Stability of singularly perturbed switched systems with time delay and impulsive effects

Nonlinear Analysis 71 (2009) 4297–4308

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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Stability of singularly perturbed switched systems with time delay and impulsive effects Mohamad S. Alwan, Xinzhi Liu ∗ Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

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Article history: Received 18 September 2008 Accepted 26 February 2009 Keywords: Impulsive switched system Singular perturbation Time delay Exponential stability Multiple Lyapunov functions

abstract This paper studies the stability properties of singularly perturbed switched systems with time delay and impulsive effects. Such systems are assumed to consist of both unstable and stable subsystems. By using the multiple Lyapunov functions technique and the dwell time approach, some stability criteria are established. Our results show that impulses do contribute in order to obtain stability properties even when the system consists of only unstable subsystems. Numerical examples are presented to verify our theoretical results. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Many physical systems can be characterized by the fact that at a certain moment of time they exhibit switching from one operating mode to another or an abrupt change in their states. Such systems are suitably modeled as hybrid systems. A hybrid system consists of a family of continuous-time dynamical subsystems and a set of variables taking values in a finite or countable set. Two special kinds of hybrid system are considerably important, namely, switched systems and impulsive systems. In a switched system, the system behavior is represented by multi-dynamical subsystems and a switching signal to orchestrate switching between them. This type of system appears in different areas such as robotics, power electronics, multimedia, automated highway systems, and air traffic management systems. Stability and stabilizability problems of such systems have gained increasing attention (see [1–6]). It has been shown in [5] that when subsystems are exponentially stable and there exists a constant time called the dwell time, the activation time between jump discontinuities, the switched system is exponentially stable under any switching law. An extension has been made when a switched system incorporates both unstable and stable subsystems [6]. This case occurs when subsystems are viewed as closed-loop systems and, on some time intervals, some controllers are not available, meaning that unstable subsystems are given as open-loop systems, while the stable ones are represented by closed-loop systems. In an impulsive system, the dynamics are usually characterized by a pair of equations, i.e. a differential equation that describes a continuous evolutionary process and a difference equation that governs the discrete impulsive actions. Impulsive systems have applications in various fields such as physics, biology, engineering, population dynamics, aeronautics [7–9], and secure communications [10,11]. The theory of impulsive differential equations is richer than the corresponding theory of differential equations without impulses. For instance, the initial value problem of such equations may not have solutions even when the corresponding differential equations do; some fundamental properties such as the continuous dependence on initial condition, continuation of solutions, or stability may be violated or need new interpretation. On the other hand, under some conditions, impulses stabilize some systems even when the underlying systems are unstable [12,13], or make continuation of solutions possible. For more motivation,

∗

Corresponding author. Tel.: +1 519 888 4567/36007; fax: +1 519 746 4319. E-mail address: [email protected] (X. Liu).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.131

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interested readers may consult [9]. The stability of impulsive systems has received a great deal of work [7,8,14,9,11,12, 15,13]. The field of impulsive systems is currently very active since the applications of this theory have been popularly increasing. In some other cases, impulsive effects may arise as a result of switching that leads us to impulsive switched systems [16]. Examples of these systems include biological systems such as biological neural networks and bursting rhythm models. Other applications of these systems occurring by design include mechanical systems, automotive industry, aircraft, air traffic control and chaotic based secure communication. Few researchers have studied these systems including Alwan et al. [17], Guan et al. [14], Lakshmiknatham and Liu [16], and Wang et al. [13]. In a wide class of large-scale interconnected systems such as in power systems, large economies or even in networks one encounters dynamics with different speeds or multiple time scales. The singular perturbation technique is an adequate tool to describe such systems. During the last century, these systems were extensively studied by many researchers and, as a result, there have been many works available in literature(see for instance [19–22] and references therein). A large portion of those works was devoted to the stability problem (see [23–26]). On the other hand, time delay is unavoidable in many physical systems. The study of time delayed systems is usually more challenging than that of ordinary systems [27], and the presence of delay, even in first order systems, may cause undesirable performance such as oscillations or chaotic behaviors [28]. A small delay may hurt the stability of some systems [29]. Most of the work has focused on the stability problem of these systems (see [30–33] and references therein.) Delayed singularly perturbed systems are investigated in [34–36]. Most of the works focus mainly on the stability problem and finding an upper bound for the perturbation parameter so that a system under consideration is stable. Alwan et al. [18] have explored the stability property of switched singularly perturbed systems with time delay. In this paper, we shall investigate the stability problem of impulsive switched delay singularly perturbed systems and obtain some stability criteria utilizing Lyapunov functions and differential inequalities. The rest of this paper is organized as follows; in Section 2, we define some notations that shall be used throughout this paper. In Section 3, we establish some stability properties of systems consisting of stable and unstable subsystems and other consist of all unstable subsystems. Multiple Lyapunov function technique and dwell-time condition are used to analyze stability. Finally, our results are concluded in Section 4. 2. Problem formulation Consider the following impulsive switched delay systems x˙ = fσ (t ) (t , x, xt , z , zt ),

