems 201 (2012) 1 – 19 www.elsevier.com/locate/fss
Augmented Lyapunov–Krasovskii functional approaches to robust stability criteria for uncertain Takagi–Sugeno fuzzy systems with time-varying delays夡 O.M. Kwona , M.J. Parka , S.M. Leeb , Ju H. Parkc,∗ a School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 361-763, Republic of Korea b School of Electronics Engineering, Daegu University, Jinryang-eup, Gyeongsan 712-714, Republic of Korea c Department of Electrical Engineering, Yeungnam University, 214-1 Dae-dong, Kyongsan 712-749, Republic of Korea
Received 4 February 2011; received in revised form 21 December 2011; accepted 22 December 2011 Available online 29 December 2011
Abstract This paper considers the problem of robust stability analysis for uncertain Takagi–Sugeno (T–S) fuzzy systems with time-varying delays. By constructing an augmented Lyapunov–Krasovskii functional and utilizing Finsler’s Lemma, a novel criterion for delaydependent robust stability of T–S fuzzy model with time-varying delay is derived in terms of linear matrix inequalities (LMIs). Also, a further improved stability criterion is proposed by utilizing free weighting techniques. Finally, three numerical examples are included to show the superiority of the proposed criteria. © 2012 Elsevier B.V. All rights reserved. Keywords: Fuzzy system models; T–S fuzzy systems; Time-varying delay; Stability; Lyapunov method; LMI
1. Introduction Since T–S fuzzy model [1] was first introduced, the stability analysis of the model has been paid much attentions because it is very important to check the stability of certain dynamic systems when T–S fuzzy control scheme is applied. The main advantage of T–S fuzzy model is that it can be applied as an efficient tool in approximating complex nonlinear systems. It combines the flexible fuzzy logic theory and rigorous mathematical theory of linear system into an unified framework to approximate nonlinear dynamics of complex systems [2]. On the other hand, time-delays often encounter in many industrial and engineering systems such as chemical processes, rolling mill systems, neural
夡 This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0009273). ∗ Corresponding author. Tel.: +82 52 8102491; fax: +82 53 8104767. E-mail addresses: [email protected] (O.M. Kwon), [email protected] (M.J. Park), [email protected] (S.M. Lee), [email protected] (J.H. Park). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.12.014
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
networks, networked control systems and so on [3]. It is well known that time-delays can cause poor performance or instability. Therefore, the problem of delay-dependent stability and stabilization for T–S fuzzy systems with time-delays has been received great efforts by many researchers in recent years [2,4–13] because delay-dependent approaches are generally less conservative than delay-independent ones when the sizes of time-delays are small [14]. The main issue of delay-dependent stability and stabilization is to reduce the conservatism of stability and stabilization criteria. One of the important index for checking the conservatism is to enlarge the feasible region of the criteria or to get maximum delay bounds for guaranteeing the system stability. Hence, how to construct Lyapunov–Krasovskii functional and obtain the stability and stabilization conditions by calculating upper bounds of time-derivative of Lyapunov–Krasovskii functional play important roles to reduce the conservatism of the conditions. In this regard, Jeung et al. [6] proposed a delay-dependent control method by utilizing a descriptor model transformation and Park’s inequality [15]. Tian and Peng [7] derived new delay-dependent stability and stabilization criteria for uncertain T–S fuzzy systems with interval time-varying delays. The main contribution of the work [7] was that the proposed criteria allow fast time-varying delays, which mean the criteria do not need the information on derivative of the time delay. Improved robust stability and stabilization for uncertain T–S fuzzy time-delay systems without integral inequality technique [16] and model transformation method were studied in [8]. Based on a fuzzy Lyapunov–Krasovskii functional and fuzzy free weighting matrices, the stability and stabilization problems for T–S fuzzy systems with timevarying delay were investigated in [9]. In Lien et al. [10], a simple analytic method with fewer free-weighted matrices was proposed for stability analysis of uncertain T–S fuzzy systems with interval time-varying delay. Recently, delaydependent stability analysis and H∞ control for uncertain T–S fuzzy systems with interval time-varying delays were conducted in [12]. Very recently, in [13], further improvements on delay-dependent stability analysis and synthesis of T–S fuzzy systems with time-varying delay is made by considering ignored terms in estimating upper bounds of the derivative of Lyapunov–Krasovskii functional. However, there are still rooms for further improvements to enlarge the feasible region of stability criteria for T–S fuzzy systems with time-varying delays. In this paper, we revisit the problem of stability analysis for uncertain T–S fuzzy systems with time-varying delays. First, by constructing an augmented Lyapunov–Krasovskii functional and utilizing Finsler’s lemma, a delay-dependent stability condition without the use of free weighting matrices is derived in terms of LMIs, which will be introduced in Theorem 1. Inspired by the works of [17–20], a triple integral form of Lyapunov–Krasovskii functional is utilized. t t−h(t) Also, new integral forms of states (1/ h(t)) t−h(t) x(s) ds and (1/(h U − h(t))) t−h U x(s) ds are taken as augmented vectors to include further information of time-varying delays. And based on the results of Theorem 1, a further improved stability criterion is proposed by utilizing free-weighting matrices. Through three numerical examples, the reduction of the conservatism of Theorem 1 will be shown by comparing with the results proposed in other literature. Finally, it will be presented that further enlargement of delay bounds can be made by a new result, Theorem 2. Notation. Rn is the n-dimensional Euclidean space, and Rm×n denotes the set of m × n real matrix. Cn,h = C([−h, 0], Rn ) stands for the Banach space of continuous functions mapping the interval [−h, 0] into Rn , with the topology of uniform convergence. For symmetric matrices X and Y, X > Y (respectively, X ⱖ Y ) means that the matrix X − Y is positive definite (respectively, nonnegative). In , 0n and 0m×n denotes n × n identity matrix, n × n and m × n zero matrices, respectively. · refers to the Euclidean vector norm and the induced matrix norm. diag{· · ·} denotes the block diagonal matrix. represents the elements below the main diagonal of a symmetric matrix. For a given matrix X ∈ Rm×n , such that rank(X ) = r , we define X ⊥ ∈ Rn×(n−r ) as the right orthogonal complement of X; i.e., X X ⊥ = 0. X [h(t)] ∈ Rm×n means that the elements of the matrix X includes the value of h(t); e.g., X [h U ] ≡ X [h(t)=h U ] . 2. Problem statements Consider the following uncertain T–S fuzzy systems with time-varying delays: Plant rule l : IF 1 (t) is l1 , . . ., and s (t) is ls , THEN x(t) ˙ = (Al + Al (t))x(t) + (Adl + Adl (t))x(t − h(t)), l = 1, 2, . . . , r., x(s) = (s), ∀s ∈ [−h U , 0], h U > 0,
(1)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
3
where i (t)(i = 1, . . . , s) are the premise variables, and li (i = 1, . . . , s) are fuzzy sets, r is the number of IF–THEN rules, x(t) ∈ Rn is the state vector, (s) ∈ Cn,h is a given continuous vector valued initial function, Al ∈ Rn×n and Adl ∈ Rn×n are known constant matrices with appropriate dimensions, and Al (t) and Adl (t) are the system uncertainties of the form: [Al (t) Adl (t)] = D F(t)[El E dl ] (l = 1, . . . , r ),
(2)
in which the time-varying nonlinear function F(t) satisfies F T (t)F(t) ⱕ I, ∀t ⱖ 0.
(3)
The time-varying delay h(t) is a continuous function that satisfies ˙ ⱕ h D, 0 ⱕ h(t) ⱕ h U , h(t)
(4)
where h U is a positive scalar and h D is an any constant one. Using the center-average defuzzifier, product interference and singleton fuzzifier, the global dynamics of the T–S fuzzy system (1) can be inferred as follows: r wl ((t))[(Al + Al (t))x(t) + (Adl + Adl (t))x(t − h(t))] r x(t) ˙ = l=1 l=1 wl ((t)) r = l ((t))[(Al + Al (t))x(t) + (Adl + Adl (t))x(t − h(t))] (5) l=1
with (t) = [1 (t), . . . , s (t)], wl ((t)) =
s i=1
wl ((t)) , li (i (t)), l ((t)) = r l=1 wl ((t))
(6)
where li (i (t)) is the grade of membership value of i (t) in li . It is assumed that wl ((t)) ⱖ 0,
r
wl ((t)) > 0.
(7)
l=1
Then, we have the following conditions: l ((t)) ⱖ 0,
r
l ((t)) = 1.
(8)
l=1
For the sake of simplicity, let us define Aˆ =
r
l ((t))Al ,
Aˆ d =
l=1
Eˆ =
r
r
l ((t))Adl ,
l=1
l ((t))El ,
l=1
Eˆ d =
r
l ((t))E dl .
(9)
l=1
Then, system (5) can be rewritten as: ˆ x(t) ˙ = Ax(t) + Aˆ d x(t − h(t)) + Dp(t), p(t) = F(t)q(t), q(t) = Eˆ x(t) + Eˆ d (t − h(t)). The aim of this paper is to derive novel robust delay-dependent stability conditions for system (10).
(10)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Before deriving our main results, we give the following facts and lemmas. Fact 1 (Schur Complement, Boyd et al. [21]). Given constant matrices 1 , 2 , 3 with 1 = 1T and 0 < 2 , then 1 + 3T −1 2 3 < 0 if and only if −2 3 1 3T < 0 or < 0. (11) 3 −2 3T 1 Fact 2 (Park [15]). For any real vectors a, b and any matrix Q > 0 with appropriate dimensions, we have ± 2a T b ⱕ a T Qa + b T Q −1 b.
(12)
Lemma 1 (Finsler’s lemma, de Oliveira and Skelton [22]). Let ∈ Rn , = T ∈ Rn×n , and B ∈ Rm×n such that rank(B) < n. The following statements are equivalent: (i) T < 0, ∀B = 0, 0, T (ii) B ⊥ B ⊥ < 0. (iii) ∃L ∈ Rn×m : + LB + B T LT < 0. Lemma 2. For any constant matrix M ∈ Rn×n , M = M T > 0 and ⱕ s ⱕ , the following inequalities hold:
T
T x˙ (s)M x(s) ˙ ds ⱖ x(s) ˙ ds M x(s) ˙ ds , ( − )
T
( − )2 T x˙ (u)M x(u) ˙ du ds ⱖ x(u) ˙ du ds M x(u)u ˙ ds . 2 s s s Proof. According to Jensen’s inequality in [16], one can obtain (13). Moreover, the following inequality holds
T
( − s) x˙ T (u)M x(u) ˙ du ⱖ x(u) ˙ du M x(u) ˙ du . s
s
(13)
(14)
(15)
s
By using Fact 1, Eq. (15) is equivalent to the following inequality:
T ˙ du s x˙ T (u) du s x˙ (u)M x(u)
ⱖ 0. ˙ du ( − s)M −1 s x(u)
(16)
Integration of inequality (16) from to yields
T ˙ du ds s x˙ T (u) du ds s x˙ (u)M x(u)
ⱖ 0. −1 ds ˙ du ds s x(u) ( − s)M
(17)
Therefore, the inequality (17) is equivalent to the inequality (14) according to Fact 1. This completes our proof. Lemma 3. For any scalar h(t) ⱖ 0 and any constant matrix Q ∈ Rn×n , Q = Q T > 0, the following inequality holds: t t t h 2 (t) T (t)X Q −1 X T (t) + 2T (t)X x˙ T (s)Q x(s) ˙ ds ⱕ (x(t) − x(s)) ds, (18) − 2 t−h(t) s t−h(t) where
T (t) = x T (t) x T (t − h(t)) x T (t − h U ) x˙ T (t) x˙ T (t − h U ) 1 h U − h(t)
t−h(t)
x T (s) ds p T (t) ,
t−h U
and X is free weighting matrix with appropriate dimensions.
1 h(t)
t
x T (s) ds
t−h(t)
(19)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Proof. From Fact 2, the following inequality holds t t t t (X T (t))T x(u) ˙ du ds ⱕ [T (t)X Q −1 X T (t) + x˙ T (u)Q x(u)] ˙ du ds. −2 t−h(t) s
5
(20)
t−h(t) s
From (20), we obtain t t − x˙ T (s)Q x(u) ˙ du ds t−h(t) s
t
ⱕ
t
T (t)X Q −1 X T (t) du ds + 2
t−h(t) s
=
h 2 (t) 2
T (t)X Q −1 X T (t) + 2T (t)X
t
t
(X T (t))T x(u) ˙ du ds
t−h(t) s t
(x(t) − x(s)) ds.
