Circuits Syst Signal Process DOI 10.1007/s00034-015-0194-2
Finite-Time Stability of Time-Delay Switched Systems with Delayed Impulse Effects Lijun Gao1 · Yingying Cai1
Received: 24 May 2015 / Revised: 27 October 2015 / Accepted: 28 October 2015 © Springer Science+Business Media New York 2015
Abstract This paper investigates the finite-time stability problem for a class of timedelay switched systems with nonlinear perturbation and delayed impulse effects. Based on the Lyapunov function method and the technique of inequalities, stability criteria are established to guarantee that the state trajectory of the system does not exceed a certain threshold over a pre-specified finite time interval. Compared with existing results for related problems, the obtained results can be applied to a larger class of hybrid delayed systems, including those in which all of the subsystems are stable and unstable. Two examples are given to demonstrate the validity of the main results. Keywords Finite-time stability · Impulsive switched systems · Delayed impulse · Switching signal · Lyapunov function
1 Introduction Switched system, which is used to model many physical or man-made systems displaying switching features, has been extensively studied in the past years [10,25,41]. The system, typically, contains a set of subsystems described by continuous- or discretetime dynamics and a rule controlling the switching between them. This class of system constitutes an extremely active field of current scientific research. Generally, stability and stabilization problems are the main concerns in the field of switched systems. Lyapunov function techniques have been proved to be effective for dealing with stability and stabilization problems for switched systems [2,4,5,19,22,24,28]. For stability
B 1
Lijun Gao
[email protected] Department of Automation, Qufu Normal University, Rizhao 276826, Shandong, People’s Republic of China
Circuits Syst Signal Process
under arbitrary switching, the common Lyapunov function is required for all subsystems. For stability under constrained switching, the multiple Lyapunov function proposed in [33] is a powerful and effective tool. For time-dependent switched systems, dwell time and average dwell time approaches have been used to investigate stabilization problems [1,31]. For more details of recent results of basic problems in stability and stabilization for switched systems, readers are referred to [16,24] and the references cited therein. Impulsive phenomena exist widely in many evolution processes in which states are changed abruptly at certain moments of time, such as physics, engineering, and information science [7,20,36]. Moreover, impulsive control has become an important control strategy, especially while the controlled plan cannot endure continuous control input. When a dynamical system is subject to both switching and impulsive effects, such a system can be adequately described by an impulsive switched system [7]. In recent years, impulsive switched systems have drawn much attention and many useful conclusions have been reached [7,15,20,36]. Time delay is unavoidable in practice and often leads to poor performance, even instability of the systems [6,38]. It should be mentioned that delays are only assumed to appear in continuous dynamics [17,29,39,40]. In fact, during transmission of the impulse information, input delays are often encountered. For example, in the application of networked control systems, the output is transmitted through a digital communication network [12–21]. Obviously, computation time and network-induced delays result in sensor-to-controller delay and controller-to-actuator delay. In this case, the k-th input update time reaching the destination may be greater than the k-th sampling time, which causes a delay in the discrete impulse dynamics. Therefore, it is essential to consider the effects of delayed impulses because they may have destabilizing effects. To date, for stability analysis of switched systems, the most interesting studies have considered Lyapunov asymptotic stability and exponential stability [26,32,34], which is defined on an infinite time interval [3,9,11,23,27]. However, in many practical cases, we are interested in stability over a fixed finite time interval. This type of stability is called finite-time stability (FTS) if the state trajectory does not exceed a prescribed region during a fixed time interval [3]. As a composite concept, FTS has attracted much attention and some efficient results have been derived [8,14,23,37]. For example, in [23], the finite-time boundedness and stabilization problems for a class of switched time-delay systems were studied using the linear matrix inequality (LMI) method [18]. However, impulse effects at the switching points have not been considered. Based on the average dwell time approach and the Lyapunov–Krasovskii functional technique, the finite-time stability problem for discrete-time impulsive switched nonlinear systems with time-varying delays is considered. However, the results are only applied to the case in which every subsystem is stable. Recently, Wang et al. [8] investigated the FTS property for a class of nonlinear impulsive switched systems using the average dwell time approach. It should be mentioned that, to date, FTS property analysis has not been considered for nonlinear impulsive switched systems with delayed impulses [30,35]. Motivated by the aforementioned works and the practical background, the issue of finite-time stability of impulsive switched systems in the presence of both nonlinear perturbation and delayed impulse effects is considered. To summarize: (1) This paper
Circuits Syst Signal Process
considers impulsive switched nonlinear systems, which are a type of complex hybrid system of which little research has been conducted to date. The complexity introduced by the switching rule and delayed impulses makes it difficult to check the FTS property but worthwhile in the sense that it may draw much attention to this field. (2) Based on the Lyapunov function method, several criteria are established to guarantee that the state trajectory remains in a bounded region of the state space over a pre-specified finite time interval. (3) Compared with existing research, we do not impose the restriction that all subsystems must be stable; that is, even if all the subsystems are unstable, which is our main contribution, we can still ensure that the whole system is finite-time stable.
2 Model of Nonlinear Impulsive Switched Systems and Main Results Consider a class of continuous-time nonlinear impulsive switched systems as follows: x(t) ˙ = Aσ (t) (t)x(t) + f σ (t) (x(t)), t = tk x(tk ) = Jk tk− , x(tk− ) x(t0 ) = x0
(1)
where t ∈ R + , x(t) ∈ R n is the state variable, t0 ≥ 0 is the initial time, Aσ (t) ∈ C[Jt0 , R n×n ], and f ∈ C[R n , R n ]. The index function σ (t) : (0, T ) → . S = [1, . . . , s] is the switched signal, which is represented by i k according to σ (t) = i k , t ∈ (tk−1 , tk ], where s > 0 is the number of subsystems and Ai (t) is a matrix, and f i (x(t)) represent the nonlinear perturbations; moreover, they are piecewise continuous vector value functions, with f i (0) ≡ 0, t ∈ R + , and ensuring the existence and uniqueness of solutions for (1), Jk is a bounded sequence that satisfies Jk (tk , x(tk )) ≤ βk x(tk−1 ), βk is defined as a parameter on the interval (tk−1 , tk ]. Definition 1 Given an initial time t0 , three positive scalars c1 , c2 , and T with c1 < c2 , a positive definite matrix Γ , and a positive definite matrix-valued function Π defined over [0, T ], with Π (0) ≤ Γ . Then, the switched system (1) is said to be FTS with respect to (t0 , c1 , c2 , T, Γ, Π ) in the presence of impulse effects and nonlinear perturbation if x0T Γ x0 ≤ c1 ⇒ x T (t)Π (t − t0 )x(t) < c2 , ∀ t0 < t ≤ t0 + T.
(2)
2.1 Main Results Assuming that there exist some positive definite and symmetric matrices Pi such that the following conditions hold: – There exist functions αi (t) ∈ C[R, R] and ai (t) ∈ C[R, R + ] such that λ(AiT (t)Pi + Pi Ai (t)) ≤ αi (t) and 2x T Pi f i (x) ≤ ai (t)x2 . – Define qi (t) = λmin1(Pi ) (αi (t) + ai (t)). t – tk qi (s)ds ≥ 0, for t ∈ [tk−1 , tk ]. – Assume
Circuits Syst Signal Process K
βi x0
i=1
λmax (Pk+1 ) tt qk (s) ds ≤ e k λmin (Pk+1 )
c2 . λmax (Π )
(3)
Theorem 1 Suppose the aforementioned assumptions are satisfied. Then, the switched system (1) is FTS for [t0 , t0 + T ] with respect to (c1 , c2 , T, Γ, Π ). Proof Let Vi (t) = x T (t)Pi x(t), i = 1, 2, . . . , s, where x(t) is the solution of (1). Then, for t = tk , the derivation of Vi along (1) is V˙i (t) = x˙ T (t)Pi x(t) + x T (t)Pi x(t) ˙ T T =x Ai (t)Pi + Pi Ai (t) x + 2x T Pi f i (x(t)) ≤ αi (t)x T x + ai (t)x2 αi (t) T ai (t) T ≤ x Pi x + x Pi x λmin (Pi ) λmin (Pi ) αi (t) + ai (t) T x Pi x = λmin (Pi ) = qi (t)Vi . Thus, for t ∈ [t0 , t1 ], along the trajectory of system (1), it can be concluded that t q (s)ds V˙1 (t) ≤ q1 (t)V1 (t), V1 (t) ≤ V1 (t0 )e t0 1 . Due to λmin (P1 )x(t)2 ≤ x T (t)P1 x(t) ≤ x T (t0 )P1 x(t0 )e ≤ λmax (P1 x(t0 )2 e we have
