Research Article | Open Access

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

**Academic Editor:**Tamas Kalmar-Nagy

#### Abstract

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.

#### 1. 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 [1â€“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 [6â€“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], delay-dependent 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 , spectral norm by , symmetric positive (negative) definite by , maximal eigenvalue by , and minimal eigenvalue by . means that matrix is symmetric positive semidefinite. For two sets and , means that the set of all points in that are not in . For a vector , we denote the Euclidean norm by and . denotes the identity matrix. denotes *n*-dimensional real space.

#### 2. Problem Formulations and Main Results

Consider the following switched neutral system with mixed time delays:
where , is state at time *t* defined by , , is a switching signal which is a piecewise constant function and may depend on or , , taking its values in the finite set , and time-varying delay satisfies , , , , . Matrices , , and , , are constant, and the initial vector , where is the set of differentiable functions from to .

Now we define some functions , , that will be used to represent our system: The switched system in (2.1) can be rewritten as follows: where is defined in (2.2) and .

Lemma 2.1 (see [14]). *Let , , and be real matrices of appropriate dimensions with satisfying , then
**
if and only if there exists a scalar such that
*

Lemma 2.2 (Schur complement of [15]). *For a given matrix with , , the following conditions are equivalent:*(1)*,*(2)*, *

*Assumption 2.3. *Assume that there exists a convex combination such that is Hurwitz, where and

Since is Hurwitz, there exist positive definite matrices and satisfying Define some domains

From the similar proof of [7], it is easy to show . Construct some domains

We can obtain and , , where is an empty set. If Assumption 2.3 is satisfied, then the following results can be derived: Define the following switching function:

*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 by the designed switching signal, if there are two positive constants and 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 , , , , and , there exist some matrices , such that the following LMI conditions hold for all :
**
where
**
Then the system (2.1) is globally exponentially stabilizable with convergence rate by the switching signal given in (2.10).*

*Proof. *Define the Lyapunov functional
where . The time derivatives of along the trajectories of system (2.3) under the switching function (2.10) satisfy
By the condition (2.9) and switching function (2.10), we obtain
where , , are defined in (2.12), . From (2.16) with , we have
where
From (2.14), we have
where . From (2.19), we can obtain
where . Since and , we can choose a sufficiently small positive constant satisfying . By the Razumikhin-like approach of [14], we have
This completes the proof.

Consider the following uncertain switched neutral system with mixed time delays: where and are some perturbed matrices and satisfy the following condition: where , , and , , are some given constant matrices with appropriate dimensions, and , , are unknown matrices representing the parameter perturbation which satisfy The uncertain switched system in (2.22a)â€“(2.22c) can be rewritten as follows: where is defined in (2.2) and .

Now we consider the exponential stability for uncertain switched system (2.22a)â€“(2.22c).

Theorem 2.6. *Assume that for , , , , and , there exist constants , , and some matrices , such that the following LMI conditions hold for all :
**
where , , are defined in (2.12)
**
Then the system (2.22a)â€“(2.22c) is globally exponentially stabilizable with convergence rate by the switching signal given in (2.10).*

*Proof. *The time derivatives of in (2.14) along the trajectories of system (2.22a)â€“(2.22c) under the switching function (2.9) satisfy
where
By Lemmas 2.1 and 2.2, the condition in (2.25) is equivalent to By the same derivation of Theorem 2.5, this proof can be completed.

If we choose the convergence rate , we can obtain the following delay-independent condition for the global asymptotic stability of system (2.22a)â€“(2.22c).

Corollary 2.7. *Assume that for , , , and , there exist constants , , some matrices , such that the following LMI conditions hold for all :
**
where
**
Then the system (2.22a)â€“(2.22c) is globally asymptotically stabilizable by the switching signal given in (2.10).*

If , Corollary 2.7 can be reduced to the following corollary.

Corollary 2.8. *Assume that for some constants , , there exist constants , , some matrices , such that the following LMI conditions hold for all :
**
where
**
Then the system (2.22a)â€“(2.22c) is globally asymptotically stabilizable by the switching signal given in (2.10).*

*Assumption 2.9. *Assume that there exists a convex combination , some positive definite matrices and , some matrices , , such that
where and

Define some domains From the similar proof of [7], it is easy to show . Construct some domains We can obtain and , , where is an empty set. If Assumption 2.9 is satisfied, then the following results can be derived: Define the following switching function:

*Remark 2.10. *In [6, 7, 10], their assumption is given by
where . We can see that our Assumption 2.9 is more flexible with , . The main difference of Assumptions 2.3 and 2.9 is that some matrices are introduced in Assumption 2.9. These matrices play a key role to derive the delay-dependent results.

Theorem 2.11. *Assume that for , , , , and , there exist constants , , some matrices , and some matrices , , , , , and , , such that the following LMI conditions hold for all :
**
where
**
Then the system (2.22a)â€“(2.22c) is globally exponentially stabilizable with convergence rate by the switching signal given in (2.37).*

*Proof. *Define the Lyapunov functional
where , . The time derivatives of along the trajectories of system (2.24) satisfy
where . By the inequality in [1, page 322], we have
By system (2.24) and Leibniz-Newton formula, we have
By the conditions (2.42)â€“(2.44), we obtain the following result:
where
By Lemmas 2.1 and 2.2, the condition in (2.39) is equivalent to in (2.45). From and by the similar derivation of Theorem 2.5, the proof can be completed.

If , Theorem 2.11 can be reduced to the following corollary.

Corollary 2.12. *Assume that for , , , there exist constants , , some matrices , some matrices , , , , and , , such that the following LMI conditions hold for all :
**
where
**
Then the system (2.22a)â€“(2.22c) with is globally exponentially stabilizable with convergence rate by the switching signal given in (2.37).*

If , , Corollary 2.12 can be reduced to the following corollary.

Corollary 2.13. *Assume that for , , , there exist some matrices , some matrices , , , , and , , such that the following LMI conditions hold for all :
**
where , , , are defined in Corollary 2.12. Then the system (2.22a)â€“(2.22c) with is globally exponentially stabilizable with convergence rate by the switching signal given in (2.37).*

*Remark 2.14. *By setting in Theorems 2.5â€“2.11 and Corollaries 2.7â€“2.13, the global asymptotic stability for system (2.22a)â€“(2.22c) can be guaranteed.

#### 3. Numerical Examples

*Example 3.1. *Consider the system (2.22a)â€“(2.22c) and the following parameters:

By Corollary 2.7, a feasible solution of LMI (2.29) with (3.1), , , and is given by
Select the switching signal by
where , ,
The switching regions and are sketched in Figure 1. The system (2.22a)â€“(2.22c) with 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)â€“(2.22c) with (3.1).

*Example 3.2. *Consider the system (2.22a)â€“(2.22c) and the following parameters [7]:

By Corollary 2.13, some comparisons with the obtained results for switched system (2.22a)â€“(2.22c) with (3.6) are made in Table 2. The results of this paper provide a larger allowable upper bound for time delay to guarantee the global asymptotic stability of system (2.22a)â€“(2.22c) with (3.6) by the switching signal (2.37).

*Example 3.3. *Consider the following switched system with input time delay [7]:
where
The feedback control is given by with
For the given feedback control (3.9), system (3.7) can be rewritten as
where . 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 and . The convex combination parameters of our results are chosen by and .