(21)
t−h(t)
Therefore, the inequality (21) is equivalent to the inequality (18). This completes our proof. 3. Main results In this section, we propose new delay-dependent stability criteria for system (10). Before introducing our main results, the notations of several matrices are defined for simplicity: ˆ = [ Aˆ Aˆ d 0n − In 0n 0n 0n D], l = [Al Adl 0n − In 0n 0n 0n D] (l = 1, . . . , r ), ˆ = [ Eˆ Eˆ d 0n 0n 0n 0n 0n 0n ], l = [El E dl 0n 0n 0n 0n 0n 0n ] (l = 1, . . . , r ), T T 11 = R13 + R13 + N11 + G + h U Q 2 − 2Q 3 , 13 = −R13 + R23 ,
14 = R11 + N12 , 22 = −(1 − h D )G − 2Q 3 , T 2 33 = −R23 − R23 − N11 , 35 = −N12 + R22 , 44 = N22 + (h U /2)Q 3 ,
77 = −h U Q 2 − 2Q 3 , ⎡ 11 0n 13 14 R12 2Q 3 h U R33 ⎢ 0 0 0 0 2Q 3 22 n n n n ⎢ ⎢ T ⎢ 33 R12 35 0n −h U R33 ⎢ ⎢ 44 0n 0n h U R13 ⎢ 1 = ⎢ ⎢ −N22 0n h U R23 ⎢ ⎢ −2Q 0n 3 ⎢ ⎢ ⎣ 77 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 2 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0n 0n 0n 0n 0n 0n 0n 0n 0n
R33 0n
0n 0n 0n −R33 0n 0n R13
0n R23 −Q 2
⎤
−R33 0n 0n 0n ⎥ ⎥ ⎥ R33 0n ⎥ ⎥ −R13 0n ⎥ ⎥ ⎥, −R23 0n ⎥ ⎥ 0n 0n ⎥ ⎥ ⎥ Q 2 0n ⎦
0n
0n
⎤
0n ⎥ ⎥ ⎥ 0n ⎥ ⎥ 0n ⎥ ⎥ ⎥, 0n ⎥ ⎥ 0n ⎥ ⎥ ⎥ 0n ⎦ − In
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
[h(t)] = 1 + h(t)2 , ⎡ −2Q 1 2Q 1 0n 0n ⎢ −3Q 1 Q 1 0n ⎢ ⎢ 1 ⎢ −Q 1 0n 1 = × hU ⎢ ⎢ 3 hU Q 1 ⎣ ⎡
Q 1 −Q 1
⎢ ⎢ ⎢ 1 2 = 2 × ⎢ hU ⎢ ⎢ ⎣
0n
0n
0n×4n
⎥ 0n×4n ⎥ ⎥ 0n×4n ⎥ ⎥, ⎥ 0n×4n ⎦ 04n
0n 0n×4n
⎤
⎤
⎥ Q 1 0n 0n×4n ⎥ ⎥ −Q 1 0n 0n×4n ⎥ ⎥, ⎥ 0n 0n×4n ⎦ 04n
[h(t)] = 1 + h(t)2 .
(22)
Now, we have the following theorem. Theorem 1. For given scalars h U > 0 and h D , the system (10) is asymptotically stable for 0 ⱕ h(t) ⱕ h U and ˙ ⱕ h D , if there exist symmetric positive definite matrices R = [Ri j ]3×3 ∈ R3n×3n , N = [Ni j ]2×2 ∈ R2n×2n , h(t) G ∈ Rn×n , Q i (i = 1, 2, 3) ∈ Rn×n satisfying the following LMIs: T ⊥ ⊥ l 08n×n l 08n×n ( p) + ( p) lT < 08n , l = 1, . . . , r, (23) 0n×7n In 0n×7n In − In where l( p) and ( p) for p = 1, 2 are the vertices of l[h(t)] and [h(t)] , respectively, and l⊥ is the right orthogonal complement of l . Proof. For positive definite matrices R = [Ri j ]3×3 , N = [Ni j ]2×2 , G, Q i (i = 1, 2, 3), let us choose the following Lyapunov–Krasovskii’s functional candidate as: V = V1 + V2 + V3 + V4 + V5 + V6 , where
(24)
⎡
⎤T ⎡ ⎤ ⎤⎡ x(t) x(t) R11 R12 R13 ⎢ ⎢ ⎥ ⎥ V1 = ⎣ x(t − h U ) ⎦ ⎣ R22 R23 ⎦ ⎣ x(t − h U ) ⎦ , t t R33 t−h U x(s) ds t−h U x(s) ds
V2 =
t
x(s) x(s) ˙
t−h U
V3 = V4 = V5 = V6 =
t
R
T
N11 N12 x(s) ds, x(s) ˙ N22 N
T
x (s)Gx(s) ds, t−h(t) t t t−h U s t t
x˙ T (u)Q 1 x(u) ˙ du ds, x T (u)Q 2 x(u) du ds,
t−h U s t t t t−h U s
u
x˙ T (v)Q 3 x(v) ˙ dv du ds.
(25)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
The time-derivative of V1 can be obtained as: ⎡ ⎤T ⎡ ⎤ x(t) R11 R12 R13 ⎡ ⎢ ⎥ ⎢ ⎥ V˙1 = 2 ⎣ x(t − h U ) ⎦ ⎣ R22 R23 ⎦ ⎣ 1 ⎡ ⎢ ⎢ =2⎢ ⎢ ⎣
⎤T
R33
⎤
x(t) ˙
⎦, x(t ˙ − hU ) x(t) − x(t − h U )
⎤ ⎡ x(t) ˙ ⎥ x(t − h U ) ⎥ ⎥ ⎥ 2 ⎢ t x(t ˙ − hU ) ⎦, ⎣ 1 ⎥ x(s) ds h(t) t−h(t) ⎦ x(t) − x(t − h U ) t−h(t) 1 h U −h(t) t−h U x(s) ds x(t)
where
1 = h(t)
1 h(t)
x(s) ds + (h U − h(t))
t
7
t−h(t)
1 h U − h(t)
(26)
t−h(t)
x(s) ds ,
t−h U
⎤ ⎡ ⎤ I n 0n 0n R11 R12 R13 ⎥ ⎢0 I 0n ⎥ ⎢ ⎢ n n ⎥ 2 = ⎢ ⎥ × ⎣ R22 R23 ⎦ . ⎦ ⎣ 0n 0n h(t)In R33 0n 0n (h U − h(t))In ⎡
(27)
By calculating the time-derivative of V2 , we have T N11 N12 x(t) x(t) V˙2 = x(t) ˙ N22 x(t) ˙ −
x(t − h U ) x(t ˙ − hU )
T
N11 N12 N22
x(t − h U ) . x(t ˙ − hU )
(28)
We can obtain an upper bound of the time-derivative of V3 as follows: V˙3 ⱕ x T (t)Gx(t) − (1 − h D )x T (t − h(t))Gx(t − h(t)).
(29)
Calculating the time-derivative of V4 leads to t V˙4 = h U x˙ T (t)Q 1 x(t) ˙ − x˙ T (s)Q 1 x(s) ˙ ds t−h U
= h U x˙ T (t)Q 1 x(t) ˙ −
t
t−h(t)
x˙ T (s)Q 1 x(s) ˙ ds −
t−h(t)
x˙ T (s)Q 1 x(s) ˙ ds.
t−h U
By using −1 = −
h(t) h U − h(t) − hU hU
and Lemma 2, an upper bound of the first integral term of V˙4 can be obtained as t − x˙ T (s)Q 1 x(s) ˙ ds t−h(t)
=−
h(t) hU
h(t) ⱕ− hU
t
x˙ T (s)Q 1 x(s) ˙ ds −
t−h(t)
h U − h(t) hU
t
x˙ T (s)Q 1 x(s) ˙ ds
t−h(t)
(h U − h(t))h(t) x˙ T (s)Q 1 x(s) ˙ ds − 2 hU t−h(t) t
t
t−h(t)
x˙ T (s)Q 1 x(s) ˙ ds
(30)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
ⱕ−
1 hU
=
x(s) ˙ ds
x(t) x(t − h(t))
x(s) ˙ ds
t−h(t)
T
⎡ ⎢ ⎢ ⎣
−
x(s) ˙ ds
t−h(t)
T
t
t
Q1
t−h(t)
h U − h(t) − 2 hU
T
t
x(s) ˙ ds
Q1
2 hU
t
−
h(t) 2 hU
t−h(t)
2 hU
Q1
−
2 hU
⎤
Q1 ⎥ x(t) ⎥
⎦ x(t − h(t)) . − h(t) Q 1 h2
−
h(t) 2 hU
(31)
U
With the similar method introduced above, an upper bound of other integral term of V˙4 can be estimated as: t−h(t) x˙ T (s)Q 1 x(s) ˙ ds − t−h U
h U − h(t) t−h(t) T x˙ (s)Q 1 x(s) ˙ ds − x˙ (s)Q 1 x(s) ˙ ds hU t−h U t−h U (h U − h(t))h(t) t−h(t) T h U − h(t) t−h(t) T ⱕ− x˙ (s)Q 1 x(s) ˙ ds − x˙ (s)Q 1 x(s) ˙ ds 2 hU hU t−h U t−h U t−h(t) t−h(t)
T
h(t) ⱕ− 2 x(s) ˙ ds Q1 x(s) ˙ ds hU t−h U t−h U t−h(t) t−h(t)
T
1 − x(s) ˙ ds Q1 x(s) ˙ ds hU t−h U t−h U
⎡ ⎤ h(t) T − 1 + h(t) Q 1 + Q 1 1 2 2 hU hU hU ⎢ ⎥ x(t − h(t)) x(t − h(t)) ⎢ ⎥
= ⎣ ⎦ x(t − h U ) . x(t − h U ) h(t) − 1 + h2 Q 1 h(t) =− hU
t−h(t)
T
(32)
U
Then, from (31) and (32), the upper bound of V˙4 can be rewritten as: V˙4 ⱕ T (t)[h(t)] (t),
(33)
where [h(t)] is defined in (22). Using Lemma 2, an upper bound of the time-derivative of V5 can be calculated as t t−h(t) V˙5 = h U x T (t)Q 2 x(t) − x T (s)Q 2 x(s) ds − x T (s)Q 2 x(s) ds t−h(t)
t−h U
ⱕ h U x (t)Q 2 x(t) t
T
t 1 − x(s) ds Q2 x(s) ds h(t) t−h(t) t−h(t) t−h(t) t−h(t)
T
1 − x(s) ds Q2 x(s) ds h U − h(t) t−h U t−h U T
= h U x T (t)Q 2 x(t)
T
t t 1 1 − x(s) ds h(t)Q 2 x(s) ds h(t) t−h(t) h(t) t−h(t)
T
t−h(t) t−h(t) 1 1 x(s) ds (h U − h(t))Q 2 x(s) ds . − h U − h(t) t−h U h U − h(t) t−h U
(34)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
9
By calculating V˙6 , one can obtain t t h2 ˙ − x˙ T (u)Q 3 x(u) ˙ du ds V˙6 = U x˙ T (t)Q 3 x(t) 2 t−h U s t t t−h(t) t h2 ˙ − x˙ T (u)Q 3 x(u) ˙ du ds − x˙ T (u)Q 3 x(u) ˙ du ds = U x˙ T (t)Q 3 x(t) 2 t−h(t) s t−h U s t t t−h(t) t−h(t) h2 ˙ − x˙ T (u)Q 3 x(u) ˙ du ds − x˙ T (u)Q 3 x(u) ˙ du ds ⱕ U x˙ T (t)Q 3 x(t) 2 t−h(t) s t−h U s ⱕ
2 hU ˙ x˙ T (t)Q 3 x(t) 2
T
t t 2 − 2 x(s) ds Q 3 h(t)x(t) − x(s) ds h(t)x(t) − h (t) t−h(t) t−h(t)
T t−h(t) 2 − − h(t))x(t − h(t)) − x(s) ds Q3 (h U (h U − h(t))2 t−h U
t−h(t) × (h U − h(t))x(t − h(t)) − x(s) ds t−h U
=
2 hU x˙ T (t)Q 3 x(t) ˙ 2
T
t t 1 1 −2 x(t) − x(s) ds Q 3 x(t) − x(s) ds h(t) t−h(t) h(t) t−h(t)
T t−h(t) 1 −2 x(t − h(t)) − x(s) ds Q3 h U − h(t) t−h U
t−h(t) 1 × x(t − h(t)) − x(s) ds , h U − h(t) t−h U
(35)
where Lemma 2 was utilized in (35). Since the following inequality holds from (3) and (10), p T (t) p(t) ⱕ q T (t)q(t),
(36)
there exists a positive scalar satisfying the following inequality:
[q T (t)q(t) − p T (t) p(t)] ⱖ 0.
(37)
Note that ˆ T (t), ˆ q T (t)q(t) = T (t)
(38)
ˆ was defined in (22). where From (26)–(38) and S-procedure [21], the time-derivative of V has a new upper bound as: ˆ T )(t), ˆ V˙ ⱕ T (t)([h(t)] + [h(t)] +
(39)
where (t) was defined in (19), and [h(t)] and [h(t)] were in (22), respectively. Also, the system (10) with the augmented vector (t) can be rewritten as: ˆ 0n×1 = (t), ˆ is defined in (22). where
(40)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Therefore, an asymptotic stability condition for system (10) can be ˆ T )(t) ˆ T (t)([h(t)] + [h(t)] + <0 ˆ subject to 0n×1 = (t).
(41)
Then, by utilizing the statements of (i) and (iii) of Lemma 1, for any matrix L(t) with appropriate dimension, the condition (41) is equivalent to ˆ T ˆ + L(t) ˆ + ˆ T LT (t) < 0. [h(t)] + [h(t)] + By using Fact 1, inequality (42) is equivalent to ˆT ˆ + ˆ T LT (t) [h(t)] + [h(t)] + L(t) < 09n . − In The above inequality (43) can be represented as r [h(t)] + [h(t)] + L(t) l + lT LT (t) lT l ((t)) < 09n . − In
(42)
(43)
(44)
l=1
At each lth rule, multiplying on the left side and right side of the inequality (44) by ⊥ ⊥ T l 08n×n ( l ) 07n×n and 0n×8n In 0n×7n In leads to ⊥ T ( l ) ([h(t)] + [h(t)] + L(t) l + lT LT (t)) l⊥ ( l⊥ )T lT < 08n (l = 1, . . . , r ), − In which can be rewritten by ⊥ T ⊥ l 08n×n l 08n×n [h(t)] + [h(t)] lT < 08n (l = 1, . . . , r ). 0n×7n In 0n×7n In − In
(45)
(46)
By the convexity of h(t), we have
1 (t) = 1 −
h(t) h(t) , 2 (t) = . hU hU
(47)
Let us define H = {H (t) : 0 ⱕ h(t) ⱕ h U }.
