Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2011, Article ID 431751, 20 pages
http://dx.doi.org/10.1155/2011/431751
Research Article

Stochastic Finite-Time Guaranteed Cost Control of Markovian Jumping Singular Systems

1College of Science, Henan University of Technology, Zhengzhou 450001, China
2Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China

Received 17 April 2011; Revised 2 August 2011; Accepted 8 August 2011

Academic Editor: Alexei Mailybaev

Copyright © 2011 Yingqi Zhang 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 problem of stochastic finite-time guaranteed cost control is investigated for Markovian jumping singular systems with uncertain transition probabilities, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, the definitions of stochastic singular finite-time stability, stochastic singular finite-time boundedness, and stochastic singular finite-time guaranteed cost control are presented. Then, sufficient conditions on stochastic singular finite-time guaranteed cost control are obtained for the family of stochastic singular systems. Designed algorithms for the state feedback controller are provided to guarantee that the underlying stochastic singular system is stochastic singular finite-time guaranteed cost control in terms of restricted linear matrix equalities with a fixed parameter. Finally, numerical examples are given to show the validity of the proposed scheme.

1. Introduction

Singular systems are also referred to as descriptor systems or generalized state-space systems and describe a larger family of dynamic systems. The singular systems are applied to handle mechanical systems, electric circuits, interconnected systems, and so forth; see more practical examples in [1, 2] and the references therein. Many control problems have been extensively investigated, and results in state-space systems have been extended to singular systems, such as stability, stabilization, and robust control; for instance, see the references in [310]. Meanwhile, Markovian jumping systems are referred to as a special family of hybrid systems and stochastic systems, which are very appropriate to model plants whose structure is subject to random abrupt changes, such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes; see the references in [11, 12]. The existing results for Markovian jumping systems include a large of variety of problems such as stochastic Lyapunov stability [1316], sliding mode control [17, 18], the 𝐻 control [19, 20], the 𝐻 filtering [12, 21], and so forth; for more results, the readers are to refer to [2224] and the references therein.

In many practical applications, on the other hand, many concerned problems are the practical ones which described system state which does not exceed some bound over a time interval. Compared with classical Lyapunov asymptotical stability on which most results in the literature concentrated, finite-time stability (FTS) or short-time stability was studied to deal with the transient behavior of systems in finite time interval. Some earlier results on FTS can be found in [2528]. Some appealing results were obtained to guarantee finite-time stability, finite-time boundedness, and finite-time stabilization of different systems including linear systems, nonlinear systems, and stochastic systems; for instance, see the papers in [2935] and the references therein. However, to date and to the best of our knowledge, the problems of stochastic singular finite-time guaranteed cost control analysis of stochastic singular systems have not been investigated, although some studies on stochastic singular systems have been conduced recently; see the references [811, 15, 18]. We investigate finite-time guaranteed cost control of one class of continuous-time stochastic singular systems. Our results are totally different from those previous results. This motivates us for the study.

It is well known that linear matrix inequalities (LMIs) have viewed as a powerful formulation and design technique for a variety of linear control problems. Thus reducing a control design problem to an LMI can be considered as a practical solution to this problem [36]. At present, it is an important tool to address stability and stabilization, roust control, the 𝐻 filtering, guaranteed cost control, and so forth; see the references [2, 411, 13, 15] and the references therein. The novelty of our study is that stochastic finite-time stability, stochastic finite-time bounded and stochastic finite-time guaranteed cost control are investigated for one family of Markovian jumping singular systems with uncertain transition probabilities, parametric uncertainties, and time-varying norm-bounded disturbance. The main contribution of this paper is that sufficient conditions on stochastic singular finite-time guaranteed cost control are obtained for the class of stochastic singular systems and, a state feedback controller is designed to guarantee that the underlying stochastic singular system is stochastic singular finite-time guaranteed cost control in terms of restrict LMIs with a fixed parameter.

The rest of this paper is organized as follows. In Section 2 the problem formulation and some preliminaries are introduced. The results on stochastic singular finite-time guaranteed cost control are given in Section 3. Section 4 presents numerical examples to demonstrate the validity of the proposed methodology. Some conclusions are drawn in Section 5.

Notations
Throughout the paper, 𝑛 and 𝑛×𝑚 denote the sets of 𝑛 component real vectors and 𝑛×𝑚 real matrices, respectively. The superscript 𝑇 stands for matrix transposition or vector. 𝐸{} denotes the expectation operator with respect to some probability measure 𝒫. In addition, the symbol denotes the transposed elements in the symmetric positions of a matrix, and diag{} stands for a block-diagonal matrix. 𝜆min(𝑃) and 𝜆max(𝑃) denote the smallest and the largest eigenvalue of matrix 𝑃, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation

Let the dynamics of the class of Markovian jumping singular systems be described by the following:𝐸𝑟𝑡𝐴𝑟̇𝑥(𝑡)=𝑡𝑟+Δ𝐴𝑡𝑥𝐵𝑟(𝑡)+𝑡𝑟+Δ𝐵𝑡𝑢𝐺𝑟(𝑡)+𝑡𝑟+Δ𝐺𝑡𝑤(𝑡),(2.1) where 𝑥(𝑡)𝑛 is system state, 𝑢(𝑡)𝑚 is system input, 𝐸(𝑟𝑡) is a singular matrix with rank𝐸(𝑟𝑡)=𝑟𝑖<𝑛; {𝑟𝑡,𝑡0} is continuous-time Markovian stochastic process taking values in a finite space ={1,2,,𝑁} with transition matrix Γ=(𝜋𝑖𝑗)𝑁×𝑁 and the transition probabilities are described as follows:𝑃𝑖𝑗𝑟=𝑃𝑟𝑡+Δ=𝑗𝑟𝑡=𝜋=𝑖𝑖𝑗Δ+𝑜(Δ),if𝑖𝑗,1+𝜋𝑖𝑗Δ+𝑜(Δ),if𝑖=𝑗,(2.2) where limΔ0𝑜(Δ)/Δ=0, 𝜋𝑖𝑗 satisfies 𝜋𝑖𝑗0(𝑖𝑗), and 𝜋𝑖𝑖=𝑁𝑗=1,𝑗𝑖𝜋𝑖𝑗 for all 𝑖,𝑗; Δ𝐴(𝑟𝑡), Δ𝐵(𝑟𝑡), and Δ𝐺(𝑟𝑡) are uncertain matrices and satisfy𝑟Δ𝐴𝑡𝑟,Δ𝐵𝑡𝑟,Δ𝐺𝑡𝑟=𝐹𝑡Δ𝑟𝑡𝐸1𝑟𝑡,𝐸2𝑟𝑡,𝐸3𝑟𝑡,(2.3) where Δ(𝑟𝑡) is an unknown, time-varying matrix function and satisfiesΔ𝑇𝑟𝑡Δ𝑟𝑡𝐼,𝑟𝑡.(2.4) Moreover, the disturbance 𝑤(𝑡)𝑝 satisfies𝑇0𝑤𝑇(𝑡)𝑤(𝑡)𝑑𝑡<𝑑2,𝑑>0,(2.5) and the matrices 𝐴(𝑟𝑡), 𝐵(𝑟𝑡), and 𝐺(𝑟𝑡) are coefficient matrix and of appropriate dimension for all 𝑟𝑡. In addition, we make the following assumption on uncertain transition probabilities in stochastic singular system (2.1).

Assumption 1. The jump rates of the visited modes from a given mode 𝑖 are assumed to satisfy 0<𝜋𝑖𝜋𝑖𝑗𝜋𝑖,𝑖,𝑗,𝑖𝑗,(2.6) where 𝜋𝑖 and 𝜋𝑖 are known parameters for each mode or may represent the lower and upper bounds when all the jump rates are known, that is, 0<𝜋𝑖=min𝑖,𝑗𝜋𝑖𝑗0,𝑖𝑗𝜋𝑖=max𝑖,𝑗𝜋𝑖𝑗.0,𝑖𝑗(2.7) Moreover, let 𝑁𝑖 denote the number of visited modes from 𝑖 including the mode itself.
Consider a state feedback controller 𝑢𝑟(𝑡)=𝐾𝑡𝑥𝑟𝑡,(2.8) where {𝐾(𝑟𝑡),𝑟𝑡=𝑖} is a set of matrices to be determined later. The system (2.1) with the controller (2.8) can be written by the form of the control system as follows: 𝐸𝑟𝑡̇𝑥(𝑡)=𝐴𝑟𝑡𝑥(𝑡)+𝐺𝑟𝑡𝑤(𝑡),(2.9) where 𝐴(𝑟𝑡)=𝐴(𝑟𝑡)+Δ𝐴(𝑟𝑡)+[𝐵(𝑟𝑡)+Δ𝐵(𝑟𝑡)]𝐾(𝑟𝑡) and 𝐺(𝑟𝑡)=𝐺(𝑟𝑡)+Δ𝐺(𝑟𝑡).

Definition 2.1 (see [12, regular and impulse free]). (i) The singular system with Markovian jumps (2.9) with 𝑢(𝑡)=0 is said to be regular in time interval [0,𝑇] if the characteristic polynomial det(𝑠𝐸(𝑟𝑡)𝐴(𝑟𝑡)) is not identically zero for all 𝑡[0,𝑇].
(ii) The singular systems with Markovian jumps (2.9) with 𝑢(𝑡)=0 is said to be impulse free in time interval [0,𝑇], if deg(det(𝑠𝐸(𝑟𝑡)𝐴(𝑟𝑡)))=rank(𝐸(𝑟𝑡)) for all 𝑡[0,𝑇].

Definition 2.2 (stochastic singular finite-time stability (SSFTS)). The closed-loop singular system with Markovian jumps (2.9) with 𝑤(𝑡)=0 is said to be SSFTS with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡)), with 𝑐1<𝑐2 and 𝑅(𝑟𝑡)>0, if the stochastic system is regular and impulse free in time interval [0,𝑇] and satisfies 𝐸𝑥𝑇(0)𝐸𝑇𝑟𝑡𝑅𝑟𝑡𝐸𝑟𝑡𝑥(0)𝑐21𝑥𝐸𝑇(𝑡)𝐸𝑇𝑟𝑡𝑅𝑟𝑡𝐸𝑟𝑡𝑥(𝑡)<𝑐22[].,𝑡0,𝑇(2.10)

Definition 2.3 (stochastic singular finite-time boundedness (SSFTB)). The closed-loop singular system with Markovian jumps (2.9) which satisfies (2.5) is said to be SSFTB with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡),𝑑), with 𝑐1<𝑐2 and 𝑅(𝑟𝑡)>0, if the stochastic system is regular and impulse free in time interval [0,𝑇] and condition (2.10) holds.

Remark 2.4. SSFTB implies that not only is the dynamical mode of the stochastic singular system finite-time bounded but also the whole mode of the stochastic singular system is finite-time bounded in that the static mode is regular and impulse free.