ε˙z = gσ (t ) (t , x, xt , z , zt ), ∆x = Bk x(t − ), t = tk ∆z = Ck z (t − ), t = tk

t 6= tk t 6= tk

(1)

where x ∈ Rm , z ∈ Rn are respectively the slow and fast states of the system, and ε is a small positive parameter. For t0 ≥ 0 and S = {1, 2, . . . , N } with N > 1 being the number of subsystems, σ : [t0 , ∞) → S , which is represented by {ik } according to [tk−1 , tk ) → ik ∈ S , is a piecewise constant function called the switching signal. The role of the switching signal is to switch between the vector fields in the right-hand side of (1). In fact ik (or i for simplicity of notation) means the ith subsystem is activated on the subinterval [tk−1 , tk ). The discontinuities of σ form a strictly increasing sequence of switching or impulsive times {tk }∞ k=1 which satisfy tk−1 < tk and limk→∞ tk = ∞, i.e. the impulses are a consequences of the switchings. For any switching signal σ , we denote by T + (t0 , t ) and T − (t0 , t ) the total activation time of unstable and stable subsystems respectively over the time interval [t0 t ). Later, we will show that, for any t0 , if the ratio of T − (t0 , t ) to T + (t0 , t ) is no smaller than a specific positive constant, exponential stability is guaranteed. The set of switching signals satisfying the above condition may be denoted by Sd [TD ]. We assume that x(tk+ ) = x(tk ), meaning that the solution is rightcontinuous. ∆y = y(t ) − y(t − ), where y(t − ) = lims→t − y(s), represents the difference between the state just before and after the impulse action. The vector field functions fi and gi are as defined in Ballinger and Liu [37], and fi (t , 0, 0, 0, 0) ≡ 0 and gi (t , 0, 0, 0, 0) ≡ 0. Definition 1. Let x(t ) be a function mapping [t0 − τ , ∞) into Rn . Then for any fixed t ∈ [t0 , ∞), a new function xt mapping [−τ , 0] into Rn is defined as follows xt (θ ) = x(t + θ ),

for θ ∈ [−τ , 0].

For all t ≥ t0 , let x(t ) and z (t ) represent the solutions of (1) with the initial conditions xt0 and zt0 , respectively. Denote by Cτ = PC ([−τ , 0], Rn ), with τ > 0, representing a time delay, the set of piece-wise continuous functions from [−τ , 0] to Rn . If φ ∈ Cτ , the τ -norm of this function is defined by kφt kτ = sup−τ ≤θ ≤0 kφ(θ )k, where k · k is the Euclidean norm on Rn . We also define xt − ∈ PCP ([−τ , 0], Rn ) by xt − (s) = x(t + s) for −τ ≤ s < 0 and xt − (s) = x(t − ) for s = 0. n Let kAk = max1≤j≤n i=1 |aij | be the norm of the n × n matrix A = (aij ) and AT be the transpose of a matrix A. A matrix T P = P is said to be positive definite if all its eigenvalues are positive. Denote by λmin (P ) and λmax (P ) the minimum and

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maximum eigenvalue of P, respectively. For i ∈ S , if Ai is Hurwitz,1 then there exist positive definite matrices Pi and Qi satisfying Lyapunov equation ATi Pi + Pi Ai = −Qi . Define Vi (x) = xT Pi x. Then, we have, for all i ∈ S ,

λmin (Pi )kxk2 ≤ Vi (x) ≤ λmax (Pi )kxk2 and, for any i, j ∈ S , Vj (x) ≤ µVi (x), where µ = λM /λm ≥ 1 with λM = max{λmax (Pi ), ∀i ∈ S } and λm = min{λmin (Pi ), ∀i ∈ S }. Definition 2. The trivial solution of system (1) is said to be exponentially stable if there exist positive constants K , and λ such that

  kx(t )k + kz (t )k ≤ K kxt0 kτ + kzt0 kτ e−λ(t −t0 ) ,

t ≥ t0

for all x(t ) and z (t ), the solutions of system (1). Remark. Every subsystem in (1) is viewed as an interconnected system. A proper way to deal with such a system is to decompose it into small isolated subsystems and study the stability of each individual subsystems; namely, we initially ignore the interconnection between the subsystems. In the next step, we combine our results from the first step to draw conclusion about the stability of the interconnected system. For doing so, we introduce a special kind of matrix called an M-matrix, which plays an important role in analyzing the stability of large-scale interconnected systems. Definition 3 ([33]). An n × n matrix S = [sij ] with sij ≤ 0, i 6= j is said to be an M-matrix if its leading (successive) principal minors are positive. The following is a delay-version comparison lemma that will be frequently used to calculate the growth and decay rates of unstable and stable subsystems. Lemma ([38,35]). Consider the following differential inequality y˙ (t ) ≤ A(t )y(t ) + B(t )

