Advances in Mathematical Physics

Volume 2013 (2013), Article ID 592483, 20 pages

http://dx.doi.org/10.1155/2013/592483

## Stability Analysis for Neutral Delay Markovian Jump Systems with Nonlinear Perturbations and Partially Unknown Transition Rates

Department of Auto, School of Information Science and Technology, University of Science and Technology of China, Anhui 230027, China

Received 1 May 2013; Accepted 13 June 2013

Academic Editor: Shao-Ming Fei

Copyright © 2013 Xinghua Liu and Hongsheng Xi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of exponential stability for the uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations is investigated in this paper. This study starts from the corresponding nominal systems with known and partially unknown transition rates, respectively. By constructing a novel augmented Lyapunov functional which contains triple-integral terms and fully utilizes the bound of the delay, the delay-range-dependent and rate-dependent exponential stability criteria are developed by the Lyapunov theory, reciprocally convex lemma, and free weighting matrices. Then, the results about nominal systems are extended to the uncertain case. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods.

#### 1. Introduction

Neutral time-delay systems have been the focus of the research community, which are often encountered in such practical situations as distributed networks, population ecology, processes including steam or heat exchanges [1], and robots in contact with rigid environments [2], and so forth. The existence of time delay may cause the instability of the systems, thus making the stability analysis of time-delay systems an interesting topic. Existing results can be roughly classified into two categories, delay-independent criteria and delay-dependent criteria, where the latter is generally regarded as less conservative. In addition, it should be pointed out that the stability of neutral time-delay systems is more difficult to tackle since the derivative of the delayed state is involved. The situation is similar as singular systems [3, 4], whose stability problem is more complicated than that for regular systems because more factors need to be considered. In the past decades, considerable attention has been devoted to the robust delay-independent stability and delay-dependent stability of linear neutral systems, which are mainly obtained based on the Lyapunov-Krasovskii (L-K) method [5–8]. Furthermore, when nonlinear perturbations or parameter uncertainties appear in neutral systems, some results on stability analysis have been also presented [9–14]. Various techniques have been proposed in these papers, for example, model transformation techniques, the improved bounding techniques, and matrix decomposition approaches. In particular, He et al. [14] propose a new method for dealing with time-delay systems, which employs free weighting matrices to express the relationships between the terms in the Newton-Leibniz formula and has brought novel results. However, many complex systems with uncertainties and neutral types as well as time-varying or state-dependent delays are still inviting further investigation.

It is noted that many practical dynamics such as solar thermal central receivers, robotic manipulator systems, aircraft control systems, and economic systems, experience abrupt changes in their structures, caused by phenomena such as component failures or repairs, changes in subsystem interconnections, and sudden environmental changes. This class of systems can be described as Markovian jump systems (MJSs) where the abrupt variation in the structures and parameters can be naturally represented by the jumps in MJSs. Since its first introduction by Krasovskii and Lidskii in 1961, MJSs have received much attention, and considerable progress has been made; see, for example, [15–22] and references therein for more details. In view of these results, although related research has made good achievement, we have to admit that the transition probabilities in the jumping process determine the system behavior to a large extent. However, the likelihood of obtaining such available knowledge is actually questionable, and the cost is probably expensive. Thus, it is significant and necessary, from control perspectives, to further study more general jump systems with partly unknown transition probabilities. Recently, many results on the Markovian jump systems with partly unknown transition probabilities are obtained [23–27]. Most of these improved results just require some free matrices or the knowledge of the known elements in transition rate matrix, such as the bounds or structures of uncertainties, and some else of the unknown elements do not need to be considered. It is a great progress on the analysis of Markovian jump systems. However, few of these results are concerned with neutral delay systems. It is urgent and significant to consider the problem of delay-dependent exponential stability for Markovian jumping neutral systems with partially unknown transition rates. Besides, to the best of the authors’ knowledge, the neutral Markovian jump systems have not been fully investigated, and it is very challenging, especially when nonlinear perturbations exist. These facts thus motivate our study.