Definition 2.5 (see [11, 13]). Let 𝑉(𝑥(𝑡),𝑟𝑡=𝑖,𝑡>0) be the stochastic function; define its weak infinitesimal operator 𝕃 of stochastic process {(𝑥(𝑡),𝑟𝑡=𝑖),𝑡0} by 𝕃𝑉𝑥(𝑡),𝑟𝑡=𝑖,𝑡=𝑉𝑡(𝑥(𝑡),𝑖,𝑡)+𝑉𝑥(𝑥(𝑡),𝑖,𝑡)̇𝑥(𝑡,𝑖)+𝑁𝑗=1𝜋𝑖𝑗𝑉(𝑥(𝑡),𝑗,𝑡).(2.11) Associated with this system (2.9) is the cost function 𝐽𝑇𝑟𝑡=𝐸𝑇0𝑥𝑇(𝑡)𝑅1𝑟𝑡𝑥(𝑡)+𝑢𝑇(𝑡)𝑅2𝑟𝑡,𝑢(𝑡)𝑑𝑡(2.12) where 𝑅1(𝑟𝑡) and 𝑅2(𝑟𝑡) are two given symmetric positive definite matrices for all 𝑟𝑡=𝑖.

Definition 2.6. There exists a controller (2.8) and a scalar 𝜓0 such that the closed-loop stochastic singular system with Markovian jumps (2.9) is SSFTB with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡),𝑑) and the value of the cost function (2.12) satisfies 𝐽𝑇(𝑟𝑡)<𝜓0 for all 𝑟𝑡; then stochastic singular system (2.9) is said to be stochastic singular finite-time guaranteed cost control. Moreover, 𝜓0 is said to be a stochastic singular guaranteed cost bound, and the designed controller (2.8) is said to be a stochastic singular finite-time guaranteed cost controller for stochastic singular system (2.9).
In the paper, our main aim is to concentrate on designing a state feedback controller of the form (2.8) that renders the closed-loop stochastic singular system with Markovian jumps (2.9) stochastic singular finite-time guarantee cost control.

Lemma 2.7 (see [36]). For matrices 𝑌, 𝐷, and 𝐻 of appropriate dimensions, where 𝑌 is a symmetric matrix, then 𝑌+𝐷𝐹(𝑡)𝐻+𝐻𝑇𝐹𝑇(𝑡)𝐷𝑇<0(2.13) holds for all matrix 𝐹(𝑡) satisfying 𝐹𝑇(𝑡)𝐹(𝑡)𝐼 for all 𝑡, if and only if there exists a positive constant 𝜖, such that the following equality holds: 𝑌+𝜖𝐷𝐷𝑇+𝜖1𝐻𝑇𝐻<0.(2.14)

3. Main Results

This section deals with the guaranteed cost SSFTB analysis and design for the closed-loop singular system with Markovian jumps (2.9).

Theorem 3.1. The closed-loop singular system with Markovian jumps (2.9) is SSFTB with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡),𝑑), if there exist a scalar 𝛼0, a set of nonsingular matrices {𝑃(𝑖),𝑖} with 𝑃(𝑖)𝑛×𝑛, sets of symmetric positive definite matrices {𝑄1(𝑖),𝑖} with Q1(𝑖)𝑛×𝑛, {𝑄2(𝑖),𝑖} with 𝑄2(𝑖)𝑝×𝑝, and for all 𝑟𝑡=𝑖 such that𝐸(𝑖)𝑃𝑇(𝑖)=𝑃(𝑖)𝐸𝑇(𝑖)0,(3.1a)𝐴(𝑖)𝑃𝑇(𝑖)+𝑃(𝑖)𝐴𝑇(𝑖)+Γ(𝑖)𝐺(𝑖)𝑄2(𝑖)<0,(3.1b)𝑃1(𝑖)𝐸(𝑖)=𝐸𝑇(𝑖)𝑅1/2(𝑖)𝑄1(𝑖)𝑅1/2(𝑖)𝐸(𝑖),(3.1c)max𝑖𝜆max𝑄1𝑐(𝑖)21+max𝑖𝜆max𝑄2𝑑(𝑖)2<min𝑖𝜆min𝑄1𝑐(𝑖)22𝑒𝛼𝑇(3.1d) hold, where Γ(𝑖)=𝑁𝑗=1𝜋𝑖𝑗𝑃(𝑖)𝑃1(𝑗)𝐸(𝑗)𝑃𝑇(𝑖)+𝑃(𝑖)[𝑅1(𝑖)+𝐾𝑇(𝑖)𝑅2(𝑖)𝐾(𝑖)]𝑃𝑇(𝑖)𝛼𝐸(𝑖)𝑃𝑇(𝑖). Moreover, a stochastic singular finite-time guaranteed cost bound for the stochastic singular system can be chosen as 𝜓0=𝑒𝛼𝑇max𝑖{𝜆max(𝑄1(𝑖))𝑐21}+max𝑖{𝜆max(𝑄2(𝑖))𝑑2}.

Proof. Firstly, we prove that the singular system with Markovian jumps (2.9) is regular and impulse free in time interval [0,𝑇]. By Schur complement and noting condition (3.1b), we have 𝐴(𝑖)𝑃𝑇(𝑖)+𝑃(𝑖)𝐴𝑇𝜋(𝑖)+𝑖𝑖𝛼𝐸(𝑖)𝑃𝑇(𝑖)<𝑁𝑗=1,𝑗1𝜋𝑖𝑗𝑃(𝑖)𝑃1(𝑗)𝐸(𝑗)𝑃𝑇(𝑖)0.(3.2) Now, we choose nonsingular matrices 𝑀(𝑖) and 𝑁(𝑖) such that 𝐼𝑀(𝑖)𝐸(𝑖)𝑁(𝑖)=diag𝑟𝑖,0,𝑀(𝑖)𝐴𝐴(𝑖)𝑁(𝑖)=11(𝑖)𝐴12𝐴(𝑖)21(𝑖)𝐴22(,𝑖)𝑀(𝑖)𝑃(𝑖)𝑁𝑇(𝑃𝑖)=11(𝑖)𝑃12𝑃(𝑖)21(𝑖)𝑃22.(𝑖)(3.3) Then, we have 𝐸(𝑖)=𝑀1𝐼(𝑖)diag𝑟𝑖𝑁,01(𝑖),𝑃(𝑖)=𝑀1𝑃(𝑖)11(𝑖)𝑃12𝑃(𝑖)21(𝑖)𝑃22(𝑁𝑖)𝑇(𝑖).(3.4) From (3.1a) and (3.4), one can obtain 𝑀1𝐼(𝑖)diag𝑟𝑖𝑁,01𝑀(𝑖)1𝑃(𝑖)11(𝑖)𝑃12𝑃(𝑖)21(𝑖)𝑃22𝑁(𝑖)𝑇(𝑖)𝑇=𝑀1𝑃(𝑖)11(𝑖)𝑃12𝑃(𝑖)21(𝑖)𝑃22𝑁(𝑖)𝑇𝑀(𝑖)1(𝑖)diag{𝐼𝑟𝑖,0}𝑁1(𝑖)𝑇0.(3.5) Computing the above condition (3.5) and noting that 𝑃(𝑖) is nonsingular matrix, one can obtain from (3.3) and (3.4) that 𝑃11(𝑖)=𝑃𝑇11(𝑖)0, 𝑃21(𝑖)=0 and det(𝑃22(𝑖))0 for all 𝑖. Thus, we have 𝐸(𝑖)𝑃𝑇(𝑖)=𝑃(𝑖)𝐸𝑇(𝑖)=𝑀1𝑃(𝑖)11𝑀(𝑖)000𝑇(𝑖)0.(3.6) Pre- and post-multiplying (3.2) by 𝑀(𝑖) and 𝑀𝑇(𝑖), respectively, and noting the equality (3.6), this results in the following matrix inequality: 𝐴22(𝑖)𝑃𝑇22(𝑖)+𝑃22(𝑖)𝐴𝑇22(𝑖)<0,(3.7) where the star will not be used in the following discussion. By Schur complement, we have 𝐴22(𝑖)𝑃T22(𝑖)+𝑃22(𝑖)𝐴𝑇22(𝑖)<0. Therefore 𝐴22(𝑖) is nonsingular, which implies that the closed-loop continuous-time singular system with Markovian jumps (2.9) is regular and impulse free in time interval [0,𝑇].
Let us consider the quadratic Lyapunov-Krasovskii functional candidate as 𝑉(𝑥(𝑡),𝑖)=𝑥𝑇(𝑡)𝑃1(𝑖)𝐸(𝑖)𝑥(𝑡) for stochastic singular system (2.9). Computing 𝕃𝑉 the derivative of 𝑉(𝑥(𝑡),𝑖) along the solution of system (2.9) and noting the condition (3.1a), we obtain 𝕃𝑉(𝑥(𝑡),𝑖)=𝜉𝑇𝑃(𝑡)1(𝑖)𝐴(𝑖)+𝐴𝑇(𝑖)𝑃𝑇(𝑖)+𝑁𝑗=1𝜋𝑖𝑗𝑃1(𝑗)𝐸(𝑗)𝛼𝑃1(𝑖)𝐸(𝑖)𝑃1(𝑖)𝜉𝐺(𝑖)0(𝑡),(3.8) where 𝜉(𝑡)=[𝑥𝑇(𝑡),𝑤𝑇(𝑡)]𝑇. Pre- and postmultiplying (3.1b) by diag{𝑃1(𝑖),𝐼} and diag{𝑃𝑇(𝑖),𝐼}, respectively, we obtain 𝑃1(𝑖)𝐴(𝑖)+𝐴𝑇(𝑖)𝑃𝑇(𝑖)+𝑁𝑗=1𝜋𝑖𝑗𝑃1(𝑗)𝐸(𝑗)+𝑅1(𝑖)+𝐾𝑇(𝑖)𝑅2(𝑖)𝐾(𝑖)𝛼𝑃1(𝑖)𝐸(𝑖)𝑃1(𝑖)𝐺(𝑖)𝑄2(𝑖)<0.