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Global mean square exponential stability and stabilization of uncertain switched delay systems with Lévy noise and flexible switching signals
´ noise and flexible switching Global mean square exponential stability and stabilization of uncertain switched delay systems with Levy signals
Journal Pre-proof
Global mean square exponential stability and stabilization of ´ noise and flexible uncertain switched delay systems with Levy switching signals Ling Liu, Xiangwu Ding, Wuneng Zhou, Xiaoli Li PII: DOI: Reference:
S0016-0032(19)30648-9 https://doi.org/10.1016/j.jfranklin.2018.12.037 FI 4143
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Journal of the Franklin Institute
Received date: Revised date: Accepted date:
8 May 2018 30 September 2018 3 December 2018
Please cite this article as: Ling Liu, Xiangwu Ding, Wuneng Zhou, Xiaoli Li, Global mean square exponential stability and stabilization of uncertain switched delay systems with ´ Levy noise and flexible switching signals, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2018.12.037
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Global mean square exponential stability and stabilization of uncertain switched delay systems with L´evy noise and flexible switching signals Ling Liua , Xiangwu Dingb,∗, Wuneng Zhoua,∗, Xiaoli Lia a College
of Information Sciences and Technology, Donghua University, Shanghai, 200051, China. of Computer Sciences and Technology, Donghua University, Shanghai, 200051, China.
b College
Abstract This paper focuses on the stability analysis and stabilization problem for a class of uncertain switched delay systems with L´evy noise and flexible switching signals which unify the high-frequency switching and low-frequency switching. By employing the theory of switched systems, mathematical induction and stochastic analysis technique, some sufficient conditions in form of algebraic inequalities are derived to guarantee the stability and stabilization of such systems. Different from dwell time and average dwell time, the proposed switching rule constrained the partial dwelltime shows that the switching number in the same time interval can be more elastic. Finally, numerical examples are implemented to illustrate the effectiveness of the theoretical results. Keywords: Switched system; Flexible switching rule; L´evy noise; Partial dwell time
1. Introduction Switched systems, which contain lots of subsystems demonstrated by continuous or discrete differential equations and a rule that orchestrates switching among these subsystems, have caught lots of attention from researchers owing to their enormous application in various areas [1]-[14]. In practice, many evolutionary processes are always accompanied by environment changed suddenly and external perturbations abruptly, which lead to the fact that one fixed model can’t simulate the actual phenomenon precisely. And the peculiarity of switched systems can reveal the real world effectively. It has been shown that the occurrence of infinite different subsystems and abundant switching rules such that the states of switched systems are very complex and interesting. Switched systems with some stable subsystems and a special switching signal will become unstable while these systems with some unstable subsystems and a smart rule will become stable. Therefore, it is of great significance to analyze the dynamical characters of switching rules combining with the features of subsystems to reach the goal we want. From the theoretical analysis point, there are two classical problems with switched systems: 1) the stability of switched systems with arbitrarily switching rules and 2) the stability of switched systems with constrained switching signals. The first problem means that how to deal with the subsystems to guarantee the stability of the resulting systems with all permissible switching rules. Unlike the first one, the second problem always needs to manage the switching signals to assure the stability of the resulting systems with some given subsystems. In this paper, we are bent on the second problem. It is noticed that lots of literature have researched the stability of switched systems with constrained switching rules [15]-[27], such as random switching [15]-[16], dwell-time switching [19], [23]-[25]. By establishing a relationship between the sampling period and the average dwell time, the stabilization problem of switched linear neutral systems has been studied in [23]. In [24], the authors proposed an event-triggered sampling mechanism to deal with the stability of switched systems. In view of the fact that the active subsystem and the candidate controller are not always simultaneous, [25] researched the asynchronously switched control problem for a class of switched systems. Additionally, [26] introduced L´evy noise to switched system and developed the stability of such system. Most of the ∗ Corresponding
authors Email address: [email protected] (Wuneng Zhou )
Preprint submitted to
October 3, 2019
existing results are supposed that the switching numbers in the same time interval are constrained by a common upper bound, that is, subsystems cannot jump from one to another frequently under these assumptions. In the real world, however, switching frequency will be high in some time intervals and low in some other time intervals due to the occurrence of the intermittently or irregularly switching. For instance, in power systems, sudden noise or unexpected environment change will give rise to the existence of instantaneously switching. Therefore, it is important to consider the flexible frequency switching signals into the switched systems. In other words, dropping the conditions about the bound of dwell-time or average dwell-time admits of no delay. Some innovative works are devoted to this point [28]-[31]. In [29], switching signals which beyond the average dwell time regime have been proposed. Moreover, switching signals developed in [30] allow the number of switches to grow faster than an affine function of the length of a time interval. [31] investigated the stability of discrete-time switched systems via a mode-dependent average dwell time approach. Whereas, the value of switching dwell time is needed to estimate by applying these criteria. Since the occurrence of irregularly switching, it is not easy to measure the value of switching dwell time. Thus, it is necessary to find more easy-to-use tools to analyze such issues. On the other hand, time-delay, caused by the finite switching speeds or slow processing of information transmission between different neurons in practical networks, is well known to be a usual phenomenon in nature, society, and technology. It is necessary to deal with the characteristics of dynamic systems with constant or time-vary delays, the relevant results can be found in many existing literatures [32]-[39]. Furthermore, systems taking into account L´evy noise which consists of Gaussian process and Poisson point process can illustrate the real developing process accurately such as bond yields and others [40]-[51]. Besides, the appearance of packet dropouts, time-delay and external perturbation in real networks may result in uncertainties [52]-[57]. Thus, it is natural and significant to study the problem of stability and stabilization of switched delay systems involving uncertainties. Up to now, there are few results about uncertain switched systems with L´evy noise, especially uncertain switched systems with flexible switching signals, which inspire us to study this topic. Moreover, the terms of stochastic disturbance make the analytical approach totally different, and seeking for an effective method to dispose of the irregularly switching is still an open problem. In this paper, we aim at studying the stability and stabilization of uncertain switched delay systems with L´evy noise in the presence of flexible frequency switching. Together with the theory of switched systems and some mathematical analysis techniques, some sufficient criteria ensuring the stability of such systems are developed. The results in this paper show that even there is some high-frequency switching in some time intervals, an uncertain switched system is mean square exponentially stable as long as the partial dwell time satisfies certain conditions, which relaxes the restriction on common bound of average dwell time. As a whole, the main contributions of this paper are listed as follows: 1) A class of switching signals which constrain partial dwell time for analyzing the dynamic behavior of uncertain switched delay systems is proposed. 2) Taking into account the L´evy noise and time-delay for the uncertain switched delay systems with flexible switching signals. 3) The stability and stabilization criteria in form of some inequalities are derived by applying the generalized Itˆo formula and mathematical induction. Moreover, the analytical approach utilized in this paper can be extended to more general switched delay systems. The rest of this paper is organized as follows. Section 2 gives some notations, definitions, assumptions, and lemmas. In Section 3, main results are stated by applying the theory of switched systems and some mathematical analysis techniques. Numerical examples are given to show the validity of the results in Section 4. Finally, concluding remarks are made in Section 5. 2. Preliminaries and model description Denote N and N + as the set of natural numbers and non-negative integers, respectively. Denote < and <+ as the set of real numbers and non-negative real numbers, respectively. Denote
the maximum or minimum eigenvalue of matrix A. A > 0 or A < 0 denotes that the matrix A is a symmetric and positive or negative definite matrix. Let Cτ = C([−τ, 0], Rn ), τ > 0, and for each φ ∈ Cτ , the norm is defined by 2
kφkτ = sup−τ≤s≤0 |φ(s)|. For h(t) : <+ → <, denote D+ h(t) = lim sup s→0+
h(t + s) − h(t) . s
Let (Ω, F , {F }0 , P) be a complete probability space with a filtration {Ft }t≥0 which is right continuous, and F0 contains all P-null sets. B(t) = (B1 (t), · · · , Bm (t))T is an m-dimensional Ft -adaptive Brownian motion defined on the complete probability space. N(t, $) means an Ft adapted Poisson random measure on [0, +∞) × < with compensator e $), and N(t, e $) satisfies N(dt, e martingale measure N(t, d$) = N(dt, d$) − π(d$)dt, π(d$)dt is a Possion point process. In this paper, we consider a class of uncertain switched delay systems with L´evy noise described by dx(t) = x(s) =
h i R τ2 −Aσ(t) (t)x(t) + Bσ(t) (t)x(t − τ1 ) + Cσ(t) (t) 0 x(t − s)ds + Uσ(t) (t) dt + %σ(t) (t, x(t), x(t − τ))dB(t) R + < ισ(t) (t, x(t), x(t − τ), $)N(dt, d$), t ≥ 0, φ(s), s ∈ [−τ, 0],
(1) where x(t) ∈ Rn is the state vector, σ : [0, +∞) → M = {1, 2, · · · , m} is a right-continuous, piecewise constant function called the switching rule, m is the number of subsystems. The switching time sequences {t p }, p ∈ N, satisfying 0 = t0 ≤ t1 ≤ · · · ≤ t p → +∞, σ(t) = z p , t ∈ [t p , t p+1 ), p ∈ N, represents that the subsystem z p is activated during [t p , t p+1 ). Denote G0 = {t0 , t1 , · · · , t p , · · · } as the set of switching sequences, p ∈ N. G ⊂ G0 means a class of switching sequences and S(G) denotes the set of all orderer pairs (z p , z p+1 ) which indicates that subsystem z p switches to subsystem z p+1 with the switching sequences G, z p , z p+1 ∈ M, p ∈ N. {(Ai (t), Bi (t), Ci (t)) : i ∈ M} is a collection of matrices corresponding to the individual subsystem of systemR (1), φ(s) ∈ Cτ , τ = max{τ1 , τ2 }, τ, τ1 , τ2 τ2 are positive constant delays, denote xτ = x(t − τ), xτ1 = x(t − τ1 ), xτ2 = 0 x(t − s)ds. Ai (t) = Ai + QiA Fi (t)ΓiA , C C C Bi (t) = Bi + QiB Fi (t)ΓiB , Ci (t) = Ci + Qi Fi (t)Γi , where Ai , Bi , Ci , QiA , QiB , Qi , ΓiA , ΓiB , ΓCi are real known matrices and Fi (t) is an unknown real matrix with FiT (t)Fi (t) ≤ In , i ∈ M. For any i ∈ M, %i (t, x(t), xτ ) : < ×
(2)
for any solution x(t) = x(t, φ) of system (1) with initial value φ(s) ∈ Cτ , s ∈ [−τ, 0], where α is called the convergence rate. Definition 2 System (1) is said to be globally mean square exponentially stabilized over the class S(G), G ⊆ G0 , if there exists an appropriate feedback control law such that the closed-loop switched system (1) is globally mean square exponentially stable. Definition 3 For any given constants θ > 0, a > 0, b > 0, d > 0, define function g(ξ, γ) : <+ × <+ → < as follows g(ξ, γ) = −γ + θ − ξ a exp(γτ1 ) + b exp(γτ2 ) + d exp(γτ) .
Remark 1 It’s obvious that g(ξ, γ) is a continuous function. For every fixed second argument, it’s monotonous nonincreasing. 3
Assumption 1 There exist appropriate dimension positive definition matrices Λ1z p and Λ2z p , z p ∈ M, p ∈ N, such that trace[%Tzp (t, x(t), xτ )%z p (t, x(t), xτ )] ≤ xT (t)Λ1z p x(t) + xτT Λ2z p xτ , t ∈ [t p , t p+1 ). Assumption 2 There exist appropriate dimension positive definition matrices Ξ1z p and Ξ2z p , z p ∈ M, p ∈ N, such that ιTzp (t, x(t), xτ )ιz p (t, x(t), xτ ) ≤ xT (t)Ξ1z p x(t) + xτT Ξ2z p xτ , t ∈ [t p , t p+1 ). And the characteristic measure π(d$)dt satisfies π(d$)dt = λ0 µ(d$)dt, ˆ where λ0 is the intensity of Poisson distribution and µˆ is the probability distribution of random variable $. Assumption 3 For each pair (i, j) ∈ S(G), G ⊆ G0 , there exist µi, j ≥ 1 such that : (3)
Pi ≤ µi, j P j , i, j ∈ M, and µ = max {µi, j }, G ⊆ G0 . For given constant N ∈ N + , assume that (i, j)∈S(G)
µ(N) = max+ {µ(n, N)} < ∞,
(4)
n∈N
κ−1 Y
µ(n, κ) =
j=0
where µ0 = µz0 ,z−1 = 1, κ = 1, 2, · · · , N.
µzκ(n−1)+ j , zκ(n−1)+ j−1 , n ∈ N + ,
(5)
Remark 2 Under Assumptions 1-3 which are standard for analyzing the stability of systems with random noise [40][44], system (1) admits trivial solution. Assumption 3 is useful for dealing with switched systems [27]-[30], and (4), (5) are prepared for the introduction of partial dwell-time method in the main results. µ0 = 1 means system (1) will be not switching during [−τ, t1 ). Now we are going to present some lemmas which will be helpful for the results. Lemma 1 ([36]) Let u, v ∈
uT v + vT u ≤ ηuT u + η−1 vT v,
for any η > 0.
Lemma 2 ([13]) If S ∈ Rn×n is a symmetric positive definite matrix and Q ∈ Rn×n is a symmetric matrix, then λmin (S −1 Q)xT S x ≤ xT Qx ≤ λmax (S −1 Q)xT S x, x ∈ Rn . Lemma 3 ([33]) Given any appropriate dimension real matrix P > 0 and a vector function h(·) : [s1 , s2 ] →
s2 s1
h(s)ds
!T
P
Z
s2 s1
! Z h(s)ds ≤ (s2 − s1 )
s2
hT (s)Ph(s)ds.
s1
3. Main results In this section, we are going to deduce criteria of global mean square exponential stability and construct linear feedback controllers for system (1) with L´evy noise and flexible switching signals. 4
For system (1), the generalized Itˆo formula of function V(t, x(t), σ(t)) ∈ C2,1 (<+ ×
where the operator LV(t, x(t), xτ , σ(t)) : <+ ×
=
1 V x (t, x(t), σ(t)) fσ(t) (t, x(t), xτ ) + trace[%Tσ(t) (t, x(t), xτ )V xx (t, x(t), σ(t))%σ(t) (t, x(t), xτ )] 2 R + < [V(t, x(t) + ισ (t, x(t), xτ ), σ(t)) − V(t, x(t), σ(t))]π(d$) + Vt (t, x(t), σ(t)),
(7)
fσ(t) (t, x(t), xτ ) = −Aσ(t) (t)x(t) + Bσ(t) (t)xτ1 + Cσ(t) (t)xτ2 , Vt (t, x(t), σ(t)) =
∂V(t, x(t), σ(t)) , ∂t
∂V(t, x(t), σ(t)) ∂V(t, x(t), σ(t)) ∂V(t, x(t), σ(t)) V x (t, x(t), σ(t)) = , ,··· , ∂x1 ∂x2 ∂xn ! ∂2 V(t, x(t), σ(t)) . V xx (t, x(t), σ(t)) = ∂xk ∂xl n×n
!
Moreover, for any p ∈ N and t ∈ [t p , t p+1 ), we denote σ(t) by z p , z p ∈ M. Let ∆h > 0 be small enough such that t + ∆h ∈ [t p , t p+1 ). Then, we can obtain EV(t + ∆h, x(t + ∆h), z p ) − EV(t, x(t), z p ) =
Z
t
t+∆h
ELV(s, x(s), z p )ds.
Since ELV(t, x(t), σ(t)) is continuous in the interval t ∈ [t p , t p+1 ), it follows that D+ EV(t, x(t), z p ) = ELV(t, x(t), z p ), t ∈ [t p , t p+1 ), p ∈ N.
(8)
3.1. Global mean square exponential stability In this subsection, we are going to analyze the global mean square exponential stability of system (1). Firstly, a preliminary theorem which plays a key role in the proof of the main results is given. Theorem 1 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. For any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0 and positive constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0 such the following inequalities hold: Pi ≤ ci In , (9) 1 C 3 4 A A T B (Γi ) Pi Q i Pi Qi Ωi + Ωi + Ωi + θi Pi Pi Qi A T −1 (Q ) P −η I 0 0 0 i i 3 n < 0, A Γi 0 −η3 In 0 0 (10) B T −1 (Q ) P 0 0 −η I 0 i i 4 n (QCi )T Pi 0 0 0 −η−1 5 In θ − (a + b + d) > 0,
(11)
where θ = min{θi , i ∈ M}, Ω1i = −Pi Ai − ATi Pi , Ω3i = ci Λ1i , Ω4i = λ0 [(ci + 1)Ξ1i + Pi Pi ], a = max{ai , i ∈ M}, ai = −1 −1 −1 B T B −1 2 −1 −1 −1 C T C µ[η−1 1 λmax ((Pi ) )+η4 λmax ((Pi ) (Γi ) Γi )], b = max{bi , i ∈ M}, bi = µ[η2 (τ2 ) λmax ((Pi ) )+η5 λmax ((Pi ) (Γi ) Γi )], −1 2 −1 2 d = max{di , i ∈ M}, di = µ[ci λmax ((Pi ) Λi ) + (ci + 1)λ0 λmax ((Pi ) Ξi )], i ∈ M. 5
Consider a function V(x, σ(t)) =
(
xT (t)Pσ(t) x(t), t ∈ [−τ, 0], exp(γt)xT (t)Pσ(t) x(t), t ≥ 0,
(12)
where γ ∈ (0, θ) is a solution of g(1, γ) > 0. If there exists tˆ ∈ [t p , t p+1 ) for some p ∈ N satisfying V(tˆ, σ(tˆ)) , 0, V(t, σ(t)) ≤ V(tˆ, σ(tˆ)), ∀t ∈ [tˆ − τ, tˆ], then D+ EV(t, σ(t))|t=tˆ < 0. (13) Proof. For convenience, we denote σ(t) by z p , z p ∈ M, t ∈ [t p , t p+1 ), p ∈ N and then V(t, z p ) = exp(γt)xT (t)Pz p x(t). Based on (7) and system (1), we can obtain
LV(t, σ(t))
= 2xT (t)Pz p [−Az p (t)x(t) + Bz p (t)xτ1 + Cz p (t)xτ2 ] exp(γt) + γ exp(γt)xT Pz p x(t) +trace{%Tzp (t, x(t), xτ )Pz p %z p (t, x(t), xτ )} exp(γt) o R n + < [x(t) + ιz p (t, x, xτ , $)]T Pz p [x(t) + ιz p (t, x, xτ , $)] − xT (t)Pz p x(t) exp(γt)π(d$) ≤ exp(γt) xT (t)Ω1z p x(t) + 2xT (t)Pz p Bz p xτ1 + 2xT (t)Pz p Cz p xτ2 ] + γV(t, z p ) + exp(γt) − 2xT (t)Pz p QzAp Fz p (t)ΓzAp x(t) + 2xT (t)Pz p QzBp Fz p (t)ΓzBp xτ1 + 2xT (t)Pz p QCzp Fz p (t)ΓCi xτ2 ] +cz p trace{%Tzp (t, x(t), xτ )%z p (t, x(t), xτ )} exp(γt) o R n + < (cz p + 1)ιTzp (t, x, xτ , $)ιz p (t, x, xτ , $) + xT (t)Pz p Pz p x(t) exp(γt)π(d$),
Ω1z p
(14)
ATzp Pz p .
where = −Pz p Az p − From Lemmas 1- 3, it yields that 2xT (t)Pz p Bz p xτ1 + 2xT (t)Pz p Cz p xτ2 ≤ ≤ ≤
T T T η1 xT (t)Pz p Bz p (Bz p )T Pz p x(t) + η−1 1 xτ1 xτ1 + η2 x (t)Pz p C z p (C z p ) Pz p x(t) R τ2 T R τ2 +η−1 x(t − s)ds x(t − s)ds 2 0 0 R τ2 T R τ2 −1 −1 xT (t)Ω21 Pz p 0 x(t − s)ds z p x(t) + η2 λmax ((Pz p ) ) 0 x(t − s)ds
−1 T +η−1 1 λmax ((Pz p ) )xτ1 Pz p xτ1
−1 −1 xT (t)Ω21 z p x(t) + η2 λmax ((Pz p ) )τ2 −1 T +η−1 1 λmax ((Pz p ) )xτ1 Pz p xτ1 ,
R τ2 0
(15)
xT (t − s)Pz p x(t − s)ds
−2xT (t)Pz p QzAp Fz p (t)ΓzAp x(t) + 2xT (t)Pz p QzBp Fz p (t)ΓzBp xτ1 + 2xT (t)Pz p QCzp Fz p (t)ΓCi xτ2 ≤
T A T A T B B T η3 xT (t)Pz p QzAp (QzAp )T Pz p x(t) + η−1 3 x (t)(Γz p ) Γz p x(t) + η4 x (t)Pz p Qz p (Qz p ) Pz p x(t)
≤
−1 −1 B T B T xT (t)Ω22 z p x(t) + η4 λmax ((Pz p ) (Γz p ) Γz p )xτ1 Pz p xτ1 R τ2 −1 C T C T +η−1 5 λmax ((Pz p ) (Γz p ) Γz p )τ2 0 x (t − s)Pz p x(t − s)ds.
T B T B T C C T −1 T C T C +η−1 4 xτ1 (Γz p ) Γz p xτ1 + η5 x (t)Pz p Qz p (Qz p ) Pz p x(t) + η5 xτ2 (Γz p ) Γz p xτ2
(16)
T T 22 A A T −1 A T A B B T where Ω21 z p = η1 Pz p Bz p (Bz p ) Pz p + η2 Pz p C z p (C z p ) Pz p , Ωz p = η3 Pz p Qz p (Qz p ) Pz p + η3 (Γz p ) Γz p + η4 Pz p Qz p (Qz p ) Pz p +
η5 Pz p QCzp (QCzp )T Pz p , and ηi , i = 1, · · · , 5, are positive constants. 6
From Assumptions 1-2, we have cz p trace{%Tzp (t, x(t), xτ )%z p (t, x(t), xτ )} exp(γt) ≤ ≤
≤ ≤ ≤
cz p [xT (t)Λ1z p x(t) + xτT Λ2z p xτT ] exp(γt)
(17)
xT (t)Ω3z p x(t) exp(γt) + cz p λmax ((Pz p )−1 Λ2z p )xτT Pz p xτT exp(γt)
o R n (cz p + 1)ιTzp (t, x, xτ , $)ιz p (t, x, xτ , $) + xT (t)Pz p Pz p x(t) exp(γt)π(d$) < n o R (cz p + 1)[xT (t)Ξ1z p x(t) + xτT Ξ2z p xτ ] + xT (t)Pz p Pz p x(t) exp(γt) < λ0 ν(d$) xT (t)Ω4z p x(t) exp(γt) + (cz p + 1)λ0 xτT Ξ2z p xτ exp(γt)
(18)
xT (t)Ω4z p x(t) exp(γt) + (cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p )xτT Pz p xτ exp(γt),
where Ω3z p = cz p Λ1z p , Ω4z p = λ0 [(cz p + 1)Ξ1z p + Pz p Pz p ]. Then, taking (15)-(18) to (14), we get LV(t, σ(t))|t=tˆ ≤
exp(γtˆ)xT (tˆ) Ω1z p + Ω2z p + Ω3z p + Ω4z p x(tˆ) + γV(tˆ, z p ) −1 −1 −1 B T B +[η−1 1 λmax ((Pz p ) ) + η4 λmax ((Pz p ) (Γz p ) Γz p )] ×xT (tˆ − τ1 )Pz p x(tˆ − τ1 ) exp(γ(tˆ − τ1 + τ1 )) −1 C T C +[η−1 λ ((P )−1 ) + η−1 5 λmax ((Pz p ) (Γz p ) Γz p )]τ2 R τ22 max z p × 0 xT (tˆ − s)Pz p x(tˆ − s) exp(γ(tˆ − s + s))ds +cz p λmax ((Pz p )−1 Λ2z p )xT (tˆ − τ)Pz p x(tˆ − τ) exp(γ(tˆ − τ + τ)) +(cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p )xT (tˆ − τ)Pz p x(tˆ − τ) exp(γ(tˆ − τ + τ)).
(19)
22 where Ω2z p = Ω21 z p + Ωz p . In light of Assumption 3, the following inequalities can be obtained
xT (tˆ − τ)Pz p x(tˆ − τ)
≤ µz p ,σ(tˆ−τ) xT (tˆ − τ)Pσ(tˆ−τ) x(tˆ − τ) ≤ µxT (tˆ − τ)Pσ(tˆ−τ) x(tˆ − τ),
(20)
xT (tˆ − τ1 )Pz p x(tˆ − τ1 ) ≤ µxT (tˆ − τ1 )Pσ(tˆ−τ1 ) x(tˆ − τ1 ),
(21)
xT (tˆ − τ2 )Pz p x(tˆ − τ2 ) ≤ µxT (tˆ − τ2 )Pσ(tˆ−τ2 ) x(tˆ − τ2 ),
(22)
where constants µz p ,σ(tˆ−τ) and µ are defined in Assumption 3. From (20)-(22), one gets that
7
LV(t, σ(t))|t=tˆ ≤
≤ ≤
xT (tˆ)Ωz p x(tˆ) exp(γtˆ) + γV(tˆ, z p ) h i −1 −1 −1 B T B +µ η−1 1 λmax ((Pz p ) ) + η4 λmax ((Pz p ) (Γz p ) Γz p ) exp(γτ1 )
×xT (tˆ − τ1 )Pσ(tˆ−τ1 ) x(tˆ − τ1 ) exp(γ(tˆ − τ1 )) h i −1 −1 −1 C T C +µ η−1 2 λmax ((Pz p ) ) + η5 λmax ((Pz p ) (Γz p ) Γz p ) τ2 exp(γτ2 ) R τ2 × 0 xT (tˆ − s)Pσ(tˆ)−s x(tˆ − s) exp(γ(tˆ − s))ds h i +µ cz p λmax ((Pz p )−1 Λ2z p ) + (cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p ) exp(γτ)xT (tˆ − τ)Pz p x(tˆ − τ)
(23)
−θz p exp(γtˆ)xT (tˆ)Pz p x(tˆ) + γV(tˆ, z p ) + az p exp(γτ1 )V(tˆ − τ, σ(tˆ − τ))
+bz p exp(γτ2 )τ2 V(tˆ − τ2 , σ(tˆ − τ2 )) + dz p exp(γτ)τ2 V(tˆ − τ, σ(tˆ − τ)) h i − −γ + θz p − az p exp(γτ1 ) − bz p exp(γτ2 ) − dz p exp(γτ) V(tˆ, z p ),
where Ωz p = Ω1z p + Ω2z p + Ω3z p + Ω4z p , and −1 −1 −1 B T B az p = µ[η−1 1 λmax ((Pz p ) ) + η4 λmax ((Pz p ) (Γz p ) Γz p )], 2 −1 −1 −1 C T C bz p = µ[η−1 2 (τ2 ) λmax ((Pz p ) ) + η5 λmax ((Pz p ) (Γz p ) Γz p )],
From (11), we have which implies
h i dz p = µ cz p λmax ((Pz p )−1 Λ2z p ) + (cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p ) . LV(t, σ(t))|t=tˆ ≤ −g(1, γ)V(tˆ, z p ) < 0, D+ EV(t, σ(t))|t=tˆ < 0.
This completes the proof of Theorem 1. Remark 3 From (11), it is noticed that g(1, 0) > 0 and g(1, θ) < 0. Thus, it is possible that g(1, γ) > 0 holds for γ ∈ (0, θ), which can be computed by Matlab. In theoretical, g(1, γ) > 0 shows the range of time delay which may be not too large. Due to the fact the dynamics of systems have been influenced by both of time delay and flexible switching signals, it is difficult to measure the relationship between the affection of time delay and the switching signals. Remark 4 Different from [9]-[12] which focus on the dynamic behavior of switched systems in the presence of arbitrary switching signals, the objective of this paper is to analyse the stability and stabilization of uncertain switched systems under constrained switching signals. It aims to establish a bridge between the flexible switching signals and subsystems to keep the system stable. Whereas, as the result of the existence of high-frequency and low-frequency switching signals, the restriction on the dynamic behavior of subsystems is still stronger. In term of Theorem 1, some corollaries can be obtained as follows. Corollary 1 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. For any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, and constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (9), (10) and (11) hold. Given function V(t, σ(t)) which defined in (12) with γ ∈ (0, θ), θ = min{θi }, ensuring that i∈M
g(ξ, γ) > 0. For some p ∈ N, if there exist ς1 , ς2 ∈ [t p , t p+1 ), ς1 < ς2 satisfying V(ς, ˆ σ(ς)) ˆ , 0 for some ςˆ ∈ [ς1 , ς2 ],
8
V(s, σ(s)) ≤ ξV(ς, ˆ σ(ς)), ˆ ∀s ∈ [ςˆ − τ, ς], ˆ then D+ EV(t, σ(t))|t=ςˆ < 0.
