Exponential Stability and Guaranteed Cost of Switched Linear Systems with Mixed Time-Varying Delays

This paper deals with the problems of exponential stability and guaranteed cost of switched linear systems with mixed time delays. Based on Lyapunov functional method, we present new delaydependent conditions that guarantee both the exponential stability and an upper bound for some performance index. The criteria are delay-dependent conditions and are given in terms of linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness of the results.


Introduction
A switched system is a dynamical system that includes several subsystems and a logical rule that orchestrates switching between these subsystems at each instant of time 1 .The logical rule generates switching signals to determine which subsystem will be activated on a certain time interval.In fact, switching systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models in manufacturing, communication networks automotive engine control, chemical processes, and so on see, for example, 2, 3 .There have been many studies on stability of switching systems 4-6 .
The stability analysis of switched time delay has attracted a lot of attention 7-9 .The main approach for stability analysis relies on the use of Lyapunov Krasovskii functionals and LMI approach for constructing common Lyapunov function and switching rules 10-12 .In 13 , the asymptotic stability of switched linear time delay symmetric systems has been studied.In 11 , a switching system composed by a finite number of linear point time delay differential equations has been studied, and it has been shown that the asymptotic stability may be achieved by using a common Lyapunov function method and minimum switching rule.The results of 11 have been extended in 14 to linear switching system with 2 ISRN Applied Mathematics discrete and distributed delays.The delays considered are time invariant, and the conditions are given in terms of maximal and minimal eigenvalues of certain matrices.The exponential stability problem was considered in 15 for switching linear systems with impulsive effects by using the matrix measure concept.The exponential stability of time delay systems has been considered in 16, 17 , and switching linear systems with mixed delays have been studied in 18 .In 19 , sufficient stability conditions have been obtained by using a generalisation of Halanay's inequality.
In this paper, we will focus on exponential stability of switched linear systems with mixed time delays.The subsystems considered are continuous with time-varying delays.The derived conditions are delay dependent and are given in terms of LMIs.The conditions guarantee both the exponential stability and an upper bound of a performance index.The approach is based on Lyapunov Krasovskii functional and allows us to avoid explicitly substitution of the dynamic system equation in the derivative of the Lyapunov function.Furthermore, we use an extended variable to completely avoid bounding treatment of the weighted cross-product of the instantaneous state and the delayed state.As a consequence, the conservatism of the conditions is expected to be reduced with respect to earlier works e.g. 10, 16, 20 .The conditions allow us to compute easily both the stability factor and the decay rate of the solution.Numerical computations are performed for illustration.
In this paper, we are interested in establishing conditions guaranteeing the asymptotic stability of switched system 2.1 and finding the least upper bound for the cost function given by where Y > 0. Assume that there exists a Hurwitz linear convex combination A of A i , that is, Then for a positive definite matrix Z 1 , there exists a positive definite matrix R 11 satisfying Hence for x / 0, we can write Which implies that for at least one i.So for a positive definite matrix Z, construct N region of R n as follows: The switching rule is chosen as follows: S R : Step 0: let x t φ t .
Step 1: set σ x Step 2: stay in the ith mode as long as x t in Ω i .
Step 3: if x t hits the boundary of Ω i , go to Step 1 to determine the next mode.

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Definition 2.1.Given α > 0, the system 2.1 is α-exponentially stable if there exist a switching rule σ and a constant β ≥ 1 such that every solution x t, φ of the system satisfies the following inequality: x t, φ ≤ βe −αt φ , t ≥ 0.

2.12
The following lemmas will be useful.
Lemma 2.3.For any constant matrix M M T ∈ R n×n , M > 0, scalar γ ≥ η t > 0, vector function ω : 0, γ → R n such that the integrations in the following are well defined, then

Main Results
Let, where " * " denotes the symmetric part in a symmetric matrix and,

and F 4 satisfying.
i There exist 0

3.6
The switching rule is given by S R , and the solution x t, φ of the system satisfies x t, φ ≤ βe −αt φ , t ≥ 0.

3.7
Furthermore, the cost function in 2.5 satisfies

3.8
Proof.Let x t : {x t s , s ∈ −r, 0 } and consider the following Lyapunov Krasovskii functional: x T t θ Sx t θ dθ ds.