(48)
Then, the vertex set Hvex is defined as: Hvex = {H : h(t) = 0 or h U }.
(49)
There are two elements in Hvex , and we can enumerate elements in Hvex by Hk , where k = 1, 2: H1 = 0, H2 = h U .
(50)
Since [h(t)] and [h(t)] are affinely dependent on h(t) ∈ H , respectively, [h(t)] and [h(t)] can be expressed as follows: [h(t)] + [h(t)] = (1 + 1 ) + h(t)(2 + 2 ) = (1 + 1 ) + [ 1 (t)H1 + 2 (t)H2 ](2 + 2 ) = (1 + 1 ) + [ 1 (t) · 0 + 2 (t) · h U ](2 + 2 ) = (1 + 1 ) + 1 (t)(0)(2 + 2 ) + 2 (t)(h U )(2 + 2 ), where i (i = 1, 2) and i (i = 1, 2) are defined in (22).
(51)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
11
Note that 1 (t) + 2 (t) = 1 for h(t), and i (t) ⱖ 0 (i = 1, 2). Therefore, by utilizing the properties of convex-hull, inequality (46) holds if
l⊥ 0n×7n
l⊥ 0n×7n
08n×n In 08n×n In
T
T
[0] + [0] lT − In
l⊥ 0n×7n
[h U ] + [h U ] lT − In
08n×n In
l⊥ 0n×7n
< 08n , l = 1, . . . , r ,
08n×n In
(52)
< 08n , l = 1, . . . , r .
(53)
Therefore, if LMIs (23) hold, then the inequality (41) holds, which means that system (10) is asymptotically stable. This completes our proof. t t−h(t) Remark 1. Our augmented vector (t) includes the states of (1/ h(t)) t−h(t) x(s) ds and (1/(h U −h(t))) t−h U x(s) ds. The motivation of this idea is that more information about h(t) may be utilized in delay-dependent stability condition of (23). Remark 2. Theorem 1 is derived by utilizing the Lyapunov–Krasovskii’s functional without free-weighting matrices. By considering free-weighting matrices, a further improved delay-dependent stability criterion can be derived, which will be introduced in Theorem 2. For the sake of simplicity in Theorem 2, we define the following notations: X T = [X 1T X 2T X 3T X 4T X 5T 0n 0n 0n ], Y T = [Y1T Y2T Y3T Y4T Y5T 0n 0n 0n ], Z T = [Z 1T 0n 0n 0n 0n Z 2T 0n 0n ], W T = [0n W1T 0n 0n 0n 0n W2T 0n ], ⎡
In −In
⎢0 ⎢ n 1 = [0n×3n In 0n×4n ], 2 = ⎢ ⎣ In 0n ˆ [h(t)] = [h(t)] + 1T
hU Q 1 +
In 0n In
0n
0n 0n
0n
0n
−In 0n 0n
0n
0n
0n 0n
0n
⎤
0n ⎥ ⎥ ⎥, 0n ⎦
0n 0n −In 0n 0n 0n 0n −In 0n
2 hU Q 4 1 + [X Y h(t)Z (h U − h(t))W]2 2
+2T [X Y h(t)Z (h U − h(t))W]T ,
(54)
where [h(t)] is defined in (22). Now, we have the following theorem. Theorem 2. For given scalars h U > 0 and h D , the system (10) is asymptotically stable for 0 ⱕ h(t) ⱕ h U and ˙ ⱕ h D , if there exist symmetric positive definite matrices R = [Ri j ]3×3 ∈ R3n×3n , N = [Ni j ]2×2 ∈ R2n×2n , h(t) G ∈ Rn×n , Q i (i = 1, 2, 3, 4) ∈ Rn×n , and any matrices X i , Yi (i = 1, 2, 3, 4, 5) ∈ Rn×n , Z i ∈ Rn×n , Wi (i = 1, 2) ∈ Rn×n satisfying the following LMIs for l = 1, . . . , r : ⎤T ⎡ l⊥ 08n×n 09n×3n ⎥ ⎢ ⎦ ⎣ 0n×7n In 03n×8n I3n
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
⎤ ˆ [0] T 08n×n hU W hU Y l ⎥ ⎢ 0n 0n 0n − In ⎥ ⎢ ⎥ ⎢ ×⎢ −2Q 4 0n 0n ⎥ ⎦ ⎣ −2Q 4 0n −h U Q 1 ⎡ ⊥ ⎤ l 08n×n 09n×3n ⎦ In × ⎣ 0n×7n < 011n , (l = 1, . . . , r ), 03n×8n I3n ⎡ ⊥ ⎤T l 08n×n 0 9n×3n ⎦ ⎣ 0n×7n In 03n×8n I3n ⎤ ⎡ ˆ [h U ] T hU Z 08n×n hU X l ⎥ ⎢ 0n 0n 0n − In ⎥ ⎢ ⎥ ×⎢ −2Q 4 0n 0n ⎥ ⎢ ⎦ ⎣ −2Q 4 0n −h U Q 1 ⎡ ⊥ ⎤ l 08n×n 09n×3n ⎦ In × ⎣ 0n×7n < 011n , (l = 1, . . . , r ), 03n×8n I3n ⎡
(55)
(56)
where l , and l are defined in (22), respectively, and others are in (54). Proof. Let us choose the following Lyapunov–Krasovskii’s functional candidate with free-weighting matrices as: V =
7
Vi ,
(57)
i=1
where Vi (i = 1, . . . , 6) is the same Lyapunov–Krasovskii’s functional used in Theorem 1, and V7 =
t
t
t−h U s
t
x˙ T (v)Q 4 x(v) ˙ dv du ds.
(58)
u
Using Lemma 3, we have new upper bound of the time-derivative of V4 as: t t−h(t) V˙4 = h U x˙ T (t)Q 1 x(t) ˙ − x˙ T (s)Q 1 x(s) ˙ ds − x˙ T (s)Q 1 x(s) ˙ ds t−h(t)
t−h U
ⱕ h U x˙ (t)Q 1 x(t) ˙ T
T T +h(t)T (t)X Q −1 1 X (t) + 2 (t)X [x(t) − x(t − h(t))] T T +(h U − h(t))T (t)Y Q −1 1 Y (t) + 2 (t)Y[x(t − h(t)) − x(t − h U )],
where X and Y are defined in (54). Similarly, an upper bound of the time-derivative of V7 can be calculated as: t t t−h(t) t h2 V˙7 = U x˙ T (t)Q 4 x(t) ˙ − x˙ T (u)Q 4 x(u) ˙ du ds − x˙ T (u)Q 4 x(u) ˙ du ds 2 t−h(t) s t−h U s ⱕ
2 hU h 2 (t) T T ˙ + x˙ T (t)Q 4 x(t) (t)Z Q −1 4 Z (t) 2 2
(59)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
t 1 (h U − h(t))2 T T +2 (t)Zh(t) x(t) − x(s) ds + (t)W Q −1 4 W (t) h(t) t−h(t) 2 t−h(t) 1 +2T (t)W(h U − h(t)) x(t − h(t)) − x(s) ds , h U − h(t) t−h U
13
T
where Z and W are defined in (54), and Lemma 3 was utilized in (60). From (37), (59) and (60), the time-derivative of Vˆ has a new upper bound as: ⎞ ⎛ ˆ T ˆ + h(t)X Q −1 X T + (h U − h(t))Y Q −1 Y T ˆ [h(t)] + 1 1 ⎠ (t), V˙ ⱕ T (t) ⎝ h 2 (t) (h U −h(t))2 −1 T −1 T + 2 Z Q4 Z + W Q4 W 2
(60)
(61)
ˆ [h(t)] was defined in (54). where Therefore, an asymptotic stability condition for system (10) can be ⎛ ⎞ ˆ T ˆ + h(t)X Q −1 X T + (h U − h(t))Y Q −1 Y T ˆ[h(t)] + 1 1 ⎠ (t) < 0 T (t) ⎝ h 2 (t) (h U −h(t))2 −1 T −1 T + 2 Z Q4 Z + W Q W 4 2 ˆ subject to 0n×1 = (t),
(62)
ˆ [h(t)] are defined in (22) and (54), respectively. ˆ and where By utilizing (i) and (iii) of Lemma 1 and Fact 1, for any matrix L(t), the condition (62) is equivalent to ⎞ ⎡⎛ ⎤ ˆ [h(t)] + h(t)X Q −1 X T 1 ⎢⎜ ⎥ T ⎟ ⎢ ⎜ +(h U − h(t))Y Q −1 ⎥ 1 Y ⎟ ⎟ ⎢⎜ ⎥ ⎟ ⎢⎜ T ⎥ h 2 (t) −1 T ⎟ l ⎥ ⎢⎜ + 2 Z Q4 Z ⎟ ⎢⎜ ⎥ < 09n (l = 1, . . . , r ). ⎢ ⎜ (h U −h(t))2 ⎥ −1 T ⎟ W Q W ⎢⎝ + ⎥ ⎠ 4 2 ⎢ ⎥ T T ⎣ ⎦ L(t) l + l L (t) − In
(63)
At each lth rule, by multiplying on the left side and right side of the inequality (63) by ⊥ T ⊥ ( l ) 07n×n l 08n×n and , 0n×8n In 0n×7n In the following inequality can be obtained: ⎞ ⎛ ⎡ ˆ [h(t)] + h(t)X Q −1 X T 1 ⎜ ⎢ T ⎟ ⎜ +(h U − h(t))Y Q −1 ⎢ 1 Y ⎟ ⎟ ⎜ ⎢ 2 ⎟ ⊥ ⎢ ⊥ T⎜ T ⎟ l ( l⊥ )T lT ⎢ ( l ) ⎜ + h 2(t) Z Q −1 Z 4 ⎟ ⎜ ⎢ ⎜ (h U −h(t))2 ⎢ −1 T ⎟ W Q W ⎢ ⎠ ⎝+ 4 2 ⎢ T T ⎣ L(t) l + l L (t) − In
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 08n (l = 1, . . . , r ). ⎥ ⎥ ⎥ ⎦
From Fact 1, inequality (64) is equivalent to ⎡⎡ ⎤ ˆ [h(t)] + h(t)X Q −1 X T 1 T ⊥ ⊥ ⊥ T T l ( l ) l ⎥ ⎢ ⎢ ( l ) T ⎢⎣ ⎦ +(h U − h(t))Y Q −1 1 Y ⎢ ⎢ − In ⎢ ⎢ ⎣
(64)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
h(t)( l⊥ )T Z
(h U − h(t))( l⊥ )T W
0n
0n
−2Q 4
0n
−2Q 4
⎤ ⎥ ⎥ ⎥ < 010n (l = 1, . . . , r ). ⎥ ⎦
Inequality (65) can be represented as: ⎡ ⎤T l⊥ 08n×n 09n×2n ⎥ ⎢ ⎣ 0n×7n ⎦ In 02n×8n I2n ⎤ ⎤ ⎡⎡ ˆ [h(t)] + h(t)X Q −1 X T 1 T (h U − h(t))W ⎥
l ⎥ h(t)Z ⎢⎢ T ⎥ ⎢ ⎣ +(h U − h(t))Y Q −1 ⎦ 1 Y ⎥ ⎢ 0 0n n ×⎢ ⎥ − In ⎥ ⎢ ⎦ ⎣ −2Q 4 0n −2Q 4 ⎡ ⎤ ⊥ l 08n×n 09n×2n ⎥ ⎢ × ⎣ 0n×7n ⎦ < 010n (l = 1, . . . , r ). In 02n×8n I2n
(65)
(66)
From the properties of convex-hull, inequality (66) holds for 0 ⱕ h(t) ⱕ h U if the following two inequalities are satisfied ⎤T ⎡ l⊥ 08n×n 09n×2n ⎥ ⎢ ⎦ ⎣ 0n×7n In 02n×8n I2n ⎤ ⎡ ˆ [0] + h U Y Q −1 Y T T 08n×n hU W l 1 ⎥ ⎢ ⎥ ⎢ 0n 0n − In ×⎢ ⎥ ⎦ ⎣ −2Q 4 0n −2Q 4 ⎡ ⎤ l⊥ 08n×n 09n×2n ⎥ ⎢ × ⎣ 0n×7n (67) ⎦ < 010n , In ⎡ ⎢ ⎣
l⊥
02n×8n 08n×n
I2n ⎤T
09n×2n ⎥ ⎦ 0n×7n In 02n×8n I2n ⎤ ⎡ ˆ [h U ] + h U X Q −1 X T T hU Z 08n×n l 1 ⎥ ⎢ ⎥ ⎢ 0n 0n − In ×⎢ ⎥ ⎦ ⎣ −2Q 4 0n −2Q 4 ⎡ ⎤ l⊥ 08n×n 09n×2n ⎥ ⎢ × ⎣ 0n×7n ⎦ < 010n . In 02n×8n I2n
(68)
Using Fact 1, the above inequalities (67) and (68) can be handled by the LMIs (55) and (56), which guarantees the stability of the system (10) by the Lyapunov stability method. This completes our proof.
