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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 450168, 22 pages

http://dx.doi.org/10.1155/2012/450168

## Stability Analysis for Markovian Jump Neutral Systems with Mixed Delays and Partially Known Transition Rates

^{1}School of Mathematics and Computer Science, Yunnan University of Nationalities, Kunming 650031, China^{2}School of Information Science and Engineering, Yunnan University, Kunming 650091, China^{3}Library of Yunnan, University of Nationalities, Kunming 650031, China

Received 6 May 2012; Accepted 19 September 2012

Academic Editor: Ziemowit Popowicz

Copyright © 2012 Lianglin Xiong et al. 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 delay-dependent stability problem is studied for Markovian jump neutral systems with partial information on transition probabilities, and the considered delays are mixed and model dependent. By constructing the new stochastic Lyapunov-Krasovskii functional, which combined the introduced free matrices with the analysis technique of matrix inequalities, a sufficient condition for the systems with fully known transition rates is firstly established. Then, making full use of the transition rate matrix, the results are obtained for the other case, and the uncertain neutral Markovian jump system with incomplete transition rates is also considered. Finally, to show the validity of the obtained results, three numerical examples are provided.

#### 1. Introduction

A switched system is a dynamic system consisted of a number of subsystems and a rule that manages the switching between these subsystems. In the past, a large number of excellent papers and monographs on the stability of switched systems have been published such as [1–7] and the references cited therein. Among the results for switched systems, the stabilization problem of switched neutral systems has also been explored by some researchers [8–22], and mainly two kinds of switching rule are designed in these articles. Some state-dependent switching rules are obtained assuming the convex combination of the systems matrix, see, for example, [8, 10, 20]. To reduce the conservative, the authors in [11] have investigated the stabilization for switched neutral systems without the assumption that the restraint of the convex combination on systems matrices, the dwell time, and state-dependent rules were designed. Similarly, the authors in [12] have also studied the problem of the BIBO stability for switched neutral systems. Using the Razumikhin-like approach [13] and the Leibniz-Newton formula, the global exponential stability for a class of switched neutral systems with interval-time-varying state delay and two classes of perturbations is investigated in [14], with arbitrary switching signal. Moreover, with the constructed state-dependent switching rule, the authors in [15] have investigated the global exponential stability of switched neutral systems. With the dwell time approach, the improved stability conditions for a class of switched neutral systems with mixed time-varying delays have also been obtained in [16], and the upper bound of derivative of the discrete time-varying delay is not restricted to one. In [17], the robust reliable stabilization of uncertain switched neutral systems with delayed switching has been considered. The fault estimator for switched nonlinear systems of neutral type has been designed in [9]. In [18], the authors have studied the problem of exponential stability for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations, obtaining the less conservative conditions based on the introduced free matrices. More recently, the markovian jump parameters have been considered for the analysis of switched neutral systems in [19].

In the past few decades, as a special switched system, markovian jump systems (MJSs) have been widely studied due to the fact that many dynamical systems subject to random abrupt variations can be modeled by MJSs such as manufacturing systems, networked control systems, and fault-tolerant control systems. There are a lot of useful results that have been presented in the literature, such as [23–29], and the references therein. For the analysis of MJSs, 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. Rather than having a large complexity to measure or estimate all the transition probabilities, 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 [30–37]. By introducing the free matrices based on the property of transition rate matrix, [32] gave less conservative conditions than that in [30] for Markovian jump systems with partial information on transition probability. And [33] provided with a new approach to obtain the necessary and sufficient conditions for markovian jump linear systems with incomplete transition probabilities, which may be appropriate to discuss the counterpart of delay systems. 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 need not be considered. It is a great progress on the analysis of markovian jump systems. However, a few of these papers have considered the effect of delay on the stability or stabilization for the corresponding neutral systems, except for [19]. The global exponential stability of the markovian jumping neutral systems with interval time-varying delays has been studied by [19]; however, the transition probabilities are fully known, and the constructed Lyapunov did not fully consider the effect of the transition probabilities on the integrand. To the best of the authors' knowledge, the markovian jump neutral systems have not been fully investigated, and it is very challenging. All of these motivate this paper.