(24)
Proof. Based on Theorem 1, for ςˆ ∈ [ς1 , ς2 ], it can be derived that n h io D+ EV(t, σ(t)) ≤ − −γ + θz p − ξ az p exp(γτ1 ) + bz p exp(γτ2 ) + dz p exp(γτ) EV(t, σ(t)).
(25)
If for some ςˆ ∈ [ς1 , ς2 ], V(s, σ(s)) ≤ ξV(ς, ˆ σ(ς)), ˆ ∀s ∈ [ςˆ − τ, ς], ˆ from Theorem 1, it yields that D+ EV(t, σ(t))|t=ςˆ ≤ −g(ξ, γ)V(ς, ˆ σ(ς)) ˆ < 0. This completes the proof of Corollary 1.
Corollary 2 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. For any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, and constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (9), (10) and (11) hold. Given function V(t, σ(t)) which defined in (12) with γ ∈ (0, θ), θ = min{θi }, ensuring that i∈M
g(ξ, γ) > 0. For some p ∈ N, if there exist ς1 , ς2 ∈ [t p , t p+1 ), ς1 < ς2 satisfying V(ς, σ(ς)) , 0 for any ς ∈ [ς1 , ς2 ], V(s, σ(s)) ≤ ξV(ς, σ(ς)), ∀s ∈ [ς − τ, ς], then EV(ς2 , σ(ς2 )) ≤ exp(−g(ξ, γ, z p )(ς2 , ς1 )). EV(ς1 , σ(ς1 ))
(26)
Proof. Based on Theorem 1 and (25), we can get D+ EV(t, σ(t)) ≤ −g(ξ, γ)EV(t, σ(t)) < 0, ∀ς ∈ [ς1 , ς2 ], which implies that (26) holds. This completes the proof of Corollary 2. Then, aided by Theorem 1 and Corollaries 1-2, the stability of switched system (1) is formulated in the next theorem, which mainly concentrates on partial dwell time. Theorem 2 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. Given N ∈ N + , for any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, and constants µi, j ≥ 1, θi ≥ 0, θ ≥ 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (9)-(11) hold. Moreover, G denotes the class of switching sequences tk satisfying inf {tNn − tNn−1 } > τ +
n∈Z+
lnµ(N) . g(µ(N), 0)
(27)
Then, system (1) is globally mean square exponentially stable over the class G.
Proof. Let x(t) = x(t, φ) be a solution of system (1) through (0, φ). Without loss of generality, assume that φ , 0. Choose proper constant γ0 > 0 such that g(µ(N), γ0 ) > 0 and inf {tNn − tNn−1 } > τ +
n∈Z+
lnµ(N) . g(µ(N), γ0 )
(28)
Define the following functions T t ∈ [−τ, 0], x (t)Pσ(t) x(t), V(x, σ(t)) = exp(γt)xT (t)Pσ(t) x(t), t ≥ 0, 9
(29)
κi (t) =
H(t, n, N) =
Nn−1 X
(
1, 0,
t ∈ [ti , ti+1 ), others,
κi (t)V(t, σ(t)),
t ∈ [tN(n−1) , tNn ).
i=N(n−1)
(30)
Next, we aim to prove the following inequality EH(t, n, N) ≤ ωµ(n, N),
t ∈ [tN(n−1) , tNn ),
n ∈ N+ ,
(31)
where ω = maxi∈M {λmax (Pi )}kφk2τ > 0. First we show that (31) holds for n = 1, namely EH(t, n, N) ≤ ωµ(1, N) = ω
N−1 Y
µz j z j−1 ,
j=0
t ∈ [0, tN ).
(32)
Since EV(t, σ(t)) ≤ ω, t ∈ [−τ, 0], we can obtain that EH(t, 1, 1) = EV(t, σ(t)) ≤ ω = ωµ0 = ωµ(1, 1), t ∈ [0, t1 ). If it is not true, then there exist some t ∈ (0, t1 ) such that EV(t, σ(t)) ≥ ω. Set t` = inf{t ∈ (0, t1 ) : EV(t, σ(t)) ≥ ω}. Thus, it follows that EV(t`, σ(t`)) = ω, EV(s, σ(s)) ≤ EV(t`, σ(t`)), D+ EV(t`, σ(t`)) ≥ 0,
s ∈ [t` − τ, t`],
(33)
in view of Theorem 1, (33) is contradict to Theorem 1. Then, (31) holds for n = 1, N = 1. Now, for n = 1, we assume that, for some integer p ≤ N − 1, EH(t, 1, p) ≤ ωµ(1, p), t ∈ [0, t p ). Then, the inequality EH(t, 1, p + 1) ≤ ωµ(1, p + 1) holds, t ∈ [0, t p+1 ). In fact, µ(1, p) ≤ µ(1, p + 1), t ∈ [0, tl+1 ). Therefore, it aims to show EV(t, z p ) ≤ ρµ(1, p + 1), t ∈ [t p , t p+1 ). (34) From Assumption 3, for any i, j ∈ M,
Pi ≤ µi, j P j ,
which indicates that V(tn , zn ) =
exp(γtn )xT (tn )Pzn x(tn )
=
exp(γtn− )xT (tn )Pzn x(tn− )
≤
µzn ,zn−1 exp(γtn )xT (tn )Pzn−1 x(tn )
=
µzn ,zn−1 V(tn− , zn−1 ),
(35)
for n ∈ N + . Particularly, V(t p , z p ) ≤ µz p ,z p−1 V(z−p , z p−1 ) ≤ ωµ(1, p + 1). Thus, (34) holds for t = t p . If (34) is not true for t ∈ (t p , t p+1 ), there exist some t ∈ (t p , t p+1 ) which are against (34). Let t¯ = inf{t > t p , V(t, z p ) ≥ ωµ(1, p + 1)}, then by virtue of the fact that EV(s, σ(s)) ≤ ωµ(1, p) ≤ ωµ(1, p + 1), s ∈ [t p − τ, t p ), it yields that
10
EV(s, σ(s)) ≤ EV(t¯, σ(t¯)), EV(t¯, σ(t¯)) = ωµ(1, p + 1), D+ EV(t¯, σ(t¯)) ≥ 0,
s ∈ [t¯ − τ, t¯],
which leads to a contradiction with Theorem 1. Thus, it follows that EV(t, z p ) ≤ ωµ(1, p + 1), t ∈ [t p , t p+1 ), which implies that EH(t, 1, N) ≤ ωµ(1, N), t ∈ [0, tN ). Hence, (31) holds for n = 1. Next, we show that if (31) holds for n = l, l ∈ Z+ , namely EH(t, l, N) ≤ ωµ(l, N) = ω
N−1 Y
µzN(l−1)+ j ,zN(l−1)+ j−1 ,
(36)
j=0
for t ∈ [tN(l−1) , tNl ), then (31) holds for n = l + 1, that is, EH(t, l + 1, N) ≤ ωµ(l + 1, N) = ω
N−1 Y
µzNl+ j ,zNl+ j−1 ,
(37)
j=0
for t ∈ [tNl , tN(l+1) ). First, we show a fact that there exists some tˇ ∈ [tNl−1 , tNl − τ] such that EV(tˇ, zNl−1 ) < ω. Or else, it holds that ωµ(l, N) ≥ EV(s, σ(s)) ≥ ω, s ∈ [tNl−1 , tNl − τ], which yields EV(t, σ(t)) ≤ ωµ(l, N) ≤ µ(l, N)EV(s, σ(s)), t ∈ [s − τ, s],
(38)
for s ∈ [tNl−1 , tNl − τ]. Choose ξ = µ(l, N), it follows from Corollary 2 and (28) that 1 µ(l, N)
≤
ω ωµ(l, N) EV(tNl − τ) EV(tNl−1 , σ(tNl−1 ))
≤
exp(−g(µ(l, N), γ0 )(tNl − τ − tNl−1 ))
≤
exp(−g(µ(N), γ0 )(tNl − τ − tNl−1 )) 1 , µ(N)
=
<
(39)
which is a contradiction. Next, we proof that EV(tNl − τ, zNl−1 ) < ω. If it is not true, there exists t´ ∈ (tˇ, tNl − τ] satisfies V(t´, σ(t´)) = ω ≥ EV(s, σ(s)), s ∈ [tˇ, t´] and D+ EV(t´, σ(t´)) ≥ 0. Then, EV(t, σ(t)) ≤ ωµ(l, N) = EV(t´, σ(t´))µ(l, N), t ∈ [t´ − τ, t´]. Choose ξ = µ(l, N) and it is a contradiction from Corollary 1. Finally, we have EV(t, σ(t)) < ω, t ∈ [tNl − τ, tNl ).
(40)
Suppose it is not true, one may get some t ∈ [tNl − τ, tNl ) such that V(t, σ(t)) ≥ ω. Since V(tNl − τ, zNl−1 ) < ω, let t˘ = inf{tNl − τ ≤ t < tNl , V(t, σ(t)) ≥ ω}, then, it yields that V(t˘, zNl−1 ) = ω ≥ V(s, zNl−1 ), s ∈ [tNl − τ, t˘], and D+ V(t˘, zNl−1 ) ≥ 0. On the other hand, V(t˘, zNl−1 )µ(l, N) = ωµ(l, N) ≥ V(t, σ(t)), t ∈ [t˘ − τ, t˘]. Therefore, (40) is right from Corollary 1. − Additionally, from Theorem 1 and (35), it is obvious that V(tNl , zNl ) ≤ µzNl ,zNl−1 V(tNl , zNl−1 ) ≤ µzNl ,zNl−1 ω, which 11
results in V(t, σ(t)) ≤ µNl,Nl−1 ω, t ∈ [tNl , tNl+1 ). Thus, employing the similar discussing, for any Nl ≤ k ≤ Nl + N − 1, one may get EV(t, σ(t)) ≤ ω that is EH(t, n, N) ≤ ω
k−Nl Y
µzNl+ j , zN+ j−1 , t ∈ [tk , tk+1 ),
(41)
µzNl+ j , zN+ j−1 = ωµ(l + 1, N),
(42)
j=0
N−1 Y j=0
for t ∈ [tk , tk+1 ). Hence, (31) holds for n = l + 1. And we can conclude that (31) holds for all n ∈ Z+ , which implies H(t, n, N) ≤ ωµ(N), t ≥ 0, namely Ekx(t)k ≤ ∆kφk2τ exp(−γ0 t), ∀t ≥ 0, ∀{t p } ∈ F , (43)
λmax (Pi , i ∈ M) . Therefore, system (1) under the switching signal (27) is globally mean square λmin (Pi , i ∈ M) exponentially stable. This completes the proof of Theorem 2.
where ∆ = µ(N)
Remark 5 Theorem 2 aims to design a class of switching signals which constrained the partial dwell time such that switched delay systems will be stable while the other dwell time can be chosen arbitrarily. Different from dwell time or average dwell time [23]-[25], it is noted that there exist high-frequency switching in some time intervals and switching numbers in some time domains will be flexible. Remark 6 The model proposed in this paper is just similar to [26]. In this paper we have added distributed delays and flexible switching signals into the model of [26]. Different from [26] which based on average dwell time, the analytical approach presented in Theorem 2 is novel and interesting, which can be extended to more general switched delay systems with flexible switching signals. Remark 7 There are many literatures about stability of systems with L´evy noise [40]-[45]. It has been realized that the generalized Itˆo formula and stochastic version of Lyapunov approach are powerful techniques to study them. Asking for exponential decay results in pth moment exponential stability which has highly-value in many areas, and this paper has investigated the global exponential stability in mean square of systems in case of p = 2. The results can be generalized to the pth moment exponential stability of uncertain switched systems with L´evy noise. On the other hand, lots of researches devoted themselves to the estimates of integrals of stochastic part for considering the influence of integral terms on the solution. In the future, we are going to study the state estimate and pth moment exponential stability of stochastic switched systems with flexible switching signals. 3.2. Global mean square exponential stabilization In this subsection, we will make switched system with L´evy noise and flexible switching signals achieve exponential stabilization under linear control. In the following, we use the linear feedback scheme to realize stabilization, i.e., the controller Uz p (t), z p ∈ M, t ∈ [t p , t p+1 ), p ∈ N can be presented by n o Uz p (t) = Kz p x(t) = diag kz1p , · · · , kznp x(t), (48) where kzl p , l = 1, 2, · · · , n, means the control gain. System (1) can be rewritten as follows dx(t) = −(A σ(t) (t) − Kσ(t) )x(t) + Bσ(t) (t)xτ1 + C σ(t) (t)xτ2 dt + %σ(t) (t, x(t), x(t − τ))dB(t) R + < ισ(t) (t, x(t), x(t − τ), $)N(dt, d$), t ≥ 0. x(s) = φ(s), s ∈ [−τ, 0]. 12
(49)
Similar to Theorem 1 and Theorem 2, we have the following stabilization criteria. Theorem 3 Assume that x(t) = x(t, φ) is a solution of system (49) through (0, φ) and Assumptions 1-3 hold. If for any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0 and positive constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0 such the following inequalities hold Pi ≤ ci In , (50) 3 4 5 Ωi + Ωi + Ωi + θi Pi A T (Q ) P i i ΓiA (QiB )T Pi (QCi )T Pi
Pi QiA −η−1 3 In 0 0 0
(ΓiA )T 0 −η3 In 0 0
Pi QiB 0 0 −η−1 4 In 0
θ − (a + b + d) > 0,
Pi QCi 0 0 0 −η−1 5 In
< 0,
(51)
(52)
where θ = min{θi , i ∈ M}, Ω3i = ci Λ1i , Ω4i = λ0 [(ci + 1)Ξ1i + Pi Pi ], Ω5i = −Pi [Ai − Ki ] − [Ai − Ki ]T Pi , a = −1 −1 −1 B T B −1 2 −1 max{ai , i ∈ M}, ai = µ[η−1 1 λmax ((Pi ) ) + η4 λmax ((Pi ) (Γi ) Γi )], b = max{bi , i ∈ M}, bi = µ[η2 (τ2 ) λmax ((Pi ) ) + −1 −1 C T C −1 2 −1 2 η5 λmax ((Pi ) (Γi ) Γi )], d = max{di , i ∈ M}, di = µ[ci λmax ((Pi ) Λi ) + (ci + 1)λ0 λmax ((Pi ) Ξi )], i ∈ M. Consider a function V(x, σ(t)) defined as (12). If there exists tˆ ∈ [t p , t p+1 ) for some p ∈ N satisfying V(tˆ, σ(tˆ)) , 0, V(t, σ(t)) ≤ V(tˆ, σ(tˆ)), ∀t ∈ [tˆ − τ, tˆ], then D+ EV(t, σ(t))|t=tˆ < 0. Theorem 4 Assume that x(t) = x(t, φ) is a solution of system (49) through (0, φ) and Assumptions 1-4 hold. Given N ∈ N + , if for any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, Ki , and constants µi, j ≥ 1, θi ≥ 0, θ ≥ 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (50)-(52) hold. Moreover, G denotes the class of switching sequences tk satisfying lnµ(N) . (53) inf {tNn − tNn−1 } > τ + n∈Z+ g(µ(N), 0) Then, system (1) under controller (49) is globally mean square exponentially stabilized over the class G.
