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Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations
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Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations$ Wei Kang a,b,n, Shouming Zhong b, Kaibo Shi c, Jun Cheng d a
School of Information Engineering, Fuyang Normal College, Fuyang 236041, PR China School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, PR China c School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, PR China d School of Science, Hubei University for Nationalities, Enshi 44500, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 3 August 2015 Received in revised form 23 October 2015 Accepted 5 November 2015 This paper was recommended for publication by Jeff Pieper
In this paper, the problem of finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations is investigated. By constructing a novel Lyapunov–Krasovskii functional and employing a new summation inequality named discrete Wirtinger-based inequality, reciprocally convex approach and zero equality, the improved finite-time stability criteria are derived to guarantee that the state of the system with time-varying delay does not exceed a given threshold when fixed time interval. Furthermore, the obtained conditions are formulated in forms of linear matrix inequalities which can be solved by using some standard numerical packages. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed method. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Finite-time stability Discrete-time system Time-varying delay Nonlinear perturbations
1. Introduction It is well known that the phenomenon of time delay is very common in various practical systems such as chemical systems, biological systems and networked control systems. The existence of time delay in a system may lead to instability or bad performance [1–3]. Thus, it is important to investigate the stability and control of time-delay systems and a variety of approaches and results have been developed for the stability analysis and control synthesis of time-varying delay systems [4–9], such as delay-partitioning/decomposition way [10,11], model transformation [12,13], the free-weighting matrices technique [14,15], reciprocally convex approach [16], improved derivative estimation based on the Wirtinger-based inequality [17–20]. In many practical systems, our interests are concerned on the behavior of the system over a finite-time interval. For instance, in the presence of saturation or controlling the trajectory of a space vehicle from an initial point to a final one in a prescribed time interval. That is, the state of the system does not exceed a bound over a given finite-time interval. In order to deal with the transient ☆ This work was supported by National Natural Science Foundation of China (61273015, 11401104). n Corresponding author at: School of Information Engineering, Fuyang Normal College, Fuyang 236041, PR China. E-mail address: [email protected] (W. Kang).
performance of control systems, finite-time stability (FTS) was introduced in [21]. With the development of Lyapunov function approach and linear matrix inequality techniques, a great number of results on FTS, finite-time stabilization were obtained for delayed system [22–27]. It is well-known that the finite-time stability theory plays an important role in many practical applications, especially in the study of the transient behavior system, and in those applications where larger values of the state should not be attained. For example, in [34], the authors have given a flight control example to demonstrate the practical use of the theoretic results on FTS. In [35], the authors obtain the finite-time consensus conditions for discrete-time multi-agent system with time-varying communication delays by utilizing FTS theory. It should be noted that all the above-mentioned research have been devoted to continuous-time systems. However, discrete-time plays a considerable role than the continuous-time counterparts in realistic applications. Particulary, when implementing the delayed continuous-time systems for computer simulation, it becomes necessary to develop discrete-time systems. Therefore, finite-time stability for discrete-time systems with time-varying delay is a significant issue and has been studied by some researchers recently [28–33]. In [30], sufficient FTS conditions are obtained by using Lyapunov–Krasovskii functional method and an appropriate model transformation of the original system. Further improved results were derived in [31] by choosing a novel Lyapunov–Krasovskii functional and using free-weighting matrices technique. In
http://dx.doi.org/10.1016/j.isatra.2015.11.006 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i
W. Kang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
[32], the original is firstly transformed into two interconnected subsystems to get the less conservative results. Then, new conditions of FTS and stabilization are derived in terms of linear matrix inequalities (LMIs). Very recently, there is a novel summation inequality named discrete Wirtinger-based inequality which can be applied in obtaining less conservative results in comparison with the Jensen inequality. Thus, how to obtain less conservative FTS criteria for discrete-time systems with time-varying delay and nonlinear perturbations by using discrete Wirtinger-based inequality and reciprocally convex approach in the novel Lyapunov–Krasovskii functional is an interesting challenge, which motivates the main purpose of our study. In this paper, we concentrate on the FTS problem for discretetime delayed system with nonlinear perturbations. The main contribution of this paper lies in three aspects: firstly, compared with [30,32] and some other literature, a new Lyapunov–Kraki1 is introduced, sovskii functional involving variable ratios β which contains more information and may lead to a less conservative result; secondly, different from some existing works such as [30–33], a novel inequality named discrete Wirtinger-based inequality is employed to estimate the difference of the Lyapunov– Krasovskii functional for discrete-time system; thirdly, this paper is the first one concerned with the FTS criteria for discrete-time not only with time-varying delay but also with nonlinear perturbations. Finally, three numerical examples are given to confirm the effectiveness and less conservatism of the proposed method. Notations: Throughout this paper, the superscripts 1 and T stand for the inverse and transpose of a matrix, respectively; P 4 0 ðP Z 0; P o 0; P r 0Þ means that the matrix P is symmetric positive definite (positive-semi definite, negative definite and negativesemi definite); Rmn is the set of m n real matrices; n denotes the symmetric block in symmetric matrix; I denotes the identity matrix with compatible dimensions; Z denotes the set of integers; λmax ðQ Þ and λmin ðQ Þ denote, respectively, the maximal and the minimal eigenvalue of matrix Q.
R 40, if k A f1; 2; …; Ng sup
θ A f hM ;hM þ 1;…;0g
ψ T ðθ Þψ ðθ Þ r c 1 ;
xT ðkÞRxðkÞ r c2 :
)
Remark 1. From Definition 1, we can get finite-time stable (FTS) which is different to Lyapunov asymptotic stability, that is, a system in which FTS may not be mean-square stable, while a meansquare stable system may not be FTS. On one hand, if a system is not mean square stable, it can still be FTS by choosing a smaller N and sufficiently large c2. On the other hand, although a system is mean square stable, its state trajectory may exceed the region f xj xT ðkÞRxðkÞ r c2 g if c2 is chosen sufficiently small and N sufficiently larger. Lemma 1 (Park et al. [16]). Let f 1 ; f 2 ; …; f n : Rm ↦R have positive values in an open subset D and Rm , then, the reciprocally convex combination of fi over D satisfies X1
min P
fαi j αi 4 0
i
αi ¼ 1g i
αi
f i ðtÞ ¼
X X f i ðtÞ þ max g ij ðt Þ; gi;j ðtÞ
i
subject to (
"
g i;j Rm -R; g j;i ðt Þ ¼ g i;j ðt Þ;
f i ðtÞ g j;i ðt Þ
iaj
# ) g i;j ðt Þ Z0 : f j ðt Þ
Lemma 2 (Nam et al. [19]). For a given symmetric positive matrix X 4 0, and three given non-negative integers a; b; k satisfying a r br k, vector function yðiÞ ¼ xðiþ 1Þ xðiÞ, xðiÞ : ½k b; k a \ Z-Rn , let us denote 8 " # k a1 X > > < 1 2 xðiÞ þxðk aÞ xðk bÞ ; a o b; χ ðk; a; bÞ ¼ b a i ¼ k b > > : 2xðk aÞ; a ¼ b; then, we have ðb aÞ
k a1 X
yT ðiÞXyðiÞ r ½xðk aÞ xðk bÞT
i ¼ kb
X½xðk aÞ xðk bÞ 3Ω Z Ω; T
2. Problem statement and preliminaries In this paper, we consider the following discrete-time system with time-varying delay and nonlinear perturbations: xðk þ 1Þ ¼ AxðkÞ þ Bxðk hðkÞÞ þ f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ; xðθÞ ¼ ψ ðθÞ;
θ A f hM ; hM þ1; …; 0g;
ð1Þ
where xðkÞ ¼ ½x1 ðkÞ; x2 ðkÞ; …; xn ðkÞT A Rn is the state vector of the system, ψ ðjÞ is the initial condition, A and B are known constant matrices with appropriate dimensions, h(k) denotes the known time-varying delay and satisfies 0 o hm rhðkÞ r hM , f 1 ðk; xðkÞÞ and f 2 ðk; xðk hðkÞÞÞ are nonlinear function, and are assumed to satisfy the following conditions: J f 1 ðk; xðkÞÞ J r ρ1 J xðkÞ J ; J f 1 ðk; xðk hðkÞÞÞ J r ρ2 J xðk hðkÞÞ J ; where ρ1 and ρ2 are known constants. Define that yðkÞ ¼ xðk þ 1Þ xðkÞ, which satisfies yT ðkÞyðkÞ r δ, for k A f hM ; hM þ 1; …; 0g. In this paper, the aim is to derive sufficient conditions which guarantee the finite-time stability of discrete-time system (1), before proceeding, the following definition and some lemmas are necessarily introduced.
ð2Þ
where Ω ¼ xðk aÞ þ xðk bÞ χ ðk; a; bÞ. P aþ1 T Remark 2. Lemma 2 implies that ðb aÞ ki ¼ ðiÞXy kb y T ðiÞ r ½xðk aÞ xðk bÞ X½xðk aÞ xðk bÞ, which is exactly the Jensen inequality. Therefore, Lemma 2 is less conservative than the celebrated Jensen inequality since a positive quantity is added in the right-hand side of the inequalities. Lemma 3 (Liu et al. [4]). Let X and Y are real matrices of appropriate dimensions. For a given scalar ϵ 4 0 and vectors x; y A Rn , then 2xT XYy r ϵ 1 xT X T Xx þ ϵyT Y T Yy:
3. Main results In this section, we firstly develop the finite-time stability criterion for system (1). For the sake of simplicity of matrix and vector representation, ei A R8nn ði ¼ 1; 2; …; 8Þ are defined as block entry matrices (for example e1 ¼ ½I; 0; 0; 0; 0; 0; 0; 0T ). The other notations are defined as 8 " # kX 1 > > < 1 2 xðiÞ þ xðkÞ xðk hm Þ ; hm 4 0; υ1 ðkÞ ¼ χ ðk; 0; hm Þ ¼ hm i ¼ k hm > > : 2xðkÞ; h ¼ 0; m
Definition 1 (Zuo et al. [31]). The discrete-time system (1) is said to be finite-time stable (FTS) with respect to ðc1 ; c2 ; R; NÞ, where
υ2 ðkÞ ¼ χ ðk; hm ; hðkÞÞ
Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i
W. Kang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
¼
8 > > > <
2 3 k X hm 1 1 42 xðiÞ þ xðk hm Þ xðk hðkÞÞ5; hðkÞ hm i ¼ k hðkÞ
hm o hðkÞ;
> > > : 2xðk hm Þ;
hm ¼ hðkÞ;
υ3 ðkÞ ¼ χ ðk; hðkÞ; hM Þ ¼
8 > > > <
T
T
T
T 1 ðkÞ;
υ
T 2 ðkÞ;
υ
T T 3 ðkÞ; y ðkÞ:
Theorem 1. For given scalars β 4 1, the system (1) is finite-time stable with respect to ðc1 ; c2 ; R; NÞ if there exist symmetric positive definite matrices P A Rnn 4 0, Q 1 A Rnn 40, Q 2 A Rnn 4 0, Z 1 A Rnn 4 0, Z 2 A Rnn 4 0 and matrices S A R2n2n , U 1 A Rnn , U 2 A Rnn and scalars λi 4 0; ði ¼ 1; 2; …; 6Þ, εi 4 0; ði ¼ 1; 2; …; 6Þ such that: " # Z2 S Φ1 ¼ 4 0; ð3Þ n Z2
λ1 I r P~ r λ2 I;
ð4Þ
0 o Q~1 r λ3 I;
0 o Q~2 r λ4 I;
ð5Þ
0 o Z 1 r λ5 I;
0 o Z 2 r λ6 I;
ð6Þ
c1 μ1 þ δμ2 o β 2 6 6 6 6 6 Ξ¼6 6 6 6 6 6 4
N
kX 1
βk 1 i yT ðiÞZ 1 yðiÞ
X hm 1 kX 1
βk 1 i yT ðiÞZ 2 yðiÞ:
j ¼ hM i ¼ k þ j
Taking the forward difference of V(k) and using β 4 1, it yields that
ΔV 1 ðkÞ ¼ xT ðk þ 1ÞPxðk þ 1Þ βxT ðkÞPxðkÞ þðβ 1ÞxT ðkÞPxðkÞ ¼ ðxðkÞ þ yðkÞÞT PðxðkÞ þyðkÞÞ βxT ðkÞPxðkÞ þðβ 1ÞV 1 ðkÞ; k X
ΔV 2 ðkÞ ¼
i ¼ k hm þ 1 kX 1
þ ðβ 1Þ
kX 1
βk i xT ðiÞQ 1 xðiÞ
kX 1
k X
βk i 1 xT ðiÞQ 1 xðiÞ þ
βk i xT ðiÞQ 2 xðiÞ
i ¼ k hM þ 1 kX 1
βk i xT ðiÞQ 2 xðiÞ þðβ 1Þ
i ¼ k hM
βk i 1 xT ðiÞQ 2 xðiÞ
i ¼ k hM
¼ xT ðkÞQ 1 xðkÞ β β
βk i xT ðiÞQ 1 xðiÞ
i ¼ k hm
i ¼ k hm
ð7Þ
n
Θ1 ε1 I
n
n
0 ε2 I
n
n
n
0 0 ε3 I
n
n
n
n
ε4 I
n
n
n
n
n
0 ε5 I
n
n
n
n
n
n
Θ1
Θ2
Θ2
Θ3
Θ3
0 0 0
0 0 0
0 0 0 0 0 ε6 I
3
hm T
x ðk hm ÞQ 1 xðk hm Þ þxT ðkÞQ 2 xðkÞ
x ðk hM ÞQ 2 xðk hM Þ þ ðβ 1ÞV 2 ðkÞ;
hM T
1 X
ΔV 3 ðkÞ ¼ hm
½yT ðkÞZ 1 yðkÞ β
ð11Þ
i T
y ðk þ iÞZ 1 yðk þ iÞ
X hm 1
þ ðhM hm Þ
½yT ðkÞZ 2 yðkÞ β
i
yðk þ iÞZ 2 yðk þ iÞ
i ¼ hM
7 7 7 7 7 7 7 o 0; 7 7 7 7 5
ð8Þ
þ ðβ 1ÞV 3 ðkÞ r yT ðkÞðhm Z 1 þ ðhM hm Þ2 Z 2 ÞyðkÞ 2
kX 1
hm
βk i yT ðiÞZ 1 yðiÞ ðhM hm Þ
k X hm 1
βk i yT ðiÞZ 2 yðiÞ
i ¼ k hM
i ¼ k hm
þ ðβ 1ÞV 3 ðkÞ r yT ðkÞðhm Z 1 þ ðhM hm Þ2 Z 2 ÞyðkÞ 2
kX 1
hm β
yT ðiÞZ 1 yðiÞ ðhM hm Þβ
hm þ 1
k X hm 1
yT ðiÞZ 2 yðiÞ
i ¼ k hM
i ¼ k hm
þ ðβ 1ÞV 3 ðkÞ:
where
Ξ 1 ¼ ðe1 þ e8 ÞPðe1 þ e8 ÞT βe1 Pe1 þ e1 ðQ 1 þ Q 2 ÞeT1 βhm e2 Q 1 eT2 h 2 β M e4 Q 2 eT4 þ hm e8 Z T1 e8 þ ðhM hm Þ2 e8 Z T2 e8 "
βΠ 1
Z1
0
0
3Z 1
#
"
Π
T 1
β
hm þ 1
Π2
Z2
S
n
Z2
#
Π
Z2
0
0
3Z 2
hm
ðhM hm Þ
k X hm 1
yT ðiÞZ 2 yðiÞ r ðhM hm Þ
i ¼ k hM
ðhM hm Þ
;
k X hðkÞ 1
þ 3αT2 ðkÞZ 2 α2 ðkÞ
Proof. Define a new augmented of Lyapunov–Krasovskii functional as follows: ð9Þ
k X hm 1
yT ðiÞZ 2 yðiÞ
i ¼ k hðkÞ
i ¼ k hM
hm ðhm 1Þ ðh h Þðh þ h þ 1Þ λ5 þ βhM 1 M m M m λ6 : 2 2
V 1 ðkÞ ¼ xT ðkÞPxðkÞ;
ð13Þ
Now, we assume that hm o hðkÞ o hM , based on Lemma 2, we have
μ1 ¼ λ2 þ βhm 1 hm λ3 þ βhM 1 hM λ4 ;
where
yT ðiÞZ 1 yðiÞ r ½xðkÞ xðk hm ÞT Z 1 ½xðkÞ xðk hm Þ
3½xðkÞ þxðk hm Þ υ1 ðkÞT Z 1 ½xðkÞ þxðk hm Þ υ1 ðkÞ:
#
VðkÞ ¼ V 1 ðkÞ þ V 2 ðkÞ þ V 3 ðkÞ;
kX 1 i ¼ k hm
T 2;
1 1 1 1 1 1 P~ ¼ R 2 PR 2 ; Q~1 ¼ R 2 Q 1 R 2 ; Q~2 ¼ R 2 Q 2 R 2 ;
μ2 ¼ βhm 1
ð12Þ
By Lemma 2, we obtain
Π 1 ¼ ½e1 e2 ; e1 þ e2 e5 ; Π 2 ¼ ½e2 e3 ; e2 þ e3 e6 ; e3 e4 ; e3 þ e4 e7 ; Ξ 2 ¼ 2½e1 U 1 þ e3 U 2 þ e8 U 3 ½e1 ðA IÞT þ e3 BT e8 T ; Ξ 3 ¼ ðε1 þ ε3 þ ε5 Þρ21 e1 eT1 þ ðε2 þ ε4 þ ε6 Þρ22 e3 eT3 ; Θ1 ¼ e1 U T1 ; Θ2 ¼ e3 U T2 ; Θ3 ¼ e8 U T3 ; "
ð10Þ
i ¼ hm
λ1 c2 ;
Ξ1 þ Ξ2 þ Ξ3
Z2 ¼
1 X
þ ðhM hm Þ
hðkÞ o hM ; hM ¼ hðkÞ;
T
βk 1 i xT ðiÞQ 2 xðiÞ;
i ¼ k hM
M
ξ ðkÞ ¼ ½x ðkÞ; x ðk hm Þ; x ðk hðkÞÞ; x ðk hM Þ; υ T
i ¼ k hm
V 3 ðiÞ ¼ hm
kX 1
βk 1 i xT ðiÞQ 1 xðiÞ þ
j ¼ hm i ¼ k þ j
2 3 k X hðkÞ 1 1 42 xðiÞ þ xðk hðkÞÞ xðk hM Þ5; hM hðkÞ i ¼ kh
> > > : 2xðk hðkÞÞ;
kX 1
V 2 ðkÞ ¼
3
yT ðiÞZ 2 yðiÞ r
hM hm T ½α ðkÞZ 2 α1 ðkÞ hðkÞ hm 1
hM hm T ½α ðkÞZ 2 α3 ðkÞ þ 3αT4 ðkÞZ 2 α4 ðkÞ: hM hðkÞ 3
ð14Þ
hm Noting hðkÞ þ hhMMhðkÞ ¼ 1 and according to Lemma 1 with hM hm hm " # Z2 S 4 0, it follows from (14) that n Z2 " # k X hm 1 Z2 S yT ðkÞZ 2 yðkÞ r αT ðkÞ αðkÞ; ð15Þ ðhM hm Þ n Z2 i ¼ kh M
αT ðkÞ ¼ ½αT1 ðkÞ; αT2 ðkÞ; αT3 ðkÞ; αT4 ðkÞ, α1 ðkÞ ¼ xðk hm Þ xðk where hðkÞÞ, α2 ðkÞ ¼ xðk hm Þ þ xðk hðkÞÞ υ2 ðkÞ, α3 ðkÞ ¼ xðk hðkÞÞ x " # Z2 0 ðk hM Þ, α4 ðkÞ ¼ xðk hðkÞÞ þ xðk hM Þ υ3 ðkÞ, Z 2 ¼ . If 0 3Z 2
Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i
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4
hðkÞ ¼ hm , we have xðk hm Þ xðk hðkÞÞ ¼ 0, and xðk hm Þ þ xðk hðkÞÞ χ ðk; hm ; hðkÞÞ ¼ 0, if hðkÞ ¼ hM , we have xðk hðkÞÞ xðk hM Þ ¼ 0, and xðk hðkÞÞ þ xðk hM Þ χ ðk; hðkÞ; hM Þ ¼ 0. When hðkÞ ¼ hm or hðkÞ ¼ hM , (15) still holds using Lemma 2. Therefore, we have
"
hm þ 1
αT ðkÞ
Z2
S
n
Z2
#
ð17Þ
On the other hand, for any matrixes U 1 ; U 2 ; U 3 with appropriate dimensions, we get 2½xT ðkÞU 1 þ xT ðk hðkÞÞU 2 þ yT ðkÞU 3 ½ðA IÞxðkÞ þ Bxðk hðkÞÞ þ f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ yðkÞ ¼ 2½xT ðkÞU 1 þ xT ðk hðkÞÞU 2 þ yT ðkÞU 3 ½ðA IÞxðkÞ þ Bxðk hðkÞÞ yðkÞ þ 2½xT ðkÞU 1 þ xT ðk hðkÞÞU 2 þ yT ðkÞU 3 ½f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ ¼ 0:
ð18Þ
There exist εi ði ¼ 1; 2; …; 6Þ, applying Lemma 3, we can obtain: 2xT ðkÞU 1 ½f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ rðε1 1 þ ε2 1 ÞxT ðkÞU T1 U 1 xðkÞ
ε
ρ
2 T 2 1 xðkÞ þ 2 x ðk hðkÞÞ 2 xðk hðkÞÞ;
ð19Þ
2xT ðk hðkÞÞU 2 ½f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ r ðε3 1 þ ε4 1 ÞxT ðk hðkÞÞU T2 U 2 xðk hðkÞÞ þ ε3 xT ðkÞρ21 xðkÞ þ ε4 xT ðk hðkÞÞρ22 xðk hðkÞ;
ð20Þ
þ ε6 xT ðk hðkÞÞρ22 xðk hðkÞÞ:
ð21Þ
From (17) to (21), it can be deduced that
ΔVðkÞ ðβ 1ÞVðkÞ r ξT ðkÞðΞ 1 þ Ξ 2 þ Ξ 3 ÞξðkÞ þ ðε1 1 þ ε2 1 ÞxT ðkÞU T1 U 1 xðkÞ þ ðε3 1 þ ε4 1 ÞxT ðk hðkÞÞ U T2 U 2 xðk hðkÞÞ þ ðε5 1 þ ε6 1 ÞyT ðkÞU T3 U 3 yðkÞ:
ð22Þ
By applying Schur complements and the inequality (8), we have
ΔVðkÞ ðβ 1ÞVðkÞ o 0;
ð23Þ
i.e., VðkÞ o βVðk 1Þ:
ð24Þ
Therefore, it infers that VðkÞ o βVðk 1Þ o β Vðk 2Þ o ⋯ o β Vð0Þ: k
ð25Þ
Furthermore, the initial value of Lyapunov–Krasovskii functional can be described as 1 X
β 1 i xT ðiÞQ 1 xðiÞ
i ¼ hm 1 X
þ
β 1 i xT ðiÞQ 2 xðiÞ þ
X hm 1 X 1
β 1 i yT ðiÞZ 1 yðiÞ
j ¼ hm i ¼ j
i ¼ hM
þ
1 X 1 X
β 1 i yT ðiÞZ 2 yðiÞ r λmax ðP~ ÞxT ð0ÞRxð0Þ
j ¼ hM i ¼ j
þβ
hm 1
λmax ðQ~1 Þ
1 X
xT ðiÞRxðiÞ þ β
hM 1
λmax ðQ~2 Þ
i ¼ hm 1 X i ¼ hM
xT ðiÞRxðiÞ þ β
hm 1
λmax Z 1
1 X 1 X j ¼ hm i ¼ j
yT ðiÞyðiÞ
yT ðiÞyðiÞ ¼ ðλ2 þ β
hm 1
hm λ3
hM λ4 Þc1 þ β
hm 1 hm ðhm 1Þ
2
hM 1 ðhM hm ÞðhM þ hm þ 1Þ
2
λ5
λ6 δ ¼ μ1 c1 þ μ2 δ:
ð26Þ
ð27Þ
It can be derived from (25) to (27) and (7) that k
μ1 c1 þ μ2 δ o c2 : λmin ðP~ Þ
ð28Þ
Hence, the system (1) is finite-time stable. This completes the proof. □ In the following, we consider the system with time-varying delay xðk þ 1Þ ¼ AxðkÞ þ Bxðk hðkÞÞ xðθÞ ¼ ψ ðθÞ; θ A f hM ; hM þ 1; …; 0g:
ð29Þ
Based on Theorem 1, the following criterion can be easily derived. Corollary 1. For given scalars β 4 1, the system (29) is finite-time stable with respect to ðc1 ; c2 ; R; NÞ if there exist symmetric positive definite matrices P A Rnn 4 0, Q 1 A Rnn 4 0, Q 2 A Rnn 4 0, Z 1 A Rnn 4 0, Z 2 A Rnn 4 0 and matrices S A R2n2n , U 1 A Rnn , U 2 A Rnn and scalars λi 4 0ði ¼ 1; 2; …; 6Þ, such that:
Ξ ¼ Ξ 1 þ Ξ 2 o0; Φ1 ¼
r ðε5 1 þ ε6 1 ÞyT ðkÞU T3 U 3 yðkÞ þ ε5 xT ðkÞρ21 xðkÞ
Vð0Þ ¼ xT ð0ÞPxð0Þ þ
hM 1
"
2yT ðkÞU 3 ½f 1 ðk; xðkÞÞ þf 2 ðk; xðk hðkÞÞÞ
2
þβ
xT ðkÞRxðkÞ r β
ΔVðkÞ ðβ 1ÞVðkÞ r ξT ðkÞΞ 1 ξðkÞ:
X hm 1 X 1
VðkÞ Z λmin ðP~ ÞxT ðkÞRxðkÞ:
From (10) to (16), one can obtain
þ ε1 x ðkÞρ
λmax Z 2
Note that
αðkÞ þ ðβ 1ÞV 3 ðkÞ: ð16Þ
T
hM 1
j ¼ hM i ¼ j
þβ
ΔV 3 ðkÞ r yT ðkÞðh2m Z 1 þ ðhM hm Þ2 Z 2 ÞyðkÞ β½xðkÞ xðk hm ÞT Z 1 ½xðkÞ xðk hm Þ 3β ½xðkÞ þ xðk hm Þ υ1 ðkÞT Z 1 ½xðkÞ þ xðk hm Þ υ1 ðkÞ β
þβ
Z2
S
n
Z2
ð30Þ
# 4 0;
λ1 I r P~ r λ2 I;
ð31Þ ð32Þ
0 o Q~1 r λ3 I;
0 o Q~2 r λ4 I;
ð33Þ
0 o Z 1 r λ5 I;
0 o Z 2 r λ6 I;
ð34Þ
c1 μ1 þ δμ2 o β
N
ð35Þ
λ1 c 2 ;
where Ξ 1 ; Ξ 2 ; Z 2 ; P~ ; Q~1 ; Q~2 ; μ1 ; μ2 are defined in Theorem 1. Remark 3. The less conservatism of the proposed finite-time stable criteria over [31] relies on how to deal with the term P1 Pk 1 k1i T y ðiÞZ 1 yðiÞ and the term j ¼ hm i ¼ kþj β P hm 1 Pk 1 k1i T y ðiÞZ 2 yðiÞ. Discrete Wirtinger-based i ¼ kþj β j ¼ hM P1 inequality (Lemma 3) is employed to handle the term j ¼ hm Pk 1 k1i T β y ðiÞZ yðiÞ instead of using discrete Jensen 1 i ¼ kþj inequality in [31]. Based on reciprocally convex approach and Discrete Wirtinger-based inequality, the new inequality (15) is P h m 1 Pk 1 k1i T introduced to handle the term y ðiÞZ 2 yðiÞ i ¼ kþj β j ¼ hM instead of using free-weighting matrices method in [31]. Remark 4. In this paper, we construct a new Lyapunov–Krasovskii ki1 functional involving variable ratios β which is different from the traditional way in [30,32]. By doing this, no inequality enlargement is required to obtain ΔV ðkÞ o ðβ 1ÞVðkÞ. However, in [30,32], ΔV ðkÞ is enlarged by ΔVðkÞ o ðβ 1ÞV 1 ðkÞ o ðβ 1ÞVðkÞ. It shows that our proposed method contains more information of the system and yields less conservatism than the existing ones. Remark 5. In Corollary 1 for given N; β; c1 and c2, if hm is fixed, hM can be viewed as an optimization parameter. We can use the
Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i
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5
2.5
following optimization algorithm to obtain the maximum value of hM:
2
max hM P; Q 1 ; Q 2 ; Z 1 ; Z 2 ; S; λi ði ¼ 1; 2; …; 6Þ s:t: Inequalities ð30Þ to ð35Þ:
ð36Þ
State response
1.5 1 0.5 0 −0.5 −1
4. Numerical examples
−1.5
In this section, we present three examples to demonstrate the effectiveness of our results.