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
15
Remark 3. If G = 0 in both Theorems 1 and 2, one can easily obtain delay-dependent robust stability criteria for ˙ uncertain T–S fuzzy systems with fast time-varying delays, which do not need the information of h(t). Remark 4. In the field of delay-dependent stability, one of major concerns is to get maximum delay bounds with fewer decision variables [23–25]. By utilizing Finsler’s lemma, one can eliminate free variables which were used in zero T equalities in the works [24,25]. From Lemma 1, one can check that the B ⊥ B ⊥ < 0 is equivalent to the existence of L such that + LB + B T LT < 0 holds. Insertion of such an additional matrix L does not play a role to reduce the T conservatism of B ⊥ B ⊥ < 0. It only increases the number of decision variables. Therefore, our proposed stability criteria are derived in the form of (ii) in Lemma 1. Furthermore, inspired by the work of [17–20], our approach utilized the triple integral forms of Lyapunov–Krasovskii’s functional. However, unlike the method of [20], different upper bound of V˙7 was taken to utilize more information about h(t) as mentioned in Remark 1, which may lead to less conservative results. 4. Numerical examples Example 1. Revisit an example studied in [7]: Rule 1: If (x2 (t)/0.5) is about 0, then x(t) ˙ = (A1 + A1 (t))x(t) + (Ad1 + Ad1 (t))x(t − h(t)) + B1 u(t); Rule 2: If (x2 (t)/0.5) is about or −, then x(t) ˙ = (A2 + A2 (t))x(t) + (Ad2 + Ad2 (t))x(t − h(t)) + B2 u(t), where 0 1 0 1 0.1 0 , A2 = , Ad1 = Ad2 = , = 0.01/, A1 = 0.1 −2 0.1 −0.5 − 1.5 0.1 −0.2 −0.03 0 0 −0.15 0.2 , E1 = E2 = B1 = B2 = , D= , 0 0.03 1 0 0.04 −0.05 −0.35 E d1 = E d2 = . 0.08 −0.45
(69)
The membership functions are
1 1 1 ((t)) = 1 − × , 1 + exp[−3(x2 (t)/0.5 − /2)] 1 + exp [−3(x2 (t)/0.5 + /2)] 2 ((t)) = 1 − 1 ((t)).
(70)
2 Assume that control law is u(t) = l=1 l ((t))K l x(t) and the controller gains K 1 and K 2 are known. Since the input matrices Bl (l = 1, 2) are same at each rule, the closed-loop system can be represented as x(t) ˙ =
2 2
l ((t))m ((t)){(Al + Al (t) + Bl K m )x(t) + (Adl + Adl (t))x(t − h(t))}.
(71)
l=1 m=1
When h D is unknown and Al (t) = Adl (t) = 0(l = 1, 2), maximum delay bounds obtained by Theorems 1 and 2 with the controller gains [7,12] are listed as Table 1. When h D = 0 and Al (t) = Adl (t) = 0(l = 1, 2), the obtained results are shown in Table 2 with the controller gains [11,12]. Lastly, when h D = 0 and Al (t) = Adl (t) is not zero, the comparison of our results with those of [11,12] is conducted as shown in Table 3. From the tables, one can see our proposed Theorems 1 and 2 significantly improve feasible region of stability criteria, which enlarge maximum delay bounds for guaranteeing the system stability. Fig. 1 shows the simulation results of system (69) when Al (t) = Adl (t) = 0, h(t) = h U = 100, K 1 = [−0.9318 0.1265], K 2 = [−0.9318 − 1.3687] and x T (0) = [1 − 0.5].
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Table 1 Upper bounds of time-delay when h D is unknown, Ai (t) = 0, and Adi (t) = 0 (Example 1). Method
hU
K1
K2
Tian et al. [7]
7.0355
[−13.9297 − 54.9242]
[−13.9297 − 55.9468]
Li et al. [12]
7.0355 7.0708
[−12.6786 − 9.3258] [−75.2272 − 34.8354]
[−12.6786 − 10.8210] [−75.2272 − 36.3306]
Theorem 1
8.6598 8.6588 8.6601
[−13.9297 − 54.9242] [−12.6786 − 9.3258] [−75.2272 − 34.8354]
[−13.9297 − 55.9468] [−12.6786 − 10.8210] [−75.2272 − 36.3306]
Theorem 2
9.9982 9.9992 9.9999
[−13.9297 − 54.9242] [−12.6786 − 9.3258] [−75.2272 − 34.8354]
[−13.9297 − 55.9468] [−12.6786 − 10.8210] [−75.2272 − 36.3306]
Table 2 Upper bounds of time-delay when h D = 0, Ai (t) = 0, and Adi (t) = 0 (Example 1). Method
hU
K1
K2
Chen et al. [11]
25.7865
[−1.2141 0.8750]
[−1.2141 − 0.6202]
Li et al. [12]
25.7865 26.8617
[−0.9318 0.1265] [−0.9211 0.1344]
[−0.9318 − 1.3687] [−0.9211 − 1.3609]
Theorem 1
∞ ∞ ∞
[−1.2141 0.8750] [−0.9318 0.1265] [−0.9211 0.1344]
[−1.2141 − 0.6202] [−0.9318 − 1.3687] [−0.9211 − 1.3609]
Theorem 2
∞ ∞ ∞
[−1.2141 0.8750] [−0.9318 0.1265]] [−0.9211 0.1344]
[−1.2141 − 0.6202] [−0.9318 − 1.3687] [−0.9211 − 1.3609]
Table 3 Upper bounds of time-delay when h D = 0 (Example 1). Method
hU
K1
K2
Chen et al. [11]
20.5640
[−1.3987 − 0.6601]
[−1.3991 − 2.1607]
Li et al. [12]
20.5640 22.4852
[−1.3778 − 1.9868] [−1.4470 − 2.0154]
[−1.3778 − 3.4820] [−1.4470 − 3.5106]
Theorem 1
∞ ∞ ∞
[−1.3987 − 0.6601] [−1.3778 − 1.9868] [−1.4470 − 2.0154]
[−1.3991 − 2.1607] [−1.3778 − 3.4820] [−1.4470 − 3.5106]
Theorem 2
∞ ∞ ∞
[−1.3987 − 0.6601] [−1.3778 − 1.9868] [−1.4470 − 2.0154]
[−1.3991 − 2.1607] [−1.3778 − 3.4820] [−1.4470 − 3.5106]
Example 2. Consider the following T–S fuzzy system: Rule 1: If x1 (t) is about 11 , then x(t) ˙ = A1 x(t) + Ad1 x(t − h(t)); Rule 2: If x1 (t) is about 21 , then x(t) ˙ = A2 x(t) + Ad2 x(t − h(t)),
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19 1
17
x1(t) x2(t)
x(t)
0.5
0
−0.5
0
20
40
60
80 100 120 140 160 180 200 Time (Seconds)
Fig. 1. State response of the system with h(t) = 100 (Example 1). Table 4 Upper bounds of time-delay with different h D (Example 2). hD
0.03
0.1
0.5
0.9
Tian et al. [7] Zuo et al. [8] Wu et al. [9] Yang et al. [13] Theorem 1 Theorem 2
0.3950 0.5429 0.5432 0.5456 0.6928 0.8369
0.3950 0.4808 0.4809 0.5030 0.6039 0.7236
0.3950 0.4746 0.4752 0.4995 0.5974 0.7154
0.3950 0.4454 0.4455 0.4988 0.5849 0.7014
where
A1 =
−3.2 0.6 , 0 −2.1
Ad1 =
1 0.9 , 0 2
A2 =
−1 0 , 1 −3
Ad2 =
0.9 0 , 1 1.6
(72)
and the membership functions for rules 1 and 2 are 1 ((t)) =
1 , 1 + exp(−2x1 (t)))
2 ((t)) = 1 − 1 ((t)).
(73)
With different h D , the obtained maximum delay bounds by applying Theorems 1 and 2 are listed in Table 4. Also, the results of [7–9,13] are included in Table 4. From this table, one con confirm that our proposed theorems provide larger delay bounds than those of [7–9,13]. Example 3. Consider the following T–S fuzzy model with the same membership functions as Example 2: x(t) ˙ =
2
l ((t)){Al x(t) + Adl x(t − h(t))}
(74)
l=1
where A1 =
−2 0 , 0 −0.9
Ad1 =
−1 0 , −1 −1
A2 =
−1 0.5 , 0 −1
Ad2 =
−1 0 . 0.1 −1
(75)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Table 5 Upper bounds of time-delay with different h D (Example 3). hD
0
0.1
Unknown
Tian et al. [7] Lien et al. [10] Li et al. [12] Theorem 1 Theorem 2
1.597 1.5973 1.5974 1.6196 1.6609
– 1.484 1.4847 1.4977 1.5332
0.721 0.831 0.982 1.0691 1.2696
The delay bounds obtained in [7,10,12] and our results are shown in Table 5. From Table 5, one can confirm both Theorems 1 and 2 give enhanced delay bounds than the ones in [7,10,12].
5. Conclusions In this paper, two delay-dependent robust stability conditions for uncertain T–S fuzzy systems with time-varying delays have been presented as Theorems 1 and 2. In Theorem 1, by constructing the augmented Lyapunov–Krasovskii functional including triple integral term and utilizing Finsler’s Lemma, the new condition for robust stability of the concerned systems were derived in terms of LMIs without including free-weighting matrices. Also, based on the result of Theorem 1, further improved results with free-weighting matrices were introduced in Theorem 2. The three numerical examples show that Theorems 1 and 2 significantly provide larger delay bounds than those of other literatures. This implies that our proposed criteria make enhancement of feasible region of stable conditions. References [1] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modelling and control, IEEE Trans. Syst. Man Cybern. 15 (1985) 116–132. [2] C. Peng, D. Yue, Y.-C. Tian, New approach on robust delay-dependent H∞ control for uncertain T–S fuzzy systems with interval time-varying delay, IEEE Trans. Fuzzy Syst. 17 (2009) 890–900. [3] S.-I. Niculescu, Delay effects on stability: a robust control approach, Lecture Notes in Control and Information Sciences, vol. 269, 2002. [4] R. Rakkiyappan, P. Balasubramaniam, R. Krishnasamy, Delay dependent stability analysis of neutral systems with mixed time-varying delays and nonlinear perturbations, J. Comput. Appl. Math. 235 (2011) 2147–2156. [5] S. Lakshmanan, T. Senthilkumar, P. Balasubramaniam, Improved results on robust stability of neutral systems with mixed time-varying delays and nonlinear perturbations, Appl. Math. Modell. 35 (2011) 5355–5368. [6] E.T. Jeung, D.C. Oh, H.B. Park, Delay-dependent control for time-delayed T–S fuzzy systems using descriptor representation, Int. J. Control Autom. Syst. 2 (2004) 182–188. [7] E. Tian, C. Peng, Delay-dependent stability analysis and synthesis of uncertain T–S fuzzy systems with time-varying delay, Fuzzy Sets Syst. 157 (2006) 544–559. [8] Z. Zuo, Y. Wang, Robust stability and stabilization for nonlinear uncertain time-delay systems via fuzzy control approach, IET Control Theory Appl. 1 (2007) 422–429. [9] H.N. Wu, H.X. Li, New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with time-varying delay, IEEE Trans. Fuzzy Syst. 15 (2007) 482–493. [10] C.H. Lien, K.W. Yu, W.D. Chen, Z.L. Wan, Y.J. Chung, Stability criteria for uncertain Takagi–Sugeno fuzzy systems with interval time-varying delay, IET Proc. Control Theory Appl. 1 (2007) 764–769. [11] B. Chen, X.P. Liu, S.C. Tong, New delay-dependent stabilization conditions of T–S systems with constant delay, Fuzzy Sets Syst. 158 (2007) 2209–2224. [12] L. Li, X. Liu, T. Chai, New approaches on H∞ control of T–S fuzzy systems with interval time-varying delay, Fuzzy Sets Syst. 160 (2009) 1669–1688. [13] Z. Yang, Y.P. Yang, New delay-dependent stability analysis and synthesis of T–S fuzzy systems with time-varying delay, Int. J. Robust Nonlinear Control 20 (2010) 313–322. [14] S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems, Int. J. Syst. Sci. 39 (2008) 1095–1113. [15] P.G. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Autom. Control 44 (1999) 876–877. [16] K. Gu, An integral inequality in the stability problem of time-delay systems, in: The 39th IEEE Conference on Decision Control, Sydney, Australia, December 2000, pp. 2805–2810. [17] Y. Ariba, F. Gouaisbaut, Delay-dependent stability analysis of linear systems with time-varying delay, in: The 46th IEEE Conference on Decision Control, New Orleans, USA, December 2007, pp. 2053–2058.
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[18] Y. Ariba, F. Gouaisbaut, An augmented model for robust stability analysis of time-varying delay systems, Int. J. Control 82 (2009) 1616–1626. [19] J. Sun, G.P. Liu, J. Chen, Delay-dependent stability and stabilization of neutral time-delay systems, Int. J. Robust Nonlinear Control 19 (2009) 1364–1375. [20] J. Sun, G.P. Liu, J. Chen, D. Rees, Improved delay-range-dependent stability criteria for linear systems with time-varying delays, Automatica 46 (2010) 466–470. [21] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. [22] M.C. de Oliveira, R.E. Skelton, Stability Tests for Constrained Linear Systems, Springer-Verlag, Berlin, 2001 pp. 241–257. [23] T. Li, L. Guo, L. Wu, Simplified approach to the asymptotical stability of linear systems with interval time-varying delays, IET Control Theory Appl. 3 (2009) 252–260. [24] O.M. Kwon, S.M. Lee, J.H. Park, Linear matrix inequality approach to new delay-dependent stability criteria for uncertain dynamic systems with time-varying delays, J. Optim. Theory Appl. 149 (2011) 630–646. [25] O.M. Kwon, Stability criteria for uncertain stochastic dynamic systems with time-varying delays, Int. J. Robust Nonlinear Control 21 (2011) 338–350.