In this paper, the exponential stability problem of neutral Markovian jump systems with mixed interval time-varying delay, nonlinear perturbations, and partially unknown transition rates is investigated. A new augmented Lyapunov functional containing triple-integral terms is constructed by dividing the delay interval into two subintervals, and then the delay-range-dependent and rate-dependent exponential stability criteria are obtained by reciprocally convex lemma and free weighting matrices. Moreover, in contrast with the recent research on uncertain transition rates, our proposed concept of the partly unknown transition rates does not require any knowledge of the unknown elements, such as the bounds or structures of uncertainties. On the basis of the obtained results about nominal systems, we further extend the criteria to the uncertain case. All the obtained results are presented in terms of LMIs that can be solved numerically.

The main contributions of this paper can be summarized as follows: the bound of the delay is fully utilized in this paper; that is, improved bounding technique is used to reduce the conservativeness. The constructed Lyapunov functional contains some triple-integral terms which is very effective in the reduction of conservativeness and has not appeared in the context of neutral Markovian jump systems with nonlinear perturbations before. The reciprocally convex lemma is used to derive the delay-range-dependent stability conditions, which can reduce the conservativeness of the investigated systems. The proposed results are applicable to the uncertain transition rates and expressed in a new representation, which are proved to be less conservative than some existing ones.

The remainder of the paper is organized as follows. Section 2 presents the problem and preliminaries. Section 3 gives the main results, which are then verified by numerical examples in Section 4. Section 5 concludes the paper.

*Notation 1. *The following notations are used throughout the paper. denotes the dimensional Euclidean space, and is the set of all matrices. (), where and are both symmetric matrices, means that is negative (positive) definite. is the identity matrix with proper dimensions. For a symmetric block matrix, we use to denote the terms introduced by symmetry. is defined to be the expectation operator with respect to the probability measure. is the Euclidean norm of vector , , while is spectral norm of matrix , . is the eigenvalue of matrix with maximum (minimum) real part. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

#### 2. Problem Statement and Preliminaries

Given a probability space where is the sample space, is the algebra of events, and is the probability measure defined on . is a homogeneous, finite-state Markovian process with right continuous trajectories taking values in a finite set , with the mode transition probability matrix being where , , and is the transition rate from mode to , and for any state or mode , it satisfies

The following uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations over the space are considered: where is the system state and is time-varying neutral delay which satisfies , . The time-varying retarded delay is such that where , , and are constant real values. The initial condition is a continuously differentiable vector-valued function. , , and are unknown nonlinear perturbations which are, with respect to the current state , the delayed state and the neutral delay state , respectively. For all , they are assumed to be bounded in magnitude as where , , and are given constants, for simplicity, , , and .

, , and are known mode-dependent constant matrices, and , are uncertainties. For simplicity of notations, we denote , , , , by , , , , for . The parametric matrix and the admissible parametric uncertainties satisfy the following condition: where , , and are known mode-dependent constant matrices with appropriate dimensions and is an unknown and time-varying matrix satisfying Particularly, the following nominal systems can be obtained for :

Before proceeding with the main results, we present the following definitions, assumptions, and lemmas.

*Assumption 1. *System matrices , are Hurwitz, and all the eigenvalues have negative real parts for each mode. , is full rank in row.

*Assumption 2. *The Markov process is irreducible, and the system mode is available at time .

*Definition 3 (see [28]). *Define operator as . is said to be stable if the homogeneous difference equation
is uniformly asymptotically stable. In this paper, that is, .

*Definition 4 (see [29]). * The system in (3) is exponentially stable with a decay rate for all , if there exist scalars and such that for all ,
where is the exponential decay rate, denotes the Euclidean norm, and

*Definition 5 (see [30]). *Define the stochastic Lyapunov-Krasovskii function of system (3) as , where its infinitesimal generator is defined as

Lemma 6 (see [31]). *For given real constant matrices , , and , with appropriate dimensions, where and . if and only if
*

Lemma 7. *For any constant matrix , continuous functions , constant scalars , and constant such that the following integrations are well defined,*(a)*(b)*

*Proof. *(a) Is directly obtained from [32]. In addition, from and , it is held that . Then,

(b) Is thus true by [33].

Lemma 8 (see [34]). *For functions , , , and with and with , matrices , ; then there exists matrix such that
**
and the following inequality holds:
*

Lemma 9 (see [35]). *For given matrices , , and with appropriate dimensions:
**
for all satisfying , if and only if there exists a scalar , such that
*

#### 3. Main Results

This section will first state the exponential stability analysis for (9) with known and partly unknown transition rates, respectively. Then, the uncertain systems described by (3) are considered. With creative Lyapunov functional and novel matrix inequalities analysis, delay-range-dependent and rate-dependent exponential stability conditions are presented.

##### 3.1. Exponential Stability for the Nominal Systems with Known Transition Rates

In this subsection, we consider the full information on the transition rates and give the following conditions to guarantee the exponential stability of the nominal systems.

Theorem 10. *For the given finite set of modes with transition rates matrix, scalars , , , , , , , , , and constant scalar satisfying , the systems described by (9) are exponentially stable with decay rate and if the operator is stable, and there exist symmetric positive matrices , , , , , , , , , , , and matrices , , , , , for any scalars , , and any matrices , with appropriate dimensions, such that the following linear matrix inequalities (22), (23), and (24) hold:
**
where
**
where is a linear operator on by
**
where are block entry matrices; that is,
*

*Proof. *Construct the following Lyapunov functional:
where

Taking as its infinitesimal generator along the trajectory of system (9), we obtain the following from Definition 5 and (28)–(34):
In view of (6), the following inequalities hold for any scalars , , and :
From (35) and (42), we have

Define

Since it is easy to see ; then from (5) we also obtain that
where is defined in Theorem 10. Notice (a) of Lemma 7; then

Notice (b) of Lemma 7; then

For , the following is held from (a) of Lemma 7:
where

By Lemma 8, there exists matrix with appropriate dimensions such that

Similarly, considering and following the same procedure, there exists matrix with appropriate dimensions such that

For , with the same matrix inequalities technique, we obtain the following:

Consider and , which are directly estimated by (a) of Lemma 7; that is,

In addition, there exist matrices with appropriate dimensions, such that the following equality holds according to (9):

Substituting (36)–(42) and (45)–(54) into (43), we obtain

On the other hand, for , the integral terms
The above equations: are disposed and estimated by Lemma 8.
are directly estimated by (a) of Lemma 7. Therefore,

With (55) and (58), the following inequality (59) is held for if (22), (23), and (24) are satisfied

From the Lyapunov functional (28) and (59), it is held that

Moreover, we have

Then, from (60) and (61), it is readily seen that
where .

Therefore, by Definition 4, the system (9) is exponentially stable with a decay rate . This completes the proof.

*Remark 11. *It is noted that the integral intervals in (61) are enlarged as follows:
Equation (61) can be obtained by letting be defined as previously.

*Remark 12. *In Theorem 10, the factors
may be enlarged as . This will lead conservative results due to the fact that cannot achieve and at the same time. While we apply Lemma 7 to these terms, the method by using reciprocally convex lemma [34] can achieve less conservative results. Moreover, for , the factor that appeared in the derivative of Lyapunov functional may be directly enlarged as . In this paper, we enlarge it as to reduce the conservativeness of the obtained criteria. In the literature [32, 36, 37], this factor is enlarged as , which only holds for .

*Remark 13. *The proposed Lyapunov functional (28) contains some triple-integral terms, which has not been used in any of the existing literatures in the same context before. Compared with the existing ones, [33] has shown that such a Lyapunov functional type is very effective in the reduction of conservatism. Besides, the information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms such as , , and .

*Remark 14. *It should be also mentioned that the result obtained in Theorem 10 is delay-range-dependent and decay rate-dependent stability condition for (9), which is less conservative than the previous ones and will be verified in Section 4. Although the large number of introduced free weighting matrices may increase the complexity of computation, utilizing the technique of free weighting matrices would reduce the conservativeness. In addition, the given results can be extended to more general systems with neutral delay . That is, . The results can be obtained by using the similar methods.

The information on the delay derivative may not be available in many cases. The following corollary is therefore given, which can be obtained from Theorem 10 by setting and .

Corollary 15. *For the given finite set of modes with transition rates matrix, scalars , , , , , , , and constant scalar satisfying , the systems described by (9) are exponentially stable with decay rate and if the operator is stable, and there exist symmetric positive matrices , , , , , , , , , , , and matrices , , , , , for any scalars , , and any matrices , with appropriate dimensions, such that linear matrix inequalities (22) and (65) hold:
*