sup

y(θ ),

t ∈ [t0 , ∞), t0 ≥ 0

t −τ ≤θ≤t

where A(t ), B(t ) be n × n matrices of continuous, bounded functions, y(t ) = supt −τ ≤θ≤t y(θ ) = (supt −τ ≤θ≤t y1 (θ ), supt −τ ≤θ≤t y2 (θ ), . . . , supt −τ ≤θ≤t yn (θ ))T .

(y1 (t ), y2 (t ), . . . , yn (t ))T ≥ 0, and

S1. Assume that A(t ) and AT (t ) + A(t ) are Hurwitz, A˙ (t ) is bounded, and the following inequality hold

λmax (AT (t ) + A(t )) + 2kB(t )k < 0. Then, there exists a positive constant ζ such that

ky(t )k ≤ kyt0 kτ e−ζ (t −t0 ) ,

t ≥ t0 .

where ζ is the unique positive solution of

ζ + λmax (AT (t ) + A(t )) + kB(t )k + kB(t )keζ τ = 0. S2. For any positive constants β1 and β2 for which kB(t )k ≤ β1 and λmax (AT (t ) + A(t )) + 2kB(t )k ≤ β2 , we have

ky(t )k ≤ kyt0 kτ eξ (t −t0 ) ,

t ≥ t0

where ξ = (β1 + β2 )/2. 3. Main results In this section, we shall establish some Lyapunov-type sufficient conditions to warrant some stability properties of linear and special class of nonlinear systems. Throughout this paper, we denote by Su = {1, 2, . . . , q} and Ss = {q+1, q+2, . . . , N } the sets of indices of the unstable and stable subsystems, respectively. 3.1. Linear Systems Consider the following impulsive switched system 1 An n × n matrix A is said to be Hurwitz if all its eigenvalues have negative real parts.

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x˙ = A11i x + A12i xt + B11i z + B12i zt ,

εi z˙ = A21i x + A22i xt + B21i z , ∆x = Bk x(t ), ∆z = Ck z (t − ), −

t 6= tk

t 6= tk

(2)

t = tk t = tk

where, for any i ∈ S = Su ∪ Ss , the matrices A11i and A12i are of dimension m × m, B11i and B12i of dimension m × n, A21i and A22i of dimension n × m, B21i of dimension n × n, and B21i is nonsingular and Hurwitz. The delayed fast variable is dropped in the fast system for simplicity of calculations. Before stating the sufficient conditions, let the uncoupled slow and fast subsystems of (2) be given by x˙ = A11i x

and εi z˙ = B21i z ,

respectively. Theorem 1. The trivial solution of (2) is exponentially stable if the following assumptions hold. A1. For i ∈ Su , A11i has eigenvalues with positive real parts and, for i ∈ Ss , A11i is Hurwitz; A2. for i ∈ S and t ∈ [tk−1 , tk ), there exist positive constants a11i , a12i , a21i , a22i , b11i , b12i , b21i , b22i such that 2xT P1i [A12i xt + B11i z + B12i zt ] ≤ a11i kxk2 + a12i kxt k2τ + b11i k(z − hi )k2 + b12i k(z − hi )t k2τ ,

−2(z − hi )T P2i h˙ i ≤ a21i kxk2 + a22i kxt k2τ + b21i k(z − hi )k2 + b22i k(z − hi )t k2τ , 1 where hi (t ) = −B− 21i [A21i x + A22i xt ] and P1i and P2i are positive definite matrices satisfying the Lyapunov equations

AT11i P1i + P1i A11i = −Q1i + 2γ ∗ P1i , AT11i P1i + P1i A11i = −Q1i , BT21i P2i + P2i B21i = −Q2i ,

i ∈ Su i ∈ Ss i∈S

for any positive definite matrices Q1i and Q2i ;

A3. (i) for all i ∈ Su , assume that λmin (e ATi + e Ai ) + ke Bi k > 0, where a   b a 11i 11 12i i ∗ 2γ +  λ1m   λ λ 1m 2m e e Bi =  a Ai =  a21i λmin (Q2i ) − εi b21i  , 22i

−

λ1m

εi λ2m

λ1m

and γ ∗ is a positive constant such that the matrix A11i − γ ∗ I is Hurwitz;

b12i



λ2m  b22  i

λ2m

(ii) for i ∈ Ss , there exist positive constants εi∗ such that −e Ai is an M-matrix and λmax (e Ai + e AT ) + 2ke Bi k < 0 where a  λ (Q ) − a   i b11i b min 1i 11i 12i 12i

−  e Ai = 

λ1M

a21i

λ1m

 λ2m λmin (Q2i ) − εi∗ b21i  , − εi∗ λ2M

 1m e Bi =  λ a 22i

λ1m

λ2m  , b22  i

λ2m

and Q1i and Q2i are defined in Assumption A2; A4. let λ+ = max{ξi : i ∈ Su }, λ− = min{ζi : i ∈ Ss } with ξi and ζi being the growth and decay rates of unstable and stable subsystems, respectively, and, for any t0 , assume that the switching signal guarantees that inf

t ≥ t0

T − (t0 , t ) T + (t0 , t )

≥

λ+ + λ∗ , λ− − λ∗

(3)

where T + (t0 , t ) and T − (t0 , t ) are defined in the previous section and λ∗ ∈ (0, λ− ). Furthermore, there exists 0 < ν < ζi such that (i) for i ∈ Su and k = 1, 2, . . . , l ln µ(αk + βk + γk + ψk ) − ν(tk − tk−1 ) ≤ 0;

(4)

(ii) for i ∈ {l + 1, l + 2, . . . , N − 1} and k = l + 1, l + 2, . . . , N − 1 ln µ(αk + βk + γk + ψk eζi τ ) + ζi τ − ν(tk − tk−1 ) ≤ 0,

(5)

where ζi is the unique positive solution of

ζi + λmax (e ATi + e Ai ) + ke Bi k + ke Bi keζi τ = 0, λ2M αk = µλ2max ([I + Bk ]), βk = (kUk k + rk + sk )rk , λ1m

γk = µ(kUk k + rk + sk )kUk k,

M.S. Alwan, X. Liu / Nonlinear Analysis 71 (2009) 4297–4308

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λ2M (kUk k + rk + sk )sk , Uk = I + Ck , λ1m −1 1 rk = max{kRik k : Rik = [I + Ck ]B− 21i A21i − B21i A21i [I + Bk ] ∀i ∈ S }, and ψk =

−1 1 sk = max{kSik k : Sik = [I + Ck ]B− 21i A22i − B21i A22i [I + Bk ] ∀i ∈ S }.

Proof. Define Vi (t ) = Vi (x(t )) = xT (t )P1i x(t ) and Wi (t ) = Wi ((z − hi )(t )) = (z − hi )T (t )P2i (z − hi )(t ). Then, the time derivative of Vi and Wi along the trajectories of x(t ) and z (t ) are (i) for i ∈ Su V˙ i (t ) ≤



2γ ∗ +

a11i



b11i

λ2m

Wi ((t )) +

a12i

b12i

kWit kτ , λ1m λ2m ˙ i (t ) = (˙z − h˙ i )T P2i (z − hi ) + (z − hi )T P2i (˙z − h˙ i ) W 1 T  1 = (A21i x + A22i xt + B21i z ) − h˙ i P2i (z − hi ) + (z − hi )T P2i (A21i x + A22i xt + B21i z ) − h˙ i ε ε a21i λmin (Q2i ) − εi b21i a22i b22i ≤ Vi ( t ) − Wi (t ) + kVit kτ + kWit kτ , λ1m εi λ2M λ1m λ2m λ1m

Vi (t ) +

kVit kτ +

(ii) for i ∈ Ss b11i a12i b12i λmin (Q1i ) − a11i Vi (t ) + Wi (t ) + kVit kτ + kWit kτ , λ1M λ2m λ1m λ2m λ (Q ) − εi∗ b21i a22i b22i a ˙ i (t ) ≤ 21i Vi (t ) − min 2i Wi (t ) + k Vi k τ + kWit kτ , W λ1m εi∗ λ2M λ1m t λ2m V˙ i (t ) ≤ −

˙ (t ) in a vector form yields, for i ∈ Su , where εi∗ ≥ i > 0. Combining V˙ (t ) and W  

V˙ (t ) ˙ (t ) W



2γ ∗ +

 ≤

a21i

a11i

λ1m

λ1m

b11i

a



12i

 V (t )  λ1m λ2m λmin (Q2i ) − εi b21i  W (t ) +  a22i − λ1m εi λ2M 



b12i



  λ2m  kVit kτ  b22i kWit kτ λ2m

and, by A3(i) and S2, there exists ξi > 0 such that Vi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )eξi (t −tk−1 ) Wi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )eξi (t −tk−1 ) . Similarly, for i ∈ Ss , we get





− V˙ (t )  ≤  ˙ (t ) W 

λmin (Q1i ) − a11i λ1M a21i

λ1m

b11i



a 12i    λ2m V ( t )  λ  +  a 1m λmin (Q2i ) − εi∗ b21i  W (t ) 22i − λ1m εi∗ λ2M

b12i

  λ2m  kVit kτ b22i  kWit kτ λ2m

and, by A3(ii) and S1, there exists ζi > 0 such that Vi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )e−ζi (t −tk−1 ) Wi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )e−ζi (t −tk−1 ) . At t = tk , we have Vi (tk ) = x(tk )T P1i x(tk )