Fuzzy Sets and Systems 201 (2012) 1 – 19 www.elsevier.com/locate/fss
Augmented Lyapunov–Krasovskii functional approaches to robust stability criteria for uncertain Takagi–Sugeno fuzzy systems with time-varying delays夡 O.M. Kwona , M.J. Parka , S.M. Leeb , Ju H. Parkc,∗ a School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 361-763, Republic of Korea b School of Electronics Engineering, Daegu University, Jinryang-eup, Gyeongsan 712-714, Republic of Korea c Department of Electrical Engineering, Yeungnam University, 214-1 Dae-dong, Kyongsan 712-749, Republic of Korea
Received 4 February 2011; received in revised form 21 December 2011; accepted 22 December 2011 Available online 29 December 2011
Abstract This paper considers the problem of robust stability analysis for uncertain Takagi–Sugeno (T–S) fuzzy systems with time-varying delays. By constructing an augmented Lyapunov–Krasovskii functional and utilizing Finsler’s Lemma, a novel criterion for delaydependent robust stability of T–S fuzzy model with time-varying delay is derived in terms of linear matrix inequalities (LMIs). Also, a further improved stability criterion is proposed by utilizing free weighting techniques. Finally, three numerical examples are included to show the superiority of the proposed criteria. © 2012 Elsevier B.V. All rights reserved. Keywords: Fuzzy system models; T–S fuzzy systems; Time-varying delay; Stability; Lyapunov method; LMI
1. Introduction Since T–S fuzzy model [1] was first introduced, the stability analysis of the model has been paid much attentions because it is very important to check the stability of certain dynamic systems when T–S fuzzy control scheme is applied. The main advantage of T–S fuzzy model is that it can be applied as an efficient tool in approximating complex nonlinear systems. It combines the flexible fuzzy logic theory and rigorous mathematical theory of linear system into an unified framework to approximate nonlinear dynamics of complex systems [2]. On the other hand, time-delays often encounter in many industrial and engineering systems such as chemical processes, rolling mill systems, neural
夡 This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0009273). ∗ Corresponding author. Tel.: +82 52 8102491; fax: +82 53 8104767. E-mail addresses: [email protected] (O.M. Kwon), [email protected] (M.J. Park), [email protected] (S.M. Lee), [email protected] (J.H. Park). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.12.014
2
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
networks, networked control systems and so on [3]. It is well known that time-delays can cause poor performance or instability. Therefore, the problem of delay-dependent stability and stabilization for T–S fuzzy systems with time-delays has been received great efforts by many researchers in recent years [2,4–13] because delay-dependent approaches are generally less conservative than delay-independent ones when the sizes of time-delays are small [14]. The main issue of delay-dependent stability and stabilization is to reduce the conservatism of stability and stabilization criteria. One of the important index for checking the conservatism is to enlarge the feasible region of the criteria or to get maximum delay bounds for guaranteeing the system stability. Hence, how to construct Lyapunov–Krasovskii functional and obtain the stability and stabilization conditions by calculating upper bounds of time-derivative of Lyapunov–Krasovskii functional play important roles to reduce the conservatism of the conditions. In this regard, Jeung et al. [6] proposed a delay-dependent control method by utilizing a descriptor model transformation and Park’s inequality [15]. Tian and Peng [7] derived new delay-dependent stability and stabilization criteria for uncertain T–S fuzzy systems with interval time-varying delays. The main contribution of the work [7] was that the proposed criteria allow fast time-varying delays, which mean the criteria do not need the information on derivative of the time delay. Improved robust stability and stabilization for uncertain T–S fuzzy time-delay systems without integral inequality technique [16] and model transformation method were studied in [8]. Based on a fuzzy Lyapunov–Krasovskii functional and fuzzy free weighting matrices, the stability and stabilization problems for T–S fuzzy systems with timevarying delay were investigated in [9]. In Lien et al. [10], a simple analytic method with fewer free-weighted matrices was proposed for stability analysis of uncertain T–S fuzzy systems with interval time-varying delay. Recently, delaydependent stability analysis and H∞ control for uncertain T–S fuzzy systems with interval time-varying delays were conducted in [12]. Very recently, in [13], further improvements on delay-dependent stability analysis and synthesis of T–S fuzzy systems with time-varying delay is made by considering ignored terms in estimating upper bounds of the derivative of Lyapunov–Krasovskii functional. However, there are still rooms for further improvements to enlarge the feasible region of stability criteria for T–S fuzzy systems with time-varying delays. In this paper, we revisit the problem of stability analysis for uncertain T–S fuzzy systems with time-varying delays. First, by constructing an augmented Lyapunov–Krasovskii functional and utilizing Finsler’s lemma, a delay-dependent stability condition without the use of free weighting matrices is derived in terms of LMIs, which will be introduced in Theorem 1. Inspired by the works of [17–20], a triple integral form of Lyapunov–Krasovskii functional is utilized. t t−h(t) Also, new integral forms of states (1/ h(t)) t−h(t) x(s) ds and (1/(h U − h(t))) t−h U x(s) ds are taken as augmented vectors to include further information of time-varying delays. And based on the results of Theorem 1, a further improved stability criterion is proposed by utilizing free-weighting matrices. Through three numerical examples, the reduction of the conservatism of Theorem 1 will be shown by comparing with the results proposed in other literature. Finally, it will be presented that further enlargement of delay bounds can be made by a new result, Theorem 2. Notation. Rn is the n-dimensional Euclidean space, and Rm×n denotes the set of m × n real matrix. Cn,h = C([−h, 0], Rn ) stands for the Banach space of continuous functions mapping the interval [−h, 0] into Rn , with the topology of uniform convergence. For symmetric matrices X and Y, X > Y (respectively, X ⱖ Y ) means that the matrix X − Y is positive definite (respectively, nonnegative). In , 0n and 0m×n denotes n × n identity matrix, n × n and m × n zero matrices, respectively. · refers to the Euclidean vector norm and the induced matrix norm. diag{· · ·} denotes the block diagonal matrix. represents the elements below the main diagonal of a symmetric matrix. For a given matrix X ∈ Rm×n , such that rank(X ) = r , we define X ⊥ ∈ Rn×(n−r ) as the right orthogonal complement of X; i.e., X X ⊥ = 0. X [h(t)] ∈ Rm×n means that the elements of the matrix X includes the value of h(t); e.g., X [h U ] ≡ X [h(t)=h U ] . 2. Problem statements Consider the following uncertain T–S fuzzy systems with time-varying delays: Plant rule l : IF 1 (t) is l1 , . . ., and s (t) is ls , THEN x(t) ˙ = (Al + Al (t))x(t) + (Adl + Adl (t))x(t − h(t)), l = 1, 2, . . . , r., x(s) = (s), ∀s ∈ [−h U , 0], h U > 0,
(1)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
3
where i (t)(i = 1, . . . , s) are the premise variables, and li (i = 1, . . . , s) are fuzzy sets, r is the number of IF–THEN rules, x(t) ∈ Rn is the state vector, (s) ∈ Cn,h is a given continuous vector valued initial function, Al ∈ Rn×n and Adl ∈ Rn×n are known constant matrices with appropriate dimensions, and Al (t) and Adl (t) are the system uncertainties of the form: [Al (t) Adl (t)] = D F(t)[El E dl ] (l = 1, . . . , r ),
(2)
in which the time-varying nonlinear function F(t) satisfies F T (t)F(t) ⱕ I, ∀t ⱖ 0.
(3)
The time-varying delay h(t) is a continuous function that satisfies ˙ ⱕ h D, 0 ⱕ h(t) ⱕ h U , h(t)
(4)
where h U is a positive scalar and h D is an any constant one. Using the center-average defuzzifier, product interference and singleton fuzzifier, the global dynamics of the T–S fuzzy system (1) can be inferred as follows: r wl ((t))[(Al + Al (t))x(t) + (Adl + Adl (t))x(t − h(t))] r x(t) ˙ = l=1 l=1 wl ((t)) r = l ((t))[(Al + Al (t))x(t) + (Adl + Adl (t))x(t − h(t))] (5) l=1
with (t) = [1 (t), . . . , s (t)], wl ((t)) =
s i=1
wl ((t)) , li (i (t)), l ((t)) = r l=1 wl ((t))
(6)
where li (i (t)) is the grade of membership value of i (t) in li . It is assumed that wl ((t)) ⱖ 0,
r
wl ((t)) > 0.
(7)
l=1
Then, we have the following conditions: l ((t)) ⱖ 0,
r
l ((t)) = 1.
(8)
l=1
For the sake of simplicity, let us define Aˆ =
r
l ((t))Al ,
Aˆ d =
l=1
Eˆ =
r
r
l ((t))Adl ,
l=1
l ((t))El ,
l=1
Eˆ d =
r
l ((t))E dl .
(9)
l=1
Then, system (5) can be rewritten as: ˆ x(t) ˙ = Ax(t) + Aˆ d x(t − h(t)) + Dp(t), p(t) = F(t)q(t), q(t) = Eˆ x(t) + Eˆ d (t − h(t)). The aim of this paper is to derive novel robust delay-dependent stability conditions for system (10).
(10)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Before deriving our main results, we give the following facts and lemmas. Fact 1 (Schur Complement, Boyd et al. [21]). Given constant matrices 1 , 2 , 3 with 1 = 1T and 0 < 2 , then 1 + 3T −1 2 3 < 0 if and only if −2 3 1 3T < 0 or < 0. (11) 3 −2 3T 1 Fact 2 (Park [15]). For any real vectors a, b and any matrix Q > 0 with appropriate dimensions, we have ± 2a T b ⱕ a T Qa + b T Q −1 b.
(12)
Lemma 1 (Finsler’s lemma, de Oliveira and Skelton [22]). Let ∈ Rn , = T ∈ Rn×n , and B ∈ Rm×n such that rank(B) < n. The following statements are equivalent: (i) T < 0, ∀B = 0, 0, T (ii) B ⊥ B ⊥ < 0. (iii) ∃L ∈ Rn×m : + LB + B T LT < 0. Lemma 2. For any constant matrix M ∈ Rn×n , M = M T > 0 and ⱕ s ⱕ , the following inequalities hold:
T
T x˙ (s)M x(s) ˙ ds ⱖ x(s) ˙ ds M x(s) ˙ ds , ( − )
T
( − )2 T x˙ (u)M x(u) ˙ du ds ⱖ x(u) ˙ du ds M x(u)u ˙ ds . 2 s s s Proof. According to Jensen’s inequality in [16], one can obtain (13). Moreover, the following inequality holds
T
( − s) x˙ T (u)M x(u) ˙ du ⱖ x(u) ˙ du M x(u) ˙ du . s
s
(13)
(14)
(15)
s
By using Fact 1, Eq. (15) is equivalent to the following inequality:
T ˙ du s x˙ T (u) du s x˙ (u)M x(u)
ⱖ 0. ˙ du ( − s)M −1 s x(u)
(16)
Integration of inequality (16) from to yields
T ˙ du ds s x˙ T (u) du ds s x˙ (u)M x(u)
ⱖ 0. −1 ds ˙ du ds s x(u) ( − s)M
(17)
Therefore, the inequality (17) is equivalent to the inequality (14) according to Fact 1. This completes our proof. Lemma 3. For any scalar h(t) ⱖ 0 and any constant matrix Q ∈ Rn×n , Q = Q T > 0, the following inequality holds: t t t h 2 (t) T (t)X Q −1 X T (t) + 2T (t)X x˙ T (s)Q x(s) ˙ ds ⱕ (x(t) − x(s)) ds, (18) − 2 t−h(t) s t−h(t) where
T (t) = x T (t) x T (t − h(t)) x T (t − h U ) x˙ T (t) x˙ T (t − h U ) 1 h U − h(t)
t−h(t)
x T (s) ds p T (t) ,
t−h U
and X is free weighting matrix with appropriate dimensions.
1 h(t)
t
x T (s) ds
t−h(t)
(19)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Proof. From Fact 2, the following inequality holds t t t t (X T (t))T x(u) ˙ du ds ⱕ [T (t)X Q −1 X T (t) + x˙ T (u)Q x(u)] ˙ du ds. −2 t−h(t) s
5
(20)
t−h(t) s
From (20), we obtain t t − x˙ T (s)Q x(u) ˙ du ds t−h(t) s
t
ⱕ
t
T (t)X Q −1 X T (t) du ds + 2
t−h(t) s
=
h 2 (t) 2
T (t)X Q −1 X T (t) + 2T (t)X
t
t
(X T (t))T x(u) ˙ du ds
t−h(t) s t
(x(t) − x(s)) ds.