In this paper, the delay-dependent stability problem of neutral Markovian jump linear systems with partly unknown transition probabilities is investigated. The obtained results are presented in the form of linear matrix inequalities, which is easily computed by the Matlab toolbox. The considered systems are more general than the systems with completely known or completely unknown transition probabilities, which can be viewed as two special cases of the ones tackled here. Moreover, in contrast with the recent research on uncertain transition probabilities, our proposed concept of the partly unknown transition probabilities does not require any knowledge of the unknown elements, such as the bounds or structures of uncertainties. In addition, the relationship between the stability criteria currently obtained for the usual MJLS and switched linear system under arbitrary switching is exposed by our proposed systems. Furthermore, the number of matrix inequalities conditions obtained in this paper is much more than some existing results due to the introduced free matrices based on the system itself and the information of transition probabilities in this paper, which may increase the complexity of computation. However, it would decrease the conservativeness for the delay-dependent stability conditions. Finally, two numerical examples are provided to illustrate the effectiveness of our results.

#### 2. Problem Statement and Preliminaries

Consider the following neutral system with markovian jump parameters: where is a right-continuous Markov process on the probability space taking values in a finite state space, with generator given by where , , for , is the transition rate from mode at time to mode at time , . , , and are known matrix functions of the markovian process, is the state vector, and is the initial condition function. and are mode-dependent delays, when , , and .

Since the transition probability depends on the transition rates for the continuous-time MJSs, the transition rates of the jumping process are considered to be partly accessible in this paper. For instance, the transition rate matrix with operation modes may be expressed as where represents the unknown transition rate. For notational clarity, , the set denotes with , , and .

Moreover, if , it is further described as where is a nonnegation integer with , and represent the known element of the set in the row of the transition rate matrix .

*Remark 2.1. *It is worthwhile to note that if which means that any information between the mode and the other modes is not accessible, then MJSs with modes can be regarded as ones with modes. It is clear that when , the system (2.1) becomes the usual assumption case.

For the sake of simplicity, the solution with is denoted by . It is known from [38] that is a Markov process with an initial state , and its weak infinitesimal generator, acting on function , is defined in [39]:

Throughout this paper, the following definition is necessary. More details refer to [23].

*Definition 2.2 (see, [32]). *The system (2.1) is said to be stochastically stable if the following holds:
for every initial condition and .

#### 3. Stability Analysis for Neutral Markovian Jump Systems

The purpose of this section is to state the stability analysis for neutral markovian jump systems with partly unknown transition rates. Throughout the paper, the matrix is assumed to be . Before giving the stability result of systems (2.1) with a partly unknown transition rate matrix (2.3), the stability of neutral markovian jump systems (2.1) with all transition probabilities known is firstly investigated. With the introduced free matrices and the novel analysis technique of matrix, the stability conditions are presented in this section.

Theorem 3.1. *The system (2.1) with a fully known transition rate matrix is stochastically stable if there exist matrices , and and any matrices with appropriate dimensions satisfying the following linear matrix inequalities:
**
with
*

* Proof. *Construct a stochastic Lyapunov functional candidate as
where
where are all positive definite matrices with appropriate dimensions to be determined. Then, for given , and the weak infinitesimal operator of the stochastic process along the evolution of are given as
According to the definition of the weak infinitesimal operator and the expression (2.2), it can be shown that
Similar to the above, we can obtain
Moreover, there exist matrices with appropriate dimensions, such that the following equality holds according to (2.1):
where

From (3.7)–(3.10) and with (3.2)-(3.3), one can obtain that
where are defined in this theorem. Therefore,
which means that systems (2.1) are stochastic stability. The proof is completed.

Based on the result of Theorem 3.1, the next theorem will relate to the stability condition of systems (2.1) with partially known transition probabilities.

Theorem 3.2. *The system (2.1) with a partly unknown transition rate matrix (2.4) is stochastically stable if there exist matrices , and and any matrices with appropriate dimensions satisfying the following linear matrix inequalities:
**
with
**
and is a given lower bound for the unknown diagonal element. *

* Proof. * For the case of the systems (2.1) with partly unknown transition probabilities, and taking into account the situation that the information of transition probabilities is not accessible completely, due to , the following zero equation holds for arbitrary matrices is satisfied:
and the inequality (3.12) can be rewritten as
where has already been defined on the above
with the elements are the same as those in , except for
and note that and for all , namely, for all . Therefore, it follows from easy computation that if , inequalities (3.20) and the formula (3.25) less than imply that
On the other hand, for the same reason, if , inequalities (3.20)-(3.21) and the formula (3.25) less than also imply that inequality (3.27) holds. Therefore,
which means that systems (2.1) with partly unknown transition probabilities are stochastically stable.

Note that the formula (3.25) less than can be represented as follows:
where is an unknown element in transition probabilities matrix, and
with
One can note that (3.29) can be separated into two cases, and .*Case I* (): it should be first noted that in this case one has . In fact, we only need to consider because means all the elements in the throw of the transition rate matrix.

Now the formula (3.29) can be rewritten as
It follows from and that
Similar to the above proof, (3.2) and (3.3) can be rewritten as (3.16) and (3.18), respectively, for this case. Accordingly, for , is equivalent to (3.14) which is satisfied for all , which also implies that, in the presence of unknown elements , the system stability is ensured if (3.14), (3.16), (3.18), and (3.20) hold.*Case II* (): for the sake of simple expression, let .

In this case, is unknown, , and , and following the same analysis of the above case, we just consider . And now the formula (3.29) can be rewritten as
Similarly, since we have
one can know that
which means that is equivalent to ,
and from the defined in Theorem 3.2, we have that , which means that may take any value between for some arbitrarily small. Then, can be further written as a convex combination
where takes value arbitrarily in . So, (3.37) holds if and only if ,
simultaneously hold. Since is arbitrarily small, (3.39) holds if and only if
which is the case in (3.40) when for . Hence, (3.37) is equivalent to (3.15). Furthermore, following the same line of this proof, (3.2) and (3.3) can be represented as (3.17) and (3.19), respectively, in this case.

Therefore, from the above discussion, in the presence of unknown elements in the transition probabilities matrix, we can readily draw a conclusion that the system (2.1) with partly known transition rates is stable if the inequalities in Theorem 3.2 are satisfied. It completes this proof.

*Remark 3.3. *In order to obtain the less conservative stability criterion of MJSs with partial information on transition probabilities, similar to [32], the free-connection weighting matrices are introduced by making use of the relationship of the transition rates among various subsystems, that is, for all , which overcomes the conservativeness of using the fixed-connection weighting matrices. However, it is difficult to decrease the conservative using free-connection matrices only based on the above equalities, but not on the systems and the themselves Newton-Leibniz formula. Moreover, this paper is inspired by [30], and the delay-dependent stability results in this theorem are the extension of [30] to delay systems to some extent. Although the large number of introduced free weighting matrices may increase the complexity of computation, using the technique of free weighting matrices would reduce the conservativeness, which would be reflected in the fifth section.

*Remark 3.4. *It should be noted that the more known elements are there in (2.3), the lower the conservative of the condition will be. In other word, the more unknown elements are there in (2.3), the lower the maximum of time delay will be in Theorem 3.2. Actually, if all transition probabilities are unknown, the corresponding system can be viewed as a switched linear system under arbitrary switching. Thus, the conditions obtained in Theorem 3.2 will thereby cover the results for arbitrary switched linear system with mixed delays. In that case, one can see that the stability condition in Theorem 3.2 becomes seriously conservative, for many constraints. Fortunately, we can use the common Lyapunov functional method to analyze the stability for the system under the assumption that all transition probabilities are not known.

For the stability analysis of the neutral markovian jump systems with all transition probabilities is not known, one can construct the following common Lyapunov functional: and following a similar line as in the proof of Theorem 3.1, we can obtain the following corollary.

Corollary 3.5. * The system (2.1) with all elements unknown in transition rate matrix (2.3) is stochastically stable if there exist positive definite matrices and any matrices with appropriate dimensions satisfying the following linear matrix inequalities:
**
with
*

#### 4. Extension to Uncertain Neutral Markov Jump Systems

In this section, we will consider the uncertain neutral Markov jump systems with partially unknown transition probabilities as follows: are known mode-dependent constant matrices with appropriate dimensions, while , are the time-varying but norm-bounded uncertainties satisfying where , and are known mode-dependent matrices with appropriate dimensions, and is the time-varying unknown matrix function with Lebesgue norm measurable elements satisfying .

Theorem 4.1. *The uncertain neutral markovian jump system (4.1) with a partly unknown transition rate matrix (2.3) is stochastically stable if there exist matrices , and and any matrices , and with appropriate dimensions satisfying the following linear matrix inequalities:
**
with
**
and is a given lower bound for the unknown diagonal element.*

*Proof. * can be written as
where
and are defined in Theorem 3.2. According to the approach in [40] with Lemma 2.4 in [41], there exists a scalar such that (4.12) are equivalents to

Introducing new variables , , , and , and with Schur's complement [42], yields inequalities (4.3). Similarly, it concludes (4.4) with the same proof. On the other hand, with the same variables substitution, we note that pre- and postmultiplying, respectively, (3.16)–(3.21) by a scalar yield (4.5)–(4.10), which completes this proof.