The proof of Theorem 3 and Theorem 4 can be completed as those of Theorems 1-2, the proof is omitted here.
4. Illustrative examples In this section, we give two numerical examples and their simulations to show the effectiveness of the developed results. Example 1 Consider a two-dimensional time delay switched system with two subsystems # " R τ2 x(t − s) x(t − τ1 ) dx(t) = −Ai (t)x(t) + Bi (t) + Ci (t) 0 ds dt + %(t, x(t), x(t − τ))dB(t) 5 4 R + < ι(t, x(t), x(t − τ), $)N(dt, d$), where τ1 = τ2 = τ = 0.2, i ∈ {1, 2} and we take ! ! 5 0 2 −1 2 A1 = , B1 = , C1 = 0 5 −2 1 −2 5 A2 = 0
! 0 1 , B2 = 6 1
! −2 1 , C2 = 1 −1
! ! 1 0.1 0 A B C , i = 1, 2, , Qi = Qi = Qi = −1 0 0.1 ! ! 0.1 0 −2 A B C , i = 1, 2, , Γi = Γi = Γi = −1 0 0.1
%(t, x(t), x(t − τ)) = 13
x(t) + x(t − τ) , 10
(54)
ι(t, x(t), x(t − τ), $) =
[x(t) + x(t − τ)]$2 . 10
To choose =
1/10 0 0 1/10
Ξ11 = Ξ21 = Ξ12 = Ξ22 =
1/10 0 0 1/10
Λ11
=
Λ21
=
Λ12
=
Λ22
!
!
, ,
and Fi (t) be any matrix satisfying FiT (t)Fi (t) ≤ In for all t ∈ <, i = 1, 2. Let B(t) and N(t, $) be all one-dimensional. The plot of L´evy noise is presented as Figure 1 and Figure 2. We choose V(t, 1) = (x12 + x22 ) exp(γ0 t), V(t, 2) = (x12 + 0.5x12 x22 + x22 ) exp(γ0 t), and obtain the following estimates: µ1,2 = 1.4, µ2,1 = 1.4, µ = 1.4, c1 = 1, c2 = 1.3. To choose ηi = 1, i = 1, · · · , 5, θ1 = 7, θ2 = 7, θ = 7, then we have γ0 = 0.2, a = 0.9006, b = 0.0335, d = 0.674, g(1, 0) = 5.3919 > 0. Considering the case of N = 3, it is easy to check that in this case g(µ(N), γ0 ) = 1.1511 > 0. Hence, based on Theorem 2, the class G, where G denotes the class of switching sequences {t p } p∈M , should satisfy inf {t3n − t3n−1 } > 0.9735.
n∈N
We choose t3n − t3n−1 = 1, n ∈ N + ,
(55)
tn − tn−1 = 0.2, n , 3k, k ∈ N + , n ∈ N + .
(56)
Thus, the conditions of Theorem 2 are all satisfied, so system (54) under the switching rule (55), (56) is globally mean square exponentially stable from Theorem 2. The execution of switching signals (55), (56) and dynamic behavior of (54) with initial states x(t) = [2, −1]T and (55), (56) are simulated as Figure 3 and Figure 4, respectively. It is noted that, even there exist high-frequency switching in some domains, the stability of system (54) can still be assured as long as the partial dwell time constrained. Example 2 Consider a three-dimensional time delay switched system with four subsystems # " R τ2 x(t − s) x(t − τ1 ) + Ci (t) 0 ds + Ui (t) dt + %(t, x(t), x(t − τ))dB(t) dx(t) = −Ai (t)x(t) + Bi (t) 4 5 R + < ι(t, x(t), x(t − τ), $)N(dt, d$),
where τ1 = τ2 = τ = 0.2, i ∈ {1, 2, 3, 4} and we take 1 0 0 1 A1 = 0 1 0 , B1 = −1 0 0 1 1 2 A2 = 0 0 1 A3 = 0 0 1 A4 = 0 0
2 2 1 3 1 , C1 = 2 1 2 1
1 1 1
2 1 1 1 −2 0 0 , B2 = 1 2 2 , C2 = 2 1 1 1 2 1 1 3 1 1 1 2 2 0 0 2 0 , B3 = 2 1 1 , C3 = 3 1 0 2 1 −1 2 1 −1 1 2 1 3 2 0 0 1 0 , B4 = 1 3 1 , C4 = 2 1 0 2 1 3 2 3 −1 0.1 0 0.25 0 0 0.25 QiA = ΓiA = 0 0.1 0 , QiB = ΓiB = QCi = ΓCi = 0 0 0 0.1 0 0 0 1 0
14
1 −1 , 2
1 1 , 1 1 3 , 1 1 1 , 1 0 0 , i = 1, 2, 3, 4, 0.25
(57)
%(t, x(t), x(t − τ)) = ι(t, x(t), x(t − τ), $) = To choose
1/10 0 1/10 Λ1i = Λ2i = 0 0 0 1/10 0 1/10 Ξ1i = Ξ2i = 0 0 0
x(t) + x(t − τ) , 10
[x(t) + x(t − τ)]$2 . 10 0 0 1/10 0 0 1/10
, i = 1, 2, 3, 4,
, i = 1, 2, 3, 4,
and Fi (t) be any matrix satisfying FiT (t)Fi (t) ≤ In for all t ∈ <, i = 1, 2, 3, 4. Let B(t) and N(t, $) be all onedimensional. The plot of L´evy noise is presented as Figure 5 and Figure 6. We choose tn − tn−1 = 0.2, n ∈ N + .
(58)
Figure 7 and Figure 8 show the execution of switching signal (58) and system (57) without external controllers, respectively. Such figures implies that system (57) with initial states x(t) = [2, −1, −2]T and Ui (t) = 0, i ∈ {1, 2, 3, 4} is unstable. Next, we choose V(t, 1) = (2x12 + 2x22 + 1.8x32 ) exp(γ0 t), V(t, 2) = (2x12 + 0.1x1 x2 + 2x22 + 1.52x32 ) exp(γ0 t), V(t, 3) = (1.4x12 +2x22 +1.34x32 ) exp(γ0 t), V(t, 4) = (1.6x12 +0.25x2 x3 +1.8x22 +1.8x32 ) exp(γ0 t) and obtain the following estimates: µ1,2 = 1.4, µ1,3 = 1.5, µ1,4 = 1.5, µ2,1 = 1.4, µ2,3 = 1.5, µ2,4 = 1.5, µ3,1 = 1, µ3,2 = 1.2, µ3,4 = 1.4, µ4,1 = 1.3, µ4,2 = 1.4, µ4,3 = 1.5, µ = 1.5, c1 = 2, c2 = 2.5, c3 = 2, c4 = 2.5. To choose ηi = 1, i = 1, 2, 3, 4, 5, θ1 = 8, θ2 = 7, θ3 = 6, θ4 = 7, θ = 6, then we have γ0 = 0.2, a = 0.8586, b = 0.0331, d = 0.6519, g(1, 0) = 4.4564 > 0. Considering the case of N = 3, then we have inf n∈N + {t3n − t3n−1 } > 1.7393 from (53). According to Theorem 3 and Theorem 4, we can choose
U1 (t) = U2 (t) = U3 (t) = U4 (t) =
−6x12 (t) − 6x22 (t) − 6x32 (t),
−5x12 (t) − 6x22 (t) − 6x32 (t),
−6x12 (t) − 5x22 (t) − 5x32 (t),
(59)
−6x12 (t) − 6x22 (t) − 6x32 (t),
t3n − t3n−1 = 1.8, n ∈ N + ,
tn − tn−1 = 0.2, n , 3k, k ∈ N + , n ∈ N + .
(60) (61)
Thus, the conditions of Theorem 2 are all satisfied, so system (57) under controllers (59) and switching rule (60), (61) is globally mean square exponentially stable from Theorem 4. The execution of switching signals (60), (61) and dynamic behavior of (57) with initial states x(t) = [2, −1, −2]T and (59) are simulated as Figure 9 and Figure 10, respectively. It is noted that, system (57) with switching rule (60), (61) is stabilized by these controllers and flexible switching rule. 5. Concluding remarks The paper has considered a class of uncertain switched delay systems with L´evy noise and frequency switching rule, and provided some sufficient conditions for global mean square exponential stability and stabilization of these 15
systems under given switching signals. The major analysis methods are the theory of switched systems and mathematical induction. Finally, two numerical examples are presented to show the effectiveness of the results. The approaches proposed in this paper may be extended to more general switched systems, such as switched systems with partial information of subsystems. Here, some related future works are given. 1) In this paper, g(1, γ) > 0 derived in Theorem 1 implies that the value of time delay maybe not too large. Thus, it is important to study the stability of uncertain switched delay systems by relaxing the condition. 2) Although this paper proposed a method to investigate the stability and stabilization of uncertain switched system with L´evy noise and flexible switching signals, sufficient conditions used to constrain the dynamic behavior of subsystems are still stronger. Hence, seeking for a more effective approach to relax the conservativeness is in our field of interest. 3) In real systems, many abrupt failure or unexpected perturbance are hard to be witnessed, which may be an obstacle to obtain accurate information from the activated model [58]-[59]. Thus, investigations of the case with partial information of subsystems and the stability issues are also part of our future extensions. Acknowledgements This work was supported by the Natural Science Foundation of China (Grant No. 61573095) and the Natural Science Foundation of Shanghai (Grant No. 16ZR1446700). References [1] D. Liberzon, A. S. Morse, Basic problems instability and design of switched systems, IEEE Control Syst. 19 (5) (1999) 59-70. [2] D. Liberzon, J. P. Hespanha, A. S. Morse, Stability of switched systems: A Lie-algebraic condition, Syst. Control Lett. 37 (3) (1999) 117-122. [3] H. Lin, P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Autom. Control 54 (2) (2009) 308-322. [4] D. Liberzon, Switching in Systems and Control, Birkhauser, 2003. [5] Y. M. Jia, Alternative proofs for improved LMI representations for the analysis and the design of continuoustime systems with polytopic type uncertainty: A predictive approach, IEEE Trans. Autom. Control 48 (8) (2003) 1413-1416. [6] W. Xie, C. Wen, Z. Li, Input-to-state stabilization of switched nonlinear systems, IEEE Trans. Autom. Control 46 (7) (2001) 1111-1116. [7] J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Trans. Autom. Control 47 (11) (2002) 1883-1887. [8] H. Ishii, T. Basar, R. Tempo, Randomized algorithms for synthesis of switching rules for multimodal systems, IEEE Trans. Autom. Control 50 (6) (2005) 754-767. [9] S. P. Huang, Z. R. Xiang, Finite-time stabilization of a class of switched stochastic nonlinear systems under arbitrary switching, Int. J. Robust Nonlinear Control 26 (10) (2016) 2136-2152. [10] L. G. Wu, R. N. Yang, P. Shi, X. J. Su, Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings, Automatica 59 (2015) 206-215. [11] D. Zhai, L. An, J. Dong, Q. Zhang, Adaptive exact tracking control for a class of uncertain nonlinear switched systems with arbitrary switchings, J. Frankl. Inst. 354 (7) (2017) 2816-2831. [12] Y. M. Jia, Robust control with decoupling performance for steering and traction of 4WS vehicles under velocityvarying motion, IEEE Trans. Control Syst. Technol. 8 (3) (2000) 554-569. [13] Z. H. Guan, D. J. Hill, X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Autom. Control 50 (7) (2005) 1058-1062. [14] P. Shi, F. Li, A survey on Markovian jump systems: modeling and design, Int. J. Control Autom. Syst. 13 (1) (2015) 1-16. [15] W. Zhou, Q. Zhu, P. Shi, H. Su, J. Fang, L. Zhou, Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters, IEEE Trans. Cybern. 44 (12) (2014) 2848-2860. 16
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Journal Pre-proof
Global mean square exponential stability and stabilization of ´ noise and flexible uncertain switched delay systems with Levy switching signals Ling Liu, Xiangwu Ding, Wuneng Zhou, Xiaoli Li PII: DOI: Reference:
S0016-0032(19)30648-9 https://doi.org/10.1016/j.jfranklin.2018.12.037 FI 4143
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
8 May 2018 30 September 2018 3 December 2018
Please cite this article as: Ling Liu, Xiangwu Ding, Wuneng Zhou, Xiaoli Li, Global mean square exponential stability and stabilization of uncertain switched delay systems with ´ Levy noise and flexible switching signals, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2018.12.037
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Global mean square exponential stability and stabilization of uncertain switched delay systems with L´evy noise and flexible switching signals Ling Liua , Xiangwu Dingb,∗, Wuneng Zhoua,∗, Xiaoli Lia a College
of Information Sciences and Technology, Donghua University, Shanghai, 200051, China. of Computer Sciences and Technology, Donghua University, Shanghai, 200051, China.
