#### Abstract

This paper is concerned with the delay-dependent control problem for continuous-time descriptor Markovian jump systems with time-varying delay. By constructing various Lyapunov-Krasovskii functionals for different subsystems, together with delay decomposition method, a new delay-dependent bounded real lemma (BRL) is derived, under which descriptor Markovian jump time-delay systems are regular, impulse-free, and stochastically stable and satisfy a prescribed performance level. Since the reciprocally convex combination approach is adopted to estimate the upper bound of the integral terms, the BRL obtained in this paper is less conservative than some existing ones. Based on the proposed BRL, a sufficient condition for the existence of state feedback controller is provided. Finally, three numerical examples are provided to demonstrate the validity of the proposed methods.

#### 1. Introduction

Over the past decades, stability problems for time-delay systems have received considerable attention. Many methods have been presented to determine the stability. In the published results, two types of criteria have been widely investigated (see [1]), that is, delay-independent one and delay-dependent one. As we know, the delay-dependent criteria are less conservative than delay-independent ones, especially when the time delay is fairly small. Therefore, in order to reduce the conservatism of stability conditions, a great number of delay-dependent results have been reported, including, descriptor modeled transformation method (e.g., [2]), parameterized modeled transformation one (e.g., [3]), free-weighting matrix one (e.g., [4]), discretized Lyapunov functional one (e.g., [5]), delay decomposition one (e.g., [6]) and so on (e.g., [7, 8]). Among the aforementioned methods, since discretized Lyapunov functional method is complicated and it is difficult to apply this method to design controller, the delay decomposition method is less conservative than others except the discretized Lyapunov functional method.

Markovian jump systems are modeled by many stochastic systems, which arise abrupt random changes in their structure, such as component failures or repairs and unexpected environmental changes. A great deal of attention has been devoted to Markovian jump systems (see [9, 10]). Recently, many efforts have been made to study the stability analysis and controller synthesis problems for Markovian jump systems with time-varying and time-invariant delay. Reference [11] divides the time-invariant delay into segments and uses the delay division method to obtain a less conservative criterion than [12]. In [13], stability and stabilization are discussed for Markovian jump systems with time-varying delay by selecting distinct Lyapunov matrices for different subsystems and introducing a triple-integral term. Reference [14] studies the analysis for Markovian jump stochastic systems with time-varying delay. By dividing the delay interval into two parts, two improved delay-dependent conditions have been presented to guarantee the stochastic stability.

Descriptor systems are also referred to as singular systems, semistate systems, which are a natural representation of dynamical systems and describe a much wider range of systems than standard state-space systems (see, e.g., [10, 15]). When descriptor systems are subject to time delay and stochastic disturbance in their structure, it is very natural to cast this model into descriptor Markovian jump time-delay systems. Very recently, a large number of researches have been published to study descriptor Markovian jump time-delay systems. By constructing more complicated Lyapunov-Krasovskii functionals, [16] studies the passivity and stochastic stability of descriptor Markovian jump systems with time-varying delay. Based on the delay decomposition method, [17] investigates the control problem for descriptor Markovian jump time-delay systems. To the best of our knowledge, the time-varying delay in most published results has not been tackled adequately, which leaves us much room to improve. All of these motivate the current study.

In this paper, we investigate control problem for continuous-time descriptor Markovian jump systems with time-varying delay. Motivated by delay decomposition method, through designing various Lyapunov-Krasovskii functionals for different subsystems, a delay-dependent BRL is established, which guarantees that the considered system is stochastically admissible and satisfies a prescribed performance level. The reciprocally convex combination method is used in the proof of the proposed results, which is one of the contributions to this paper. The other is the employed Lyapunov-Krasovskii functionals. Thanks to all of these, the obtained results are less conservative than some existing results.

The rest of this paper is organized as follows. Section 2 gives problem description and some essential lemmas. Section 3 focuses on control problem for continuous-time descriptor Markovian jump time-delay systems. A new delay-dependent BRL is proposed, and the design method for the state feedback controller, which guarantees that the resultant closed-loop system is stochastically admissible with a prescribed performance level, is provided. Section 4 provides numerical examples to illustrate the effectiveness of our methods, and Section 5 concludes the paper.

*Notation.* The notation used throughout this paper is quite standard. stands for the -dimensional Euclidean space, and represents the set of all real matrices. The superscripts and stand for matrix transposition and matrix inverse, respectively. represents the space of square integrable vector functions over the interval . is the probability space, is the sample space, is the -algebra of subsets of the sample space, and is the probability measure on . is the expectation operator with respect to some probability measure . represents the set of positive integers. The notation () implies that is a real symmetric and positive definite (semipositive definite) matrix. For simplicity, is often adopted to denote . The asterisk in matrix block implies that it can be induced by symmetric position. A block diagonal matrix with diagonal blocks will be denoted by . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

#### 2. Problem Formulation and Preliminaries

Fix the probability space and consider the following continuous-time descriptor Markovian jump systems with time-varying delay: where is the state vector, is the input vector, is the controlled output vector, and is the disturbance input vector which belongs to . is a known real matrix which may be singular, and it is generally assumed that rank . The jumping mode , which takes value in a finite set , is the continuous-time Markov process with the following mode transition probabilities: where , , and , satisfying , represent the switching rate from mode at time to mode at time for all . , , , , , , , and are known real constant matrices representing the deterministic systems for each . The considered time-varying delay satisfies where and are known constants. Throughout this paper, for each , we use and to denote matrices and .

To facilitate exposition, the unforced descriptor Markovian jump time-delay systems are given as follows:

In the sequel, we recall some indispensable definitions.

*Definition 1 (see [10]). *When the disturbance signal , for any given scalars and , consider the following statements.(1)The continuous-time descriptor Markovian jump time-delay systems (4) are said to be regular and impulse-free for any time-varying delay satisfying (3) if the pairs are regular and impulse-free for each .(2)The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically stable if there exists a scalar such that
â€‰where denotes the solution to systems (4) at time under the given initial conditions.(3)The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible if they are regular, impulse-free, and stochastically stable.

*Definition 2 (see [17]). *For a given scalar , the descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible with performance level ; if the system with is stochastically admissible and under zero initial condition, it satisfies
for any nonzero .

In this paper, the purpose is to design a mode-dependent state feedback controller for the continuous-time descriptor Markovian jump time-delay systems (1) with a prescribed scalar such that the resultant closed-loop systems are stochastically admissible with performance level . Specifically, the desired state feedback controller can be expressed as

To this end, some essential lemmas are introduced to expedite the following discussion.

Lemma 3 (see [18]). *For any constant matrix , , scalar , and vector function such that the following integration is well defined, then
*

Lemma 4 (see [8]). *Let have positive values in an open subset of . Then the reciprocally convex combination of over satisfies
**
subject to
*

Based on Lemma 4, we present the following lemma, which can be regarded as an alternative form of the reciprocally convex combination.

Lemma 5. *For any constant matrix , , and a variable satisfying , if there exists a matrix satisfying , then
**
where and are vector functions with compatible dimensions.*

* Proof. *Taking
it can be easily verified that the above choice satisfies constraint (10) in Lemma 4. Note that with the equality (9), we have
By Lemma 4, (13) and (14) yield that
This completes the proof.

Lemma 6 (see [10]). *Given any real matrices , , and with and , then one has
**
if and only if
**
or equivalently
*

Lemma 7 (see [10]). *Given matrices , and with appropriate dimensions
**
for all satisfying if and only if there exists scalar such that
*

Lemma 8 (see [19]). *Let be symmetric such that , and let be nonsingular. Then, is nonsingular and its inverse is expressed as
**
where and are full rank with and and are of full column rank and composed of bases of null space of and , respectively. is symmetric and is nonsingular such that , .*

#### 3. Main Results

In this section, the delay-dependent control problem will be studied for the descriptor Markovian jump time-delay system (1) by virtue of the delay decomposition method. To this end, by dividing the delay interval into two disjoint subintervals and , , the following delay-dependent bounded real lemma (BRL) for the system (4) is firstly proposed, which will play an indispensable role in the derivation of our main results.

