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New exponential stability criteria for neutral system with time-varying delay and nonlinear perturbations
Advances in Difference Equations volume 2014, Article number: 44 (2014)
In this paper, the problem of exponential stability for neutral system with time-varying delay and nonlinear perturbations is investigated. By using the technology of model transformations, based on a linear matrix inequality (LMI) and a generalized Lyapunov-Krasovskii function, a new criterion for exponential stability with delay dependence is obtained. Due to a new integral inequality, the result is less conservative. Finally, some numerical examples are presented to illustrate the effectiveness of the method.
Time delay is frequently viewed as a source of instability, and it is encountered in various engineering systems such as electrical circuits, chemical processes, networked control systems power systems, and other areas [1, 2]. Current efforts on the problem of the stability of time-delay systems can be divided into two categories, namely delay-independent criteria and delay-dependent criteria. A number of delay-independent sufficient conditions for the asymptotic stability of delay systems have been presented by various researchers (for example ). Also a few delay-dependent sufficient conditions have been shown in [4, 5]. So the problem of robust stability analysis for time-delay neutral systems is important both in theory and in practice and is of interest to many researchers; see [6, 7] and the references therein.
Recently, many researchers have paid a lot of attention to the problem of robust stability for delay systems with nonlinear uncertainties [2, 8–11], and many methods have been proposed to deal with nonlinear uncertainties (see for example [10, 11]). For instance, by using a descriptor model transformation and decomposition technique, some delay-dependent stability criteria are obtained in . In  the stability conditions are developed by a descriptor model transformation technique, and the nonlinear uncertainties are handled by the S-procedure. However, these results are only concerned with asymptotic stability, without providing any conditions for exponential stability and any information as regards the decay rates.
As is well known, exponential stability converges faster than others, so the issue of exponential stability for some systems with time delay has received considerable attention in recent years [11–14]. For example, Liu  has investigated the exponential stability of a general power system. Kwon and Park discussed the exponential stability of uncertain dynamic systems including state delay in .
Considering those, many researchers have studied the exponential stability analysis for neutral systems with time-varying and nonlinear perturbations [15–18]. Chen et al.  presented a new criterion for exponential stability for uncertain neutral systems with nonlinear perturbations by employing an integral inequality. Ali  investigated the exponential stability for a neutral delay differential system with nonlinear uncertainties by following a generalized eigenvalue problem approach.
Based on the above, the exponential stability of neutral system with nonlinear uncertainties is discussed in this paper, and by employing a Lyapunov-Krasovskii function, the LMI method, and a new integral inequality, a sufficient condition for exponential stability of the system is provided. Finally, some numerical examples are presented to illustrate the effectiveness of the method.
2 System description and preliminary lemma
Consider the neutral system with delay and nonlinear uncertainties described by
where is the state vector, and A, B, C, , , , and are known real parameter matrices of appropriate dimensions. is for the time-varying continuous functions satisfying , , in which d, , and are positive constants. and are given continuous vector-valued initial functions. , , and are nonlinear uncertainties and satisfy the following conditions:
Let , then system (1) is equivalent to the following:
Definition 1 Consider the linear system (2). If there exist two scalars and such that
and we let , then system (2) is exponentially stable, where ε is called the exponential convergence rate.
Before proceeding with the main results, several lemmas are necessary.
Lemma 1 For any symmetric positive-definite constant matrix , where , and , if there exists a vector function such that the following integration is well defined, then we have
Remark 1 Lemma 1 will play a key role in the derivation of a less conservative delay-dependent condition.
Lemma 2 
Let U, V, W, and M be real matrices of appropriate dimensions with M satisfying , then
for all , if and only if there exists a scalar such that
3 Main results
Theorem 1 For the prescribed scalar , system (2) is globally exponentially stable with a convergence rate, if there exist positive-definite matrices , , , for positive-definite matrices , , , and real matrix satisfying the following LMI:
Proof Let us consider the Lyapunov functional candidate to be
Then the time derivative of along the solution of system (1) gives
By Lemma 1, we get
by Lemma 2, we get
Define the extended vector
If , that is to say, , then based on the Lyapunov method, system (2) is asymptotically stable, that is to say, system (1) is asymptotically stable.
Furthermore, we prove the exponential stability of the neutral system (1).
Now, it is easy to see from (4) that there exists a scalar such that for any t, .
Moreover, by the definition of a Lyapunov functional, there exist positive scalars , , , and such that for any t we have the following inequality:
To prove the exponential stability of system (2), we define a new function by , where the scalar . Then, we find that for any t,
By interchanging the integration sequence, we get
Let be small enough such that .
Then, substituting (9) into (8) shows that there exists a scalar such that, for any t,
It can be seen that
Substituting (11) into (10) yields for any t.
From Definition 1, has exponential stability, that is, system (1) has exponential stability. This completes the proof. □
Remark 3 When and , system (1) is converted into the system in Ali . However, our system has more the nature of universality.
Remark 4 When and , system (1) reduces to the traditional power system with nonlinear perturbations (12). However, for the general power system with nonlinear perturbations one has achieved some results in [9, 16–18, 20, 21]. We have
Corollary 1 For prescribed scalar , system (12) is exponentially admissible with convergence rate ε, if there exist positive-definite matrices and positive-definite symmetric matrices , , , and a real matrix satisfying the following LMI:
4 Numerical examples
Example 1 Consider the following neutral time-delay system :
with , , , ,
Using the Matlab LMI Toolbox and Theorem 1, we obtain maximum allowable upper bounds of , which are listed in Table 1. Table 1 describes maximum allowable upper bounds of delays that guarantee the exponential stability of system (1). It can be seen that our stability condition is less conservative than the results discussed in [16, 22, 23].
By applying our criteria and using the Matlab LMI Toolbox, we have the comparative result listed in Table 2. From Table 2, we can see that our results are much less conservative than those in [16–18, 20, 21].
The exponentially stability of a neutral system with nonlinear perturbations has been solved in terms of the LMI approach. Using the Lyapunov-Krasovskii functional method, the model transformation technique, and a new integral inequality, a new criterion for the exponential stability of systems is given. The criterion is presented in terms of linear matrix inequalities, which can easily be solved by the Matlab Toolbox and will have wide application in practical engineering. Finally, numerical examples are presented to illustrate the effectiveness of the method.
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This paper is supported by the National Natural Science Foundation of China (No. 61273004). The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.
The authors declare that they have no competing interests.
LZ carried out the main part of this manuscript. YM participated in the discussion and corrected the main theorem. All authors read and approved the final manuscript.
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Cite this article
Ma, Y., Zhu, L. New exponential stability criteria for neutral system with time-varying delay and nonlinear perturbations. Adv Differ Equ 2014, 44 (2014). https://doi.org/10.1186/1687-1847-2014-44
- neutral system
- nonlinear perturbations
- exponential stability
- linear matrix inequality