(21)
t−h(t)
Therefore, the inequality (21) is equivalent to the inequality (18). This completes our proof. 3. Main results In this section, we propose new delay-dependent stability criteria for system (10). Before introducing our main results, the notations of several matrices are defined for simplicity: ˆ = [ Aˆ Aˆ d 0n − In 0n 0n 0n D], l = [Al Adl 0n − In 0n 0n 0n D] (l = 1, . . . , r ), ˆ = [ Eˆ Eˆ d 0n 0n 0n 0n 0n 0n ], l = [El E dl 0n 0n 0n 0n 0n 0n ] (l = 1, . . . , r ), T T 11 = R13 + R13 + N11 + G + h U Q 2 − 2Q 3 , 13 = −R13 + R23 ,
14 = R11 + N12 , 22 = −(1 − h D )G − 2Q 3 , T 2 33 = −R23 − R23 − N11 , 35 = −N12 + R22 , 44 = N22 + (h U /2)Q 3 ,
77 = −h U Q 2 − 2Q 3 , ⎡ 11 0n 13 14 R12 2Q 3 h U R33 ⎢ 0 0 0 0 2Q 3 22 n n n n ⎢ ⎢ T ⎢ 33 R12 35 0n −h U R33 ⎢ ⎢ 44 0n 0n h U R13 ⎢ 1 = ⎢ ⎢ −N22 0n h U R23 ⎢ ⎢ −2Q 0n 3 ⎢ ⎢ ⎣ 77 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 2 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0n 0n 0n 0n 0n 0n 0n 0n 0n
R33 0n
0n 0n 0n −R33 0n 0n R13
0n R23 −Q 2
⎤
−R33 0n 0n 0n ⎥ ⎥ ⎥ R33 0n ⎥ ⎥ −R13 0n ⎥ ⎥ ⎥, −R23 0n ⎥ ⎥ 0n 0n ⎥ ⎥ ⎥ Q 2 0n ⎦
0n
0n
⎤
0n ⎥ ⎥ ⎥ 0n ⎥ ⎥ 0n ⎥ ⎥ ⎥, 0n ⎥ ⎥ 0n ⎥ ⎥ ⎥ 0n ⎦ − In
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
[h(t)] = 1 + h(t)2 , ⎡ −2Q 1 2Q 1 0n 0n ⎢ −3Q 1 Q 1 0n ⎢ ⎢ 1 ⎢ −Q 1 0n 1 = × hU ⎢ ⎢ 3 hU Q 1 ⎣ ⎡
Q 1 −Q 1
⎢ ⎢ ⎢ 1 2 = 2 × ⎢ hU ⎢ ⎢ ⎣
0n
0n
0n×4n
⎥ 0n×4n ⎥ ⎥ 0n×4n ⎥ ⎥, ⎥ 0n×4n ⎦ 04n
0n 0n×4n
⎤
⎤
⎥ Q 1 0n 0n×4n ⎥ ⎥ −Q 1 0n 0n×4n ⎥ ⎥, ⎥ 0n 0n×4n ⎦ 04n
[h(t)] = 1 + h(t)2 .
(22)
Now, we have the following theorem. Theorem 1. For given scalars h U > 0 and h D , the system (10) is asymptotically stable for 0 ⱕ h(t) ⱕ h U and ˙ ⱕ h D , if there exist symmetric positive definite matrices R = [Ri j ]3×3 ∈ R3n×3n , N = [Ni j ]2×2 ∈ R2n×2n , h(t) G ∈ Rn×n , Q i (i = 1, 2, 3) ∈ Rn×n satisfying the following LMIs: T ⊥ ⊥ l 08n×n l 08n×n ( p) + ( p) lT < 08n , l = 1, . . . , r, (23) 0n×7n In 0n×7n In − In where l( p) and ( p) for p = 1, 2 are the vertices of l[h(t)] and [h(t)] , respectively, and l⊥ is the right orthogonal complement of l . Proof. For positive definite matrices R = [Ri j ]3×3 , N = [Ni j ]2×2 , G, Q i (i = 1, 2, 3), let us choose the following Lyapunov–Krasovskii’s functional candidate as: V = V1 + V2 + V3 + V4 + V5 + V6 , where
(24)
⎡
⎤T ⎡ ⎤ ⎤⎡ x(t) x(t) R11 R12 R13 ⎢ ⎢ ⎥ ⎥ V1 = ⎣ x(t − h U ) ⎦ ⎣ R22 R23 ⎦ ⎣ x(t − h U ) ⎦ , t t R33 t−h U x(s) ds t−h U x(s) ds
V2 =
t
x(s) x(s) ˙
t−h U
V3 = V4 = V5 = V6 =
t
R
T
N11 N12 x(s) ds, x(s) ˙ N22 N
T
x (s)Gx(s) ds, t−h(t) t t t−h U s t t
x˙ T (u)Q 1 x(u) ˙ du ds, x T (u)Q 2 x(u) du ds,
t−h U s t t t t−h U s
u
x˙ T (v)Q 3 x(v) ˙ dv du ds.
(25)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
The time-derivative of V1 can be obtained as: ⎡ ⎤T ⎡ ⎤ x(t) R11 R12 R13 ⎡ ⎢ ⎥ ⎢ ⎥ V˙1 = 2 ⎣ x(t − h U ) ⎦ ⎣ R22 R23 ⎦ ⎣ 1 ⎡ ⎢ ⎢ =2⎢ ⎢ ⎣
⎤T
R33
⎤
x(t) ˙
⎦, x(t ˙ − hU ) x(t) − x(t − h U )
⎤ ⎡ x(t) ˙ ⎥ x(t − h U ) ⎥ ⎥ ⎥ 2 ⎢ t x(t ˙ − hU ) ⎦, ⎣ 1 ⎥ x(s) ds h(t) t−h(t) ⎦ x(t) − x(t − h U ) t−h(t) 1 h U −h(t) t−h U x(s) ds x(t)
where
1 = h(t)
1 h(t)
x(s) ds + (h U − h(t))
t
7
t−h(t)
1 h U − h(t)
(26)
t−h(t)
x(s) ds ,
t−h U
⎤ ⎡ ⎤ I n 0n 0n R11 R12 R13 ⎥ ⎢0 I 0n ⎥ ⎢ ⎢ n n ⎥ 2 = ⎢ ⎥ × ⎣ R22 R23 ⎦ . ⎦ ⎣ 0n 0n h(t)In R33 0n 0n (h U − h(t))In ⎡
(27)
By calculating the time-derivative of V2 , we have T N11 N12 x(t) x(t) V˙2 = x(t) ˙ N22 x(t) ˙ −
x(t − h U ) x(t ˙ − hU )
T
N11 N12 N22
x(t − h U ) . x(t ˙ − hU )
(28)
We can obtain an upper bound of the time-derivative of V3 as follows: V˙3 ⱕ x T (t)Gx(t) − (1 − h D )x T (t − h(t))Gx(t − h(t)).
(29)
Calculating the time-derivative of V4 leads to t V˙4 = h U x˙ T (t)Q 1 x(t) ˙ − x˙ T (s)Q 1 x(s) ˙ ds t−h U
= h U x˙ T (t)Q 1 x(t) ˙ −
t
t−h(t)
x˙ T (s)Q 1 x(s) ˙ ds −
t−h(t)
x˙ T (s)Q 1 x(s) ˙ ds.
t−h U
By using −1 = −
h(t) h U − h(t) − hU hU
and Lemma 2, an upper bound of the first integral term of V˙4 can be obtained as t − x˙ T (s)Q 1 x(s) ˙ ds t−h(t)
=−
h(t) hU
h(t) ⱕ− hU
t
x˙ T (s)Q 1 x(s) ˙ ds −
t−h(t)
h U − h(t) hU
t
x˙ T (s)Q 1 x(s) ˙ ds
t−h(t)
(h U − h(t))h(t) x˙ T (s)Q 1 x(s) ˙ ds − 2 hU t−h(t) t
t
t−h(t)
x˙ T (s)Q 1 x(s) ˙ ds
(30)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
ⱕ−
1 hU
=
x(s) ˙ ds
x(t) x(t − h(t))
x(s) ˙ ds
t−h(t)
T
⎡ ⎢ ⎢ ⎣
−
x(s) ˙ ds
t−h(t)
T
t
t
Q1
t−h(t)
h U − h(t) − 2 hU
T
t
x(s) ˙ ds
Q1
2 hU
t
−
h(t) 2 hU
t−h(t)
2 hU
Q1
−
2 hU
⎤
Q1 ⎥ x(t) ⎥
⎦ x(t − h(t)) . − h(t) Q 1 h2
−
h(t) 2 hU
(31)
U
With the similar method introduced above, an upper bound of other integral term of V˙4 can be estimated as: t−h(t) x˙ T (s)Q 1 x(s) ˙ ds − t−h U
h U − h(t) t−h(t) T x˙ (s)Q 1 x(s) ˙ ds − x˙ (s)Q 1 x(s) ˙ ds hU t−h U t−h U (h U − h(t))h(t) t−h(t) T h U − h(t) t−h(t) T ⱕ− x˙ (s)Q 1 x(s) ˙ ds − x˙ (s)Q 1 x(s) ˙ ds 2 hU hU t−h U t−h U t−h(t) t−h(t)
T
h(t) ⱕ− 2 x(s) ˙ ds Q1 x(s) ˙ ds hU t−h U t−h U t−h(t) t−h(t)
T
1 − x(s) ˙ ds Q1 x(s) ˙ ds hU t−h U t−h U
⎡ ⎤ h(t) T − 1 + h(t) Q 1 + Q 1 1 2 2 hU hU hU ⎢ ⎥ x(t − h(t)) x(t − h(t)) ⎢ ⎥
= ⎣ ⎦ x(t − h U ) . x(t − h U ) h(t) − 1 + h2 Q 1 h(t) =− hU
t−h(t)
T
(32)
U
Then, from (31) and (32), the upper bound of V˙4 can be rewritten as: V˙4 ⱕ T (t)[h(t)] (t),
(33)
where [h(t)] is defined in (22). Using Lemma 2, an upper bound of the time-derivative of V5 can be calculated as t t−h(t) V˙5 = h U x T (t)Q 2 x(t) − x T (s)Q 2 x(s) ds − x T (s)Q 2 x(s) ds t−h(t)
t−h U
ⱕ h U x (t)Q 2 x(t) t
T
t 1 − x(s) ds Q2 x(s) ds h(t) t−h(t) t−h(t) t−h(t) t−h(t)
T
1 − x(s) ds Q2 x(s) ds h U − h(t) t−h U t−h U T
= h U x T (t)Q 2 x(t)
T
t t 1 1 − x(s) ds h(t)Q 2 x(s) ds h(t) t−h(t) h(t) t−h(t)
T
t−h(t) t−h(t) 1 1 x(s) ds (h U − h(t))Q 2 x(s) ds . − h U − h(t) t−h U h U − h(t) t−h U
(34)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
9
By calculating V˙6 , one can obtain t t h2 ˙ − x˙ T (u)Q 3 x(u) ˙ du ds V˙6 = U x˙ T (t)Q 3 x(t) 2 t−h U s t t t−h(t) t h2 ˙ − x˙ T (u)Q 3 x(u) ˙ du ds − x˙ T (u)Q 3 x(u) ˙ du ds = U x˙ T (t)Q 3 x(t) 2 t−h(t) s t−h U s t t t−h(t) t−h(t) h2 ˙ − x˙ T (u)Q 3 x(u) ˙ du ds − x˙ T (u)Q 3 x(u) ˙ du ds ⱕ U x˙ T (t)Q 3 x(t) 2 t−h(t) s t−h U s ⱕ
2 hU ˙ x˙ T (t)Q 3 x(t) 2
T
t t 2 − 2 x(s) ds Q 3 h(t)x(t) − x(s) ds h(t)x(t) − h (t) t−h(t) t−h(t)
T t−h(t) 2 − − h(t))x(t − h(t)) − x(s) ds Q3 (h U (h U − h(t))2 t−h U
t−h(t) × (h U − h(t))x(t − h(t)) − x(s) ds t−h U
=
2 hU x˙ T (t)Q 3 x(t) ˙ 2
T
t t 1 1 −2 x(t) − x(s) ds Q 3 x(t) − x(s) ds h(t) t−h(t) h(t) t−h(t)
T t−h(t) 1 −2 x(t − h(t)) − x(s) ds Q3 h U − h(t) t−h U
t−h(t) 1 × x(t − h(t)) − x(s) ds , h U − h(t) t−h U
(35)
where Lemma 2 was utilized in (35). Since the following inequality holds from (3) and (10), p T (t) p(t) ⱕ q T (t)q(t),
(36)
there exists a positive scalar satisfying the following inequality:
[q T (t)q(t) − p T (t) p(t)] ⱖ 0.
(37)
Note that ˆ T (t), ˆ q T (t)q(t) = T (t)
(38)
ˆ was defined in (22). where From (26)–(38) and S-procedure [21], the time-derivative of V has a new upper bound as: ˆ T )(t), ˆ V˙ ⱕ T (t)([h(t)] + [h(t)] +
(39)
where (t) was defined in (19), and [h(t)] and [h(t)] were in (22), respectively. Also, the system (10) with the augmented vector (t) can be rewritten as: ˆ 0n×1 = (t), ˆ is defined in (22). where
(40)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Therefore, an asymptotic stability condition for system (10) can be ˆ T )(t) ˆ T (t)([h(t)] + [h(t)] + <0 ˆ subject to 0n×1 = (t).
