Switching Signal Design for Global Exponential Stability of Uncertain Switched Neutral Systems

The switching signal design for global exponential stability of switched neutral systems is investigated in this paper. LMI-based delay-dependent and delay-independent criteria are proposed to guarantee the global stability via the constructed switching signal. Razumikhin-like approach is used to find the stability results. Finally, some numerical examples are illustrated to show the main results.


Introduction
It is well known that the existence of delay in a system may cause instability or bad system performance in control systems.Time-delay phenomenon appears in many practical systems, such as AIDS epidemic, aircraft stabilization, chemical engineering systems, inferred grinding model, manual control, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines.Hence stability analysis for time-delay systems has been considered in the recent years 1-3 .Neutral systems are described by functional differential equations which depend on the delays of state and state derivative.Some practical examples of neutral systems include distributed networks, heat exchanges, and processes including steam 4 .
Switched system is a class of hybrid systems which is consisting of several subsystems and uses the switching signal to specify which subsystem is activated to the system trajectories at each instant of time.Some examples for switched systems are automated highway systems, constrained robotics, power systems and power electronics, transmission and stepper motors 5 .Stability analysis of switched time-delay systems has been an attractive research topic 6-13 .It is interesting to note that the stability for each subsystem cannot imply that of the overall system under arbitrary switching signal 9 .Another interesting fact is that the stability of a switched system can be achieved by choosing the switching signal even when each subsystem is unstable 6, 7, 10 .In this paper, the switching signal design will be considered for uncertain switched neutral systems with mixed delays.The switching signal will be proposed to guarantee the stability of switched system even when each subsystem is unstable.Based on Razumikhin-like approach 11 , delaydependent and delay-independent results are provided.New and flexible LMI conditions are proposed to design the switching signal which guarantees the global exponential and asymptotic stability of uncertain switched neutral systems.Some numerical examples are provided to demonstrate the use of our results.
The notation used throughout this paper is as follows.For a matrix A, we denote the transpose by A T , spectral norm by A , symmetric positive negative definite by A > 0 A < 0 , maximal eigenvalue by λ max A , and minimal eigenvalue by λ min A .A ≤ B means that matrix B − A is symmetric positive semidefinite.For two sets X and Y , X − Y means that the set of all points in X that are not in Y .For a vector x, we denote the Euclidean norm by x and x t s sup −H≤θ≤0 x t θ 2 ẋ t θ 2 .I denotes the identity matrix.R n denotes n-dimensional real space.

Problem Formulations and Main Results
Consider the following switched neutral system with mixed time delays: where x ∈ R n , x t is state at time t defined by x t θ : x t θ , ∀θ ∈ −H, 0 , σ is a switching signal which is a piecewise constant function and may depend on t or x, σ, taking its values in the finite set {1, 2, . . ., N}, and time-varying delay satisfies 0 ≤ h t ≤ h M , ḣ t ≤ h D , h M > 0, τ > 0, H max{h M , τ}.Matrices D, A 0i , and A 1i ∈ R n×n , i 1, 2, . . ., N, are constant, and the initial vector φ ∈ C 1 , where C 1 is the set of differentiable functions from −H, 0 to R n .Now we define some functions λ i t , i 1, 2, . . ., N, that will be used to represent our system: 2.2 The switched system in 2.1 can be rewritten as follows: where λ i t is defined in 2.2 and N i 1 λ i t 1, ∀t ≥ 0.
Lemma 2.1 see 14 .Let U, V , W, and M be real matrices of appropriate dimensions with M satisfying M M T , then if and only if there exists a scalar ε > 0 such that Since F is Hurwitz, there exist positive definite matrices P and Q satisfying Define some domains From the similar proof of 7 , it is easy to show N i 1 Ω i R n .Construct some domains

2.8
We can obtain N i 1 Ω i R n and Ω i ∩ Ω j Φ, i / j, where Φ is an empty set.If Assumption 2.3 is satisfied, then the following results can be derived: Define the following switching function:

2.10
Definition 2.4 see 14 .The system 2.1 with the designed switching signal is said to be the globally exponentially stabilizable with convergence rate α > 0 by the designed switching signal, if there are two positive constants α and Ψ such that Now we present a result to design the switching signal that guarantees global exponential stability of system 2.1 .Theorem 2.5.Assume that for D < 1, 0 < α < − ln D /τ, 0 ≤ α i ≤ 1, i 1, 2, . . ., N, and N i 1 α i 1, there exist some n × n matrices P, Q, R 1 , R 2 > 0, such that the following LMI conditions hold for all i 1, 2, . . ., N: where

2.13
Then the system 2.1 is globally exponentially stabilizable with convergence rate α by the switching signal given in 2.10 .
Proof.Define the Lyapunov functional

2.14
where P, R 1 , R 2 > 0. The time derivatives of V x t along the trajectories of system 2.3 under the switching function 2.10 satisfy

2.15
By the condition 2.9 and switching function 2.10 , we obtain where Ξ i , i 1, 2, . . ., N, are defined in 2.12 , X T x T t x T t − τ x T t − h t .From 2.16 with Ξ i < 0, we have 2.17 where
Consider the following uncertain switched neutral system with mixed time delays: where ΔA 0i t and ΔA 1i t are some perturbed matrices and satisfy the following condition: where M i , N 0i , and N 1i , i 1, 2, . . ., N, are some given constant matrices with appropriate dimensions, and F i t , i 1, 2, . . ., N, are unknown matrices representing the parameter perturbation which satisfy The uncertain switched system in 2.22a -2.22c can be rewritten as follows: where λ i t is defined in 2.2 and N i 1 λ i t 1, ∀t ≥ 0. Now we consider the exponential stability for uncertain switched system 2.22a -2.22c .