x(t) ≤
t
t0
t t0
q1 (s)ds
q1 (s)ds
t λmax (P1 ) q (s)ds x(t0 )2 e t0 1 , t ∈ [t0 , t1 ]. λmin (P1 )
Similarly, we can obtain x(tk ) ≤ Jk (tk− , x(tk− )) ≤ βk x(tk−1 ) ≤ βk βk−1 x(tk−2 ) ≤ · · · ≤
k
βi x0 ,
i=1
Therefore,
2
k t λmax (Pk+1 ) q (s)ds βi x0 2 e tk k+1 x(t) ≤ λmin (Pk+1 ) i=1 k λmax (Pk+1 ) tt qk+1 (s)ds e k = βi x0 , λmin (Pk+1 ) i=1
(4)
Circuits Syst Signal Process
which demonstrates that x0 ≤
c1 , λmin (Γ )
(5)
from inequality (2). Using (3), (4), and (5), one can get x(t) ≤
k i=1
βi
λmax (Pk+1 ) tt qk+1 (s)ds e k λmin (Pk+1 )
c1 ≤ λmin (Γ )
c2 , λmax (Π )
that is, x(t) ≤
c2 . λmax (Π )
(6)
In general, x T (t)Π x(t) ≤ λmax (Π )x(t)2 .
(7)
Finally, combining (6) and (7) results in the following inequality x T (t)Π x(t) ≤ c2 . Therefore, the impulse switched system (1) is FTS for the finite time interval [t0 , T ].
3 Model of a Nonlinear Delay Switched System with Linear Impulse and Main Results Consider a class of nonlinear delay switched systems with linear impulses x(t) ˙ = Aσ (t) (t)x(t) + f σ (t) (x(t), x(t − r (t))), t = tk
x(tk ) = Dk x(tk− ) x(t0 ) = x0
(8)
where A(t) ∈ C[Jt0 , R n×n ], f ∈ C[R n ×Cr , R n ], Dk ∈ R n×n , k ∈ N ∗ , 0 ≤ r (t) ≤ r . . The index function σ (t) : (0, T ) → s = [1, . . . , s] is the switched signal, s >0 is the number of subsystems, Ai (t) is a matrix, and f i (x(t), x(t − r (t))) is the nonlinear perturbation satisfying f i (x(t), x(t − r (t))) ≤ γ x(t) + β(t), i ∈ s, with γ > 0 a T constant and β(t) ≥ 0 a Lebegue function such that 0 eλτ β(τ )dτ < ∞. Moreover, Dk (k = 1, 2, . . .) is a constant matrix on the interval (tk−1 , tk ], which represents the impulse effect at the switching time. Remark 1 Function r (t) can be any inverse function satisfying 0 ≤ r (t) ≤ r , where r is a specific constant.
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Definition 2 Given four positive scalars c1 , c2 , T, and r , where c1 < c2 and r is a given delay, two matrices P1 , Γ > 0, and Π (t) > 0, which satisfies Γ ≤ P1 , the nonlinear delay switched system (8) is said to be FTS with respect to (c1 , c2 , T, Γ, Π, γ , β(t), r ) in the presence of impulse effects and nonlinear perturbation if x0T Γ x0 ≤ c1 ⇒ x T (t)Π x(t) < c2 , ∀ 0 < t ≤ T.