b College
Abstract This paper focuses on the stability analysis and stabilization problem for a class of uncertain switched delay systems with L´evy noise and flexible switching signals which unify the high-frequency switching and low-frequency switching. By employing the theory of switched systems, mathematical induction and stochastic analysis technique, some sufficient conditions in form of algebraic inequalities are derived to guarantee the stability and stabilization of such systems. Different from dwell time and average dwell time, the proposed switching rule constrained the partial dwelltime shows that the switching number in the same time interval can be more elastic. Finally, numerical examples are implemented to illustrate the effectiveness of the theoretical results. Keywords: Switched system; Flexible switching rule; L´evy noise; Partial dwell time
1. Introduction Switched systems, which contain lots of subsystems demonstrated by continuous or discrete differential equations and a rule that orchestrates switching among these subsystems, have caught lots of attention from researchers owing to their enormous application in various areas [1]-[14]. In practice, many evolutionary processes are always accompanied by environment changed suddenly and external perturbations abruptly, which lead to the fact that one fixed model can’t simulate the actual phenomenon precisely. And the peculiarity of switched systems can reveal the real world effectively. It has been shown that the occurrence of infinite different subsystems and abundant switching rules such that the states of switched systems are very complex and interesting. Switched systems with some stable subsystems and a special switching signal will become unstable while these systems with some unstable subsystems and a smart rule will become stable. Therefore, it is of great significance to analyze the dynamical characters of switching rules combining with the features of subsystems to reach the goal we want. From the theoretical analysis point, there are two classical problems with switched systems: 1) the stability of switched systems with arbitrarily switching rules and 2) the stability of switched systems with constrained switching signals. The first problem means that how to deal with the subsystems to guarantee the stability of the resulting systems with all permissible switching rules. Unlike the first one, the second problem always needs to manage the switching signals to assure the stability of the resulting systems with some given subsystems. In this paper, we are bent on the second problem. It is noticed that lots of literature have researched the stability of switched systems with constrained switching rules [15]-[27], such as random switching [15]-[16], dwell-time switching [19], [23]-[25]. By establishing a relationship between the sampling period and the average dwell time, the stabilization problem of switched linear neutral systems has been studied in [23]. In [24], the authors proposed an event-triggered sampling mechanism to deal with the stability of switched systems. In view of the fact that the active subsystem and the candidate controller are not always simultaneous, [25] researched the asynchronously switched control problem for a class of switched systems. Additionally, [26] introduced L´evy noise to switched system and developed the stability of such system. Most of the ∗ Corresponding
authors Email address: [email protected] (Wuneng Zhou )
Preprint submitted to
October 3, 2019
existing results are supposed that the switching numbers in the same time interval are constrained by a common upper bound, that is, subsystems cannot jump from one to another frequently under these assumptions. In the real world, however, switching frequency will be high in some time intervals and low in some other time intervals due to the occurrence of the intermittently or irregularly switching. For instance, in power systems, sudden noise or unexpected environment change will give rise to the existence of instantaneously switching. Therefore, it is important to consider the flexible frequency switching signals into the switched systems. In other words, dropping the conditions about the bound of dwell-time or average dwell-time admits of no delay. Some innovative works are devoted to this point [28]-[31]. In [29], switching signals which beyond the average dwell time regime have been proposed. Moreover, switching signals developed in [30] allow the number of switches to grow faster than an affine function of the length of a time interval. [31] investigated the stability of discrete-time switched systems via a mode-dependent average dwell time approach. Whereas, the value of switching dwell time is needed to estimate by applying these criteria. Since the occurrence of irregularly switching, it is not easy to measure the value of switching dwell time. Thus, it is necessary to find more easy-to-use tools to analyze such issues. On the other hand, time-delay, caused by the finite switching speeds or slow processing of information transmission between different neurons in practical networks, is well known to be a usual phenomenon in nature, society, and technology. It is necessary to deal with the characteristics of dynamic systems with constant or time-vary delays, the relevant results can be found in many existing literatures [32]-[39]. Furthermore, systems taking into account L´evy noise which consists of Gaussian process and Poisson point process can illustrate the real developing process accurately such as bond yields and others [40]-[51]. Besides, the appearance of packet dropouts, time-delay and external perturbation in real networks may result in uncertainties [52]-[57]. Thus, it is natural and significant to study the problem of stability and stabilization of switched delay systems involving uncertainties. Up to now, there are few results about uncertain switched systems with L´evy noise, especially uncertain switched systems with flexible switching signals, which inspire us to study this topic. Moreover, the terms of stochastic disturbance make the analytical approach totally different, and seeking for an effective method to dispose of the irregularly switching is still an open problem. In this paper, we aim at studying the stability and stabilization of uncertain switched delay systems with L´evy noise in the presence of flexible frequency switching. Together with the theory of switched systems and some mathematical analysis techniques, some sufficient criteria ensuring the stability of such systems are developed. The results in this paper show that even there is some high-frequency switching in some time intervals, an uncertain switched system is mean square exponentially stable as long as the partial dwell time satisfies certain conditions, which relaxes the restriction on common bound of average dwell time. As a whole, the main contributions of this paper are listed as follows: 1) A class of switching signals which constrain partial dwell time for analyzing the dynamic behavior of uncertain switched delay systems is proposed. 2) Taking into account the L´evy noise and time-delay for the uncertain switched delay systems with flexible switching signals. 3) The stability and stabilization criteria in form of some inequalities are derived by applying the generalized Itˆo formula and mathematical induction. Moreover, the analytical approach utilized in this paper can be extended to more general switched delay systems. The rest of this paper is organized as follows. Section 2 gives some notations, definitions, assumptions, and lemmas. In Section 3, main results are stated by applying the theory of switched systems and some mathematical analysis techniques. Numerical examples are given to show the validity of the results in Section 4. Finally, concluding remarks are made in Section 5. 2. Preliminaries and model description Denote N and N + as the set of natural numbers and non-negative integers, respectively. Denote < and <+ as the set of real numbers and non-negative real numbers, respectively. Denote
the maximum or minimum eigenvalue of matrix A. A > 0 or A < 0 denotes that the matrix A is a symmetric and positive or negative definite matrix. Let Cτ = C([−τ, 0], Rn ), τ > 0, and for each φ ∈ Cτ , the norm is defined by 2
kφkτ = sup−τ≤s≤0 |φ(s)|. For h(t) : <+ → <, denote D+ h(t) = lim sup s→0+
h(t + s) − h(t) . s
Let (Ω, F , {F }0 , P) be a complete probability space with a filtration {Ft }t≥0 which is right continuous, and F0 contains all P-null sets. B(t) = (B1 (t), · · · , Bm (t))T is an m-dimensional Ft -adaptive Brownian motion defined on the complete probability space. N(t, $) means an Ft adapted Poisson random measure on [0, +∞) × < with compensator e $), and N(t, e $) satisfies N(dt, e martingale measure N(t, d$) = N(dt, d$) − π(d$)dt, π(d$)dt is a Possion point process. In this paper, we consider a class of uncertain switched delay systems with L´evy noise described by dx(t) = x(s) =
h i R τ2 −Aσ(t) (t)x(t) + Bσ(t) (t)x(t − τ1 ) + Cσ(t) (t) 0 x(t − s)ds + Uσ(t) (t) dt + %σ(t) (t, x(t), x(t − τ))dB(t) R + < ισ(t) (t, x(t), x(t − τ), $)N(dt, d$), t ≥ 0, φ(s), s ∈ [−τ, 0],
(1) where x(t) ∈ Rn is the state vector, σ : [0, +∞) → M = {1, 2, · · · , m} is a right-continuous, piecewise constant function called the switching rule, m is the number of subsystems. The switching time sequences {t p }, p ∈ N, satisfying 0 = t0 ≤ t1 ≤ · · · ≤ t p → +∞, σ(t) = z p , t ∈ [t p , t p+1 ), p ∈ N, represents that the subsystem z p is activated during [t p , t p+1 ). Denote G0 = {t0 , t1 , · · · , t p , · · · } as the set of switching sequences, p ∈ N. G ⊂ G0 means a class of switching sequences and S(G) denotes the set of all orderer pairs (z p , z p+1 ) which indicates that subsystem z p switches to subsystem z p+1 with the switching sequences G, z p , z p+1 ∈ M, p ∈ N. {(Ai (t), Bi (t), Ci (t)) : i ∈ M} is a collection of matrices corresponding to the individual subsystem of systemR (1), φ(s) ∈ Cτ , τ = max{τ1 , τ2 }, τ, τ1 , τ2 τ2 are positive constant delays, denote xτ = x(t − τ), xτ1 = x(t − τ1 ), xτ2 = 0 x(t − s)ds. Ai (t) = Ai + QiA Fi (t)ΓiA , C C C Bi (t) = Bi + QiB Fi (t)ΓiB , Ci (t) = Ci + Qi Fi (t)Γi , where Ai , Bi , Ci , QiA , QiB , Qi , ΓiA , ΓiB , ΓCi are real known matrices and Fi (t) is an unknown real matrix with FiT (t)Fi (t) ≤ In , i ∈ M. For any i ∈ M, %i (t, x(t), xτ ) : < ×
(2)
for any solution x(t) = x(t, φ) of system (1) with initial value φ(s) ∈ Cτ , s ∈ [−τ, 0], where α is called the convergence rate. Definition 2 System (1) is said to be globally mean square exponentially stabilized over the class S(G), G ⊆ G0 , if there exists an appropriate feedback control law such that the closed-loop switched system (1) is globally mean square exponentially stable. Definition 3 For any given constants θ > 0, a > 0, b > 0, d > 0, define function g(ξ, γ) : <+ × <+ → < as follows g(ξ, γ) = −γ + θ − ξ a exp(γτ1 ) + b exp(γτ2 ) + d exp(γτ) .
Remark 1 It’s obvious that g(ξ, γ) is a continuous function. For every fixed second argument, it’s monotonous nonincreasing. 3
Assumption 1 There exist appropriate dimension positive definition matrices Λ1z p and Λ2z p , z p ∈ M, p ∈ N, such that trace[%Tzp (t, x(t), xτ )%z p (t, x(t), xτ )] ≤ xT (t)Λ1z p x(t) + xτT Λ2z p xτ , t ∈ [t p , t p+1 ). Assumption 2 There exist appropriate dimension positive definition matrices Ξ1z p and Ξ2z p , z p ∈ M, p ∈ N, such that ιTzp (t, x(t), xτ )ιz p (t, x(t), xτ ) ≤ xT (t)Ξ1z p x(t) + xτT Ξ2z p xτ , t ∈ [t p , t p+1 ). And the characteristic measure π(d$)dt satisfies π(d$)dt = λ0 µ(d$)dt, ˆ where λ0 is the intensity of Poisson distribution and µˆ is the probability distribution of random variable $. Assumption 3 For each pair (i, j) ∈ S(G), G ⊆ G0 , there exist µi, j ≥ 1 such that : (3)
Pi ≤ µi, j P j , i, j ∈ M, and µ = max {µi, j }, G ⊆ G0 . For given constant N ∈ N + , assume that (i, j)∈S(G)
µ(N) = max+ {µ(n, N)} < ∞,
(4)
n∈N
κ−1 Y
µ(n, κ) =
j=0
where µ0 = µz0 ,z−1 = 1, κ = 1, 2, · · · , N.
µzκ(n−1)+ j , zκ(n−1)+ j−1 , n ∈ N + ,
(5)
Remark 2 Under Assumptions 1-3 which are standard for analyzing the stability of systems with random noise [40][44], system (1) admits trivial solution. Assumption 3 is useful for dealing with switched systems [27]-[30], and (4), (5) are prepared for the introduction of partial dwell-time method in the main results. µ0 = 1 means system (1) will be not switching during [−τ, t1 ). Now we are going to present some lemmas which will be helpful for the results. Lemma 1 ([36]) Let u, v ∈
uT v + vT u ≤ ηuT u + η−1 vT v,
for any η > 0.
Lemma 2 ([13]) If S ∈ Rn×n is a symmetric positive definite matrix and Q ∈ Rn×n is a symmetric matrix, then λmin (S −1 Q)xT S x ≤ xT Qx ≤ λmax (S −1 Q)xT S x, x ∈ Rn . Lemma 3 ([33]) Given any appropriate dimension real matrix P > 0 and a vector function h(·) : [s1 , s2 ] →
s2 s1
h(s)ds
!T
P
Z
s2 s1
! Z h(s)ds ≤ (s2 − s1 )
s2
hT (s)Ph(s)ds.
s1
3. Main results In this section, we are going to deduce criteria of global mean square exponential stability and construct linear feedback controllers for system (1) with L´evy noise and flexible switching signals. 4
For system (1), the generalized Itˆo formula of function V(t, x(t), σ(t)) ∈ C2,1 (<+ ×
where the operator LV(t, x(t), xτ , σ(t)) : <+ ×
=
1 V x (t, x(t), σ(t)) fσ(t) (t, x(t), xτ ) + trace[%Tσ(t) (t, x(t), xτ )V xx (t, x(t), σ(t))%σ(t) (t, x(t), xτ )] 2 R + < [V(t, x(t) + ισ (t, x(t), xτ ), σ(t)) − V(t, x(t), σ(t))]π(d$) + Vt (t, x(t), σ(t)),
(7)
fσ(t) (t, x(t), xτ ) = −Aσ(t) (t)x(t) + Bσ(t) (t)xτ1 + Cσ(t) (t)xτ2 , Vt (t, x(t), σ(t)) =
∂V(t, x(t), σ(t)) , ∂t
∂V(t, x(t), σ(t)) ∂V(t, x(t), σ(t)) ∂V(t, x(t), σ(t)) V x (t, x(t), σ(t)) = , ,··· , ∂x1 ∂x2 ∂xn ! ∂2 V(t, x(t), σ(t)) . V xx (t, x(t), σ(t)) = ∂xk ∂xl n×n
!
Moreover, for any p ∈ N and t ∈ [t p , t p+1 ), we denote σ(t) by z p , z p ∈ M. Let ∆h > 0 be small enough such that t + ∆h ∈ [t p , t p+1 ). Then, we can obtain EV(t + ∆h, x(t + ∆h), z p ) − EV(t, x(t), z p ) =
Z
t
t+∆h
ELV(s, x(s), z p )ds.
