Exponential Stability for a Class of Switched Nonlinear Systems with Mixed Time-Varying Delays via an Average Dwell-Time Method

The problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied. The constraint on the derivative of the time-varying delay is not required which allows the time delay to be a fast time-varying function. We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems. Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton are formulated. New delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.


Introduction
The switched systems are an important class of hybrid systems.They are described by a family of continuous or discrete-time subsystems and a rule that orchestrates the switching between the subsystems.Recently, switched systems have attracted much attention due to the widespread application in control, chemical engineering processing 1 , communication networks, traffic control 2, 3 , and control of manufacturing systems 4-6 .A switched nonlinear system with time delay is called switched nonlinear delay system, where delay may be contained in the system state, control input, or switching signals.In 7-9 , some stability properties of switched linear delay systems composed of both stable and unstable subsystems have been studied by using an average dwell-time approach and piecewise Lyapunov functions.It is shown that when the average dwell time is sufficiently large and the total activation time of the unstable subsystems is relatively small compared with that of the Hurwitz stable subsystems, global exponential stability is guaranteed.The concept of dwell time was extended to average dwell time by Hespanha and Morse 10 with switching among stable subsystems.Furthermore, 11 generalized the results to the case where stable and unstable subsystems coexist.
The stability analysis of nonlinear time-delay systems has received increasing attention.Time-delay systems are frequently encountered in various areas such as chemical engineering systems, biological modeling, and economics.The stability analysis for nonlinear time-delay systems has been investigated extensively.Various approaches to such problems have been proposed, see 12-14 and the references therein.It is well known that the existences of time delay in a system may cause instability and oscillations system.Thus, the stability analysis of nonlinear time-delay systems has received considerable attention for the last few decades, see 15, 16 .Recently, the exponential stability problems for time-delay systems have been studied by many researchers, see for examples 17-21 .The problems have been dealt with for various control areas such as exponential stabilization for linear time-delay systems 22 , exponential stabilization for uncertain time-delay systems 21, 23 .Hien and Phat 23 presented exponential stability and stabilization conditions for a class of uncertain linear system with time-varying delay, based on an improved Lyapunov-Krasovskii functional combined with Leibniz-Newton formula.The robust stability conditions are derived in terms of LMIs, but the time-varying delays are required to be differentiable and the lower bound is restricted to zero.However, in most cases, these conditions are difficult to satisfy.Therefore, in this paper, we will employ some new techniques so that the above conditions can be removed.
Stability analysis of nonlinear systems with distributed delays is of both practical and theoretical importance.For some systems, delay phenomena may not be simply considered as delays in the velocity terms and/or discrete delays in the states.Therefore, it is desirable to extend the system model to include distributed delays.Practical applications, modeled by systems with distributed delays, can be found in 24-26 .In 27 , Gao et al. studied the problem of linear switching systems with discrete and continuous time delay.By constructing Lyapunov functional under a condition on the time delay, we show that it stabilizes the system for sufficiently small delays.
In this paper, the problem of exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays is studied.The discrete time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying delay are available.However, the delay function is not necessary to be differentiable.We study the stability properties of switched nonlinear systems consisting of both stable and unstable subsystems.Average dwell-time approached and improved piecewise Lyapunov functional combined with Leibniz-Newton's formula, and new delay-dependent sufficient conditions for the exponential stabilization of the switched systems are first established in terms of LMIs.A numerical example is also given to illustrate the effectiveness of the proposed method.
The rest of this paper is organized as follows.In Section 2, we give notations, definition, propositions, and lemma which will be used in the proof of the main results.Delay-dependent sufficient conditions for the exponential stability of switched nonlinear time-varying delay systems are presented in Section 3. A Numerical example illustrated the obtained results is given in Section 4. The paper ends with conclusions in Section 5 and cited references.

Preliminaries
The following notation will be used in this paper.R denotes the set of all real nonnegative numbers.R n denotes the n-dimensional space and the vector norm • .M n×r denotes the space of all n × r matrices.
We also let A T denote the transpose of matrix A : A is symmetric if A A T : I denotes the identity matrix.λ A denotes the set of all eigenvalues of A. λ max A max{Re λ : λ ∈ λ A } : x t : {x t s : s ∈ −h, 0 }, x t sup s∈ −h,0 x t s : C 0, t , R n denotes the set of all R n -valued continuous functions on 0, t .L 2 0, t , R m denotes the set of all the R m -valued square integrable functions on 0, t : matrix A is called semipositive definite

2.2
The time delays h t and k t denoted the discrete time-varying delay and distributed timevarying delay, respectively.They are assumed to satisfy the following conditions: The nonlinear perturbation satisfies the following growth condition: We introduce the following technical well-known propositions, which will be used in the proof of our results.Definition 2.1 see 7 .For any switching signal σ t and any T 2 > T 1 ≥ 0, let N σ T 1 , T 2 denote the number of switching of σ t over the interval T 1 , T 2 .For given holds, then T a is call average dwell time and N 0 is called chattering bound.As commonly used in the literature, for convenience, we choose N 0 0 in this paper.Proposition 2.2 Cauchy inequality .For any symmetric positive definite matrix N ∈ M n×n and x, y ∈ R n one has ±2x T y ≤ x T Nx y T N −1 y.