(9)
3.1 Main Results Assuming that there exist some positive definite and symmetric matrices Pi such that the following conditions hold: – There exist functions α(t) ∈ C[R, R] and a1 (t), a2 (t) ∈ C[R, R+ ] such that λ(AiT (t)Pi + Pi Ai (t)) ≤ αi (t), and 2x T Pi f (x, y) ≤ a1 (t)x 2 + a2 (t)y 2 . t +r tk a2 (s) – Define that lk2 = tkk e s qi0 (η)dη λmin (Pi ) ds, Mk = Dk + lk max{Dk , 1}, 1 tk +1 qi (s)ds (Pi ) 2 tk e , where q0 (η) = (αi (η) + a1 (η)/λmin (Pi )), and βk = λλmax min (Pi ) s 1 qi (s) = λmin (Pi (t)) (a2 (s + r ) + α(s) + a1 (s)) s+r q0 (η)ds. t – tk qi (s)ds ≥ 0, for t ∈ [tk−1 , tk ]. – Assume K
Mjβj MN
j=1
c1 (l0 + D0 ) ≤ λmin (Γ )
c2 . λmax (Π )
(10)
Theorem 2 Suppose the aforementioned assumptions are satisfied. If there exist a constant M and a function θ (t) such that the function δ(t) = M f (x(t + r ), x(t)) + θ (t) > 0 holds, then the nonlinear delay switched system is FTS for [t0 , T ] with respect to (c1 , c2 , T, Γ, Π, Υ, β(t), r ). Proof Let Vi (t) = x T (t)Pi x(t), where x(t) is the solution of (8). Then, for t = tk , the derivation of Vi along (1) is V˙i (t) = x˙ T (t)Pi x(t) + x T (t)Pi x(t) ˙ = x T (AiT (t)Pi + Pi Ai (t))x + 2x T Pi f (t, x(t), x(t − r )) ≤ αi (t)x T x + a1 (t)x2 + a2 (t)x(t − r )2 a2 (t) . = q0 (t)Vi + λmin Vi (t − r ) Thus, for t ∈ [t0 , t1 ], V1 (t) ≤ V1 (t0 )e
t t0
q0 (s)ds
+
t t
e
t0
s
q0 (η)dη
×
a2 (s) V1 (s − r )ds. λmin (P1 )
Circuits Syst Signal Process
For t ∈ [t0 , t0 + r ], V1 (t) ≤ V1 (t0 )e ≤ V1 (t0 )e
t t0
t t0
q0 (s)ds
q0 (s)ds
+ V1 (t0 ) ×
t t
e
s
q0 (η)dη
t0 t0 +r t
+ V1 (t0 ) ×
e
s
a2 (s) ds λmin (P1 )
q0 (η)dη
a2 (s) ds λmin (P1 )
q0 (η)dη
a2 (s) ds λmin (P1 )
t0
and t ∈ [t0 + r, t1 ], V1 (t) ≤ V1 (t0 )e
t t0
q0 (s)ds
+ V1 (t0 ) ×
t0 +r t
e
s
t0 t
a2 (s) V1 (s − r )ds + e s q0 (η)dη λmin (P1 ) t0 +r t0 +r t t a2 (s) q (s)ds ds ≤ V1 (t0 )e t0 0 + V1 (t0 ) × e s q0 (η)dη λ min (P1 ) t0 t−r t a2 (s + r ) + V1 (s)ds. e s q0 (η)dη λmin (P1 ) t0 Multiplying e
−
V1 (t)e
−
t t0
t t0
t
q0 (s)ds
q0 (s)ds