−2.5
−2
Example 1. Consider the system model (1) with the following parameters: 0:8 0 0:1 0 1 0 A¼ ; B¼ ; R¼ ; 0:05 0:9 0:2 0:1 0 1
δ ¼ 1:1;
hm ¼ 2;
; c1 ¼ 4:1;
c2 ¼ 60;
In addition, for ρ1 ¼ 0:01, ρ2 ¼ 0:02 and hM ¼8, by applying Theorem 1 with the Matlab LMI tool box, we can get a set of feasible solution as follows: 23:1696 1:7226 0:8720 0:2284 P¼ ; Q1 ¼ ; 1:7226 6:6256 0:2284 0:0974 5:8021 0:1393 Q2 ¼ ; 0:1393 1:2909 20:1333 0:9076 4:9611 0:6728 Z1 ¼ ; Z2 ¼ ; 0:9076 2:8276 0:6728 0:9583 4:3737 1:0229 S11 ¼ ; 0:0309 0:6657 0:0438 0:0064 0:6605 0:0709 S12 ¼ ; S21 ¼ ; 0:0060 0:0032 0:3704 0:2276 0:4369 0:1650 S22 ¼ ; 0:0696 0:0220 34:1250 2:6149 3:4139 13:4608 U1 ¼ ; U2 ¼ ; 0:9076 9:8739 2:4938 3:0804 211:2394 68:6456 U3 ¼ ; 71:2365 74:8856
λ1 ¼ 6:3590; λ2 ¼ 24:6618; λ3 ¼ 1:1354; λ4 ¼ 6:5435; λ5 ¼ 26:3734; λ6 ¼ 5:3023; ε1 ¼ 794:1291; ε2 ¼ 448:8131; ε3 ¼ 394:8690; ε4 ¼ 217:9974; ε5 ¼ 1549:1; ε6 ¼ 896:9325: In order to verify the stability properties, we choose initial values as follows:
ψ ðθÞθ A f 8; 7;…;0g ¼ ½ψ ð 8Þ; ψ ð 7Þ; …; ψ ð0Þ
¼
1
2
2
2
⋯
2
2
2
2
2
2
⋯
2
2
ð37Þ
Obviously, it can be seen that the initial values satisfy the following conditions: sup
ψ T ðθÞψ ðθÞ r α ¼ 4:1;
θ A f 8; 7;…;0g
sup
θ A f 8; 7;…; 1g
½ψ ðθ þ 1Þ ψ ðθÞT ½ψ ðθ þ 1Þ ψ ðθÞ r δ ¼ 1:1 Figs. 1 and 2show the state responses x(k) and the evolution of xT ðkÞxðkÞ of the system (1) for an initial value ψ ð0Þ ¼ ½ 2; 2. It can
20
30
40
50
60
70
80
90
Fig. 1. State responses x(k) of the system (1) with time-varying delay 2 r hðkÞ r 8.
9 8
N ¼ 90:
In this example, when ρ1 ¼ 0:01 and ρ2 ¼ 0:01, by Theorem 1, we can get the upper bound of the time-varying delay is τM ¼ 9. We can obtain upper bound of τM for different ρ1 and ρ2, which are summarized in Table 1. It can be found from Table 3 that the magnitude of nonlinear perturbations increase, the maximum allowable delay also decreases.
10
t
7 6 State reponse
β ¼ 1:001;
0
5 4 3 2 1 0 −1
0
10
20
30
40
50
60
70
80
90
t
Fig. 2. State responses xT ðkÞxðkÞ of the system (1).
Table 1 Allowable bounds of τM for different ρ1, ρ2. Theorem Theorem Theorem Theorem Theorem
1 1 1 1 1
(ρ1 ¼ 0; ρ2 ¼ 0) (ρ1 ¼ 0:01; ρ2 ¼ 0:01) (ρ1 ¼ 0:01; ρ2 ¼ 0:02) (ρ1 ¼ 0:02; ρ2 ¼ 0:02) (ρ1 ¼ 0:03; ρ2 ¼ 0:03)
12 9 8 7 6
be seen that the state responses satisfy xT ðkÞxðkÞ r 60;
k A f1; 2; …; 90g:
Therefore, it concludes that the system (1) with time-varying delay 2 r hðkÞ r 8 is finite-time stable with respect to ð4:1; 60; I; 90Þ. Example 2. Consider the system model (29) with the following parameters: 0:6 0 0:1 0 1 0 A¼ ; B¼ ; R¼ ; 0:35 0:7 0:2 0:1 0 1 β ¼ 1:001; δ ¼ 1:1; c1 ¼ 2:1; c2 ¼ 80; N ¼ 80: In this example, when hm ¼2, by Theorem 2, we can get the upper bound of the time-varying delay is τM ¼ 13. In order to compare the results with those in [30–32], we give Table 2 in the following. From Table 2 it can be seen that our results are less conservative than the ones in [30–32]. The explanation for this is given in Remarks 2, 3 and 4. In order to verify our results, we choose initial values as follows:
ψ ðθÞθ A f 13; 12;…;0g ¼ ½ψ ð 13Þ; ψ ð 12Þ; …; ψ ð0Þ
¼
0
1
1
1
⋯
1
1
1
1
1
1
⋯
1
1
ð38Þ
Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i
W. Kang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Table 2 Allowable bounds of τM for different τm. h
2
3
4
5
6
Theorem 1 [30] Theorem 1 [31] Theorem 1 [32] Corollary 1
6 8 10 13
6 8 11 14
7 9 12 15
8 10 13 16
9 11 14 17
1.5 1 State response
β ¼ 1:14;
δ ¼ 1:1;
c1 ¼ 3;
model (29) with the following 0:01 0:01 0:1
τm ¼ 2;
0:1
3
7 0:1 5; 0:01
2
1 6 R¼40 0
0 1
3 0 7 0 5;
0
1
τM ¼ 5:
In this example, when N ¼ 2, by corollary, we can get the smallest eligible values of parameter c2. In order to compare the results with those in [31–33], we give the following in Table 3 when N ¼ 5; 10; 20. From Table 2 it can be seen that the values of c2 are smaller than those in [31–33]. Thus, the approach in this paper is less conservative than those in [31–33].
2
0.5 0 −0.5
5. Conclusions
−1 −1.5 −2
0
10
20
30
40
50
60
70
80
t
Fig. 3. State responses x(k) of the system (29) with time-varying delay 2 r hðkÞ r 13. 3 2.5 2 State response
Example 3. Consider the system parameters: 3 2 2 0:2 0:1 0:1 0:1 7 6 6 A ¼ 4 0:2 0:1 0:4 5; B ¼ 4 0:1 0:2 1 0:2 0:1
1.5 1 0.5
In this paper, the problem of finite-time stability for discretetime system with time-varying delay and nonlinear perturbations has been investigated. By choosing a novel Lyapunov–Krasovskii ki1 functional with power function β , based on discrete Wirtinger-based inequality and reciprocally convex method, less conservative stability criteria are established in the forms of LMIs. Three numerical examples are given to demonstrate the effectiveness of the proposed results. Finally, it should be worth noting that the proposed method in this paper can be greatly applicable in many other cases, such as discrete-time Markovian jumps systems, singular systems and switch systems. Besides, other method such as the delay-partitioning technique can be employed to further reduce the conservativeness of the obtained results, which will be investigated in our future work.
0 −0.5 −1
References 0
10
20
30
40
50
60
70
80
t
Fig. 4. State responses xT ðkÞxðkÞ of the system (29).
Table 3 The smallest eligible value of the parameter c2. N
5
10
20
Theorem 1 [31] Theorem 1 [32] Theorem 6 [33] Corollary 1
32 56 15 8
57 125 37 17
140 518 119 69
Obviously, it can be seen that the initial values satisfy the following conditions: sup
ψ T ðθÞψ ðθÞ r α ¼ 2:1;
θ A f 13; 12;…;0g
sup
θ A f 13; 12;…; 1g
½ψ ðθ þ 1Þ ψ ðθÞT ½ψ ðθ þ 1Þ ψ ðθÞ r δ ¼ 1:1 Figs. 3 and 4 show the state responses x(k) of the system (29) and the evolution of xT ðkÞxðkÞ for an initial value ψ ð0Þ ¼ ½ 1; 1. It can be seen that, the state responses satisfy xT ðkÞxðkÞ r80;
k A f1; 2; …; 80g:
Hence, it concludes that the system (29) with time-varying delay 2 r hðkÞ r 13 is finite-time stable with respect to ð2:1; 80; I; 80Þ.