, N, and
such that the following LMI conditions hold for all i 1, 2, . . ., N:

2.25
where Ξ jki , j, k 1, 2, 3, are defined in 2.12 Then the system 2.22a -2.22c is globally exponentially stabilizable with convergence rate α by the switching signal given in 2.10 .
Proof.The time derivatives of V x t in 2.14 along the trajectories of system 2.22a -2.22c under the switching function 2.9 satisfy where By Lemmas 2.1 and 2.2, the condition Ξ i < 0 in 2.25 is equivalent to Ξ i < 0. By the same derivation of Theorem 2.5, this proof can be completed.
If we choose the convergence rate α 0, we can obtain the following delayindependent condition for the global asymptotic stability of system 2.22a -2.22c ., N, and N i 1 α i 1, there exist constants ε i > 0, i 1, 2, . . ., N, some n × n matrices P, Q, R 1 , R 2 > 0, such that the following LMI conditions hold for all i 1, 2, . . ., N:

2.30
Then the system 2.22a -2.22c is globally asymptotically stabilizable by the switching signal given in 2.10 .
If D 0, Corollary 2.7 can be reduced to the following corollary.

2.32
Then the system 2.22a -2.22c is globally asymptotically stabilizable by the switching signal given in 2.10 .
Assumption 2.9.Assume that there exists a convex combination F N i 1 α i A 0i , some positive definite matrices P and Q, some matrices S i , i 1, 2, . . ., N, such that where 0 ≤ α i ≤ 1 and N i 1 α i 1.
Define some domains

2.34
From the similar proof of 7 , it is easy to show N i 1 Ω i R n .Construct some domains

2.35
We can obtain N i 1 Ω i R n and Ω i ∩ Ω j Φ, i / j, where Φ is an empty set.If Assumption 2.9 is satisfied, then the following results can be derived:

2.36
Define the following switching function:

2.40
Then the system 2.22a -2.22c is globally exponentially stabilizable with convergence rate α by the switching signal given in 2.37 .
Proof.Define the Lyapunov functional

2.41
where The time derivatives of V x t along the trajectories of system 2.24 satisfy where S i PS i .By the inequality in 1, page 322 , we have

2.43
By system 2.24 and Leibniz-Newton formula, we have

2.44
By the conditions 2.42 -2.44 , we obtain the following result: where

2.46
By Lemmas 2.1 and 2.2, the condition Σ i < 0 in 2.39 is equivalent to Σ i < 0 in 2.45 .From Σ i < 0 and by the similar derivation of Theorem 2.5, the proof can be completed.
If D 0, Theorem 2.11 can be reduced to the following corollary.

2.48
Then the system 2.22a -2.22c with D 0 is globally exponentially stabilizable with convergence rate α by the switching signal given in 2.37 .

3.3
Select the switching signal by

3.5
The switching regions Ω 1 and Ω 2 are sketched in Figure 1.The system 2.22a -2.22c with h D 0.2 and 3.1 is globally asymptotically stabilizable by the switching signal 3.4 .Some comparisons are made in Table 1.The result of this paper provides a major improvement to guarantee the global asymptotic stability of system 2.22a -2.22c with 3.1 .For the given feedback control 3.9 , system 3.7 can be rewritten as ẋ t A 0σ x t A 1σ x t − h t , t ≥ 0, 3.10 where A 1σ B σ K σ .As shown in Table 3, the results obtained in this paper provide larger allowable time delay bounds guaranteeing the global stability of system 3.7 with 3.9 by switching signal 2.37 .In 7, 10 , the convex combination parameters are chosen by α 1 1/3 and α 2 2/3.The convex combination parameters of our results are chosen by α 1 0.1 and α 2 0.9.

Conclusions
In this paper, the switching signal design for global exponential stability of uncertain switched neutral systems with mixed time delays has been considered.LMI and Razumikhinlike approaches are used to derive delay-dependent and delay-independent stability criteria.The results obtained in this paper are less conservative than the previous ones for the numerical examples investigated in this paper.

Example 3 . 1 .
Consider the system 2.22a -2.22c and the following parameters:

Table 1 :
Comparison with other previous results.

Table 2 :
Comparison with other previous results.By Corollary 2.13, some comparisons with the obtained results for switched system 2.22a -2.22c with 3.6 are made in Table2.The results of this paper provide a larger allowable upper bound for time delay to guarantee the global asymptotic stability of system 2.22a -2.22c with 3.6 by the switching signal 2.37 .

Table 3 :
Comparison with other previous results.Allowable time-varying delay bounds retaining global asymptoticand exponential stability of the system 3.7 with 3.9