*Remark 4.2. *It should be mentioned that Theorem 4.1 is an extension of (2.1) to uncertain neutral markovian jump systems (4.1) with incomplete transition descriptions. In fact, this technology is frequently adopted in dealing with the robust stability analysis of uncertain systems.

#### 5. Examples

In order to show the effectiveness of the approaches presented in the above sections, two numerical examples are provided.

*Example 5.1. * Consider the MJLS (2.1) with four operation modes whose state matrices are listed as follows:
The partly transition rate matrix is considered as

Our purpose here is to check the stability of the above system for the two different cases of transition probabilities. For Case I, it is clear to see that and are not valued, one can set , and let , and . Solving the inequalities in Theorem 3.2 using LMI toolbox, the maximum of the time delay can be computed as . However, in Case II, the maximum of the time delay can be computed as by Theorem 3.1. It is easily seen that the more transition probabilities knowledge we have, the larger the maximum of delay can be obtained for ensuring stability. This shows the trade-off between the cost of obtaining transition probabilities and the system performance.

Furthermore, when the transition probabilities are not fully known, as the delay for one of the subsystems decreases, the maximum of other delays may increase. However, when all transition probabilities are fully known, the conclusion may be on the opposite in some interval. In fact, the above observation is in accordance with the actual. Then, we assume that , and let , and , be different with , and with the same computation in Theorem 3.1, as shown in Table 1. However, just according to the approach of Theorem 3 in [32], not introducing some other free matrices and some other skills, we cannot find the feasible solutions which contain time delay to verify the stability of the system. Therefore, this example shows that the stability criterion in this paper gives much less conservative delay-dependent stability conditions. This example also shows that the approach presented in this paper is effectiveness.

*Example 5.2. * Consider the above MJLS in Example 5.1 with partially unknown transition probabilities of Case I, and the uncertain structure matrices are described by (4.2) where
In this case, we check the robust stability result provided by Theorem 4.1. One can also set , and let , and . Solving the inequalities in Theorem 4.1 by using LMI toolbox, the maximum of the time delay can be computed as . Some of the feasible solutions can be obtained as follows:
In a word, this example shows that the robust stability condition of Theorem 4.1 is feasible. It is also approved that the approach provided in this paper is effectiveness.

*Example 5.3. * Consider the MJLS (2.1) with two operation modes whose state matrices are listed as follows:
The partly transition rate matrix is considered as

The above matrix (5.8) implies that systems (2.1) with all transition probabilities are not known, which viewed the systems as switched systems with arbitrary switching. Given that , and , according to the approach of Theorem 3.2, we cannot find the feasible solutions. However, using the Matlab LMI toolbox, we solve the LMI in Corollary 3.5, and the feasible solutions can be obtained as follows:
For the case of all transition probabilities that are unknown, this example also shows that Corollary 3.5 is less conservative than Theorem 3.2 on the stability analysis for the neutral markovian jump system.

#### 6. Conclusion

The delay-dependent stability for neutral markovian jump systems with partly known transition probabilities has been investigated. Based on a new class of stochastic Lyapunov-Krasovskii functionals constructed, and combined with the technique of analysis for matrix inequalities, some new stability criteria are obtained. The main contribution of this paper contains the following two-fold: one is the extension of delay-dependent stability conditions for markovian jump delay systems to markovian neutral jump systems; the other is the new method presented to decrease the conservative brought by the markovian jump with partly known transition probabilities. The future work is to investigate the systems with mode-dependent interval mixed time delays and the systems with unsynchronised control input. Three examples have shown the effectiveness of the conditions presented in this paper.

#### Nomenclature

: | -dimensional real space |

: | Set of all real by matrices |

or: | Transpose of vector (or matrix ) |

: | (, resp.) Matrix is symmetric positive (negative, resp.) definite |

: | (, resp.) Matrix is symmetric positive (negative, resp.) semidefinite |

: | The elements below the main diagonal of a symmetric block matrix |

: | , . |

#### Acknowledgments

This work was supported by the Mathematical Tianyuan Foundation of China (Grant no. 11126305), the Natural Science Foundation of Yunnan Province (Grant no. 2011FZ172), the Natural Science Foundation of Yunnan Province (Grant no. 2011FZ169), the Natural Science Foundation of Yunnan Province (Grant no. 2009CD019), and the Youth Foundation of Yunnan University of Nationalities (Grant no. 11QN07).

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