on both sides of the last inequality yields
t0 +r
≤ V1 (t0 ) + V1 (t0 ) +
t0 t−r t0
e
s+r
q0 (η)dη
t0
a2 (s) t0 q0 (η)dη es ds λmin (P1 )
a2 (s) V1 (s)ds λmin (P1 )
≤ V1 (t0 ) + V1 (t0 )lo2 t−r s a2 (s + r ) tt q0 (η)dη e 0 + e s+r qi0 (η)dη V1 (s)ds. λmin (P1 ) t0 Let y1 (t) = V1 (t)e
−
t t0
q0 (s)ds
for t ∈ [t0 + r, t1 ],
y1 (t) ≤
V1 (t0 ) + V1 (t0 )lo2
+
t s
e
s+r
q0 (η)dη a2 (s
+ r) y1 (s)ds. λmin (P1 )
t0
The last inequality holds for all t ∈ [t0 , t1 ], that is, y1 (t) ≤ V1 (t0 ) + V1 (t0 )lo2 +
t s
e
s+r
q0 (η)dη a2 (s
t0
+ r) y1 (s)ds. λmin (P1 )
Then, the Gronwall–Bellman inequality implies that s q (η)dη a (s+r ) t s+r 2 0 e λmin (P1 ) ds , t ∈ [t0 , t1 ]. y1 (t) ≤ V1 (t0 ) + V1 (t0 )lo2 e t0
Circuits Syst Signal Process
Therefore,
x(t) ≤ e
1 t 2 t0
qi (s)ds
≤e
1 t 2 t0
qi (s)ds
V1 (t0 ) + V1 (t0 )l02 λmin (P1 ) λmax (P1 ) [x(t0 ) + xt0 l0 ], t ∈ [t0 , t1 ]. λmin (P1 )
Similarly, we have
x(t) ≤ e
1 t 2 tk
qi (s)ds
λmax (Pk+1 ) × [x(tk ) + xtk lk ], t ∈ [tk , tk+1 ]. λmin (Pk+1 )
Furthermore, x(tk ) ≤ Dk x(t ˜ k ), and xtk = suptk −r ≤t≤tk x(t) = max{x(tk ), suptk −r ≤t
≤ max{Dk x(tk− ), suptk −r ≤t
≤ max{Dk , 1}x(t ˜ k ), where x(t ˜ k ) = [x(tk−1 ) + xtk−1 lk−1 ]βk−1 . Then, x(t ˜ k ) = [x(tk−1 ) + xtk−1 lk−1 ]βk−1 ≤ [Dk−1 + lk−1 max{Dk−1 , 1}] × x(t ˜ k−1 )βk−1 = Mk−1 x(t ˜ k−1 )βk−1 ≤ Mk−1 βk−1 Mk−2 βk−2 x(t ˜ k−1 ) ≤ · · · ≤
K
M j β j x(t ˜ 1 ).
j=1
Thus, for t ∈ [tk , tk+1 ], x(t) ≤ ≤
1 t λmax (Pk+1 ) q (s)ds [x(tk ) + xtk lk ]e 2 tk i λmin (Pk+1 ) K 1 t λmax (Pk+1 ) q (s)ds Mk M j β j x(t ˜ 1 )e 2 tk i λmin (Pk+1 ) j=1
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≤ x(t ˜ k+1 ) ≤
K
M j β j x(t ˜ 1 )
j=1
=
K
M j β j Me
t1 t0
δ(s)ds
(x(t0 ) + xt0 l0 ).
j=1
So, we have x(t) ≤
K
M j β j Me
t1 t0
δ(s)ds
(x(t0 ) + xt0 l0 ).
j=1
Given that δ(t) > 0 and t1 ≤ T , there exists a large enough constant N such that t1
e t0 δ(s)ds < N holds for [t0 , T ]. Noting that x(tk ) = Dk x(tk− ), x(t0 ) = D0 x0 , x(t0 )+ xt0 l0 = x0 (D0 + l0 ), it follows that x(t) ≤
K
M j β j M N xt0 (l0 + D0 ).