Theorem 9. *For given scalars , , and , the descriptor Markovian jump time-delay system (4) is stochastically admissible with performance level for any satisfying (3), if there exist a set of positive definite matrices , , , , , , , , , and matrices , , , , satisfying the following LMIs for each :**where , , and
*

*Proof. *Assume that there exist matrices such that LMIs (22)â€“(27) hold. With these conditions in hand, we firstly show the regularity and absence of impulse of the system (4). To this end, it is from [10] that there exist a couple of nonsingular matrices and such that for each

Decomposing the associated matrices with a compatible dimension, we have
where and are arbitrarily nonsingular matrices.

From the block in the left-hand matrices of (22) or (23), and observing that , , , and are positive definite, it can be easily seen that
where is the function of and and makes no difference to the subsequent proof.

Pre- and postmultiplying (31) by and , respectively, together with (29) and (30), we get
According to [10], we can know that the pairs are regular and impulse-free for each . Therefore, by Definition 1, the system (4) is regular and impulse-free.

Next, we will show the stochastic stability of the system (4). To this end, we define a new process by ; then from [12], we can know is a Markov process with initial state . Now, construct the stochastic Lyapunov functional candidate to be
where

Let be the weak infinitesimal generator of the random process acting on [12]. Then for each , it can be shown that

Next, we will estimate the upper bound of the integral terms in and .*Case **1* (). The upper bound can be evaluated for the last five integral terms in , which can be shown as

Likewise, by applying Jensen inequality of Lemma 3 and reciprocally convex combination method of Lemma 5, the upper bound for the two definite integrals in can be shown as

Combining (35)â€“(39), it yields that
where , and
where and

From the Schur complement lemma of Lemma 5, it can be easily verified that holds if (22) holds for .*Case **2* (). The upper bound can be evaluated for the last five integral terms in , which can be shown as

Likewise, by applying Jensen inequality of Lemma 3 and reciprocally convex combination method of Lemma 5, the upper bound for the two definite integrals in can be shown as

Combining (35)â€“(37) and (43)-(44), it yields that
where , and

From the Schur complement lemma of Lemma 5, it can be easily verified that holds if (23) holds for .

Based on the above analysis, we can know that there exists a scalar such that for each
Then, by Dynkinâ€™s formula, we can have for each and
which implies that

Following a similar line in [12], we can find that the system (4) is stochastically stable under the condition that . By Definition 1, the system (4) is stochastically admissible.

In the following, we will establish the performance index of the system (4). Using the stochastic Lyapunov functional (33), under the zero initial condition, it is from (22)â€“(27) that
where ; for ,
and for ,

By Schur complement lemma of Lemma 6, together with (22)â€“(27), we have that for all . Consequently, under the zero initial condition, we have that (6) holds for any nonzero . Therefore, by Definition 2, the descriptor Markovian jump time-delay system (4) is stochastically admissible with performance level for any time delay satisfying (3). This completes the proof.

*Remark 10. *By virtue of the novel Lyapunov-Krasovskii functional, Theorem 9 provides a delay-dependent and delay-derivative-dependent criterion to guarantee the system (4) to be stochastically admissible with performance level. The novelty of Theorem 9 lies in the new Lyapunov-Krasovskii functional and the reciprocally convex combination method. By dividing the delay interval into two subintervals and the following mode-dependent definite integrals
and mode-dependent double integrals
are introduced. Obviously, the above integral terms take the mode into full consideration and contain the mode-independent one [12] as a special case. Since is restricted to the subinterval or and various Lyapunov-Krasovskii functionals are selected for different subsystems, our method can lead to reduction of conservatism.

For the unforced system (4), if disturbance signal , then we have the following time-delay systems:

Based on Theorem 9, the following stochastic admissibility result can be given.

Corollary 11. *For given scalars , , the system (55) is stochastically admissible if there exist a set of positive definite matrices , , , , , , , , and matrices , , , , satisfying the following LMIs for each :
*