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Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations$ Wei Kang a,b,n, Shouming Zhong b, Kaibo Shi c, Jun Cheng d a
School of Information Engineering, Fuyang Normal College, Fuyang 236041, PR China School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, PR China c School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, PR China d School of Science, Hubei University for Nationalities, Enshi 44500, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 3 August 2015 Received in revised form 23 October 2015 Accepted 5 November 2015 This paper was recommended for publication by Jeff Pieper
In this paper, the problem of finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations is investigated. By constructing a novel Lyapunov–Krasovskii functional and employing a new summation inequality named discrete Wirtinger-based inequality, reciprocally convex approach and zero equality, the improved finite-time stability criteria are derived to guarantee that the state of the system with time-varying delay does not exceed a given threshold when fixed time interval. Furthermore, the obtained conditions are formulated in forms of linear matrix inequalities which can be solved by using some standard numerical packages. Finally, three numerical examples are given to show the effectiveness and less conservatism of the proposed method. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Finite-time stability Discrete-time system Time-varying delay Nonlinear perturbations
1. Introduction It is well known that the phenomenon of time delay is very common in various practical systems such as chemical systems, biological systems and networked control systems. The existence of time delay in a system may lead to instability or bad performance [1–3]. Thus, it is important to investigate the stability and control of time-delay systems and a variety of approaches and results have been developed for the stability analysis and control synthesis of time-varying delay systems [4–9], such as delay-partitioning/decomposition way [10,11], model transformation [12,13], the free-weighting matrices technique [14,15], reciprocally convex approach [16], improved derivative estimation based on the Wirtinger-based inequality [17–20]. In many practical systems, our interests are concerned on the behavior of the system over a finite-time interval. For instance, in the presence of saturation or controlling the trajectory of a space vehicle from an initial point to a final one in a prescribed time interval. That is, the state of the system does not exceed a bound over a given finite-time interval. In order to deal with the transient ☆ This work was supported by National Natural Science Foundation of China (61273015, 11401104). n Corresponding author at: School of Information Engineering, Fuyang Normal College, Fuyang 236041, PR China. E-mail address: [email protected] (W. Kang).
performance of control systems, finite-time stability (FTS) was introduced in [21]. With the development of Lyapunov function approach and linear matrix inequality techniques, a great number of results on FTS, finite-time stabilization were obtained for delayed system [22–27]. It is well-known that the finite-time stability theory plays an important role in many practical applications, especially in the study of the transient behavior system, and in those applications where larger values of the state should not be attained. For example, in [34], the authors have given a flight control example to demonstrate the practical use of the theoretic results on FTS. In [35], the authors obtain the finite-time consensus conditions for discrete-time multi-agent system with time-varying communication delays by utilizing FTS theory. It should be noted that all the above-mentioned research have been devoted to continuous-time systems. However, discrete-time plays a considerable role than the continuous-time counterparts in realistic applications. Particulary, when implementing the delayed continuous-time systems for computer simulation, it becomes necessary to develop discrete-time systems. Therefore, finite-time stability for discrete-time systems with time-varying delay is a significant issue and has been studied by some researchers recently [28–33]. In [30], sufficient FTS conditions are obtained by using Lyapunov–Krasovskii functional method and an appropriate model transformation of the original system. Further improved results were derived in [31] by choosing a novel Lyapunov–Krasovskii functional and using free-weighting matrices technique. In
http://dx.doi.org/10.1016/j.isatra.2015.11.006 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i
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2
[32], the original is firstly transformed into two interconnected subsystems to get the less conservative results. Then, new conditions of FTS and stabilization are derived in terms of linear matrix inequalities (LMIs). Very recently, there is a novel summation inequality named discrete Wirtinger-based inequality which can be applied in obtaining less conservative results in comparison with the Jensen inequality. Thus, how to obtain less conservative FTS criteria for discrete-time systems with time-varying delay and nonlinear perturbations by using discrete Wirtinger-based inequality and reciprocally convex approach in the novel Lyapunov–Krasovskii functional is an interesting challenge, which motivates the main purpose of our study. In this paper, we concentrate on the FTS problem for discretetime delayed system with nonlinear perturbations. The main contribution of this paper lies in three aspects: firstly, compared with [30,32] and some other literature, a new Lyapunov–Kraki1 is introduced, sovskii functional involving variable ratios β which contains more information and may lead to a less conservative result; secondly, different from some existing works such as [30–33], a novel inequality named discrete Wirtinger-based inequality is employed to estimate the difference of the Lyapunov– Krasovskii functional for discrete-time system; thirdly, this paper is the first one concerned with the FTS criteria for discrete-time not only with time-varying delay but also with nonlinear perturbations. Finally, three numerical examples are given to confirm the effectiveness and less conservatism of the proposed method. Notations: Throughout this paper, the superscripts 1 and T stand for the inverse and transpose of a matrix, respectively; P 4 0 ðP Z 0; P o 0; P r 0Þ means that the matrix P is symmetric positive definite (positive-semi definite, negative definite and negativesemi definite); Rmn is the set of m n real matrices; n denotes the symmetric block in symmetric matrix; I denotes the identity matrix with compatible dimensions; Z denotes the set of integers; λmax ðQ Þ and λmin ðQ Þ denote, respectively, the maximal and the minimal eigenvalue of matrix Q.
R 40, if k A f1; 2; …; Ng sup
θ A f hM ;hM þ 1;…;0g
ψ T ðθ Þψ ðθ Þ r c 1 ;
xT ðkÞRxðkÞ r c2 :
)
Remark 1. From Definition 1, we can get finite-time stable (FTS) which is different to Lyapunov asymptotic stability, that is, a system in which FTS may not be mean-square stable, while a meansquare stable system may not be FTS. On one hand, if a system is not mean square stable, it can still be FTS by choosing a smaller N and sufficiently large c2. On the other hand, although a system is mean square stable, its state trajectory may exceed the region f xj xT ðkÞRxðkÞ r c2 g if c2 is chosen sufficiently small and N sufficiently larger. Lemma 1 (Park et al. [16]). Let f 1 ; f 2 ; …; f n : Rm ↦R have positive values in an open subset D and Rm , then, the reciprocally convex combination of fi over D satisfies X1
min P
fαi j αi 4 0
i
αi ¼ 1g i
αi
f i ðtÞ ¼
X X f i ðtÞ þ max g ij ðt Þ; gi;j ðtÞ
i
subject to (
"
g i;j Rm -R; g j;i ðt Þ ¼ g i;j ðt Þ;
f i ðtÞ g j;i ðt Þ
iaj
# ) g i;j ðt Þ Z0 : f j ðt Þ
Lemma 2 (Nam et al. [19]). For a given symmetric positive matrix X 4 0, and three given non-negative integers a; b; k satisfying a r br k, vector function yðiÞ ¼ xðiþ 1Þ xðiÞ, xðiÞ : ½k b; k a \ Z-Rn , let us denote 8 " # k a1 X > > < 1 2 xðiÞ þxðk aÞ xðk bÞ ; a o b; χ ðk; a; bÞ ¼ b a i ¼ k b > > : 2xðk aÞ; a ¼ b; then, we have ðb aÞ
k a1 X
yT ðiÞXyðiÞ r ½xðk aÞ xðk bÞT
i ¼ kb
X½xðk aÞ xðk bÞ 3Ω Z Ω; T
2. Problem statement and preliminaries In this paper, we consider the following discrete-time system with time-varying delay and nonlinear perturbations: xðk þ 1Þ ¼ AxðkÞ þ Bxðk hðkÞÞ þ f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ; xðθÞ ¼ ψ ðθÞ;
θ A f hM ; hM þ1; …; 0g;
ð1Þ
where xðkÞ ¼ ½x1 ðkÞ; x2 ðkÞ; …; xn ðkÞT A Rn is the state vector of the system, ψ ðjÞ is the initial condition, A and B are known constant matrices with appropriate dimensions, h(k) denotes the known time-varying delay and satisfies 0 o hm rhðkÞ r hM , f 1 ðk; xðkÞÞ and f 2 ðk; xðk hðkÞÞÞ are nonlinear function, and are assumed to satisfy the following conditions: J f 1 ðk; xðkÞÞ J r ρ1 J xðkÞ J ; J f 1 ðk; xðk hðkÞÞÞ J r ρ2 J xðk hðkÞÞ J ; where ρ1 and ρ2 are known constants. Define that yðkÞ ¼ xðk þ 1Þ xðkÞ, which satisfies yT ðkÞyðkÞ r δ, for k A f hM ; hM þ 1; …; 0g. In this paper, the aim is to derive sufficient conditions which guarantee the finite-time stability of discrete-time system (1), before proceeding, the following definition and some lemmas are necessarily introduced.