(3.9) Noting that 𝑅1(𝑖) and 𝑅2(𝑖) are two symmetric positive definite matrices for all 𝑖, thus, from (3.8) and (3.9), we have 𝐸{𝕃𝑉(𝑥(𝑡),𝑖)}<𝛼𝐸{𝑉(𝑥(𝑡),𝑖)}+𝑤𝑇(𝑡)𝑄2(𝑖)𝑤(𝑡).(3.10) Further, (3.10) can be rewritten as 𝐸𝑒𝛼𝑡𝕃𝑉(𝑥(𝑡),𝑖)<𝑒𝛼𝑡𝑤𝑇(𝑡)𝑄2(𝑖)𝑤(𝑡).(3.11) Integrating (3.11) from 0 to 𝑡, with 𝑡[0,𝑇] and noting that 𝛼0, we obtain 𝑒𝛼𝑡𝑉𝐸{𝑉(𝑥(𝑡),𝑖)}<𝐸𝑥(0),𝑖=𝑟0+𝑡0𝑒𝛼𝜏𝑤𝑇(𝜏)𝑄2(𝑖)𝑤(𝜏)𝑑𝜏.(3.12) Noting that 𝛼0, 𝑡[0,𝑇], and condition (3.1c), we have 𝐸𝑥𝑇(𝑡)𝑃1(𝑖)𝐸(𝑖)𝑥(𝑡)=𝐸{𝑉(𝑥(𝑡),𝑖)}<𝑒𝛼𝑡𝐸𝑉𝑥(0),𝑖=𝑟0+𝑒𝛼𝑡𝑡0𝑒𝛼𝜏𝑤𝑇(𝜏)𝑄2(𝑖)𝑤(𝜏)𝑑𝜏𝑒𝛼𝑡max𝑖𝜆max𝑄1𝑐(𝑖)21+max𝑖𝜆max𝑄2𝑑(𝑖)2.(3.13) Taking into account that 𝐸𝑥𝑇(𝑡)𝑃1𝑥(𝑖)𝐸(𝑖)𝑥(𝑡)=𝐸𝑇(𝑡)𝐸𝑇(𝑖)𝑅1/2(𝑖)𝑄1(𝑖)𝑅1/2(𝑖)𝐸(𝑖)𝑥(𝑡)min𝑖𝜆min𝑄1𝐸𝑥(𝑖)𝑇(𝑡)𝐸𝑇,(𝑖)𝑅(𝑖)𝐸(𝑖)𝑥(𝑡)(3.14) we obtain 𝐸𝑥𝑇(𝑡)𝐸𝑇(𝑖)𝑅(𝑖)𝐸(𝑖)𝑥(𝑡)max𝑖𝜆max𝑄11E𝑥(𝑖)𝑇(𝑡)𝑃(𝑖)𝐸(𝑖)𝑥(𝑡)<𝑒𝛼𝑇max𝑖𝜆max𝑄1𝑐(𝑖)21+max𝑖𝜆max𝑄2𝑑(𝑖)2min𝑖𝜆min𝑄1.(𝑖)(3.15) Therefore, it follows that condition (3.1d) implies 𝐸{𝑥𝑇(𝑡)𝐸𝑇(𝑟𝑡)𝑅(𝑟𝑡)𝐸(𝑟𝑡)𝑥(𝑡)}𝑐22 for all 𝑡[0,𝑇].
Once again from (3.8) and (3.9), we can easily obtain 𝕃𝑉(𝑥(𝑡),𝑖)<𝛼𝑉(𝑥(𝑡),𝑖)+𝑤𝑇(𝑡)𝑄2𝑥(𝑖)𝑤(𝑡)𝑇(𝑡)𝑅1(𝑖)𝑥(𝑡)+𝑢𝑇(𝑡)𝑅2(𝑖)𝑢(𝑡).(3.16) Further, (3.16) can be represented as 𝕃𝑒𝛼𝑡𝑉(𝑥(𝑡),𝑖)<𝑒𝛼𝑡𝑤𝑇(𝑡)𝑄2(𝑖)𝑤(𝑡)𝑒𝛼𝑡𝑥𝑇(𝑡)𝑅1(𝑖)𝑥(𝑡)+𝑢𝑇(𝑡)𝑅2(𝑖)𝑢(𝑡).(3.17) Integrating (3.17) from 0 to 𝑇, we have 𝑇0𝑒𝛼𝑡𝑥𝑇(𝑡)𝑅1(𝑖)𝑥(𝑡)+𝑢𝑇(𝑡)𝑅2<(𝑖)𝑢(𝑡)𝑑𝑡𝑇0𝑒𝛼𝑡𝑤𝑇(𝑡)𝑄2(𝑖)𝑤(𝑡)𝑑𝑡𝑇0𝕃𝑒𝛼𝑡𝑉(𝑥(𝑡),𝑖)𝑑𝑡.(3.18) Using the Dynkin formula and the fact that the system (2.9) is regular and impulse free, we obtain 𝐸𝑇0𝑒𝛼𝑡𝑥𝑇(𝑡)𝑅1(𝑖)𝑥(𝑡)+𝑢𝑇(𝑡)𝑅2<(𝑖)𝑢(𝑡)𝑑𝑡𝑇0𝑒𝛼𝑡𝑤𝑇(𝑡)𝑄2(𝑖)𝑤(𝑡)𝑑𝑡𝐸𝑇0𝕃𝑒𝛼𝑡.𝑉(𝑥(𝑡),𝑖)𝑑𝑡(3.19) Noting that 𝛼0 and 𝑅1(𝑖) and 𝑅2(𝑖) are two given symmetric positive definite matrices for all 𝑖, thus, we have 𝐽𝑇(𝑖)=𝐸𝑇0𝑥𝑇(𝑡)𝑅1(𝑖)𝑥(𝑡)+𝑢𝑇(𝑡)𝑅2(𝑖)𝑢(𝑡)𝑑𝑡𝑒𝛼𝑇𝐸𝑇0𝑒𝛼𝑡𝑥𝑇(𝑡)𝑅1(𝑖)𝑥(𝑡)+𝑢𝑇(𝑡)𝑅2(𝑖)𝑢(𝑡)d𝑡<𝑒𝛼𝑇𝑇0𝑒𝛼𝑡𝑤𝑇(𝑡)𝑄2(𝑖)𝑤(𝑡)𝑑𝑡𝐸𝑇0𝕃𝑒𝛼𝑡𝑉(𝑥(𝑡),𝑖)𝑑𝑡𝑒𝛼𝑇max𝑖𝜆max𝑄1𝑐(𝑖)21+max𝑖𝜆max𝑄2𝑑(𝑖)2.(3.20) Thus, one can obtain that the cost function 𝐽𝑇(𝑖)<𝜓0=𝑒𝛼𝑇max𝑖𝜆max𝑄1𝑐(𝑖)21+max𝑖𝜆max𝑄2𝑑(𝑖)2(3.21) holds for all 𝑖. This completes the proof of the theorem.

Corollary 3.2. The singular system with Markovian jumps (2.9) with 𝑤(𝑡)=0 is SSFTS with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡)), if there exist a scalar 𝛼0, a set of nonsingular matrices {𝑃(𝑖),𝑖} with 𝑃(𝑖)𝑛×𝑛, a set of symmetric positive definite matrices {𝑄1(𝑖),𝑖} with 𝑄1(𝑖)𝑛×𝑛, and for all 𝑟𝑡=𝑖 such that (3.1a), (3.1c) and𝐴(𝑖)𝑃𝑇(𝑖)+𝑃(𝑖)𝐴𝑇(𝑖)+Γ(𝑖)<0,(3.22a)max𝑖𝜆max𝑄1(𝑐𝑖)21<min𝑖𝜆min𝑄1(𝑐𝑖)22𝑒𝛼𝑇(3.22b) hold, where Γ(𝑖)=𝑁𝑗=1𝜋𝑖𝑗𝑃(𝑖)𝑃1(𝑗)𝐸(𝑗)𝑃𝑇(𝑖)+𝑃(𝑖)[𝑅1(𝑖)+𝐾𝑇(𝑖)𝑅2(𝑖)𝐾(𝑖)]𝑃𝑇(𝑖)𝛼𝐸(𝑖)𝑃𝑇(𝑖). Moreover, a guaranteed cost bound for stochastic singular system can be chosen as 𝜓0=max{𝑒𝛼𝑇𝜆max(𝑄1(𝑖))𝑐21,𝑖}.