Since ELV(t, x(t), σ(t)) is continuous in the interval t ∈ [t p , t p+1 ), it follows that D+ EV(t, x(t), z p ) = ELV(t, x(t), z p ), t ∈ [t p , t p+1 ), p ∈ N.
(8)
3.1. Global mean square exponential stability In this subsection, we are going to analyze the global mean square exponential stability of system (1). Firstly, a preliminary theorem which plays a key role in the proof of the main results is given. Theorem 1 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. For any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0 and positive constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0 such the following inequalities hold: Pi ≤ ci In , (9) 1 C 3 4 A A T B (Γi ) Pi Q i Pi Qi Ωi + Ωi + Ωi + θi Pi Pi Qi A T −1 (Q ) P −η I 0 0 0 i i 3 n < 0, A Γi 0 −η3 In 0 0 (10) B T −1 (Q ) P 0 0 −η I 0 i i 4 n (QCi )T Pi 0 0 0 −η−1 5 In θ − (a + b + d) > 0,
(11)
where θ = min{θi , i ∈ M}, Ω1i = −Pi Ai − ATi Pi , Ω3i = ci Λ1i , Ω4i = λ0 [(ci + 1)Ξ1i + Pi Pi ], a = max{ai , i ∈ M}, ai = −1 −1 −1 B T B −1 2 −1 −1 −1 C T C µ[η−1 1 λmax ((Pi ) )+η4 λmax ((Pi ) (Γi ) Γi )], b = max{bi , i ∈ M}, bi = µ[η2 (τ2 ) λmax ((Pi ) )+η5 λmax ((Pi ) (Γi ) Γi )], −1 2 −1 2 d = max{di , i ∈ M}, di = µ[ci λmax ((Pi ) Λi ) + (ci + 1)λ0 λmax ((Pi ) Ξi )], i ∈ M. 5
Consider a function V(x, σ(t)) =
(
xT (t)Pσ(t) x(t), t ∈ [−τ, 0], exp(γt)xT (t)Pσ(t) x(t), t ≥ 0,
(12)
where γ ∈ (0, θ) is a solution of g(1, γ) > 0. If there exists tˆ ∈ [t p , t p+1 ) for some p ∈ N satisfying V(tˆ, σ(tˆ)) , 0, V(t, σ(t)) ≤ V(tˆ, σ(tˆ)), ∀t ∈ [tˆ − τ, tˆ], then D+ EV(t, σ(t))|t=tˆ < 0. (13) Proof. For convenience, we denote σ(t) by z p , z p ∈ M, t ∈ [t p , t p+1 ), p ∈ N and then V(t, z p ) = exp(γt)xT (t)Pz p x(t). Based on (7) and system (1), we can obtain
LV(t, σ(t))
= 2xT (t)Pz p [−Az p (t)x(t) + Bz p (t)xτ1 + Cz p (t)xτ2 ] exp(γt) + γ exp(γt)xT Pz p x(t) +trace{%Tzp (t, x(t), xτ )Pz p %z p (t, x(t), xτ )} exp(γt) o R n + < [x(t) + ιz p (t, x, xτ , $)]T Pz p [x(t) + ιz p (t, x, xτ , $)] − xT (t)Pz p x(t) exp(γt)π(d$) ≤ exp(γt) xT (t)Ω1z p x(t) + 2xT (t)Pz p Bz p xτ1 + 2xT (t)Pz p Cz p xτ2 ] + γV(t, z p ) + exp(γt) − 2xT (t)Pz p QzAp Fz p (t)ΓzAp x(t) + 2xT (t)Pz p QzBp Fz p (t)ΓzBp xτ1 + 2xT (t)Pz p QCzp Fz p (t)ΓCi xτ2 ] +cz p trace{%Tzp (t, x(t), xτ )%z p (t, x(t), xτ )} exp(γt) o R n + < (cz p + 1)ιTzp (t, x, xτ , $)ιz p (t, x, xτ , $) + xT (t)Pz p Pz p x(t) exp(γt)π(d$),
Ω1z p
(14)
ATzp Pz p .
where = −Pz p Az p − From Lemmas 1- 3, it yields that 2xT (t)Pz p Bz p xτ1 + 2xT (t)Pz p Cz p xτ2 ≤ ≤ ≤
T T T η1 xT (t)Pz p Bz p (Bz p )T Pz p x(t) + η−1 1 xτ1 xτ1 + η2 x (t)Pz p C z p (C z p ) Pz p x(t) R τ2 T R τ2 +η−1 x(t − s)ds x(t − s)ds 2 0 0 R τ2 T R τ2 −1 −1 xT (t)Ω21 Pz p 0 x(t − s)ds z p x(t) + η2 λmax ((Pz p ) ) 0 x(t − s)ds
−1 T +η−1 1 λmax ((Pz p ) )xτ1 Pz p xτ1
−1 −1 xT (t)Ω21 z p x(t) + η2 λmax ((Pz p ) )τ2 −1 T +η−1 1 λmax ((Pz p ) )xτ1 Pz p xτ1 ,
R τ2 0
(15)
xT (t − s)Pz p x(t − s)ds
−2xT (t)Pz p QzAp Fz p (t)ΓzAp x(t) + 2xT (t)Pz p QzBp Fz p (t)ΓzBp xτ1 + 2xT (t)Pz p QCzp Fz p (t)ΓCi xτ2 ≤
T A T A T B B T η3 xT (t)Pz p QzAp (QzAp )T Pz p x(t) + η−1 3 x (t)(Γz p ) Γz p x(t) + η4 x (t)Pz p Qz p (Qz p ) Pz p x(t)
≤
−1 −1 B T B T xT (t)Ω22 z p x(t) + η4 λmax ((Pz p ) (Γz p ) Γz p )xτ1 Pz p xτ1 R τ2 −1 C T C T +η−1 5 λmax ((Pz p ) (Γz p ) Γz p )τ2 0 x (t − s)Pz p x(t − s)ds.
T B T B T C C T −1 T C T C +η−1 4 xτ1 (Γz p ) Γz p xτ1 + η5 x (t)Pz p Qz p (Qz p ) Pz p x(t) + η5 xτ2 (Γz p ) Γz p xτ2
(16)
T T 22 A A T −1 A T A B B T where Ω21 z p = η1 Pz p Bz p (Bz p ) Pz p + η2 Pz p C z p (C z p ) Pz p , Ωz p = η3 Pz p Qz p (Qz p ) Pz p + η3 (Γz p ) Γz p + η4 Pz p Qz p (Qz p ) Pz p +
η5 Pz p QCzp (QCzp )T Pz p , and ηi , i = 1, · · · , 5, are positive constants. 6
From Assumptions 1-2, we have cz p trace{%Tzp (t, x(t), xτ )%z p (t, x(t), xτ )} exp(γt) ≤ ≤
≤ ≤ ≤
cz p [xT (t)Λ1z p x(t) + xτT Λ2z p xτT ] exp(γt)
(17)
xT (t)Ω3z p x(t) exp(γt) + cz p λmax ((Pz p )−1 Λ2z p )xτT Pz p xτT exp(γt)
o R n (cz p + 1)ιTzp (t, x, xτ , $)ιz p (t, x, xτ , $) + xT (t)Pz p Pz p x(t) exp(γt)π(d$) < n o R (cz p + 1)[xT (t)Ξ1z p x(t) + xτT Ξ2z p xτ ] + xT (t)Pz p Pz p x(t) exp(γt) < λ0 ν(d$) xT (t)Ω4z p x(t) exp(γt) + (cz p + 1)λ0 xτT Ξ2z p xτ exp(γt)
(18)
xT (t)Ω4z p x(t) exp(γt) + (cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p )xτT Pz p xτ exp(γt),
where Ω3z p = cz p Λ1z p , Ω4z p = λ0 [(cz p + 1)Ξ1z p + Pz p Pz p ]. Then, taking (15)-(18) to (14), we get LV(t, σ(t))|t=tˆ ≤
exp(γtˆ)xT (tˆ) Ω1z p + Ω2z p + Ω3z p + Ω4z p x(tˆ) + γV(tˆ, z p ) −1 −1 −1 B T B +[η−1 1 λmax ((Pz p ) ) + η4 λmax ((Pz p ) (Γz p ) Γz p )] ×xT (tˆ − τ1 )Pz p x(tˆ − τ1 ) exp(γ(tˆ − τ1 + τ1 )) −1 C T C +[η−1 λ ((P )−1 ) + η−1 5 λmax ((Pz p ) (Γz p ) Γz p )]τ2 R τ22 max z p × 0 xT (tˆ − s)Pz p x(tˆ − s) exp(γ(tˆ − s + s))ds +cz p λmax ((Pz p )−1 Λ2z p )xT (tˆ − τ)Pz p x(tˆ − τ) exp(γ(tˆ − τ + τ)) +(cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p )xT (tˆ − τ)Pz p x(tˆ − τ) exp(γ(tˆ − τ + τ)).
(19)
22 where Ω2z p = Ω21 z p + Ωz p . In light of Assumption 3, the following inequalities can be obtained
xT (tˆ − τ)Pz p x(tˆ − τ)
≤ µz p ,σ(tˆ−τ) xT (tˆ − τ)Pσ(tˆ−τ) x(tˆ − τ) ≤ µxT (tˆ − τ)Pσ(tˆ−τ) x(tˆ − τ),
(20)
xT (tˆ − τ1 )Pz p x(tˆ − τ1 ) ≤ µxT (tˆ − τ1 )Pσ(tˆ−τ1 ) x(tˆ − τ1 ),
(21)
xT (tˆ − τ2 )Pz p x(tˆ − τ2 ) ≤ µxT (tˆ − τ2 )Pσ(tˆ−τ2 ) x(tˆ − τ2 ),
(22)
where constants µz p ,σ(tˆ−τ) and µ are defined in Assumption 3. From (20)-(22), one gets that
7
LV(t, σ(t))|t=tˆ ≤
≤ ≤
xT (tˆ)Ωz p x(tˆ) exp(γtˆ) + γV(tˆ, z p ) h i −1 −1 −1 B T B +µ η−1 1 λmax ((Pz p ) ) + η4 λmax ((Pz p ) (Γz p ) Γz p ) exp(γτ1 )
×xT (tˆ − τ1 )Pσ(tˆ−τ1 ) x(tˆ − τ1 ) exp(γ(tˆ − τ1 )) h i −1 −1 −1 C T C +µ η−1 2 λmax ((Pz p ) ) + η5 λmax ((Pz p ) (Γz p ) Γz p ) τ2 exp(γτ2 ) R τ2 × 0 xT (tˆ − s)Pσ(tˆ)−s x(tˆ − s) exp(γ(tˆ − s))ds h i +µ cz p λmax ((Pz p )−1 Λ2z p ) + (cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p ) exp(γτ)xT (tˆ − τ)Pz p x(tˆ − τ)
(23)
−θz p exp(γtˆ)xT (tˆ)Pz p x(tˆ) + γV(tˆ, z p ) + az p exp(γτ1 )V(tˆ − τ, σ(tˆ − τ))
+bz p exp(γτ2 )τ2 V(tˆ − τ2 , σ(tˆ − τ2 )) + dz p exp(γτ)τ2 V(tˆ − τ, σ(tˆ − τ)) h i − −γ + θz p − az p exp(γτ1 ) − bz p exp(γτ2 ) − dz p exp(γτ) V(tˆ, z p ),
where Ωz p = Ω1z p + Ω2z p + Ω3z p + Ω4z p , and −1 −1 −1 B T B az p = µ[η−1 1 λmax ((Pz p ) ) + η4 λmax ((Pz p ) (Γz p ) Γz p )], 2 −1 −1 −1 C T C bz p = µ[η−1 2 (τ2 ) λmax ((Pz p ) ) + η5 λmax ((Pz p ) (Γz p ) Γz p )],
From (11), we have which implies
h i dz p = µ cz p λmax ((Pz p )−1 Λ2z p ) + (cz p + 1)λ0 λmax ((Pz p )−1 Ξ2z p ) . LV(t, σ(t))|t=tˆ ≤ −g(1, γ)V(tˆ, z p ) < 0, D+ EV(t, σ(t))|t=tˆ < 0.
This completes the proof of Theorem 1. Remark 3 From (11), it is noticed that g(1, 0) > 0 and g(1, θ) < 0. Thus, it is possible that g(1, γ) > 0 holds for γ ∈ (0, θ), which can be computed by Matlab. In theoretical, g(1, γ) > 0 shows the range of time delay which may be not too large. Due to the fact the dynamics of systems have been influenced by both of time delay and flexible switching signals, it is difficult to measure the relationship between the affection of time delay and the switching signals. Remark 4 Different from [9]-[12] which focus on the dynamic behavior of switched systems in the presence of arbitrary switching signals, the objective of this paper is to analyse the stability and stabilization of uncertain switched systems under constrained switching signals. It aims to establish a bridge between the flexible switching signals and subsystems to keep the system stable. Whereas, as the result of the existence of high-frequency and low-frequency switching signals, the restriction on the dynamic behavior of subsystems is still stronger. In term of Theorem 1, some corollaries can be obtained as follows. Corollary 1 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. For any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, and constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (9), (10) and (11) hold. Given function V(t, σ(t)) which defined in (12) with γ ∈ (0, θ), θ = min{θi }, ensuring that i∈M
g(ξ, γ) > 0. For some p ∈ N, if there exist ς1 , ς2 ∈ [t p , t p+1 ), ς1 < ς2 satisfying V(ς, ˆ σ(ς)) ˆ , 0 for some ςˆ ∈ [ς1 , ς2 ],
8
V(s, σ(s)) ≤ ξV(ς, ˆ σ(ς)), ˆ ∀s ∈ [ςˆ − τ, ς], ˆ then D+ EV(t, σ(t))|t=ςˆ < 0.
(24)
Proof. Based on Theorem 1, for ςˆ ∈ [ς1 , ς2 ], it can be derived that n h io D+ EV(t, σ(t)) ≤ − −γ + θz p − ξ az p exp(γτ1 ) + bz p exp(γτ2 ) + dz p exp(γτ) EV(t, σ(t)).