ð2Þ
where Ω ¼ xðk aÞ þ xðk bÞ χ ðk; a; bÞ. P aþ1 T Remark 2. Lemma 2 implies that ðb aÞ ki ¼ ðiÞXy kb y T ðiÞ r ½xðk aÞ xðk bÞ X½xðk aÞ xðk bÞ, which is exactly the Jensen inequality. Therefore, Lemma 2 is less conservative than the celebrated Jensen inequality since a positive quantity is added in the right-hand side of the inequalities. Lemma 3 (Liu et al. [4]). Let X and Y are real matrices of appropriate dimensions. For a given scalar ϵ 4 0 and vectors x; y A Rn , then 2xT XYy r ϵ 1 xT X T Xx þ ϵyT Y T Yy:
3. Main results In this section, we firstly develop the finite-time stability criterion for system (1). For the sake of simplicity of matrix and vector representation, ei A R8nn ði ¼ 1; 2; …; 8Þ are defined as block entry matrices (for example e1 ¼ ½I; 0; 0; 0; 0; 0; 0; 0T ). The other notations are defined as 8 " # kX 1 > > < 1 2 xðiÞ þ xðkÞ xðk hm Þ ; hm 4 0; υ1 ðkÞ ¼ χ ðk; 0; hm Þ ¼ hm i ¼ k hm > > : 2xðkÞ; h ¼ 0; m
Definition 1 (Zuo et al. [31]). The discrete-time system (1) is said to be finite-time stable (FTS) with respect to ðc1 ; c2 ; R; NÞ, where
υ2 ðkÞ ¼ χ ðk; hm ; hðkÞÞ
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¼
8 > > > <
2 3 k X hm 1 1 42 xðiÞ þ xðk hm Þ xðk hðkÞÞ5; hðkÞ hm i ¼ k hðkÞ
hm o hðkÞ;
> > > : 2xðk hm Þ;
hm ¼ hðkÞ;
υ3 ðkÞ ¼ χ ðk; hðkÞ; hM Þ ¼
8 > > > <
T
T
T
T 1 ðkÞ;
υ
T 2 ðkÞ;
υ
T T 3 ðkÞ; y ðkÞ:
Theorem 1. For given scalars β 4 1, the system (1) is finite-time stable with respect to ðc1 ; c2 ; R; NÞ if there exist symmetric positive definite matrices P A Rnn 4 0, Q 1 A Rnn 40, Q 2 A Rnn 4 0, Z 1 A Rnn 4 0, Z 2 A Rnn 4 0 and matrices S A R2n2n , U 1 A Rnn , U 2 A Rnn and scalars λi 4 0; ði ¼ 1; 2; …; 6Þ, εi 4 0; ði ¼ 1; 2; …; 6Þ such that: " # Z2 S Φ1 ¼ 4 0; ð3Þ n Z2
λ1 I r P~ r λ2 I;
ð4Þ
0 o Q~1 r λ3 I;
0 o Q~2 r λ4 I;
ð5Þ
0 o Z 1 r λ5 I;
0 o Z 2 r λ6 I;
ð6Þ
c1 μ1 þ δμ2 o β 2 6 6 6 6 6 Ξ¼6 6 6 6 6 6 4
N
kX 1
βk 1 i yT ðiÞZ 1 yðiÞ
X hm 1 kX 1
βk 1 i yT ðiÞZ 2 yðiÞ:
j ¼ hM i ¼ k þ j
Taking the forward difference of V(k) and using β 4 1, it yields that
ΔV 1 ðkÞ ¼ xT ðk þ 1ÞPxðk þ 1Þ βxT ðkÞPxðkÞ þðβ 1ÞxT ðkÞPxðkÞ ¼ ðxðkÞ þ yðkÞÞT PðxðkÞ þyðkÞÞ βxT ðkÞPxðkÞ þðβ 1ÞV 1 ðkÞ; k X
ΔV 2 ðkÞ ¼
i ¼ k hm þ 1 kX 1
þ ðβ 1Þ
kX 1
βk i xT ðiÞQ 1 xðiÞ
kX 1
k X
βk i 1 xT ðiÞQ 1 xðiÞ þ
βk i xT ðiÞQ 2 xðiÞ
i ¼ k hM þ 1 kX 1
βk i xT ðiÞQ 2 xðiÞ þðβ 1Þ
i ¼ k hM
βk i 1 xT ðiÞQ 2 xðiÞ
i ¼ k hM
¼ xT ðkÞQ 1 xðkÞ β β
βk i xT ðiÞQ 1 xðiÞ
i ¼ k hm
i ¼ k hm
ð7Þ
n
Θ1 ε1 I
n
n
0 ε2 I
n
n
n
0 0 ε3 I
n
n
n
n
ε4 I
n
n
n
n
n
0 ε5 I
n
n
n
n
n
n
Θ1
Θ2
Θ2
Θ3
Θ3
0 0 0
0 0 0
0 0 0 0 0 ε6 I
3
hm T
x ðk hm ÞQ 1 xðk hm Þ þxT ðkÞQ 2 xðkÞ
x ðk hM ÞQ 2 xðk hM Þ þ ðβ 1ÞV 2 ðkÞ;
hM T
1 X
ΔV 3 ðkÞ ¼ hm
½yT ðkÞZ 1 yðkÞ β
ð11Þ
i T
y ðk þ iÞZ 1 yðk þ iÞ
X hm 1
þ ðhM hm Þ
½yT ðkÞZ 2 yðkÞ β
i
yðk þ iÞZ 2 yðk þ iÞ
i ¼ hM
7 7 7 7 7 7 7 o 0; 7 7 7 7 5
ð8Þ
þ ðβ 1ÞV 3 ðkÞ r yT ðkÞðhm Z 1 þ ðhM hm Þ2 Z 2 ÞyðkÞ 2
kX 1
hm
βk i yT ðiÞZ 1 yðiÞ ðhM hm Þ
k X hm 1
βk i yT ðiÞZ 2 yðiÞ
i ¼ k hM
i ¼ k hm
þ ðβ 1ÞV 3 ðkÞ r yT ðkÞðhm Z 1 þ ðhM hm Þ2 Z 2 ÞyðkÞ 2
kX 1
hm β
yT ðiÞZ 1 yðiÞ ðhM hm Þβ
hm þ 1
k X hm 1
yT ðiÞZ 2 yðiÞ
i ¼ k hM
i ¼ k hm
þ ðβ 1ÞV 3 ðkÞ:
where
Ξ 1 ¼ ðe1 þ e8 ÞPðe1 þ e8 ÞT βe1 Pe1 þ e1 ðQ 1 þ Q 2 ÞeT1 βhm e2 Q 1 eT2 h 2 β M e4 Q 2 eT4 þ hm e8 Z T1 e8 þ ðhM hm Þ2 e8 Z T2 e8 "
βΠ 1
Z1
0
0
3Z 1
#
"
Π
T 1
β
hm þ 1
Π2
Z2
S
n
Z2
#
Π
Z2
0
0
3Z 2
hm
ðhM hm Þ
k X hm 1
yT ðiÞZ 2 yðiÞ r ðhM hm Þ
i ¼ k hM
ðhM hm Þ
;
k X hðkÞ 1
þ 3αT2 ðkÞZ 2 α2 ðkÞ
Proof. Define a new augmented of Lyapunov–Krasovskii functional as follows: ð9Þ
k X hm 1
yT ðiÞZ 2 yðiÞ
i ¼ k hðkÞ
i ¼ k hM
hm ðhm 1Þ ðh h Þðh þ h þ 1Þ λ5 þ βhM 1 M m M m λ6 : 2 2
V 1 ðkÞ ¼ xT ðkÞPxðkÞ;
ð13Þ
Now, we assume that hm o hðkÞ o hM , based on Lemma 2, we have
μ1 ¼ λ2 þ βhm 1 hm λ3 þ βhM 1 hM λ4 ;
where
yT ðiÞZ 1 yðiÞ r ½xðkÞ xðk hm ÞT Z 1 ½xðkÞ xðk hm Þ
3½xðkÞ þxðk hm Þ υ1 ðkÞT Z 1 ½xðkÞ þxðk hm Þ υ1 ðkÞ:
#
VðkÞ ¼ V 1 ðkÞ þ V 2 ðkÞ þ V 3 ðkÞ;
kX 1 i ¼ k hm
T 2;
1 1 1 1 1 1 P~ ¼ R 2 PR 2 ; Q~1 ¼ R 2 Q 1 R 2 ; Q~2 ¼ R 2 Q 2 R 2 ;
μ2 ¼ βhm 1
ð12Þ
By Lemma 2, we obtain
Π 1 ¼ ½e1 e2 ; e1 þ e2 e5 ; Π 2 ¼ ½e2 e3 ; e2 þ e3 e6 ; e3 e4 ; e3 þ e4 e7 ; Ξ 2 ¼ 2½e1 U 1 þ e3 U 2 þ e8 U 3 ½e1 ðA IÞT þ e3 BT e8 T ; Ξ 3 ¼ ðε1 þ ε3 þ ε5 Þρ21 e1 eT1 þ ðε2 þ ε4 þ ε6 Þρ22 e3 eT3 ; Θ1 ¼ e1 U T1 ; Θ2 ¼ e3 U T2 ; Θ3 ¼ e8 U T3 ; "
ð10Þ
i ¼ hm
λ1 c2 ;
Ξ1 þ Ξ2 þ Ξ3
Z2 ¼
1 X
þ ðhM hm Þ
hðkÞ o hM ; hM ¼ hðkÞ;
T
βk 1 i xT ðiÞQ 2 xðiÞ;
i ¼ k hM
M
ξ ðkÞ ¼ ½x ðkÞ; x ðk hm Þ; x ðk hðkÞÞ; x ðk hM Þ; υ T
i ¼ k hm
V 3 ðiÞ ¼ hm
kX 1
βk 1 i xT ðiÞQ 1 xðiÞ þ
j ¼ hm i ¼ k þ j
2 3 k X hðkÞ 1 1 42 xðiÞ þ xðk hðkÞÞ xðk hM Þ5; hM hðkÞ i ¼ kh
> > > : 2xðk hðkÞÞ;
kX 1
V 2 ðkÞ ¼
3
yT ðiÞZ 2 yðiÞ r
hM hm T ½α ðkÞZ 2 α1 ðkÞ hðkÞ hm 1
hM hm T ½α ðkÞZ 2 α3 ðkÞ þ 3αT4 ðkÞZ 2 α4 ðkÞ: hM hðkÞ 3
ð14Þ
hm Noting hðkÞ þ hhMMhðkÞ ¼ 1 and according to Lemma 1 with hM hm hm " # Z2 S 4 0, it follows from (14) that n Z2 " # k X hm 1 Z2 S yT ðkÞZ 2 yðkÞ r αT ðkÞ αðkÞ; ð15Þ ðhM hm Þ n Z2 i ¼ kh M
αT ðkÞ ¼ ½αT1 ðkÞ; αT2 ðkÞ; αT3 ðkÞ; αT4 ðkÞ, α1 ðkÞ ¼ xðk hm Þ xðk where hðkÞÞ, α2 ðkÞ ¼ xðk hm Þ þ xðk hðkÞÞ υ2 ðkÞ, α3 ðkÞ ¼ xðk hðkÞÞ x " # Z2 0 ðk hM Þ, α4 ðkÞ ¼ xðk hðkÞÞ þ xðk hM Þ υ3 ðkÞ, Z 2 ¼ . If 0 3Z 2