By Lemma 2.7, Theorem 3.1, and using matrix decomposition novelty, we can obtain the following theorem.

Theorem 3.3. There exists a state feedback controller 𝑢=𝐾(𝑟𝑡)𝑥(𝑡) with 𝐾(𝑟𝑡)=𝐿𝑇(𝑟𝑡)𝑃𝑇(𝑟𝑡),𝑟𝑡=𝑖 such that the closed-loop stochastic singular system with Markovian jumps (2.9) is SSFTB with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡),𝑑), if there exist a scalar 𝛼0, a set of positive matrices {𝑋(𝑖),𝑖} with 𝑋(𝑖)𝑛×n, a set of symmetric positive definite matrices {𝑄2(𝑖),𝑖} with 𝑄2(𝑖)𝑝×𝑝, and a set of matrices {𝑌(𝑖),𝑖} with 𝑌(𝑖)𝑛×(𝑛𝑟𝑖), two sets of positive scalars {𝜎𝑖,𝑖} and {𝜖𝑖,𝑖}, for all 𝑟𝑡=𝑖 such that (3.1d) and0𝐸(𝑖)𝑃𝑇(𝑖)=𝑃(𝑖)𝐸𝑇(𝑖)=𝐸(𝑖)𝑁(𝑖)𝑋(𝑖)𝑁𝑇(𝑖)𝐸𝑇(𝑖)𝜎𝑖Ω𝐼,(3.23a)11(𝑖)𝐺(𝑖)𝑃(𝑖)𝐿(𝑖)Ω15(𝑖)𝑈𝑖𝑄2(𝑖)00𝐸𝑇3(𝑖)0𝑅11(𝑖)000𝑅21(𝑖)00𝜖𝑖𝐼0𝑊𝑖<0(3.23b) hold, where Ω11(𝑖)=𝑃(𝑖)𝐴𝑇(𝑖)+𝐿(𝑖)𝐵𝑇(𝑖)+(𝑃(𝑖)𝐴𝑇(𝑖)+𝐿(𝑖)𝐵𝑇(𝑖))𝑇+𝜖𝑖𝐹(𝑖)𝐹𝑇(𝑖)[(𝑁𝑖1)𝜋𝑖+𝛼]𝑃(𝑖)𝐸𝑇(𝑖), Ω15(𝑖)=𝑃(𝑖)𝐸𝑇1(𝑖)+𝐿(𝑖)𝐸𝑇2(𝑖), 𝑈𝑖=[𝜋𝑖𝑃(𝑖),,𝜋𝑖𝑃(𝑖)], 𝑊𝑖=diag{𝑃𝑇(1)+𝑃(1)𝜎1𝐼,,𝑃𝑇(𝑖1)+𝑃(𝑖1)𝜎𝑖1𝐼,𝑃𝑇(𝑖+1)+𝑃(𝑖+1)𝜎𝑖+1𝐼,,𝑃𝑇(𝑁) + 𝑃(𝑁)𝜎𝑁𝐼}, 𝑃(𝑖)=𝐸(𝑖)𝑁(𝑖)𝑋(𝑖)𝑁𝑇(𝑖)+𝑀1(𝑖)𝑌(𝑖)Υ𝑇(𝑖), 𝑀(𝑖)𝐸(𝑖)𝑁(𝑖)=diag{𝐼𝑟𝑖,0}, Υ(𝑖)=𝑁(𝑖)[0,𝐼𝑛𝑟𝑖]𝑇, and 𝑄1(𝑖)=𝑅1/2(𝑖)𝑀𝑇(𝑖)𝑋1(𝑖)𝑀(𝑖)𝑅1/2(𝑖). Moreover, 𝑋(𝑖) and 𝑌(𝑖) are from the form (3.35). Furthermore, a stochastic singular finite-time guaranteed cost bound for stochastic singular system can be chosen as 𝜓0=𝑒𝛼𝑇max𝑖𝜆max𝑅1/2(𝑖)𝑀𝑇(𝑖)𝑋1(𝑖)𝑀(𝑖)𝑅1/2𝑐(𝑖)21+max𝑖𝜆max𝑄2𝑑(𝑖)2.(3.24)