  ≤ λmax [I + Bk ]T P1i [I + Bk ] xT (tk− )x(tk− ) = αk Vi (tk− ), where αk = µλ2max (I + Bk ). We also have Wi (tk ) =



T

z (tk ) − hi (tk )



P2i z (tk ) − hi (tk )



oT n o n 1 −1 [ A x ( t ) + A x ] P [ A x ( t ) + A x ] = z (tk ) + B− z ( t ) + B 21 k 22 t 2 k 21 k 22 t 21i 21i i i k i i i k n oT 1 − = [I + Ck ]z (tk− ) + B− P2i 21i [A21i [I + Bk ]x(tk ) + A22i [I + Bk ]xt − ] k



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o − 1 × [I + Ck ]z (tk− ) + B− 21i [A21i [I + Bk ]x(tk ) + A22i [I + Bk ]xtk− ]   T  = z (tk− ) − hi (tk− ) UkT P2i Uk z (tk− ) − hi (tk− ) + xT (tk− )RTik P2i Rik x(tk− )  T + xTt − SikT P2i Sik xt − − 2 z (tk− ) − hi (tk− ) UkT P2i Rik x(tk− ) k k  T − 2 z (tk− ) − hi (tk− ) UkT P2i Sik xt − − 2xT (tk− )RTik P2i Sik xt − n

k

k

≤ kUk k2 · kP2i k · kz (tk− ) − hi (tk− )k2 + kRik k2 · kP2i k · kx(tk− )k2   + kSik k2 · kP2i k · kxt − k2τ + kUk k · kP2i k · kRik k kz (tk− ) − hi (tk− )k2 + kx(tk− )k2 k     + kUk k · kP2i k · kSik k kz (tk− ) − hi (tk− )k2 + kxt − k2τ + kRik k · kP2i k · kSik k kx(tk− )k2 + kxt − k2τ k k  n kU k o k R k k S k k ik ik ≤ λmax (P2i ) kUk k + kRik k + kSik k Wi (tk− ) + Vi (tk− ) + kVi kτ λmin (P2i ) λmin (P1i ) λmin (P1i ) tk− o  n kU k rk sk k Wi (tk− ) + Vi (tk− ) + k Vi − k τ ≤ λ2M kUk k + rk + sk λ2m λ1m λ1m tk = γk Wi (tk− ) + βk Vi (tk− ) + ψk kVi − kτ , t k













where βk = λ2M kUk k+ rk + sk λ k , γk = λ2M kUk k+ rk + sk λ k , ψk = λ2M kUk k+ rk + sk λ k , rk = max{kRik k; ∀i ∈ S }, 1m 2m 1m and sk = max{kSik k; ∀i ∈ S }. For instance, if we run an unstable subsystem on the first interval and a stable one on the second interval, we get, respectively, r









kU k

s

V1 (t ) ≤ kV1t0 kτ + kW1t0 kτ eξ1 (t −t0 ) , V2 (t ) ≤ kV2t1 kτ + kW2t1 kτ e−ζ2 (t −t1 ) , where

  kV2t1 kτ ≤ α1 µ kV1t0 kτ + kW1t0 kτ eξ1 (t1 −t0 ) ,   kW2t1 kτ ≤ µ(β1 + γ1 + ψ1 ) kV1t0 kτ + kW1t0 kτ eξ1 (t1 −t0 ) . Therefore, generally, one may have VN (t ) ≤

l Y

µ(αi + βi + γi + ψi )eξi (ti −ti−1 ) ×

i =1

Making use of A4, we have



∗ VN (t ) ≤ kV1t0 kτ + kW1t0 kτ e−(λ −ν)(t −t0 ) .

Similarly, we have



Y j =l +1

  × kV1t0 kτ + kW1t0 kτ e−ζN (t −tN −1 ) 

N −l−1



∗ WN (t ) ≤ kV1t0 kτ + kW1t0 kτ e−(λ −ν)(t −t0 ) .

Then, there exists K1 such that ∗ −ν)(t −t )/2 0

kx(t )k ≤ K1 (kxt0 kτ + kzt0 kτ )e−(λ

,

and by the fact that 1

1/2

kz k − khi k ≤ kz − hi k ≤ √ W λ2m i

,

there exists K2 such that ∗ −ν)(t −t )/2 0

kz (t )k ≤ K2 (kxt0 kτ + kzt0 kτ )e−(λ

.

µ(αj + βj + γj + ψj eζj τ )eζj τ e−ζj (tj −tj−1 )

M.S. Alwan, X. Liu / Nonlinear Analysis 71 (2009) 4297–4308

4303

Fig. 1. Impulsive switched system with unstable and stable subsystems.