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hðkÞ ¼ hm , we have xðk hm Þ xðk hðkÞÞ ¼ 0, and xðk hm Þ þ xðk hðkÞÞ χ ðk; hm ; hðkÞÞ ¼ 0, if hðkÞ ¼ hM , we have xðk hðkÞÞ xðk hM Þ ¼ 0, and xðk hðkÞÞ þ xðk hM Þ χ ðk; hðkÞ; hM Þ ¼ 0. When hðkÞ ¼ hm or hðkÞ ¼ hM , (15) still holds using Lemma 2. Therefore, we have
"
hm þ 1
αT ðkÞ
Z2
S
n
Z2
#
ð17Þ
On the other hand, for any matrixes U 1 ; U 2 ; U 3 with appropriate dimensions, we get 2½xT ðkÞU 1 þ xT ðk hðkÞÞU 2 þ yT ðkÞU 3 ½ðA IÞxðkÞ þ Bxðk hðkÞÞ þ f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ yðkÞ ¼ 2½xT ðkÞU 1 þ xT ðk hðkÞÞU 2 þ yT ðkÞU 3 ½ðA IÞxðkÞ þ Bxðk hðkÞÞ yðkÞ þ 2½xT ðkÞU 1 þ xT ðk hðkÞÞU 2 þ yT ðkÞU 3 ½f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ ¼ 0:
ð18Þ
There exist εi ði ¼ 1; 2; …; 6Þ, applying Lemma 3, we can obtain: 2xT ðkÞU 1 ½f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ rðε1 1 þ ε2 1 ÞxT ðkÞU T1 U 1 xðkÞ
ε
ρ
2 T 2 1 xðkÞ þ 2 x ðk hðkÞÞ 2 xðk hðkÞÞ;
ð19Þ
2xT ðk hðkÞÞU 2 ½f 1 ðk; xðkÞÞ þ f 2 ðk; xðk hðkÞÞÞ r ðε3 1 þ ε4 1 ÞxT ðk hðkÞÞU T2 U 2 xðk hðkÞÞ þ ε3 xT ðkÞρ21 xðkÞ þ ε4 xT ðk hðkÞÞρ22 xðk hðkÞ;
ð20Þ
þ ε6 xT ðk hðkÞÞρ22 xðk hðkÞÞ:
ð21Þ
From (17) to (21), it can be deduced that
ΔVðkÞ ðβ 1ÞVðkÞ r ξT ðkÞðΞ 1 þ Ξ 2 þ Ξ 3 ÞξðkÞ þ ðε1 1 þ ε2 1 ÞxT ðkÞU T1 U 1 xðkÞ þ ðε3 1 þ ε4 1 ÞxT ðk hðkÞÞ U T2 U 2 xðk hðkÞÞ þ ðε5 1 þ ε6 1 ÞyT ðkÞU T3 U 3 yðkÞ:
ð22Þ
By applying Schur complements and the inequality (8), we have
ΔVðkÞ ðβ 1ÞVðkÞ o 0;
ð23Þ
i.e., VðkÞ o βVðk 1Þ:
ð24Þ
Therefore, it infers that VðkÞ o βVðk 1Þ o β Vðk 2Þ o ⋯ o β Vð0Þ: k
ð25Þ
Furthermore, the initial value of Lyapunov–Krasovskii functional can be described as 1 X
β 1 i xT ðiÞQ 1 xðiÞ
i ¼ hm 1 X
þ
β 1 i xT ðiÞQ 2 xðiÞ þ
X hm 1 X 1
β 1 i yT ðiÞZ 1 yðiÞ
j ¼ hm i ¼ j
i ¼ hM
þ
1 X 1 X
β 1 i yT ðiÞZ 2 yðiÞ r λmax ðP~ ÞxT ð0ÞRxð0Þ
j ¼ hM i ¼ j
þβ
hm 1
λmax ðQ~1 Þ
1 X
xT ðiÞRxðiÞ þ β
hM 1
λmax ðQ~2 Þ
i ¼ hm 1 X i ¼ hM
xT ðiÞRxðiÞ þ β
hm 1
λmax Z 1
1 X 1 X j ¼ hm i ¼ j
yT ðiÞyðiÞ
yT ðiÞyðiÞ ¼ ðλ2 þ β
hm 1
hm λ3
hM λ4 Þc1 þ β
hm 1 hm ðhm 1Þ
2
hM 1 ðhM hm ÞðhM þ hm þ 1Þ
2
λ5
λ6 δ ¼ μ1 c1 þ μ2 δ:
ð26Þ
ð27Þ
It can be derived from (25) to (27) and (7) that k
μ1 c1 þ μ2 δ o c2 : λmin ðP~ Þ
ð28Þ
Hence, the system (1) is finite-time stable. This completes the proof. □ In the following, we consider the system with time-varying delay xðk þ 1Þ ¼ AxðkÞ þ Bxðk hðkÞÞ xðθÞ ¼ ψ ðθÞ; θ A f hM ; hM þ 1; …; 0g:
ð29Þ
Based on Theorem 1, the following criterion can be easily derived. Corollary 1. For given scalars β 4 1, the system (29) is finite-time stable with respect to ðc1 ; c2 ; R; NÞ if there exist symmetric positive definite matrices P A Rnn 4 0, Q 1 A Rnn 4 0, Q 2 A Rnn 4 0, Z 1 A Rnn 4 0, Z 2 A Rnn 4 0 and matrices S A R2n2n , U 1 A Rnn , U 2 A Rnn and scalars λi 4 0ði ¼ 1; 2; …; 6Þ, such that:
Ξ ¼ Ξ 1 þ Ξ 2 o0; Φ1 ¼
r ðε5 1 þ ε6 1 ÞyT ðkÞU T3 U 3 yðkÞ þ ε5 xT ðkÞρ21 xðkÞ
Vð0Þ ¼ xT ð0ÞPxð0Þ þ
hM 1
"
2yT ðkÞU 3 ½f 1 ðk; xðkÞÞ þf 2 ðk; xðk hðkÞÞÞ
2
þβ
xT ðkÞRxðkÞ r β
ΔVðkÞ ðβ 1ÞVðkÞ r ξT ðkÞΞ 1 ξðkÞ:
X hm 1 X 1
VðkÞ Z λmin ðP~ ÞxT ðkÞRxðkÞ:
From (10) to (16), one can obtain
þ ε1 x ðkÞρ
λmax Z 2
Note that
αðkÞ þ ðβ 1ÞV 3 ðkÞ: ð16Þ
T
hM 1
j ¼ hM i ¼ j
þβ
ΔV 3 ðkÞ r yT ðkÞðh2m Z 1 þ ðhM hm Þ2 Z 2 ÞyðkÞ β½xðkÞ xðk hm ÞT Z 1 ½xðkÞ xðk hm Þ 3β ½xðkÞ þ xðk hm Þ υ1 ðkÞT Z 1 ½xðkÞ þ xðk hm Þ υ1 ðkÞ β
þβ
Z2
S
n
Z2
ð30Þ
# 4 0;
λ1 I r P~ r λ2 I;
ð31Þ ð32Þ
0 o Q~1 r λ3 I;
0 o Q~2 r λ4 I;
ð33Þ
0 o Z 1 r λ5 I;
0 o Z 2 r λ6 I;
ð34Þ
c1 μ1 þ δμ2 o β
N
ð35Þ
λ1 c 2 ;
where Ξ 1 ; Ξ 2 ; Z 2 ; P~ ; Q~1 ; Q~2 ; μ1 ; μ2 are defined in Theorem 1. Remark 3. The less conservatism of the proposed finite-time stable criteria over [31] relies on how to deal with the term P1 Pk 1 k1i T y ðiÞZ 1 yðiÞ and the term j ¼ hm i ¼ kþj β P hm 1 Pk 1 k1i T y ðiÞZ 2 yðiÞ. Discrete Wirtinger-based i ¼ kþj β j ¼ hM P1 inequality (Lemma 3) is employed to handle the term j ¼ hm Pk 1 k1i T β y ðiÞZ yðiÞ instead of using discrete Jensen 1 i ¼ kþj inequality in [31]. Based on reciprocally convex approach and Discrete Wirtinger-based inequality, the new inequality (15) is P h m 1 Pk 1 k1i T introduced to handle the term y ðiÞZ 2 yðiÞ i ¼ kþj β j ¼ hM instead of using free-weighting matrices method in [31]. Remark 4. In this paper, we construct a new Lyapunov–Krasovskii ki1 functional involving variable ratios β which is different from the traditional way in [30,32]. By doing this, no inequality enlargement is required to obtain ΔV ðkÞ o ðβ 1ÞVðkÞ. However, in [30,32], ΔV ðkÞ is enlarged by ΔVðkÞ o ðβ 1ÞV 1 ðkÞ o ðβ 1ÞVðkÞ. It shows that our proposed method contains more information of the system and yields less conservatism than the existing ones. Remark 5. In Corollary 1 for given N; β; c1 and c2, if hm is fixed, hM can be viewed as an optimization parameter. We can use the
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5
2.5
following optimization algorithm to obtain the maximum value of hM:
2
max hM P; Q 1 ; Q 2 ; Z 1 ; Z 2 ; S; λi ði ¼ 1; 2; …; 6Þ s:t: Inequalities ð30Þ to ð35Þ:
ð36Þ
State response
1.5 1 0.5 0 −0.5 −1
4. Numerical examples
−1.5
In this section, we present three examples to demonstrate the effectiveness of our results.