(41)
Then, by utilizing the statements of (i) and (iii) of Lemma 1, for any matrix L(t) with appropriate dimension, the condition (41) is equivalent to ˆ T ˆ + L(t) ˆ + ˆ T LT (t) < 0. [h(t)] + [h(t)] + By using Fact 1, inequality (42) is equivalent to ˆT ˆ + ˆ T LT (t) [h(t)] + [h(t)] + L(t) < 09n . − In The above inequality (43) can be represented as r [h(t)] + [h(t)] + L(t) l + lT LT (t) lT l ((t)) < 09n . − In
(42)
(43)
(44)
l=1
At each lth rule, multiplying on the left side and right side of the inequality (44) by ⊥ ⊥ T l 08n×n ( l ) 07n×n and 0n×8n In 0n×7n In leads to ⊥ T ( l ) ([h(t)] + [h(t)] + L(t) l + lT LT (t)) l⊥ ( l⊥ )T lT < 08n (l = 1, . . . , r ), − In which can be rewritten by ⊥ T ⊥ l 08n×n l 08n×n [h(t)] + [h(t)] lT < 08n (l = 1, . . . , r ). 0n×7n In 0n×7n In − In
(45)
(46)
By the convexity of h(t), we have
1 (t) = 1 −
h(t) h(t) , 2 (t) = . hU hU
(47)
Let us define H = {H (t) : 0 ⱕ h(t) ⱕ h U }.
(48)
Then, the vertex set Hvex is defined as: Hvex = {H : h(t) = 0 or h U }.
(49)
There are two elements in Hvex , and we can enumerate elements in Hvex by Hk , where k = 1, 2: H1 = 0, H2 = h U .
(50)
Since [h(t)] and [h(t)] are affinely dependent on h(t) ∈ H , respectively, [h(t)] and [h(t)] can be expressed as follows: [h(t)] + [h(t)] = (1 + 1 ) + h(t)(2 + 2 ) = (1 + 1 ) + [ 1 (t)H1 + 2 (t)H2 ](2 + 2 ) = (1 + 1 ) + [ 1 (t) · 0 + 2 (t) · h U ](2 + 2 ) = (1 + 1 ) + 1 (t)(0)(2 + 2 ) + 2 (t)(h U )(2 + 2 ), where i (i = 1, 2) and i (i = 1, 2) are defined in (22).
(51)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
11
Note that 1 (t) + 2 (t) = 1 for h(t), and i (t) ⱖ 0 (i = 1, 2). Therefore, by utilizing the properties of convex-hull, inequality (46) holds if
l⊥ 0n×7n
l⊥ 0n×7n
08n×n In 08n×n In
T
T
[0] + [0] lT − In
l⊥ 0n×7n
[h U ] + [h U ] lT − In
08n×n In
l⊥ 0n×7n
< 08n , l = 1, . . . , r ,
08n×n In
(52)
< 08n , l = 1, . . . , r .
(53)
Therefore, if LMIs (23) hold, then the inequality (41) holds, which means that system (10) is asymptotically stable. This completes our proof. t t−h(t) Remark 1. Our augmented vector (t) includes the states of (1/ h(t)) t−h(t) x(s) ds and (1/(h U −h(t))) t−h U x(s) ds. The motivation of this idea is that more information about h(t) may be utilized in delay-dependent stability condition of (23). Remark 2. Theorem 1 is derived by utilizing the Lyapunov–Krasovskii’s functional without free-weighting matrices. By considering free-weighting matrices, a further improved delay-dependent stability criterion can be derived, which will be introduced in Theorem 2. For the sake of simplicity in Theorem 2, we define the following notations: X T = [X 1T X 2T X 3T X 4T X 5T 0n 0n 0n ], Y T = [Y1T Y2T Y3T Y4T Y5T 0n 0n 0n ], Z T = [Z 1T 0n 0n 0n 0n Z 2T 0n 0n ], W T = [0n W1T 0n 0n 0n 0n W2T 0n ], ⎡
In −In
⎢0 ⎢ n 1 = [0n×3n In 0n×4n ], 2 = ⎢ ⎣ In 0n ˆ [h(t)] = [h(t)] + 1T
hU Q 1 +
In 0n In
0n
0n 0n
0n
0n
−In 0n 0n
0n
0n
0n 0n
0n
⎤
0n ⎥ ⎥ ⎥, 0n ⎦
0n 0n −In 0n 0n 0n 0n −In 0n
2 hU Q 4 1 + [X Y h(t)Z (h U − h(t))W]2 2
+2T [X Y h(t)Z (h U − h(t))W]T ,
(54)
where [h(t)] is defined in (22). Now, we have the following theorem. Theorem 2. For given scalars h U > 0 and h D , the system (10) is asymptotically stable for 0 ⱕ h(t) ⱕ h U and ˙ ⱕ h D , if there exist symmetric positive definite matrices R = [Ri j ]3×3 ∈ R3n×3n , N = [Ni j ]2×2 ∈ R2n×2n , h(t) G ∈ Rn×n , Q i (i = 1, 2, 3, 4) ∈ Rn×n , and any matrices X i , Yi (i = 1, 2, 3, 4, 5) ∈ Rn×n , Z i ∈ Rn×n , Wi (i = 1, 2) ∈ Rn×n satisfying the following LMIs for l = 1, . . . , r : ⎤T ⎡ l⊥ 08n×n 09n×3n ⎥ ⎢ ⎦ ⎣ 0n×7n In 03n×8n I3n
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
⎤ ˆ [0] T 08n×n hU W hU Y l ⎥ ⎢ 0n 0n 0n − In ⎥ ⎢ ⎥ ⎢ ×⎢ −2Q 4 0n 0n ⎥ ⎦ ⎣ −2Q 4 0n −h U Q 1 ⎡ ⊥ ⎤ l 08n×n 09n×3n ⎦ In × ⎣ 0n×7n < 011n , (l = 1, . . . , r ), 03n×8n I3n ⎡ ⊥ ⎤T l 08n×n 0 9n×3n ⎦ ⎣ 0n×7n In 03n×8n I3n ⎤ ⎡ ˆ [h U ] T hU Z 08n×n hU X l ⎥ ⎢ 0n 0n 0n − In ⎥ ⎢ ⎥ ×⎢ −2Q 4 0n 0n ⎥ ⎢ ⎦ ⎣ −2Q 4 0n −h U Q 1 ⎡ ⊥ ⎤ l 08n×n 09n×3n ⎦ In × ⎣ 0n×7n < 011n , (l = 1, . . . , r ), 03n×8n I3n ⎡
(55)
(56)
where l , and l are defined in (22), respectively, and others are in (54). Proof. Let us choose the following Lyapunov–Krasovskii’s functional candidate with free-weighting matrices as: V =
7
Vi ,
(57)
i=1
where Vi (i = 1, . . . , 6) is the same Lyapunov–Krasovskii’s functional used in Theorem 1, and V7 =
t
t
t−h U s
t
x˙ T (v)Q 4 x(v) ˙ dv du ds.
(58)
u
Using Lemma 3, we have new upper bound of the time-derivative of V4 as: t t−h(t) V˙4 = h U x˙ T (t)Q 1 x(t) ˙ − x˙ T (s)Q 1 x(s) ˙ ds − x˙ T (s)Q 1 x(s) ˙ ds t−h(t)
t−h U
ⱕ h U x˙ (t)Q 1 x(t) ˙ T
T T +h(t)T (t)X Q −1 1 X (t) + 2 (t)X [x(t) − x(t − h(t))] T T +(h U − h(t))T (t)Y Q −1 1 Y (t) + 2 (t)Y[x(t − h(t)) − x(t − h U )],
where X and Y are defined in (54). Similarly, an upper bound of the time-derivative of V7 can be calculated as: t t t−h(t) t h2 V˙7 = U x˙ T (t)Q 4 x(t) ˙ − x˙ T (u)Q 4 x(u) ˙ du ds − x˙ T (u)Q 4 x(u) ˙ du ds 2 t−h(t) s t−h U s ⱕ
2 hU h 2 (t) T T ˙ + x˙ T (t)Q 4 x(t) (t)Z Q −1 4 Z (t) 2 2
(59)
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
t 1 (h U − h(t))2 T T +2 (t)Zh(t) x(t) − x(s) ds + (t)W Q −1 4 W (t) h(t) t−h(t) 2 t−h(t) 1 +2T (t)W(h U − h(t)) x(t − h(t)) − x(s) ds , h U − h(t) t−h U
13
T
where Z and W are defined in (54), and Lemma 3 was utilized in (60). From (37), (59) and (60), the time-derivative of Vˆ has a new upper bound as: ⎞ ⎛ ˆ T ˆ + h(t)X Q −1 X T + (h U − h(t))Y Q −1 Y T ˆ [h(t)] + 1 1 ⎠ (t), V˙ ⱕ T (t) ⎝ h 2 (t) (h U −h(t))2 −1 T −1 T + 2 Z Q4 Z + W Q4 W 2
(60)
(61)
ˆ [h(t)] was defined in (54). where Therefore, an asymptotic stability condition for system (10) can be ⎛ ⎞ ˆ T ˆ + h(t)X Q −1 X T + (h U − h(t))Y Q −1 Y T ˆ[h(t)] + 1 1 ⎠ (t) < 0 T (t) ⎝ h 2 (t) (h U −h(t))2 −1 T −1 T + 2 Z Q4 Z + W Q W 4 2 ˆ subject to 0n×1 = (t),
(62)
ˆ [h(t)] are defined in (22) and (54), respectively. ˆ and where By utilizing (i) and (iii) of Lemma 1 and Fact 1, for any matrix L(t), the condition (62) is equivalent to ⎞ ⎡⎛ ⎤ ˆ [h(t)] + h(t)X Q −1 X T 1 ⎢⎜ ⎥ T ⎟ ⎢ ⎜ +(h U − h(t))Y Q −1 ⎥ 1 Y ⎟ ⎟ ⎢⎜ ⎥ ⎟ ⎢⎜ T ⎥ h 2 (t) −1 T ⎟ l ⎥ ⎢⎜ + 2 Z Q4 Z ⎟ ⎢⎜ ⎥ < 09n (l = 1, . . . , r ). ⎢ ⎜ (h U −h(t))2 ⎥ −1 T ⎟ W Q W ⎢⎝ + ⎥ ⎠ 4 2 ⎢ ⎥ T T ⎣ ⎦ L(t) l + l L (t) − In
(63)
At each lth rule, by multiplying on the left side and right side of the inequality (63) by ⊥ T ⊥ ( l ) 07n×n l 08n×n and , 0n×8n In 0n×7n In the following inequality can be obtained: ⎞ ⎛ ⎡ ˆ [h(t)] + h(t)X Q −1 X T 1 ⎜ ⎢ T ⎟ ⎜ +(h U − h(t))Y Q −1 ⎢ 1 Y ⎟ ⎟ ⎜ ⎢ 2 ⎟ ⊥ ⎢ ⊥ T⎜ T ⎟ l ( l⊥ )T lT ⎢ ( l ) ⎜ + h 2(t) Z Q −1 Z 4 ⎟ ⎜ ⎢ ⎜ (h U −h(t))2 ⎢ −1 T ⎟ W Q W ⎢ ⎠ ⎝+ 4 2 ⎢ T T ⎣ L(t) l + l L (t) − In
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 08n (l = 1, . . . , r ). ⎥ ⎥ ⎥ ⎦
From Fact 1, inequality (64) is equivalent to ⎡⎡ ⎤ ˆ [h(t)] + h(t)X Q −1 X T 1 T ⊥ ⊥ ⊥ T T l ( l ) l ⎥ ⎢ ⎢ ( l ) T ⎢⎣ ⎦ +(h U − h(t))Y Q −1 1 Y ⎢ ⎢ − In ⎢ ⎢ ⎣
(64)
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
h(t)( l⊥ )T Z
(h U − h(t))( l⊥ )T W
0n
0n
−2Q 4
0n
−2Q 4
⎤ ⎥ ⎥ ⎥ < 010n (l = 1, . . . , r ). ⎥ ⎦
Inequality (65) can be represented as: ⎡ ⎤T l⊥ 08n×n 09n×2n ⎥ ⎢ ⎣ 0n×7n ⎦ In 02n×8n I2n ⎤ ⎤ ⎡⎡ ˆ [h(t)] + h(t)X Q −1 X T 1 T (h U − h(t))W ⎥
l ⎥ h(t)Z ⎢⎢ T ⎥ ⎢ ⎣ +(h U − h(t))Y Q −1 ⎦ 1 Y ⎥ ⎢ 0 0n n ×⎢ ⎥ − In ⎥ ⎢ ⎦ ⎣ −2Q 4 0n −2Q 4 ⎡ ⎤ ⊥ l 08n×n 09n×2n ⎥ ⎢ × ⎣ 0n×7n ⎦ < 010n (l = 1, . . . , r ). In 02n×8n I2n
(65)
(66)
From the properties of convex-hull, inequality (66) holds for 0 ⱕ h(t) ⱕ h U if the following two inequalities are satisfied ⎤T ⎡ l⊥ 08n×n 09n×2n ⎥ ⎢ ⎦ ⎣ 0n×7n In 02n×8n I2n ⎤ ⎡ ˆ [0] + h U Y Q −1 Y T T 08n×n hU W l 1 ⎥ ⎢ ⎥ ⎢ 0n 0n − In ×⎢ ⎥ ⎦ ⎣ −2Q 4 0n −2Q 4 ⎡ ⎤ l⊥ 08n×n 09n×2n ⎥ ⎢ × ⎣ 0n×7n (67) ⎦ < 010n , In ⎡ ⎢ ⎣
l⊥
02n×8n 08n×n
I2n ⎤T
09n×2n ⎥ ⎦ 0n×7n In 02n×8n I2n ⎤ ⎡ ˆ [h U ] + h U X Q −1 X T T hU Z 08n×n l 1 ⎥ ⎢ ⎥ ⎢ 0n 0n − In ×⎢ ⎥ ⎦ ⎣ −2Q 4 0n −2Q 4 ⎡ ⎤ l⊥ 08n×n 09n×2n ⎥ ⎢ × ⎣ 0n×7n ⎦ < 010n . In 02n×8n I2n
(68)
Using Fact 1, the above inequalities (67) and (68) can be handled by the LMIs (55) and (56), which guarantees the stability of the system (10) by the Lyapunov stability method. This completes our proof.