Hence, ∗ −ν)(t −t )/2 0

kx(t )k + kz (t )k ≤ K (kxt0 kτ + kzt0 kτ )e−(λ

.

where K = K1 + K2 . This shows that the trivial solution of (2) is exponential stable.



Example 1. Consider the impulsive switched system (2) with the following unstable and stable subsystems x˙ = x + 4z (t − 1),

ε˙z = x(t − 1) − z , x˙ = −5x + z (t − 1) ε˙z = x(t − 1) − z , and for any k, Bk = −1/2 and Ck = −1/2. Here, the switching signal σ takes values in the set {1, 2} alternatively. For the unstable subsystems, γ = 3, ε = 0.4, Q11 = 13, and Q21 = 1 give P11 = 3.25, P21 = 0.5, e Au =

e Bu =



0 0.31

26 0

10 0



0 −6

,



, and so A3(i) is satisfied. While for the stable subsystem, Q12 = 44 and Q22 = 8 give P12 = 4.4, P22 = 4,

and from A3(ii) we get ε ∗ = 0.15, e As =



−9 0

0



e −140 , Bs

=



0 6.15

8.8 0



. The dwell times are TDu = 1.5 and TDs = 4. Fig. 1

illustrates these results where unstable and stable subsystems are run alternatively. The set of switching or impulsive times 8 is {tk }kk= =1 = {1.5, 5.5, 7, 11, 12.5, 16.5, 18, 22}. For instance, σ (t ) = 1 (or 2) for t ∈ [0, 1.5)(or [1.5, 5.5)), respectively. In the following theorem, we show how impulses can play as a stabilizer in some impulsive systems where all subsystems are unstable. Theorem 2. Consider system (2) with S = {1, 2, . . . , N }. Assume that the following assumptions hold. A1. For any i ∈ S , A11i has eigenvalues with positive real parts; A2. A2 and A3(i) of Theorem 1 hold; A3. there exists a constant ϑ ≥ 1 such that





ln ϑµ(αi + βi + γi + ψi ) + ξi (tk+1 − tk ) ≤ 0, where µ, αi , βi , γi , ψi and ξi are defined in Theorem 1. Then, the trivial solution of (2) is stable if ϑ = 1 and asymptotically stable if ϑ > 1. Proof. Define Vi (t ) = xT P1i x and Wi (t ) = (z − hi )T P2i (z − hi ). Then, the time derivative of Vi and Wi along the trajectories of system (2) are

  a b a b ˙Vi (t ) ≤ 2γ + 11i Vi (t ) + 11i Wi (t ) + 12i kVit kτ + 12i kWit kτ λ1m λ2m λ1m λ2m a λ (Q ) − εi b21i a22i b22i ˙ i (t ) ≤ 21i Vi (t ) − min 2i W Wi (t ) + kVi kτ + kWit kτ . λ1m εi λ2M λ1m t λ2m

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Then, there exists a positive constant ξi such that Vi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )eξi (t −tk−1 ) Wi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )eξi (t −tk−1 ) . From Theorem 1, we have at t = tk Vi (tk ) ≤ αk Vi (tk− ) Wi (tk ) ≤ βk Vi (tk− ) + γk Wi (tk− ) + ψk kVi − kτ . t k

We also have, for t ∈ [tk , tk+1 ), Vi (t ) ≤ (kV1t0 kτ + kW1t0 kτ )eξ1 (t1 −t0 ) µ(α1 + β1 + γ1 + ψ1 )eξ2 (t2 −t1 )

× µ(α2 + β2 + γ2 + ψ2 )eξ2 (t3 −t2 ) · · · µ(αk + βk + γk + ψk )eξi (tk+1 −tk ) = (kV1t0 kτ + kW1t0 kτ )

1 ξ1 ( t 1 − t 0 ) e ϑµ(α1 + β1 + γ1 + ψ1 )eξ2 (t2 −t1 ) k

ϑ × ϑµ(α2 + β2 + γ2 + ψ2 )eξ2 (t3 −t2 ) · · · ϑµ(αk + βk + γk + ψk )eξi (tk+1 −tk )

≤ (kV1t0 kτ + kW1t0 kτ )

1 ξ1 ( t 1 − t 0 ) e .

ϑk

Similarly, Wi (t ) ≤ (kV1t0 kτ + kW1t0 kτ )

1 ξ1 (t1 −t0 ) e . k

ϑ

From Theorem 1, there exists a positive constant K such that K

kx(t )k + kz (t )k ≤ √

ϑk

(kxt0 kτ + kzt0 kτ )eξ1 (t1 −t0 )/2 .