3.10
It is easy to verify that where β 1 and β 2 are defined by 3.1 and 3.2 .Computing the first time derivative of V x t , we obtain

3.12
Letting ξ t x T t x T t − h t t t−h t x T s ds t t−τ t x T s ds T , taking account of 2.3 , and using the fact that for some positive definite matrices Z 1 and Z 2 , the following inequalities hold x s ds .

3.13
We can write where

3.15
And " * " denotes the symmetric part in a symmetric matrix x θ dθ − 2αV 4 x t . 3.17 T .We can easily verify that B i χ 0, where χ t ξ T t ẋT t T , and

3.18
Taking account of 3.18 , and adding and subtracting the term x T t Zx t with Z a positive definite matrix, we get

3.19
Since the condition 3.6 holds we have

3.21
By choosing the switching rule as x T t L i x t .

3.22
We have

3.23
This implies that V x t ≤ V φ e −2αt , t ≥ 0. Taking account of 3.11 , we obtain

3.25
Integrating both sides of 3.25 from 0 to T and using the initial conditions, we obtain

3.26
As the system is asymptotically stable, when T → ∞ we have

3.27
Hence, we get Remark 3.2.In order to improve the results, we can use instead of 3.18 the relation where F i is given by T .Then we can state the following result.
Theorem 3.3.For given α > 0, switched linear system 2.1 is α-exponentially stable if there exist symmetric positive definite matrices and F 4i satisfying the conditions 3.5 and 3.6 , where the matrices F k , k 0, 1, . . ., 4 are replaced with F ki , k 0, 1, . . ., 4, i 1, . . ., N. The switching rule is given by S R , the solution x t, φ of the system satisfies 3.7 , and the cost function in 2.5 satisfies 3.8 .
If h t / 0 and τ t 0, the system 2.2 is reduced to the system as follows: In this case, we have the following corollary.

3.32
And where " * " denotes the symmetric part and

3.33
Remark 3.5.In 16 , the results are given in terms of a set of generalized Lyapunov equations type, and in 18 , the results are expressed in terms of generalized algebraic Riccati equation type, while in this paper, the results are expressed in terms of linear matrix inequalities.

Examples
In this section, two examples will be presented for illustration and comparison.

4.1
We can see that each subsystem is unstable.Applying the results of 18 , it is found that the system is asymptotically stable with decay rate α 0.5.Applying the results of 14 , by choosing Q 8 * I, and applying Theorem 3.1, the critical delay of asymptotic stability without decay rate is found as 0.5722.Therefore, with the results of 14 , we can't conclude about stability of the considered system.Note that both 14, 18 study the systems with constant delays.It is well known that the results for time varying delays are more conservative than those with constant delays.Applying our results, we let μ h 0 and μ τ 0, by Theorem 3.

4.4
It can be seen that Ω 1 ∪ Ω 2 R 2 \ {0}, therefore, the switching regions are given by: Ω and the switching rule is given by Moreover the solution of the system satisfies x t, φ ≤ 3.6297e −0.984t φ , t ≥ 0. For the initial conditions x 1 t e t 1 and x 2 t 0 for t ∈ −1, 0 , we obtain a guaranteed cost J * 14.5632.When we apply Theorem 3. The state trajectories are depicted in Figure 1 for h t 0.5 s and in Figure 2 for h t 250 s.It is clear that as the delay increases as the states require much time for convergence to zeros.Now, letting h 0.5 s, α 0.4, λ 1 0.3, λ 2 0.7, and the initial conditions x 1 t e t 1 and x 2 t 0 for t ∈ −0.5, 0 , we obtain a guaranteed cost J * 80.004.

Conclusion
By using Lyapunov Krasovskii approach, the problems of exponential stability and guaranteed cost are investigated for switched linear systems with mixed time delays.Via a designed switching rule, the results allow to the bounds that characterise the exponential stability, that is, the stability factor and the decay rate for the solution.The results are delay dependent and are expressed in terms of linear matrix inequalities.Some numerical examples are given to illustrate these results presented in this paper that have significant improvement over existing ones.
Qx t − 3, we obtain the stability with decay rate α 1.186, stability factor β 2.3345, and the guaranteed cost J * 19.244.that the system is asymptotically stable independent of delay.This shows the improvements of our approach.For h 250 s, we have the following results: