Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 420420, 13 pages

http://dx.doi.org/10.1155/2013/420420

## Exponential Stability in Mean Square of Singular Markovian Jump System with Saturating Actuators and Time-Varying Delay

^{1}School of Mathematics, Hefei University of Technology, Hefei 230009, China^{2}Department of Mathematics and Physics, Hefei University, Hefei 230601, China

Received 12 March 2013; Revised 15 June 2013; Accepted 4 July 2013

Academic Editor: Jian Li

Copyright © 2013 Fangqing Ding and Xianfa Jiao. 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

This paper investigates exponential stability in mean square of singular Markovian jump systems with saturating actuators and time-varying delay. The statistical property of the Markov process is fully used to derive the differential of the function. By using a delay decomposition method, a mode-dependent Lyapunov-Krasovskii function is established. A sufficient condition is proposed for exponential stability in mean square of the system designing the memoryless state feedback. A numerical example shows that the approach proposed is effective.

#### 1. Introduction

Markovian jump systems are a special class of hybrid systems with both modes and state variables, which is described by a set of time-delayed linear systems with the transitions among models determined by a Markov chain in a finite mode set. In recent years, Markovian jump systems have received much attention, and many problems have been solved, including stability analysis, state feedback, and output feedback controller design [1, 2]. Both time-delay and saturating controls are commonly encountered in various engineering systems, which often lead to poor performance and instability of a control system. Therefore, much attention has been paid to the study on the stability problem of systems with saturating actuator. The problem of robust stabilization of uncertain time-delay systems with a saturating actuator was addressed by Niculescu et al. [3]. Liu has obtained the condition of stability for a class of time-varying delay systems with saturating actuators according to the usage of the linear matrix inequality and Leibniz-Newton formula [4]. For the systems with time-varying delays, the reported results are generally based on the assumption that the derivative of time-varying delays is less than one [3, 4]. Such restriction is very conservative. Furthermore, singular system model is a natural presentation of dynamic systems and can describe a larger class of systems than regular ones, such as large-scale systems, power systems, and constrained control systems, electrical circuits, power systems, and economics [5]. For a singular time-delay system, it is important to develop conditions which guarantee that the given singular system is not only stable but also regular and impulse free. Wu et al. discussed the stability problem on uncertain singular Markovian jump time-delay systems [6, 7]. Ma et al. investigated the robust stochastic stability problem for discrete-time uncertain singular Markov jump systems with actuator saturation [8]. Boukas et al. established an LMI condition for the singular time-delay systems with Markovian jump to be regular, impulse free, and stochastically stable. However, the result of [9] is delay independent. Generally speaking, delay-independent conditions are more conservative than delay-dependent ones, especially for small time delays. To the best of our knowledge, the robust stability for singular Markov jump systems with actuator saturation and time-varying delay has not been investigated in the literature; this problem is important in both theory and practice.

In this paper, we are concerned with stability of singular Markovian jump system with saturating actuators and time-varying delay. A new Lyapunov function can be constructed by using a delay decomposition method and considering the statistical property of Markovian process. In terms of LMI approach, stability conditions are proposed to guarantee the considered systems to be regular, impulse free, and exponentially stable in mean square. Free weighting matrix and reduction method are not introduced in the derivation of the stability criterion. Thus, the results reduce conservation.

#### 2. Problem Formulation and Assumption

Consider the singular markovian jump system with saturating actuators and time-varying delay as follows: where and are the system states and inputs, respectively, , is theth component of, and is a continuous vector-valued initial function defined on the interval. The matrix may be singular, and rank. is the probability space. , , and are known real constant matrices with appropriate dimension. As , for the sake of simplicity, , , and denote the matrixes, respectively.is the time-varying delay satisfying and , are real constant scalars.

The saturating function is defined as follows: whereare known real constants.

*Assumption 1. *The markovian jumping parameter is right-continuous markovian process and takes values in finite set , and is defined by
where , is the transition rate from to and

*Definition 2. *The infinitesimal generator of the solution to system (1) is defined as

*Definition 3. *(1) The singular Markovian jump system with saturating actuators and time-varying delay
is said to be regular and impulse free, if the pairs are singular and impulse free for every .

(2) The singular Markovian jump system with saturating actuators and time-varying delay (8) is said to be exponential stable in mean square if there exist scalars, , such that

(3) The singular Markovian jump system with saturating actuators and time-varying delay (8) is said to be mean-square exponentially admissible, if it is regular, impulse free, and exponentially stable in mean square.

Lemma 4 (see [7]). *Suppose that a positive continuous function satisfies
**
then
**
where, , , and .*

Lemma 5 (see [10]). * is an diagonal matrix whose entries are either 1 or 0. Note that there are elements in. Suppose that each element in is labeled as , , and let; it is obvious that, if . Given matrix, one has
**
where denotes convex hull.*

Lemma 6 (see [11]). *Given , for a vector and a positive-definite matrix with appropriate dimensions, and the integrations concerned are well defined, then
*

#### 3. Main Results

Considering the system (1), for each region , the system (1) becomes

Taking the memoryless state feedback control for the systems (14), one has

Substituting (15) into (14), then the corresponding closed-loop systems become

Theorem 7. *For given constants , and two positive integers , , The system (16) is exponentially stable in mean square. If there exist symmetric positive matrices, , , , , , , where
**for every, such that**where
*

*Proof. * According to the definition of the saturating function and Lemma 5, we have

Substituting (19) into (16) gives

Since , there exist nonsingular matrices and , such that

Let

According to (18a), it is easy to obtain that for every . By pre- and postmultiplying by and , respectively, we have, which implies that are nonsingular for every . Thus, the pairs are regular and impulse free for every. Thus, by Definition 2, the system (8) is regular and impulse free for any time-varying delay satisfying (2).

Choose a Lyapunov-Krasovskii function candidate as
where

Moreover, the action of infinitesimal generator (7) on each term function (23) could be expressed as
According to (20), (23), and (25), we have
Similar to (33), by Lemma 6, it is easy to obtainwhereIn conclusion, one has
where

using (18b) and (18c), it is easy to obtain

Since are nonsingular for every , we set

It is easy to get

where

Denote that

Let

Then, for every , system (20) is a restricted system equivalent to

To prove the exponential stability in mean square, we define a new function as
where and then by Dynkin’s formula, we find that for every ,

It can be seen from (23), (45), and (46) that if is chosen small enough, a constant can be found such that for ,

Hence, for any
where . To study the exponential stability in mean square of , we apply the similar analysis method of [12] and define

Then, according to (48), we have a constant ; it can be found that when ,

We construct a function as

By premultiplying the second equation of (44) with , we obtain that

Adding (52) to (51) yields that
where is any positive scalar.

By pre- and postmultiplying by and, respectively, we get a constant, such that

On the other hand, since can be chosen arbitrarily, is chosen small enough, such that . Then, we can always find a scalar , such that

It follows from (51), (53), and (54) that

It is clear that the above inequalities (55) and (56) imply that

Thus, we get from (57) that

Let; from the above inequality and the fact (50) and (58), we have
where, .

Therefore, applying Lemma 4 to the above inequality yields that

We can find from (50) and the above inequality that the system (16) is exponentially stable in mean square.

*Remark 8. *It should be pointed out that sometimes too many free variables cannot reduce the conservatism of the obtained results. In the theorem, the system is exponentially stable in mean square without resorting to the free-weighting matrices method, which is in contrast with [6], where the free-weighting matrices method was used.

*Remark 9. *In the proof of the theorem, the delay interval is divided into segments of equal length , such that the information of delayed states , , are all taken into account. It is clear that the Lyapunov function defined in the theorem is more general than the ones in [7].

#### 4. Numerical Examples

In this section, we provide a numerical example to verify the effectiveness of the proposed method. Consider systems (1) in with two regines . The system parameters are described as follows:

As , and (1), by calculating we have (2), by calculating we have