Clearly, if ϑ = 1, then system (2) is stable, and if ϑ > 1 and k → ∞, the system is asymptotically stable. This completes the proof.  Example 2. Consider the impulsive switched system (2) with the following unstable subsystems x˙ = x + 3z (t − 1), ε˙z = 2x(t − 1) − 2z , x˙ = x + 2z (t − 1), ε˙z = 4x(t − 1) − 2z ,

ε = 0.7, ε = 0.7,

and difference equations are ∆x = −0.97x(t ) and ∆z = −0.9z (t ). The switching signal σ takes values in {1, 2} alternatively. Taking γ = 2, Q11 = 2, Q21 = 2, give us P11 = 1 and P21 = 0.5 and if Q12 = 3, Q22 = 1, then P12 = 1.5 and P22 = 0.25, A1 ) = {−2.3571, 14}, ke B1 k = 12, and so that the growth rates so that µ = 2. We also get, for the first subsystem, λ(e AT1 + e T e e B2 k = 12; so that the growth rates are are ξ1 = {19, 10.8214}. For the second subsystem, λ(A2 + A2 ) = {−0.9286, 14}, ke ξ2 = {19, 11.5357}. The impulse parameters are, for any k, αk = 0.0018, βk = 0, γk = 0.048, ψk = 0.0672. A simple check shows that A3 holds if ϑ ∈ [1, 4.2735). Taking ϑ = 2 for instance and ξ1 = 10.8214 give tk+1 − tk ≤ 0.0702 and 30 ξ2 = 11.5357 give tk+1 − tk ≤ 0.0658. Thus, if TD = 0.0658, then the switching or impulsive times are {tk }kk= =1 = kTD . Fig. 2 shows the simulation results.

3.2. Nonlinear systems Consider the following nonlinear system x˙ = A11i + gi (x, xt , z , zt ),

ε˙z = B21i z + Bi (x, xt ), ∆x = Bk x(t ), t = tk ∆z = Ck z (t ), t = tk

t 6= tk

t 6= tk

where i ∈ S = Su ∪ Ss and the n × n matrix B21i is nonsingular and Hurwitz. In the next theorem we establish exponential stability of the trivial solution of system (6).

(6)

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Fig. 2. Impulsive switched system with all unstable subsystems.

Theorem 3. The trivial solution of (6) is exponentially stable if the following assumptions hold. A1. A1 of Theorem 1 holds; A2. (i) there exist positive constants a11i , a12i , a21i , a22i , b11i , b12i , b21i , b22i such that 2xT P1i gi (x, xt , z , zt ) ≤ a11i kxk2 + a12i kxt k2τ + b11i kz − hi k2 + b12i k(z − hi )t k2τ ,

−2(z − hi )T P2i h˙ i ≤ a21i kxk2 + a22i kxt k2τ + b21i kz − hi k2 + b22i k(z − hi )t k2τ , 1 where hi (t ) = −B− 21i Bi (x(t ), xt ), and P1i and P2i are positive definite matrices satisfying Lyapunov equations T A11i P1i + P1i A11i = −Q1i + 2γ P1i , i ∈ Su AT11i P1i + P1i A11i = −Q1i , i ∈ Ss BT21i P2i + P2i B21i = −Q2i , i∈S

for any positive definite matrices Q1i , Q2i ; (ii) there exist positive constants a, b, c such that



T

n o [I + Ck ]T P2i [I + Ck ]hi (tk− ) − hi (tk ) o oT n n + [I + Ck ]hi (tk− ) − hi (tk ) P2i [I + Ck ]hi (tk− ) − hi (tk )

2 z (tk− ) − hi (tk− )

≤ akz (tk− ) − hi (tk− )k2 + bkx(tk− )k2 + c kxt − k2τ , k

1 where hi (tk ) = −B− 21i Bi (x(tk ), xtk ); A3. A3 of Theorem 1 holds; A4. A4 of Theorem 1 holds where αk = µ1 λ2max ([I + Bk ]), βk = b/λ1m , γk = µ2 λ2max ([I + Ck ]) + a, and ψk = c /λ1m .

Proof. For t ∈ [tk−1 , tk ), define Vi (t ) = xT (t )P1i x(t ) and Wi (t ) = (z − hi )T (t )P2i (z − hi )(t ). Then, the time derivative of Vi and Wi along the trajectories of x(t ) and z (t ) are (i) for i ∈ Su ,

  a b a b ˙Vi (t ) ≤ 2γ + 11i Vi (t ) + 11i Wi (t ) + 12i kVit kτ + 12i kWit kτ λ1m λ2m λ1m λ2m a21i λmin (Q2i ) − εi b21i a22i b22i ˙ i (t ) ≤ W Vi ( t ) − Wi (t ) + kVi kτ + kWit kτ , λ1m εi λ2M λ1m t λ2m (ii) for i ∈ Ss , λmin (Q1i ) − a11i b11i a12i b12i Vi (t ) + Wi (t ) + kVit kτ + kWit kτ λ1M λ2m λ1m λ2m a λ (Q ) − εi∗ b21i a22i b22i ˙ i (t ) ≤ 21i Vi (t ) − min 2i W Wi (t ) + k Vi k τ + kWit kτ . λ1m εi∗ λ2M λ1m t λ2m V˙ i (t ) ≤ −