2.5
Proposition 2.3 see 28 .For any symmetric positive definite matrix M > 0, scalar γ > 0 and vector function ω : 0, γ → R n such that the integrations concerned are well defined, the following inequality holds:

Stability of Switched Nonlinear Time-Varying Delay Systems
First, we present a delay-dependent exponential stabilizability analysis conditions for the given nonlinear time-varying delay systems 2.1 .We consider the case when the stable and unstable subsystems coexist.We assume that the subsystem i 1 ≤ i ≤ k is stable, where the positive integer k satisfies 1 ≤ k ≤ m and the subsystem j k 1 ≤ j ≤ m is unstable.Let us set η i h 2i − h 1i , i 1, 2, . . ., m.For the system 2.1 , given α > 0, the following lemma provides a change estimation of Lyapunov-Krasovskii functional candidate: where

3.2
Lemma 3.1.For a given constant α > 0, suppose 2.3 holds.If there exist symmetric positive definite matrices P i , Q i , R i , U i , S i such that the following LMI holds: where

3.7
Then, along the trajectory of system 2.1 , one has Proof.Let Y i P −1 i , y t Y i x t .We consider the following Lyapunov-Krasovskii functional 3.1 .By taking the derivative of Lyapunov-Krasovskii functional candidate 3.1 along the trajectory of the system 2.1 we have

Vi1 y T t P i A
y T s S i y s ds − αV i7 . 3.9 Applying Proposition 2.3 and the Leibniz-Newton formula, we have

3.10
Note that ẏT s U i ẏ s ds.

ISRN Mathematical Analysis
Using Proposition 2.3 gives

3.13
Therefore, from 3.12 and 3.13 , we obtain y T s S i y s ds.

3.21
Thus, by the above differential inequality, we have For the system 2.1 and given β > 0, the following lemma provides a change estimate of Lyapunov-Krasovskii functional candidate: where e β t−τ x T τ Y j S j Y j x τ dτds.

3.24
Lemma 3.2.For a given constant β > 0, suppose 2.3 holds.If there exist symmetric positive definite matrices P j , Q j , R j , U j , S j such that the following LMI holds:

3.28
where A j − βI P j P j A j − βI T − 0.9 e βh 1j e βh 2j R j 2Q j k j S j a j b j I, 2 j e βk j S j 4 .

3.29
Then, along the trajectory of system 2.1 , one has Proof.By taking the derivative of Lyapunov-Krasovskii functional candidate 3.23 along the trajectory of the system 2.1 , we are able to do similar estimation as we did for the Lemma 3.1.We have the following: Vj t − βV j t ≤ 0, t ≥ t 0 ≥ 0.

3.31
Thus, by the above differential inequality, we have V j x t ≤ e β t−t 0 V j x t 0 , t ≥ t 0 ≥ 0.

3.32
From Lemmas 3.1 and 3.2, it is easy to show the following properties of the Lyapunov functional candidate 3.1 and 3.23 .
i There exist scalars ε 1 > 0, ε 2 > 0, such that ii There exists a constant scalar μ ≥ 1 such that V i x t ≤ μV j x t , i,j ∈ M.

3.34
iii The Lyapunov functional candidate 3.1 and 3.23 whose derivative along the trajectory of the corresponding subnetwork satisfies Now, for any piecewise constant switching signal σ t and any 0 ≤ t 0 < t, we let T − t 0 , t resp., T t 0 , t denote the total activation time of the stable subsystem resp., the ones of unstable subsystem during t 0 , t .Then, we choose a scalar α * ∈ 0, α arbitrarily to propose the following switching law: S1 : determine the switching signal σ t so that inf t≥t 0

3.39
and there exists μ ≥ 1 such that

3.40
Proof.Suppose that t ∈ t k , t k 1 .For piecewise Lyapunov functional candidate 3.1 and 3.23 along trajectory of network system 2.1 , we have ≤ e βT t 0 ,t −αT − t 0 ,t N 0 t−t 0 /T a ln μ V σ t 0 x t 0 .

3.42
Under the switching law S1 for any t 0 , t, we have Thus, V x t ≤ e N 0 ln μ e − α * − ln μ/T a t−t 0 V σ t 0 x t 0 ≤ c 0 e −2λ t−t 0 V σ t 0 x t 0 ,

3.45
Combining 3.44 and 3.45 leads to which means that solution to 2.1 is exponentially stable.The proof is thus completed.

Numerical Example
In this section, we now provide an example to show the effectiveness of the result in Theorem 3.  1 shows the trajectories of solutions x 1 t and x 2 t of the interval time-varying and distributed delays system.

Conclusions
In this paper, we have investigated the exponential stability for a class of switched nonlinear systems with discrete and distributed time-varying delays.The interval time-varying delay x 1 (t) x 1 (t) , x 2 (t) x 2 (t) function is not necessary to be differentiable which allows time-delay function to be a fast time-varying function.By using the average dwell-time approached and improved piecewise Lyapunov-Krasovskii functionals which are constructed based on and combined with Leibniz-Newton's formula, for the cases that subsystems are stable and unstable, some new delay-dependent stability criteria are derived by a set of linear matrix inequalities.Numerical examples are given to illustrate the effectiveness of our theoretical results.

Figure 1 :
Figure 1: The trajectories of x 1 t and x 2 t of the interval time-varying delay system 4.1 .
term in a matrix is denoted by * .
36holds on time interval t 0 , t .Meanwhile, we choose α * < α as the average dwell-time scheme: for any t > t 0 , For a given constant α > 0, β > 0 and time-varying delay satisfying 2.3 , suppose that the subsystem 1 ≤ i ≤ k of switched system 2.1 satisfies the conditions of Lemma 3.1, and the others satisfy the conditions of Lemma 3.2.Then the system 2.1 is exponentially stable for switching signal satisfying 3.36 , 3.37 and the state decay estimate is given by lim t → t k t and the relation k N σ t 0 , t ≤ N 0 t − t 0 /T a , we obtain k 2, . . ., m.