−2.5
−2
Example 1. Consider the system model (1) with the following parameters: 0:8 0 0:1 0 1 0 A¼ ; B¼ ; R¼ ; 0:05 0:9 0:2 0:1 0 1
δ ¼ 1:1;
hm ¼ 2;
; c1 ¼ 4:1;
c2 ¼ 60;
In addition, for ρ1 ¼ 0:01, ρ2 ¼ 0:02 and hM ¼8, by applying Theorem 1 with the Matlab LMI tool box, we can get a set of feasible solution as follows: 23:1696 1:7226 0:8720 0:2284 P¼ ; Q1 ¼ ; 1:7226 6:6256 0:2284 0:0974 5:8021 0:1393 Q2 ¼ ; 0:1393 1:2909 20:1333 0:9076 4:9611 0:6728 Z1 ¼ ; Z2 ¼ ; 0:9076 2:8276 0:6728 0:9583 4:3737 1:0229 S11 ¼ ; 0:0309 0:6657 0:0438 0:0064 0:6605 0:0709 S12 ¼ ; S21 ¼ ; 0:0060 0:0032 0:3704 0:2276 0:4369 0:1650 S22 ¼ ; 0:0696 0:0220 34:1250 2:6149 3:4139 13:4608 U1 ¼ ; U2 ¼ ; 0:9076 9:8739 2:4938 3:0804 211:2394 68:6456 U3 ¼ ; 71:2365 74:8856
λ1 ¼ 6:3590; λ2 ¼ 24:6618; λ3 ¼ 1:1354; λ4 ¼ 6:5435; λ5 ¼ 26:3734; λ6 ¼ 5:3023; ε1 ¼ 794:1291; ε2 ¼ 448:8131; ε3 ¼ 394:8690; ε4 ¼ 217:9974; ε5 ¼ 1549:1; ε6 ¼ 896:9325: In order to verify the stability properties, we choose initial values as follows:
ψ ðθÞθ A f 8; 7;…;0g ¼ ½ψ ð 8Þ; ψ ð 7Þ; …; ψ ð0Þ
¼
1
2
2
2
⋯
2
2
2
2
2
2
⋯
2
2
ð37Þ
Obviously, it can be seen that the initial values satisfy the following conditions: sup
ψ T ðθÞψ ðθÞ r α ¼ 4:1;
θ A f 8; 7;…;0g
sup
θ A f 8; 7;…; 1g
½ψ ðθ þ 1Þ ψ ðθÞT ½ψ ðθ þ 1Þ ψ ðθÞ r δ ¼ 1:1 Figs. 1 and 2show the state responses x(k) and the evolution of xT ðkÞxðkÞ of the system (1) for an initial value ψ ð0Þ ¼ ½ 2; 2. It can
20
30
40
50
60
70
80
90
Fig. 1. State responses x(k) of the system (1) with time-varying delay 2 r hðkÞ r 8.
9 8
N ¼ 90:
In this example, when ρ1 ¼ 0:01 and ρ2 ¼ 0:01, by Theorem 1, we can get the upper bound of the time-varying delay is τM ¼ 9. We can obtain upper bound of τM for different ρ1 and ρ2, which are summarized in Table 1. It can be found from Table 3 that the magnitude of nonlinear perturbations increase, the maximum allowable delay also decreases.
10
t
7 6 State reponse
β ¼ 1:001;
0
5 4 3 2 1 0 −1
0
10
20
30
40
50
60
70
80
90
t
Fig. 2. State responses xT ðkÞxðkÞ of the system (1).
Table 1 Allowable bounds of τM for different ρ1, ρ2. Theorem Theorem Theorem Theorem Theorem
1 1 1 1 1
(ρ1 ¼ 0; ρ2 ¼ 0) (ρ1 ¼ 0:01; ρ2 ¼ 0:01) (ρ1 ¼ 0:01; ρ2 ¼ 0:02) (ρ1 ¼ 0:02; ρ2 ¼ 0:02) (ρ1 ¼ 0:03; ρ2 ¼ 0:03)
12 9 8 7 6
be seen that the state responses satisfy xT ðkÞxðkÞ r 60;
k A f1; 2; …; 90g:
Therefore, it concludes that the system (1) with time-varying delay 2 r hðkÞ r 8 is finite-time stable with respect to ð4:1; 60; I; 90Þ. Example 2. Consider the system model (29) with the following parameters: 0:6 0 0:1 0 1 0 A¼ ; B¼ ; R¼ ; 0:35 0:7 0:2 0:1 0 1 β ¼ 1:001; δ ¼ 1:1; c1 ¼ 2:1; c2 ¼ 80; N ¼ 80: In this example, when hm ¼2, by Theorem 2, we can get the upper bound of the time-varying delay is τM ¼ 13. In order to compare the results with those in [30–32], we give Table 2 in the following. From Table 2 it can be seen that our results are less conservative than the ones in [30–32]. The explanation for this is given in Remarks 2, 3 and 4. In order to verify our results, we choose initial values as follows:
ψ ðθÞθ A f 13; 12;…;0g ¼ ½ψ ð 13Þ; ψ ð 12Þ; …; ψ ð0Þ
¼
0
1
1
1
⋯
1
1
1
1
1
1
⋯
1
1
ð38Þ
Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i
W. Kang et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
Table 2 Allowable bounds of τM for different τm. h
2
3
4
5
6
Theorem 1 [30] Theorem 1 [31] Theorem 1 [32] Corollary 1
6 8 10 13
6 8 11 14
7 9 12 15
8 10 13 16
9 11 14 17
1.5 1 State response
β ¼ 1:14;
δ ¼ 1:1;
c1 ¼ 3;
model (29) with the following 0:01 0:01 0:1
τm ¼ 2;
0:1
3
7 0:1 5; 0:01
2
1 6 R¼40 0
0 1
3 0 7 0 5;
0
1
τM ¼ 5:
In this example, when N ¼ 2, by corollary, we can get the smallest eligible values of parameter c2. In order to compare the results with those in [31–33], we give the following in Table 3 when N ¼ 5; 10; 20. From Table 2 it can be seen that the values of c2 are smaller than those in [31–33]. Thus, the approach in this paper is less conservative than those in [31–33].
2
0.5 0 −0.5
5. Conclusions
−1 −1.5 −2
0
10
20
30
40
50
60
70
80
t
Fig. 3. State responses x(k) of the system (29) with time-varying delay 2 r hðkÞ r 13. 3 2.5 2 State response
Example 3. Consider the system parameters: 3 2 2 0:2 0:1 0:1 0:1 7 6 6 A ¼ 4 0:2 0:1 0:4 5; B ¼ 4 0:1 0:2 1 0:2 0:1
1.5 1 0.5
In this paper, the problem of finite-time stability for discretetime system with time-varying delay and nonlinear perturbations has been investigated. By choosing a novel Lyapunov–Krasovskii ki1 functional with power function β , based on discrete Wirtinger-based inequality and reciprocally convex method, less conservative stability criteria are established in the forms of LMIs. Three numerical examples are given to demonstrate the effectiveness of the proposed results. Finally, it should be worth noting that the proposed method in this paper can be greatly applicable in many other cases, such as discrete-time Markovian jumps systems, singular systems and switch systems. Besides, other method such as the delay-partitioning technique can be employed to further reduce the conservativeness of the obtained results, which will be investigated in our future work.
0 −0.5 −1
References 0
10
20
30
40
50
60
70
80
t
Fig. 4. State responses xT ðkÞxðkÞ of the system (29).
Table 3 The smallest eligible value of the parameter c2. N
5
10
20
Theorem 1 [31] Theorem 1 [32] Theorem 6 [33] Corollary 1
32 56 15 8
57 125 37 17
140 518 119 69
Obviously, it can be seen that the initial values satisfy the following conditions: sup
ψ T ðθÞψ ðθÞ r α ¼ 2:1;
θ A f 13; 12;…;0g
sup
θ A f 13; 12;…; 1g
½ψ ðθ þ 1Þ ψ ðθÞT ½ψ ðθ þ 1Þ ψ ðθÞ r δ ¼ 1:1 Figs. 3 and 4 show the state responses x(k) of the system (29) and the evolution of xT ðkÞxðkÞ for an initial value ψ ð0Þ ¼ ½ 1; 1. It can be seen that, the state responses satisfy xT ðkÞxðkÞ r80;
k A f1; 2; …; 80g:
Hence, it concludes that the system (29) with time-varying delay 2 r hðkÞ r 13 is finite-time stable with respect to ð2:1; 80; I; 80Þ.
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Please cite this article as: Kang W, et al. Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.11.006i