Then, there exists positive constants ξi , (i ∈ Su ) such that Vi (t ) ≤ (kVi + kτ + kWi + kτ )eξi (t −tk−1 ) t k−1

t

k−1

Wi (t ) ≤ (kVi + kτ + kWi + kτ )eξi (t −tk−1 ) . t t k−1

k−1

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and ζi , (i ∈ Ss ) such that Vi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )e−ζi (t −tk−1 ) Wi (t ) ≤ (kVitk−1 kτ + kWitk−1 kτ )e−ζi (t −tk−1 ) . At t = tk , we have Vi (tk ) ≤ αk Vi (tk− ), where αk = µλ2max (I + Bk ), and Wi (tk ) =



z (tk ) − hi (tk )

T





P2i z (tk ) − hi (tk )

   T = z (tk− ) − hi (tk− ) [I + Ck ]T P2i [I + Ck ] z (tk− ) − hi (tk− )  T n o + 2 z (tk− ) − hi (tk− ) [I + Ck ]T P2i [I + Ck ]hi (tk− ) − hi (tk ) o n oT n + [I + Ck ]hi (tk− ) − hi (tk ) P2i [I + Ck ]hi (tk− ) − hi (tk )   ≤ λmax [I + Ck ]T P2i [I + Ck ] kz (tk− ) − hi (tk− )k2 + akz (tk− ) − hi (tk− )k2 + bkx(tk− )k2 + c kxt − k2τ k

= βk Vi (tk− ) + γk Wi (tk− ) + ψk kVi − kτ , t k

where βk = b/λ1m , γk = (λ2M λ2max [I + Ck ] + a)/λ2m , and ψk = c /λ1m . The rest of the proof is similar to that of Theorem 1; thus, it is omitted here.



Example 3. Consider impulsive switched system (6) with the following unstable and stable subsystems x˙ = 0.1x + sin z (t − 1)

ε˙z = 0.1x − z , x˙ = −10x + ln(1 + x2 (t − 1)) + z ε˙z = x − 2z and Bk = −1/2 and Ck = −1/2. For the unstable subsystem, we take Vu (x) = 0.5x2 , and Wu (z − h) = 0.5(z − h)2 , where h = 0.1x. One can easily  find 1.4 0 ˙ u ≤ (−2/ε + 0.12)Wu + 0.01Vu + 0.01kVu kτ + 0.1kWut kτ , e V˙ u (x) ≤ 1.4Vu + kWut kτ and W Au = 0.01 −2/ε + 0.12 , and

e Bu =



0 0.01



1 0.1

; taking ε = 0.1, the growth rate are ξ = {1.85, 4.8}. While for the stable subsystem, taking Vs (x) = 0.5x2 ,

2 ∗ ˙ ˙ Ws (z − h) =  0.5(z − h) where  h = 0.5xgive Vs ≤ −14Vs + Ws + 4kVst kτ , and Ws ≤ (−4/ε + 11.5)Ws + 5.5Vs + 2kVst kτ ;

thus, e As =

−14 5.5

1

e −4/ε ∗ + 11.5 and Bs

=

4 2

0 0

; by A3(ii), we get ε ∗ = 0.2341; if we take ε = 0.1 ∈ (0, 0.2341], the decay

rates are ζ = {1.5279, 2.4432}. The dwell times are TDu = 1.1 and TDs = 5. Fig. 3 shows these results after running unstable and stable subsystems alternatively. In the following we state sufficient conditions to guarantee stability and asymptotic stability of systems (6) with all unstable subsystems. Theorem 4. Consider the nonlinear system in (6) with S = {1, 2, . . . , N }. Assume that the following assumptions are satisfied. A1. For any i ∈ S , A11i has eigenvalues with positive real parts; A2. A2 of Theorem 3 and A3(i) of Theorem 1 hold; A4. there exists a constant ϑ ≥ 1 such that





ln ϑµ(αi + βi + γi + ψi ) + ξi (tk+1 − tk ) ≤ 0, where µ and ξi are defined in Theorem 1 and αi , βi , γi and ψi are defined in Theorem 3. Then, the trivial solution of (6) is stable if ϑ = 1 and asymptotically stable if ϑ > 1. The proof of this theorem is a consequence of the previous theorems; thus it is omitted here.

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4307

Fig. 3. Impulsive switched system with unstable and stable linear subsystems.

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