(11)
j=1
For term x0 , we can obtain x0 ≤
c1 λmin (P1 )
(12)
from (9) and the given term Γ ≤ P1 (t0 ) in the definition. Using (10), (11), and (12), one can get x(t) ≤
K
M j β j M N xt0 (l0 + D0 )
j=1
≤
K
Mjβj MN
j=1
≤
K j=1
Mjβj MN
c1 (l0 + D0 ) λmin (P1 ) c1 (l0 + D0 ) ≤ λmin (Γ )
c2 , λmax (Π )
that is, x(t) ≤
c2 . λmax (Π )
(13)
In general, x T (t)Π x(t) ≤ λmax (Π )x(t)2 .
(14)
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Combining (13) and (14), we can conclude that x T (t)Π x(t) ≤ c2 . Therefore, the delay switched system (1) is FTS for the finite time interval [t0 , T ].
4 Model of a Nonlinear Delay Switched System with Delayed Impulses and Main Results Consider a nonlinear delay switched system with delayed impulses x(t) ˙ = Aσ (t) (t)x + f (t, x(t), x(t − r (t))), t = tk
x(tk ) = Dk x(tk− ) + E k x(tk− − r1 ) x(t0 ) = x0
(15)
where A(t) ∈ C[Jt0 , R n×n ], Dk , E k ∈ R n×n , 0 < r (t) ≤ r with r ∗ = max{r, r1 }, and 0 ≤ r, r1 ≤ tk − tk−1 ≤ sup{tk − tk−1 } = constant < ∞. Definition 3 Given five positive scalars c1 , c2 , T, r, and r1 , where c1 < c2 and r is a given delay, two matrices P1 , Γ > 0 and a positive definite matrix-valued function Π > 0 that satisfies Γ ≤ P1 , delay switched system (15) with linear delay impulses is FTS with respect to (c1 , c2 , T, Γ, Π, γ , β(t), r, r1 ) in the presence of impulse effects and nonlinear perturbation if x0T Γ x0 ≤ c1 ⇒ x T (t)Π x(t) < c2 , ∀ 0 < t ≤ T.
(16)
4.1 Main Results To study system (15), we make the following assumptions: – There exist functions αi (t) ∈ C[R, R] and ai (t) ∈ C[R, R+ ] such that λ(AiT (t)Pi + Pi Ai (t)) ≤ αi (t) and 2x T Pi f i (t, x, y) ≤ ai (t)x2 , where Pi is a positive definite and symmetric matrix. – Define qi (t) = λmin1(Pi ) (αi (t) + ai (t)). t – tk qi (s)ds ≥ 0, for t ∈ (tk−1 , tk ]. Theorem 3 If the conditions also satisfy
MN
K j=1
M j ( D0 + l0 )
c1 ≤ λmin (Γ )
c2 , k ∈ {0} N ∗, λmax (Π )
(17)
then the nonlinear delay switched system with delayed impulses is FTS for [t0 , t0 + T ] with respect to (c1 , c2 , T, Γ, Π, Υ, β(t), r, r1 ). Proof According to the proof of Theorem 1, we get x(t) ≤ M[x(tk ) + xtk lk ]e
t tk
δ(s)ds
, t ∈ [tk , tk+1 ], k ∈ {0}
N ∗.
Circuits Syst Signal Process
Furthermore, x(tk ) ≤ Dk x(tk− ) + E k x(tk− − r1 )
≤ Dk x(tk− ) + E k {M[x(tk−1 ) tk −r1 tk−1 δ(s)ds + xtk−1 lk−1 ]e
= Dk x(tk− ) + E k M[x(tk−1 ) + xtk−1 lk−1 ]e = [Dk + E k ]e + xtk−1 lk−1 ]e
tk −r1 tk
tk
tk−1
= [Dk + E k ]e
δ(s)ds
tk
tk−1
δ(s)ds
e
tk −r1 tk
δ(s)ds
M[x(tk−1 )
δ(s)ds
tk −r1 tk
δ(s)ds
x(t ˜ k)
where x(t ˜ k ) = M[x(tk−1 ) + xtk−1 lk−1 ]e
tk
tk−1
δ(s)ds
and xtk = suptk −r ≤t≤tk x(t) = max xtk , suptk −r ≤t≤tk x(t) tk −r1 δ(s)ds tk x(t ˜ k ), suptk −r ≤t≤tk x(t) ≤ max Dk + E k e tk −r1 ˜ k ). ≤ max Dk + E k e tk δ(s)ds , 1 x(t Therefore, t
δ(s)ds
xt ≤ M[x(tk ) + xtk lk ]e tk t tk −r1 δ(s)ds ≤ M Dk + E k e tk δ(s)ds + lk x(t ˜ k )e tk tk −r1 × max Dk + E k e tk δ(s)ds , 1 ˜ k )e = Mk x(t
t tk
δ(s)ds
, t ∈ [tk , tk+1 ], k ∈ {0}
N ∗.