(25)
If for some ςˆ ∈ [ς1 , ς2 ], V(s, σ(s)) ≤ ξV(ς, ˆ σ(ς)), ˆ ∀s ∈ [ςˆ − τ, ς], ˆ from Theorem 1, it yields that D+ EV(t, σ(t))|t=ςˆ ≤ −g(ξ, γ)V(ς, ˆ σ(ς)) ˆ < 0. This completes the proof of Corollary 1.
Corollary 2 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. For any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, and constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (9), (10) and (11) hold. Given function V(t, σ(t)) which defined in (12) with γ ∈ (0, θ), θ = min{θi }, ensuring that i∈M
g(ξ, γ) > 0. For some p ∈ N, if there exist ς1 , ς2 ∈ [t p , t p+1 ), ς1 < ς2 satisfying V(ς, σ(ς)) , 0 for any ς ∈ [ς1 , ς2 ], V(s, σ(s)) ≤ ξV(ς, σ(ς)), ∀s ∈ [ς − τ, ς], then EV(ς2 , σ(ς2 )) ≤ exp(−g(ξ, γ, z p )(ς2 , ς1 )). EV(ς1 , σ(ς1 ))
(26)
Proof. Based on Theorem 1 and (25), we can get D+ EV(t, σ(t)) ≤ −g(ξ, γ)EV(t, σ(t)) < 0, ∀ς ∈ [ς1 , ς2 ], which implies that (26) holds. This completes the proof of Corollary 2. Then, aided by Theorem 1 and Corollaries 1-2, the stability of switched system (1) is formulated in the next theorem, which mainly concentrates on partial dwell time. Theorem 2 Assume that x(t) = x(t, φ) is a solution of system (1) through (0, φ) and Assumptions 1-3 hold. Given N ∈ N + , for any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, and constants µi, j ≥ 1, θi ≥ 0, θ ≥ 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (9)-(11) hold. Moreover, G denotes the class of switching sequences tk satisfying inf {tNn − tNn−1 } > τ +
n∈Z+
lnµ(N) . g(µ(N), 0)
(27)
Then, system (1) is globally mean square exponentially stable over the class G.
Proof. Let x(t) = x(t, φ) be a solution of system (1) through (0, φ). Without loss of generality, assume that φ , 0. Choose proper constant γ0 > 0 such that g(µ(N), γ0 ) > 0 and inf {tNn − tNn−1 } > τ +
n∈Z+
lnµ(N) . g(µ(N), γ0 )
(28)
Define the following functions T t ∈ [−τ, 0], x (t)Pσ(t) x(t), V(x, σ(t)) = exp(γt)xT (t)Pσ(t) x(t), t ≥ 0, 9
(29)
κi (t) =
H(t, n, N) =
Nn−1 X
(
1, 0,
t ∈ [ti , ti+1 ), others,
κi (t)V(t, σ(t)),
t ∈ [tN(n−1) , tNn ).
i=N(n−1)
(30)
Next, we aim to prove the following inequality EH(t, n, N) ≤ ωµ(n, N),
t ∈ [tN(n−1) , tNn ),
n ∈ N+ ,
(31)
where ω = maxi∈M {λmax (Pi )}kφk2τ > 0. First we show that (31) holds for n = 1, namely EH(t, n, N) ≤ ωµ(1, N) = ω
N−1 Y
µz j z j−1 ,
j=0
t ∈ [0, tN ).
(32)
Since EV(t, σ(t)) ≤ ω, t ∈ [−τ, 0], we can obtain that EH(t, 1, 1) = EV(t, σ(t)) ≤ ω = ωµ0 = ωµ(1, 1), t ∈ [0, t1 ). If it is not true, then there exist some t ∈ (0, t1 ) such that EV(t, σ(t)) ≥ ω. Set t` = inf{t ∈ (0, t1 ) : EV(t, σ(t)) ≥ ω}. Thus, it follows that EV(t`, σ(t`)) = ω, EV(s, σ(s)) ≤ EV(t`, σ(t`)), D+ EV(t`, σ(t`)) ≥ 0,
s ∈ [t` − τ, t`],
(33)
in view of Theorem 1, (33) is contradict to Theorem 1. Then, (31) holds for n = 1, N = 1. Now, for n = 1, we assume that, for some integer p ≤ N − 1, EH(t, 1, p) ≤ ωµ(1, p), t ∈ [0, t p ). Then, the inequality EH(t, 1, p + 1) ≤ ωµ(1, p + 1) holds, t ∈ [0, t p+1 ). In fact, µ(1, p) ≤ µ(1, p + 1), t ∈ [0, tl+1 ). Therefore, it aims to show EV(t, z p ) ≤ ρµ(1, p + 1), t ∈ [t p , t p+1 ). (34) From Assumption 3, for any i, j ∈ M,
Pi ≤ µi, j P j ,
which indicates that V(tn , zn ) =
exp(γtn )xT (tn )Pzn x(tn )
=
exp(γtn− )xT (tn )Pzn x(tn− )
≤
µzn ,zn−1 exp(γtn )xT (tn )Pzn−1 x(tn )
=
µzn ,zn−1 V(tn− , zn−1 ),
(35)
for n ∈ N + . Particularly, V(t p , z p ) ≤ µz p ,z p−1 V(z−p , z p−1 ) ≤ ωµ(1, p + 1). Thus, (34) holds for t = t p . If (34) is not true for t ∈ (t p , t p+1 ), there exist some t ∈ (t p , t p+1 ) which are against (34). Let t¯ = inf{t > t p , V(t, z p ) ≥ ωµ(1, p + 1)}, then by virtue of the fact that EV(s, σ(s)) ≤ ωµ(1, p) ≤ ωµ(1, p + 1), s ∈ [t p − τ, t p ), it yields that
10
EV(s, σ(s)) ≤ EV(t¯, σ(t¯)), EV(t¯, σ(t¯)) = ωµ(1, p + 1), D+ EV(t¯, σ(t¯)) ≥ 0,
s ∈ [t¯ − τ, t¯],
which leads to a contradiction with Theorem 1. Thus, it follows that EV(t, z p ) ≤ ωµ(1, p + 1), t ∈ [t p , t p+1 ), which implies that EH(t, 1, N) ≤ ωµ(1, N), t ∈ [0, tN ). Hence, (31) holds for n = 1. Next, we show that if (31) holds for n = l, l ∈ Z+ , namely EH(t, l, N) ≤ ωµ(l, N) = ω
N−1 Y
µzN(l−1)+ j ,zN(l−1)+ j−1 ,
(36)
j=0
for t ∈ [tN(l−1) , tNl ), then (31) holds for n = l + 1, that is, EH(t, l + 1, N) ≤ ωµ(l + 1, N) = ω
N−1 Y
µzNl+ j ,zNl+ j−1 ,
(37)
j=0
for t ∈ [tNl , tN(l+1) ). First, we show a fact that there exists some tˇ ∈ [tNl−1 , tNl − τ] such that EV(tˇ, zNl−1 ) < ω. Or else, it holds that ωµ(l, N) ≥ EV(s, σ(s)) ≥ ω, s ∈ [tNl−1 , tNl − τ], which yields EV(t, σ(t)) ≤ ωµ(l, N) ≤ µ(l, N)EV(s, σ(s)), t ∈ [s − τ, s],
(38)
for s ∈ [tNl−1 , tNl − τ]. Choose ξ = µ(l, N), it follows from Corollary 2 and (28) that 1 µ(l, N)
≤
ω ωµ(l, N) EV(tNl − τ) EV(tNl−1 , σ(tNl−1 ))
≤
exp(−g(µ(l, N), γ0 )(tNl − τ − tNl−1 ))
≤
exp(−g(µ(N), γ0 )(tNl − τ − tNl−1 )) 1 , µ(N)
=
<
(39)
which is a contradiction. Next, we proof that EV(tNl − τ, zNl−1 ) < ω. If it is not true, there exists t´ ∈ (tˇ, tNl − τ] satisfies V(t´, σ(t´)) = ω ≥ EV(s, σ(s)), s ∈ [tˇ, t´] and D+ EV(t´, σ(t´)) ≥ 0. Then, EV(t, σ(t)) ≤ ωµ(l, N) = EV(t´, σ(t´))µ(l, N), t ∈ [t´ − τ, t´]. Choose ξ = µ(l, N) and it is a contradiction from Corollary 1. Finally, we have EV(t, σ(t)) < ω, t ∈ [tNl − τ, tNl ).
(40)
Suppose it is not true, one may get some t ∈ [tNl − τ, tNl ) such that V(t, σ(t)) ≥ ω. Since V(tNl − τ, zNl−1 ) < ω, let t˘ = inf{tNl − τ ≤ t < tNl , V(t, σ(t)) ≥ ω}, then, it yields that V(t˘, zNl−1 ) = ω ≥ V(s, zNl−1 ), s ∈ [tNl − τ, t˘], and D+ V(t˘, zNl−1 ) ≥ 0. On the other hand, V(t˘, zNl−1 )µ(l, N) = ωµ(l, N) ≥ V(t, σ(t)), t ∈ [t˘ − τ, t˘]. Therefore, (40) is right from Corollary 1. − Additionally, from Theorem 1 and (35), it is obvious that V(tNl , zNl ) ≤ µzNl ,zNl−1 V(tNl , zNl−1 ) ≤ µzNl ,zNl−1 ω, which 11
results in V(t, σ(t)) ≤ µNl,Nl−1 ω, t ∈ [tNl , tNl+1 ). Thus, employing the similar discussing, for any Nl ≤ k ≤ Nl + N − 1, one may get EV(t, σ(t)) ≤ ω that is EH(t, n, N) ≤ ω
k−Nl Y
µzNl+ j , zN+ j−1 , t ∈ [tk , tk+1 ),
(41)
µzNl+ j , zN+ j−1 = ωµ(l + 1, N),
(42)
j=0
N−1 Y j=0
for t ∈ [tk , tk+1 ). Hence, (31) holds for n = l + 1. And we can conclude that (31) holds for all n ∈ Z+ , which implies H(t, n, N) ≤ ωµ(N), t ≥ 0, namely Ekx(t)k ≤ ∆kφk2τ exp(−γ0 t), ∀t ≥ 0, ∀{t p } ∈ F , (43)
λmax (Pi , i ∈ M) . Therefore, system (1) under the switching signal (27) is globally mean square λmin (Pi , i ∈ M) exponentially stable. This completes the proof of Theorem 2.
where ∆ = µ(N)
Remark 5 Theorem 2 aims to design a class of switching signals which constrained the partial dwell time such that switched delay systems will be stable while the other dwell time can be chosen arbitrarily. Different from dwell time or average dwell time [23]-[25], it is noted that there exist high-frequency switching in some time intervals and switching numbers in some time domains will be flexible. Remark 6 The model proposed in this paper is just similar to [26]. In this paper we have added distributed delays and flexible switching signals into the model of [26]. Different from [26] which based on average dwell time, the analytical approach presented in Theorem 2 is novel and interesting, which can be extended to more general switched delay systems with flexible switching signals. Remark 7 There are many literatures about stability of systems with L´evy noise [40]-[45]. It has been realized that the generalized Itˆo formula and stochastic version of Lyapunov approach are powerful techniques to study them. Asking for exponential decay results in pth moment exponential stability which has highly-value in many areas, and this paper has investigated the global exponential stability in mean square of systems in case of p = 2. The results can be generalized to the pth moment exponential stability of uncertain switched systems with L´evy noise. On the other hand, lots of researches devoted themselves to the estimates of integrals of stochastic part for considering the influence of integral terms on the solution. In the future, we are going to study the state estimate and pth moment exponential stability of stochastic switched systems with flexible switching signals. 3.2. Global mean square exponential stabilization In this subsection, we will make switched system with L´evy noise and flexible switching signals achieve exponential stabilization under linear control. In the following, we use the linear feedback scheme to realize stabilization, i.e., the controller Uz p (t), z p ∈ M, t ∈ [t p , t p+1 ), p ∈ N can be presented by n o Uz p (t) = Kz p x(t) = diag kz1p , · · · , kznp x(t), (48) where kzl p , l = 1, 2, · · · , n, means the control gain. System (1) can be rewritten as follows dx(t) = −(A σ(t) (t) − Kσ(t) )x(t) + Bσ(t) (t)xτ1 + C σ(t) (t)xτ2 dt + %σ(t) (t, x(t), x(t − τ))dB(t) R + < ισ(t) (t, x(t), x(t − τ), $)N(dt, d$), t ≥ 0. x(s) = φ(s), s ∈ [−τ, 0]. 12
(49)
Similar to Theorem 1 and Theorem 2, we have the following stabilization criteria. Theorem 3 Assume that x(t) = x(t, φ) is a solution of system (49) through (0, φ) and Assumptions 1-3 hold. If for any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0 and positive constants ci > 0, µi, j ≥ 1, θi > 0, η1 , η2 , η3 , η4 , η5 > 0 such the following inequalities hold Pi ≤ ci In , (50) 3 4 5 Ωi + Ωi + Ωi + θi Pi A T (Q ) P i i ΓiA (QiB )T Pi (QCi )T Pi
Pi QiA −η−1 3 In 0 0 0
(ΓiA )T 0 −η3 In 0 0
Pi QiB 0 0 −η−1 4 In 0
θ − (a + b + d) > 0,
Pi QCi 0 0 0 −η−1 5 In
< 0,
(51)
(52)
where θ = min{θi , i ∈ M}, Ω3i = ci Λ1i , Ω4i = λ0 [(ci + 1)Ξ1i + Pi Pi ], Ω5i = −Pi [Ai − Ki ] − [Ai − Ki ]T Pi , a = −1 −1 −1 B T B −1 2 −1 max{ai , i ∈ M}, ai = µ[η−1 1 λmax ((Pi ) ) + η4 λmax ((Pi ) (Γi ) Γi )], b = max{bi , i ∈ M}, bi = µ[η2 (τ2 ) λmax ((Pi ) ) + −1 −1 C T C −1 2 −1 2 η5 λmax ((Pi ) (Γi ) Γi )], d = max{di , i ∈ M}, di = µ[ci λmax ((Pi ) Λi ) + (ci + 1)λ0 λmax ((Pi ) Ξi )], i ∈ M. Consider a function V(x, σ(t)) defined as (12). If there exists tˆ ∈ [t p , t p+1 ) for some p ∈ N satisfying V(tˆ, σ(tˆ)) , 0, V(t, σ(t)) ≤ V(tˆ, σ(tˆ)), ∀t ∈ [tˆ − τ, tˆ], then D+ EV(t, σ(t))|t=tˆ < 0. Theorem 4 Assume that x(t) = x(t, φ) is a solution of system (49) through (0, φ) and Assumptions 1-4 hold. Given N ∈ N + , if for any i, j ∈ M, there exist matrices Pi , Ξi , Λi > 0, Ki , and constants µi, j ≥ 1, θi ≥ 0, θ ≥ 0, η1 , η2 , η3 , η4 , η5 > 0, ξ ≥ 1 such that (50)-(52) hold. Moreover, G denotes the class of switching sequences tk satisfying lnµ(N) . (53) inf {tNn − tNn−1 } > τ + n∈Z+ g(µ(N), 0) Then, system (1) under controller (49) is globally mean square exponentially stabilized over the class G.