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
15
Remark 3. If G = 0 in both Theorems 1 and 2, one can easily obtain delay-dependent robust stability criteria for ˙ uncertain T–S fuzzy systems with fast time-varying delays, which do not need the information of h(t). Remark 4. In the field of delay-dependent stability, one of major concerns is to get maximum delay bounds with fewer decision variables [23–25]. By utilizing Finsler’s lemma, one can eliminate free variables which were used in zero T equalities in the works [24,25]. From Lemma 1, one can check that the B ⊥ B ⊥ < 0 is equivalent to the existence of L such that + LB + B T LT < 0 holds. Insertion of such an additional matrix L does not play a role to reduce the T conservatism of B ⊥ B ⊥ < 0. It only increases the number of decision variables. Therefore, our proposed stability criteria are derived in the form of (ii) in Lemma 1. Furthermore, inspired by the work of [17–20], our approach utilized the triple integral forms of Lyapunov–Krasovskii’s functional. However, unlike the method of [20], different upper bound of V˙7 was taken to utilize more information about h(t) as mentioned in Remark 1, which may lead to less conservative results. 4. Numerical examples Example 1. Revisit an example studied in [7]: Rule 1: If (x2 (t)/0.5) is about 0, then x(t) ˙ = (A1 + A1 (t))x(t) + (Ad1 + Ad1 (t))x(t − h(t)) + B1 u(t); Rule 2: If (x2 (t)/0.5) is about or −, then x(t) ˙ = (A2 + A2 (t))x(t) + (Ad2 + Ad2 (t))x(t − h(t)) + B2 u(t), where 0 1 0 1 0.1 0 , A2 = , Ad1 = Ad2 = , = 0.01/, A1 = 0.1 −2 0.1 −0.5 − 1.5 0.1 −0.2 −0.03 0 0 −0.15 0.2 , E1 = E2 = B1 = B2 = , D= , 0 0.03 1 0 0.04 −0.05 −0.35 E d1 = E d2 = . 0.08 −0.45
(69)
The membership functions are
1 1 1 ((t)) = 1 − × , 1 + exp[−3(x2 (t)/0.5 − /2)] 1 + exp [−3(x2 (t)/0.5 + /2)] 2 ((t)) = 1 − 1 ((t)).
(70)
2 Assume that control law is u(t) = l=1 l ((t))K l x(t) and the controller gains K 1 and K 2 are known. Since the input matrices Bl (l = 1, 2) are same at each rule, the closed-loop system can be represented as x(t) ˙ =
2 2
l ((t))m ((t)){(Al + Al (t) + Bl K m )x(t) + (Adl + Adl (t))x(t − h(t))}.
(71)
l=1 m=1
When h D is unknown and Al (t) = Adl (t) = 0(l = 1, 2), maximum delay bounds obtained by Theorems 1 and 2 with the controller gains [7,12] are listed as Table 1. When h D = 0 and Al (t) = Adl (t) = 0(l = 1, 2), the obtained results are shown in Table 2 with the controller gains [11,12]. Lastly, when h D = 0 and Al (t) = Adl (t) is not zero, the comparison of our results with those of [11,12] is conducted as shown in Table 3. From the tables, one can see our proposed Theorems 1 and 2 significantly improve feasible region of stability criteria, which enlarge maximum delay bounds for guaranteeing the system stability. Fig. 1 shows the simulation results of system (69) when Al (t) = Adl (t) = 0, h(t) = h U = 100, K 1 = [−0.9318 0.1265], K 2 = [−0.9318 − 1.3687] and x T (0) = [1 − 0.5].
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O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19
Table 1 Upper bounds of time-delay when h D is unknown, Ai (t) = 0, and Adi (t) = 0 (Example 1). Method
hU
K1
K2
Tian et al. [7]
7.0355
[−13.9297 − 54.9242]
[−13.9297 − 55.9468]
Li et al. [12]
7.0355 7.0708
[−12.6786 − 9.3258] [−75.2272 − 34.8354]
[−12.6786 − 10.8210] [−75.2272 − 36.3306]
Theorem 1
8.6598 8.6588 8.6601
[−13.9297 − 54.9242] [−12.6786 − 9.3258] [−75.2272 − 34.8354]
[−13.9297 − 55.9468] [−12.6786 − 10.8210] [−75.2272 − 36.3306]
Theorem 2
9.9982 9.9992 9.9999
[−13.9297 − 54.9242] [−12.6786 − 9.3258] [−75.2272 − 34.8354]
[−13.9297 − 55.9468] [−12.6786 − 10.8210] [−75.2272 − 36.3306]
Table 2 Upper bounds of time-delay when h D = 0, Ai (t) = 0, and Adi (t) = 0 (Example 1). Method
hU
K1
K2
Chen et al. [11]
25.7865
[−1.2141 0.8750]
[−1.2141 − 0.6202]
Li et al. [12]
25.7865 26.8617
[−0.9318 0.1265] [−0.9211 0.1344]
[−0.9318 − 1.3687] [−0.9211 − 1.3609]
Theorem 1
∞ ∞ ∞
[−1.2141 0.8750] [−0.9318 0.1265] [−0.9211 0.1344]
[−1.2141 − 0.6202] [−0.9318 − 1.3687] [−0.9211 − 1.3609]
Theorem 2
∞ ∞ ∞
[−1.2141 0.8750] [−0.9318 0.1265]] [−0.9211 0.1344]
[−1.2141 − 0.6202] [−0.9318 − 1.3687] [−0.9211 − 1.3609]
Table 3 Upper bounds of time-delay when h D = 0 (Example 1). Method
hU
K1
K2
Chen et al. [11]
20.5640
[−1.3987 − 0.6601]
[−1.3991 − 2.1607]
Li et al. [12]
20.5640 22.4852
[−1.3778 − 1.9868] [−1.4470 − 2.0154]
[−1.3778 − 3.4820] [−1.4470 − 3.5106]
Theorem 1
∞ ∞ ∞
[−1.3987 − 0.6601] [−1.3778 − 1.9868] [−1.4470 − 2.0154]
[−1.3991 − 2.1607] [−1.3778 − 3.4820] [−1.4470 − 3.5106]
Theorem 2
∞ ∞ ∞
[−1.3987 − 0.6601] [−1.3778 − 1.9868] [−1.4470 − 2.0154]
[−1.3991 − 2.1607] [−1.3778 − 3.4820] [−1.4470 − 3.5106]
Example 2. Consider the following T–S fuzzy system: Rule 1: If x1 (t) is about 11 , then x(t) ˙ = A1 x(t) + Ad1 x(t − h(t)); Rule 2: If x1 (t) is about 21 , then x(t) ˙ = A2 x(t) + Ad2 x(t − h(t)),
O.M. Kwon et al. / Fuzzy Sets and Systems 201 (2012) 1 – 19 1
17
x1(t) x2(t)
x(t)
0.5
0
−0.5
0
20
40
60
80 100 120 140 160 180 200 Time (Seconds)
Fig. 1. State response of the system with h(t) = 100 (Example 1). Table 4 Upper bounds of time-delay with different h D (Example 2). hD
0.03
0.1
0.5
0.9
Tian et al. [7] Zuo et al. [8] Wu et al. [9] Yang et al. [13] Theorem 1 Theorem 2
0.3950 0.5429 0.5432 0.5456 0.6928 0.8369
0.3950 0.4808 0.4809 0.5030 0.6039 0.7236
0.3950 0.4746 0.4752 0.4995 0.5974 0.7154
0.3950 0.4454 0.4455 0.4988 0.5849 0.7014
where
A1 =
−3.2 0.6 , 0 −2.1
Ad1 =
1 0.9 , 0 2
A2 =
−1 0 , 1 −3
Ad2 =
0.9 0 , 1 1.6
(72)
and the membership functions for rules 1 and 2 are 1 ((t)) =
1 , 1 + exp(−2x1 (t)))
2 ((t)) = 1 − 1 ((t)).
(73)
With different h D , the obtained maximum delay bounds by applying Theorems 1 and 2 are listed in Table 4. Also, the results of [7–9,13] are included in Table 4. From this table, one con confirm that our proposed theorems provide larger delay bounds than those of [7–9,13]. Example 3. Consider the following T–S fuzzy model with the same membership functions as Example 2: x(t) ˙ =
2
l ((t)){Al x(t) + Adl x(t − h(t))}
(74)
l=1
where A1 =
−2 0 , 0 −0.9
Ad1 =
−1 0 , −1 −1
A2 =
−1 0.5 , 0 −1
Ad2 =
−1 0 . 0.1 −1
(75)
18
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Table 5 Upper bounds of time-delay with different h D (Example 3). hD
0
0.1
Unknown
Tian et al. [7] Lien et al. [10] Li et al. [12] Theorem 1 Theorem 2
1.597 1.5973 1.5974 1.6196 1.6609
– 1.484 1.4847 1.4977 1.5332
0.721 0.831 0.982 1.0691 1.2696
The delay bounds obtained in [7,10,12] and our results are shown in Table 5. From Table 5, one can confirm both Theorems 1 and 2 give enhanced delay bounds than the ones in [7,10,12].
5. Conclusions In this paper, two delay-dependent robust stability conditions for uncertain T–S fuzzy systems with time-varying delays have been presented as Theorems 1 and 2. In Theorem 1, by constructing the augmented Lyapunov–Krasovskii functional including triple integral term and utilizing Finsler’s Lemma, the new condition for robust stability of the concerned systems were derived in terms of LMIs without including free-weighting matrices. Also, based on the result of Theorem 1, further improved results with free-weighting matrices were introduced in Theorem 2. The three numerical examples show that Theorems 1 and 2 significantly provide larger delay bounds than those of other literatures. This implies that our proposed criteria make enhancement of feasible region of stable conditions. References [1] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modelling and control, IEEE Trans. Syst. Man Cybern. 15 (1985) 116–132. [2] C. Peng, D. Yue, Y.-C. Tian, New approach on robust delay-dependent H∞ control for uncertain T–S fuzzy systems with interval time-varying delay, IEEE Trans. Fuzzy Syst. 17 (2009) 890–900. [3] S.-I. Niculescu, Delay effects on stability: a robust control approach, Lecture Notes in Control and Information Sciences, vol. 269, 2002. [4] R. Rakkiyappan, P. Balasubramaniam, R. Krishnasamy, Delay dependent stability analysis of neutral systems with mixed time-varying delays and nonlinear perturbations, J. Comput. Appl. Math. 235 (2011) 2147–2156. [5] S. Lakshmanan, T. Senthilkumar, P. Balasubramaniam, Improved results on robust stability of neutral systems with mixed time-varying delays and nonlinear perturbations, Appl. Math. Modell. 35 (2011) 5355–5368. [6] E.T. Jeung, D.C. Oh, H.B. Park, Delay-dependent control for time-delayed T–S fuzzy systems using descriptor representation, Int. J. Control Autom. Syst. 2 (2004) 182–188. [7] E. Tian, C. Peng, Delay-dependent stability analysis and synthesis of uncertain T–S fuzzy systems with time-varying delay, Fuzzy Sets Syst. 157 (2006) 544–559. [8] Z. Zuo, Y. Wang, Robust stability and stabilization for nonlinear uncertain time-delay systems via fuzzy control approach, IET Control Theory Appl. 1 (2007) 422–429. [9] H.N. Wu, H.X. Li, New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with time-varying delay, IEEE Trans. Fuzzy Syst. 15 (2007) 482–493. [10] C.H. Lien, K.W. Yu, W.D. Chen, Z.L. Wan, Y.J. Chung, Stability criteria for uncertain Takagi–Sugeno fuzzy systems with interval time-varying delay, IET Proc. Control Theory Appl. 1 (2007) 764–769. [11] B. Chen, X.P. Liu, S.C. Tong, New delay-dependent stabilization conditions of T–S systems with constant delay, Fuzzy Sets Syst. 158 (2007) 2209–2224. [12] L. Li, X. Liu, T. Chai, New approaches on H∞ control of T–S fuzzy systems with interval time-varying delay, Fuzzy Sets Syst. 160 (2009) 1669–1688. [13] Z. Yang, Y.P. Yang, New delay-dependent stability analysis and synthesis of T–S fuzzy systems with time-varying delay, Int. J. Robust Nonlinear Control 20 (2010) 313–322. [14] S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems, Int. J. Syst. Sci. 39 (2008) 1095–1113. [15] P.G. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Autom. Control 44 (1999) 876–877. [16] K. Gu, An integral inequality in the stability problem of time-delay systems, in: The 39th IEEE Conference on Decision Control, Sydney, Australia, December 2000, pp. 2805–2810. [17] Y. Ariba, F. Gouaisbaut, Delay-dependent stability analysis of linear systems with time-varying delay, in: The 46th IEEE Conference on Decision Control, New Orleans, USA, December 2007, pp. 2053–2058.
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