Given that tk
δ(s)ds
x(t ˜ k ) = M[x(tk−1 ) + xtk−1 lk−1 ]e tk−1 tk−1 −r1 δ(s)ds x(t ˜ k−1 ) + lk−1 max {Dk−1 ≤ M Dk−1 + E k−1 e tk−1 tk−1 −r1 tk−1 δ(s)ds δ(s)ds , 1 × x(t ˜ k−1 )e tk−1 + E k−1 e tk−1
Circuits Syst Signal Process
tk−1 −r1 δ(s)ds = M Dk−1 + E k−1 e tk−1 + lk−1 max {Dk−1 tk−1 −r1 tk δ(s)ds δ(s)ds , 1 × x(t ˜ k−1 )e tk−1 + E k−1 e tk−1 = Mk−1 x(t ˜ k−1 )e
tk
tk−1
δ(s)ds
.
We can obtain x(t) = Mk x(t ˜ k )e ≤
k
t tk
δ(s)ds
M j x(t ˜ 1 )e
t t1
≤ Mk Mk−1 x(t ˜ k−1 )e
δ(s)ds
t tk−1
δ(s)ds
≤ M[x(t0 ) + xt0 l0 ]e
j=1
t t0
≤ ···
δ(s)ds
k
Mj.
j=1
interval, there always exists a large enough positive Because [t0 , t0 + T ] is a finite time t
t0 +T
δ(s)ds
≤ e t0 δ(s)ds ≤ N for all t ∈ [t0 , t0 + T ]. integer constant N such that e t0 From the system (15), it can be followed that x(t0 ) ≤ D0 x0 . Thus, x(t) ≤ M N
k
M j [ D0 +l0 ] x0 .
(18)
j=1
Combining (16), (17) with (18) yields x(t) ≤ M N
k
M j [ D0 +l0 ] x0
j=1
≤ MN
k
M j [ D0 +l0 ]
j=1
c1 ≤ λmin (Γ )
c2 , λmax (Π )
that is, x(t) ≤
c2 . λmax (Π )
(19)
In general, x T (t)Π x(t) ≤ λmax (Π )x(t)2 .
(20)
By combining (19) and (20), we can conclude that x T (t)Π x(t) ≤ c2 . Therefore, the nonlinear delay switched system (15) with delay impulse is FTS for the time interval [t0 , t0 + T ].
Circuits Syst Signal Process
Remark 2 Theorems 1, 2, and 3 reduce the finite-time stability conditions in the existing results which requires that all subsystems are asymptotically stable or at least a subsystem is stable. We can assume that λ(AiT (t)Pi + Pi Ai (t)) ≤ αi (t), where αi (t) have values less than a constant depending on the initial time t0 . This assumption coincides with the stability criterion of the time-varying systems. Thus, our results may be applied to all cases which reduce the conservatism.