The proof of Theorem 3 and Theorem 4 can be completed as those of Theorems 1-2, the proof is omitted here.
4. Illustrative examples In this section, we give two numerical examples and their simulations to show the effectiveness of the developed results. Example 1 Consider a two-dimensional time delay switched system with two subsystems # " R τ2 x(t − s) x(t − τ1 ) dx(t) = −Ai (t)x(t) + Bi (t) + Ci (t) 0 ds dt + %(t, x(t), x(t − τ))dB(t) 5 4 R + < ι(t, x(t), x(t − τ), $)N(dt, d$), where τ1 = τ2 = τ = 0.2, i ∈ {1, 2} and we take ! ! 5 0 2 −1 2 A1 = , B1 = , C1 = 0 5 −2 1 −2 5 A2 = 0
! 0 1 , B2 = 6 1
! −2 1 , C2 = 1 −1
! ! 1 0.1 0 A B C , i = 1, 2, , Qi = Qi = Qi = −1 0 0.1 ! ! 0.1 0 −2 A B C , i = 1, 2, , Γi = Γi = Γi = −1 0 0.1
%(t, x(t), x(t − τ)) = 13
x(t) + x(t − τ) , 10
(54)
ι(t, x(t), x(t − τ), $) =
[x(t) + x(t − τ)]$2 . 10
To choose =
1/10 0 0 1/10
Ξ11 = Ξ21 = Ξ12 = Ξ22 =
1/10 0 0 1/10
Λ11
=
Λ21
=
Λ12
=
Λ22
!
!
, ,
and Fi (t) be any matrix satisfying FiT (t)Fi (t) ≤ In for all t ∈ <, i = 1, 2. Let B(t) and N(t, $) be all one-dimensional. The plot of L´evy noise is presented as Figure 1 and Figure 2. We choose V(t, 1) = (x12 + x22 ) exp(γ0 t), V(t, 2) = (x12 + 0.5x12 x22 + x22 ) exp(γ0 t), and obtain the following estimates: µ1,2 = 1.4, µ2,1 = 1.4, µ = 1.4, c1 = 1, c2 = 1.3. To choose ηi = 1, i = 1, · · · , 5, θ1 = 7, θ2 = 7, θ = 7, then we have γ0 = 0.2, a = 0.9006, b = 0.0335, d = 0.674, g(1, 0) = 5.3919 > 0. Considering the case of N = 3, it is easy to check that in this case g(µ(N), γ0 ) = 1.1511 > 0. Hence, based on Theorem 2, the class G, where G denotes the class of switching sequences {t p } p∈M , should satisfy inf {t3n − t3n−1 } > 0.9735.
n∈N
We choose t3n − t3n−1 = 1, n ∈ N + ,
(55)
tn − tn−1 = 0.2, n , 3k, k ∈ N + , n ∈ N + .
(56)
Thus, the conditions of Theorem 2 are all satisfied, so system (54) under the switching rule (55), (56) is globally mean square exponentially stable from Theorem 2. The execution of switching signals (55), (56) and dynamic behavior of (54) with initial states x(t) = [2, −1]T and (55), (56) are simulated as Figure 3 and Figure 4, respectively. It is noted that, even there exist high-frequency switching in some domains, the stability of system (54) can still be assured as long as the partial dwell time constrained. Example 2 Consider a three-dimensional time delay switched system with four subsystems # " R τ2 x(t − s) x(t − τ1 ) + Ci (t) 0 ds + Ui (t) dt + %(t, x(t), x(t − τ))dB(t) dx(t) = −Ai (t)x(t) + Bi (t) 4 5 R + < ι(t, x(t), x(t − τ), $)N(dt, d$),
where τ1 = τ2 = τ = 0.2, i ∈ {1, 2, 3, 4} and we take 1 0 0 1 A1 = 0 1 0 , B1 = −1 0 0 1 1 2 A2 = 0 0 1 A3 = 0 0 1 A4 = 0 0
2 2 1 3 1 , C1 = 2 1 2 1
1 1 1
2 1 1 1 −2 0 0 , B2 = 1 2 2 , C2 = 2 1 1 1 2 1 1 3 1 1 1 2 2 0 0 2 0 , B3 = 2 1 1 , C3 = 3 1 0 2 1 −1 2 1 −1 1 2 1 3 2 0 0 1 0 , B4 = 1 3 1 , C4 = 2 1 0 2 1 3 2 3 −1 0.1 0 0.25 0 0 0.25 QiA = ΓiA = 0 0.1 0 , QiB = ΓiB = QCi = ΓCi = 0 0 0 0.1 0 0 0 1 0
14
1 −1 , 2
1 1 , 1 1 3 , 1 1 1 , 1 0 0 , i = 1, 2, 3, 4, 0.25
(57)
%(t, x(t), x(t − τ)) = ι(t, x(t), x(t − τ), $) = To choose
1/10 0 1/10 Λ1i = Λ2i = 0 0 0 1/10 0 1/10 Ξ1i = Ξ2i = 0 0 0
x(t) + x(t − τ) , 10
[x(t) + x(t − τ)]$2 . 10 0 0 1/10 0 0 1/10
, i = 1, 2, 3, 4,
, i = 1, 2, 3, 4,
and Fi (t) be any matrix satisfying FiT (t)Fi (t) ≤ In for all t ∈ <, i = 1, 2, 3, 4. Let B(t) and N(t, $) be all onedimensional. The plot of L´evy noise is presented as Figure 5 and Figure 6. We choose tn − tn−1 = 0.2, n ∈ N + .
(58)
Figure 7 and Figure 8 show the execution of switching signal (58) and system (57) without external controllers, respectively. Such figures implies that system (57) with initial states x(t) = [2, −1, −2]T and Ui (t) = 0, i ∈ {1, 2, 3, 4} is unstable. Next, we choose V(t, 1) = (2x12 + 2x22 + 1.8x32 ) exp(γ0 t), V(t, 2) = (2x12 + 0.1x1 x2 + 2x22 + 1.52x32 ) exp(γ0 t), V(t, 3) = (1.4x12 +2x22 +1.34x32 ) exp(γ0 t), V(t, 4) = (1.6x12 +0.25x2 x3 +1.8x22 +1.8x32 ) exp(γ0 t) and obtain the following estimates: µ1,2 = 1.4, µ1,3 = 1.5, µ1,4 = 1.5, µ2,1 = 1.4, µ2,3 = 1.5, µ2,4 = 1.5, µ3,1 = 1, µ3,2 = 1.2, µ3,4 = 1.4, µ4,1 = 1.3, µ4,2 = 1.4, µ4,3 = 1.5, µ = 1.5, c1 = 2, c2 = 2.5, c3 = 2, c4 = 2.5. To choose ηi = 1, i = 1, 2, 3, 4, 5, θ1 = 8, θ2 = 7, θ3 = 6, θ4 = 7, θ = 6, then we have γ0 = 0.2, a = 0.8586, b = 0.0331, d = 0.6519, g(1, 0) = 4.4564 > 0. Considering the case of N = 3, then we have inf n∈N + {t3n − t3n−1 } > 1.7393 from (53). According to Theorem 3 and Theorem 4, we can choose
U1 (t) = U2 (t) = U3 (t) = U4 (t) =
−6x12 (t) − 6x22 (t) − 6x32 (t),
−5x12 (t) − 6x22 (t) − 6x32 (t),
−6x12 (t) − 5x22 (t) − 5x32 (t),
(59)
−6x12 (t) − 6x22 (t) − 6x32 (t),
t3n − t3n−1 = 1.8, n ∈ N + ,
tn − tn−1 = 0.2, n , 3k, k ∈ N + , n ∈ N + .
(60) (61)
Thus, the conditions of Theorem 2 are all satisfied, so system (57) under controllers (59) and switching rule (60), (61) is globally mean square exponentially stable from Theorem 4. The execution of switching signals (60), (61) and dynamic behavior of (57) with initial states x(t) = [2, −1, −2]T and (59) are simulated as Figure 9 and Figure 10, respectively. It is noted that, system (57) with switching rule (60), (61) is stabilized by these controllers and flexible switching rule. 5. Concluding remarks The paper has considered a class of uncertain switched delay systems with L´evy noise and frequency switching rule, and provided some sufficient conditions for global mean square exponential stability and stabilization of these 15
systems under given switching signals. The major analysis methods are the theory of switched systems and mathematical induction. Finally, two numerical examples are presented to show the effectiveness of the results. The approaches proposed in this paper may be extended to more general switched systems, such as switched systems with partial information of subsystems. Here, some related future works are given. 1) In this paper, g(1, γ) > 0 derived in Theorem 1 implies that the value of time delay maybe not too large. Thus, it is important to study the stability of uncertain switched delay systems by relaxing the condition. 2) Although this paper proposed a method to investigate the stability and stabilization of uncertain switched system with L´evy noise and flexible switching signals, sufficient conditions used to constrain the dynamic behavior of subsystems are still stronger. Hence, seeking for a more effective approach to relax the conservativeness is in our field of interest. 3) In real systems, many abrupt failure or unexpected perturbance are hard to be witnessed, which may be an obstacle to obtain accurate information from the activated model [58]-[59]. Thus, investigations of the case with partial information of subsystems and the stability issues are also part of our future extensions. Acknowledgements This work was supported by the Natural Science Foundation of China (Grant No. 61573095) and the Natural Science Foundation of Shanghai (Grant No. 16ZR1446700). References [1] D. Liberzon, A. S. Morse, Basic problems instability and design of switched systems, IEEE Control Syst. 19 (5) (1999) 59-70. [2] D. Liberzon, J. P. Hespanha, A. S. Morse, Stability of switched systems: A Lie-algebraic condition, Syst. Control Lett. 37 (3) (1999) 117-122. [3] H. Lin, P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Autom. Control 54 (2) (2009) 308-322. [4] D. Liberzon, Switching in Systems and Control, Birkhauser, 2003. [5] Y. M. Jia, Alternative proofs for improved LMI representations for the analysis and the design of continuoustime systems with polytopic type uncertainty: A predictive approach, IEEE Trans. Autom. Control 48 (8) (2003) 1413-1416. [6] W. Xie, C. Wen, Z. Li, Input-to-state stabilization of switched nonlinear systems, IEEE Trans. Autom. Control 46 (7) (2001) 1111-1116. [7] J. Daafouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach, IEEE Trans. Autom. Control 47 (11) (2002) 1883-1887. [8] H. Ishii, T. Basar, R. Tempo, Randomized algorithms for synthesis of switching rules for multimodal systems, IEEE Trans. Autom. Control 50 (6) (2005) 754-767. [9] S. P. Huang, Z. R. Xiang, Finite-time stabilization of a class of switched stochastic nonlinear systems under arbitrary switching, Int. J. Robust Nonlinear Control 26 (10) (2016) 2136-2152. [10] L. G. Wu, R. N. Yang, P. Shi, X. J. Su, Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings, Automatica 59 (2015) 206-215. [11] D. Zhai, L. An, J. Dong, Q. Zhang, Adaptive exact tracking control for a class of uncertain nonlinear switched systems with arbitrary switchings, J. Frankl. Inst. 354 (7) (2017) 2816-2831. [12] Y. M. Jia, Robust control with decoupling performance for steering and traction of 4WS vehicles under velocityvarying motion, IEEE Trans. Control Syst. Technol. 8 (3) (2000) 554-569. [13] Z. H. Guan, D. J. Hill, X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Autom. Control 50 (7) (2005) 1058-1062. [14] P. Shi, F. Li, A survey on Markovian jump systems: modeling and design, Int. J. Control Autom. Syst. 13 (1) (2015) 1-16. [15] W. Zhou, Q. Zhu, P. Shi, H. Su, J. Fang, L. Zhou, Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters, IEEE Trans. Cybern. 44 (12) (2014) 2848-2860. 16
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18
8
3 2
7
1
Random jump amplitude
6 0
B(t)
-1 -2 -3
5
4
3 -4 2
-5 -6
0
2
4
6
8
10
12
14
16
18
1
20
2
4
6
8
10
Time(second)
12
14
16
18
20
Time(second)
Figure 1: Brownian motion
Figure 2: Poisson point process with normally distributed jump 3
2
x 1 &x2
1
0
-1
-2
-3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time(second)
Figure 3: An execution of switching signal with (55), (56)
Figure 4: Trajectory of system (54) with (55), (56)
19
5
0
2
-0.5
1
-1
Random jump amplitude
0
B(t)
-1
-2
-3
-1.5 -2 -2.5 -3 -3.5
-4 -4 -5
-6
-4.5
0
2
4
6
8
10
12
14
16
18
-5
20
2
4
6
8
10
Time
12
14
16
18
20
Time
Figure 5: Brownian motion
Figure 6: Poisson point process with normally distributed jump
4.5
3
4 2 3.5 1
x 1 &x2 &x3
{1,2,3,4}
3
2.5
0
2 -1 1.5 -2 1
0.5
0
1
2
3
4
5
6
7
8
9
-3
10
0
5
10
Time(second)
Figure 7: An execution of switching signal with (58)
15
20
25
Time(second)
Figure 8: Trajectory of system (57) with Ui (t) = 0, i ∈ {1, 2, 3, 4} and (58) 3
2
x 1 &x2 &x3
1
0
-1
-2
-3
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time(second)
Figure 9: An execution of switching signal with (60), (61)
Figure 10: Trajectory of system (57) with (59) and (60), (61)
20