5 Numerical Examples We present two examples to illustrate the effectiveness of the proposed results. 5.1 Example 1 Consider a nonlinear impulsive switched system: x(t) ˙ = Aσ (t) (t)x(t) + f σ (t) (x(t)), t = tk x(tk ) = Jk tk− , x(tk− ) x(t0 ) = x0
(21)
with ⎛
⎞ −10 5t 6.5 A1 (t) = ⎝ 2 −5.5 −1.25 ⎠ , −9 4 −8.5 f 1 (x) = (0, −x1 x3 , x1 x2 )T ,
⎛
⎞ −3 4 2.8 A2 (t) = ⎝ −2 −t 1 ⎠ , 5 −4 −4.8 T 1 1 f 2 (x) = −x2 x3 , x1 x3 , x1 x2 . 2 2
For a given finite interval [0, 5], impulse instants: tk = 0.2, 0.6, 1.1, x0 = (0.1, 0.2, 0.3)T , c1 = 31, and c2 = 32, we must analyze the finite-time stability of such a system with ⎛
⎞ ⎛ ⎞ 0.01 0 0 200 Π = ⎝ 0 0.02 0 ⎠ , Γ = ⎝ 0 5 0 ⎠ . 0 0 0.02 006 If Pi = I , then 2x T Pi f i (x) = 0 ≤ ai (t)x2 , i = 1, 2, ai (t) = 0; αi (t) ≥ λ(AiT (t)Pi + Pi Ai (t)) = t 3 + 48t 2 + 684.1875t + 2785, qi (t) = αi (t), Let β1 = 0.2, and β2 = 0.4. Then, the conditions of Theorem 1 are satisfied. The simulation results are shown in Fig. 1. 5.2 Example 2 Consider a nonlinear switched impulsive system as
Circuits Syst Signal Process 0.5 x1 x2 x3
0.4
0.3
x
0.2
0.1
0
−0.1
−0.2
0
1
2
3
4
5
t
Fig. 1 State response of Example 1
x(t) ˙ = Aσ (t) (t)x(t) + f σ (t) (t, x(t), x(t − r (t))), t = tk x(tk ) = Dk (tk− ) + E k x(tk− − r1 ) x(t0 ) = x0
(22)
with
A1 (t) =
1−|sint| 2
0.1t
−0.1t
1−|sint| 2
f1 = f2 =
,
A2 (t) =
1−|sint| 2
−0.1t
(sin(t)x1 (t) + x2 (t − 0.2)) (sin(t)x2 (t) + x1 (t − 0.2))
0.1t 1−|sint| 2
,
Dk = 0.1k I and E k = 0.3k I , here, r = 0.2, r1 = 0.1, x0 = [0.02 0.01]T , because αi (t) = 1− | sin(t) |, a1 (t) = 21 + | sin(t) |, a2 = 21 , then 1
tk+1
1
tk+1
s s+r (α(η)+a1 (η))dη
ds βk = e 2 tk q1 (s)ds = e 2 tk (a2 (s+r )+α(s)+a1 (s))e = 1.592. Given the finite interval [0, 5], impulse instants tk = 0.6, 1.1. Then, c1 = 17 and c2 = 32. We will analyze the finite-time stability of such a system with
Π=
30 0.03 0 . , Γ = 05 0 0.05
Moreover, qi0 (t) = qi (t) = αi (t). Let β1 = 0.1, β2 = 0.6. Then, the conditions of Theorem 3 are satisfied. The simulation results are shown in Fig. 2.
Circuits Syst Signal Process −3
20
x 10
x1 x2 15
x
10
5
0
−5
0
1
2
3
4
5
t
Fig. 2 State response of Example 2
6 Conclusions In this study, we explore the finite-time stability of delay switched systems with impulse and nonlinear perturbation. By employing the Lyapunov function and the inequalities technique, results are established to guarantee that the nonlinear delay impulsive switched system is FTS. Unlike existing results for related problems, the results obtained in this study can be applied to a large class of hybrid delayed systems, including stable and unstable subsystems. Acknowledgments The authors would like to thank the Editor and the reviewers for their valuable comments to improve the quality of the manuscript. This work is supported by NNSF of China under Grants 61273091, 61273123, 61304066, Shandong Provincial Scientific Research Reward Foundation for Excellent Young and Middle-aged Scientists of China under grant BS2011DX013, BS2012SF008, and Taishan Scholar Project of Shandong Province of China.
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