Proof. We firstly prove that condition (3.23b) implies condition (3.1b). By condition (3.23a), we have 𝑃1(𝑗)𝐸(𝑗)𝜎𝑗𝑃1(𝑗)𝑃𝑇(𝑗),𝑗.(3.25) Using Assumption 1, we obtain𝜋𝑖𝑖𝑃(𝑖)𝐸𝑇(𝑖)=𝑁𝑗=1,𝑗𝑖𝜋𝑖𝑗𝑃(𝑖)𝐸𝑇𝑁(𝑖)𝑖𝜋1𝑖𝑃(𝑖)𝐸𝑇(𝑖),(3.26a)𝑁𝑗=1,𝑗𝑖𝜋𝑖𝑗𝜎𝑗𝑃1(𝑗)𝑃𝑇(𝑗)𝑁𝑗=1,𝑗𝑖𝜋𝑖𝜎𝑗𝑃1(𝑗)𝑃𝑇(𝑗).(3.26b) Thus the inequality 𝑁𝑗=1,𝑗𝑖𝜋𝑖𝑗𝑃(𝑖)𝑃1(𝑗)𝐸(𝑗)𝑃𝑇(𝑖)𝑁𝑗=1,𝑗𝑖𝜋𝑖𝑗𝜎𝑗𝑃(𝑖)𝑃1(𝑗)𝑃𝑇(𝑗)𝑃𝑇(𝑖)𝑁𝑗=1,𝑗𝑖𝜋𝑖𝜎𝑗𝑃(𝑖)𝑃1(𝑗)𝑃𝑇(𝑗)𝑃𝑇(𝑖)𝑈𝑖𝑉𝑖1𝑈𝑇𝑖(3.27) holds, where 𝑈𝑖=[𝜋𝑖𝑃(𝑖),,𝜋𝑖𝑃(𝑖)] and 𝑉𝑖𝜎=diag11𝑃𝑇(1)𝑃(1),,𝜎1𝑖1𝑃𝑇(𝑖1)𝑃(𝑖1),𝜎1𝑖+1𝑃𝑇(𝑖+1)𝑃(𝑖+1),,𝜎𝑁1𝑃𝑇.(𝑁)𝑃(𝑁)(3.28) Noting that the inequality 𝜎𝑖1𝑃𝑇(𝑖)𝑃(𝑖)𝑃𝑇(𝑖)+𝑃(𝑖)𝜎𝑖𝐼 holds for each 𝑖. Thus we have 𝑁𝑗=1,𝑗𝑖𝜋𝑖𝑗𝑃(𝑖)𝑃1(𝑗)𝐸(𝑗)𝑃𝑇(𝑖)𝑈𝑖𝑊𝑖1𝑈𝑇𝑖,(3.29) where 𝑊𝑖=diag{𝑃𝑇(1)+𝑃(1)𝜎1𝐼,,𝑃𝑇(𝑖1)+𝑃(𝑖1)𝜎𝑖1𝐼,𝑃𝑇(𝑖+1)+𝑃(𝑖+1)𝜎𝑖+1𝐼,,𝑃𝑇(𝑁) + 𝑃(𝑁)𝜎𝑁𝐼}.
Therefore, a sufficient condition for (3.1b) to guarantee is that Ξ(𝑖)=Ψ(𝑖)𝐺(𝑖)𝑄2(𝑖)<0,(3.30) where Ψ(𝑖)=𝐴(𝑖)𝑃𝑇(𝑖)+𝑃(𝑖)𝐴𝑇(𝑖)+𝑃(𝑖)[𝑅1(𝑖)+𝐾𝑇(𝑖)𝑅2(𝑖)𝐾(𝑖)]𝑃𝑇(𝑖)+𝑈𝑖𝑊𝑖1𝑈𝑇𝑖[(𝑁𝑖1)𝜋𝑖+𝛼]𝐸(𝑖)𝑃𝑇(𝑖).
Noting that Ξ(𝑖)=Ψ0𝑅(𝑖)+𝑃(𝑖)1(𝑖)+𝐾𝑇(𝑖)𝑅2𝑃(𝑖)𝐾(𝑖)𝑇(𝑖)+𝑈𝑖𝑊𝑖1𝑈𝑇𝑖𝐺(𝑖)𝑄2(𝑖)+Ξ1(𝑖),(3.31) where Ξ1(𝑖)=(Δ𝐴(𝑖)+Δ𝐵(𝑖)𝐾(𝑖))𝑃𝑇(𝑖)+(Δ𝐴(𝑖)+Δ𝐵(𝑖)𝐾(𝑖))𝑃𝑇(𝑖)𝑇=0𝐸Δ𝐺(𝑖)0𝐹(𝑖)Δ(𝑖)12(𝑖)𝑃𝑇(𝑖)𝐸3+(𝑖)𝑃(𝑖)𝐸𝑇12𝐸(𝑖)𝑇3Δ(𝑖)𝑇(𝐹𝑖)𝑇,(𝑖)0(3.32) and Ψ0(𝑖)=𝐴(𝑖)𝑃𝑇𝐴(𝑖)+𝑃(𝑖)𝑇(𝑖)[(𝑁𝑖1)𝜋𝑖+𝛼]𝑃(𝑖)𝐸𝑇(𝑖), 𝐸12(𝑖)=𝐸1(𝑖)+𝐸2(𝑖)𝐾(𝑖),𝐴(𝑖)=𝐴(𝑖)+𝐵(𝑖)𝐾(𝑖).
By Lemma 2.7, we have Ξ(𝑖)Ψ0𝑅(𝑖)+𝑃(𝑖)1(𝑖)+𝐾𝑇(𝑖)𝑅2𝑃(𝑖)𝐾(𝑖)𝑇(𝑖)+𝑈𝑖𝑊𝑖1𝑈𝑇𝑖𝐺(𝑖)𝑄2(𝑖)+𝜖𝑖𝐹(𝑖)𝐹𝑇(𝑖)00+𝜖𝑖1𝑃(𝑖)𝐸𝑇12𝐸(𝑖)𝑇3𝐸(𝑖)12(𝑖)𝑃𝑇(𝑖)𝐸3=(𝑖)Ψ0(𝑖)+𝜖𝑖𝐹(𝑖)𝐹𝑇(𝑖)𝐺(𝑖)𝑄2(𝑖)Λ𝑖Γ1(𝑖)Λ𝑇𝑖,=Ξ(𝑖),(3.33) where Γ(𝑖)=diag{𝑅11(𝑖),𝑅21(𝑖),𝜖𝑖𝐼,𝑊𝑖} and Λ𝑖=𝑃(𝑖)𝑃(𝑖)𝐾𝑇(𝑖)𝑃(𝑖)𝐸𝑇12(𝑖)𝑈𝑖00𝐸𝑇3(𝑖)0.
By Schur complement, Ξ(𝑖)<0 holds if and only if the following inequality: Ω11(𝑖)𝐺(𝑖)𝑃(𝑖)𝑃(𝑖)𝐾𝑇(𝑖)𝑃(𝑖)𝐸𝑇12(𝑖)𝑈𝑖𝑄2(𝑖)00𝐸𝑇3(𝑖)0𝑅11(𝑖)000𝑅21(𝑖)00𝜖𝑖𝐼0𝑊𝑖<0(3.34) holds, where Ω11(𝑖)=𝐴(𝑖)𝑃𝑇𝐴(𝑖)+𝑃(𝑖)𝑇(𝑖)+𝜖𝑖𝐹(𝑖)𝐹𝑇(𝑖)[(𝑁𝑖1)𝜋𝑖+𝛼]𝑃(𝑖)𝐸𝑇(𝑖) and 𝐴(𝑖)=𝐴(𝑖)+𝐵(𝑖)𝐾(𝑖).
Thus, let 𝐿(𝑖)=𝑃(𝑖)𝐾𝑇(𝑖), and noting that 𝐸(𝑖)𝑃𝑇(𝑖)=𝑃(𝑖)𝐸𝑇(𝑖), from (3.34), it is easy to obtain that condition (3.23b) implies condition (3.1b).
Noting that 𝐸(𝑖) is a singular matrix with rank𝐸(𝑖)=𝑟𝑖 for every fixed 𝑟𝑡=𝑖, thus there exist two nonsingular matrices 𝑀(𝑖) and 𝑁(𝑖), which satisfy 𝑀(𝑖)𝐸(𝑖)𝑁(𝑖)=diag{𝐼𝑟𝑖,0}. Let 𝑃(𝑖)=𝑀(𝑖)𝑃(𝑖)𝑁𝑇(𝑖); by the proof of Theorem 3.1, we obtain that 𝑃(𝑖) is of the following form 𝑃11(𝑖)𝑃12(𝑖)0𝑃22(𝑖), where 𝑃11(𝑖)0, 𝑃12(𝑖)𝑟×(𝑛𝑟𝑖), 𝑃22(𝑖)(𝑛𝑟𝑖)×(𝑛𝑟𝑖). Denote Υ(𝑖)=𝑁(𝑖)[0,𝐼𝑛𝑟𝑖]𝑇. Then we have rankΥ(𝑖)=𝑛𝑟𝑖,𝐸(𝑖)Υ(𝑖)=0 and 𝑃(𝑖)=𝑀1𝑃(𝑖)11(𝑖)𝑃12(𝑖)0𝑃22(𝑁𝑖)𝑇=𝑀(𝑖)1(𝐼𝑖)𝑟𝑖0𝑁001(𝑖)𝑁(𝑖)𝑋(𝑖)𝑁𝑇(+𝑀𝑖)1(𝑖)𝑌(𝑖)0𝐼𝑛𝑟𝑖𝑁𝑇(𝑖)=𝐸(𝑖)𝑁(𝑖)𝑋(𝑖)𝑁𝑇(𝑖)+𝑀1(𝑖)𝑌(𝑖)Υ𝑇(𝑖),(3.35) where 𝑋(𝑖)=diag{𝑃11(𝑖),Θ(𝑖)} and 𝑌(𝑖)=[𝑃𝑇12(𝑖),𝑃𝑇22(𝑖)]𝑇.
Let 𝑄1(𝑖)=𝑅1/2(𝑖)𝑀𝑇(𝑖)𝑋1(𝑖)𝑀(𝑖)𝑅1/2(𝑖), one can see that 𝑃(𝑖)=𝐸(𝑖)𝑁(𝑖)𝑋(𝑖)𝑁𝑇(𝑖)+𝑀1(𝑖)𝑌(𝑖)Υ𝑇(𝑖) satisfies 𝑃(𝑖)𝐸𝑇(𝑖)=𝐸(𝑖)𝑃𝑇(𝑖)=𝐸(𝑖)𝑁(𝑖)𝑋(𝑖)𝑁𝑇(𝑖)𝐸𝑇(𝑖) and (3.1c) holds.
From the proof of Theorem 3.1 and noting that 𝑄1(𝑖)=𝑅1/2(𝑖)𝑀𝑇(𝑖)𝑋1(𝑖)𝑀(𝑖)𝑅1/2(𝑖), we can obtain 𝐽𝑇(𝑖)<𝜓0=𝑒𝛼𝑇{max𝑖{𝜆max(𝑅1/2(𝑖)𝑀𝑇(𝑖)𝑋1(𝑖)𝑀(𝑖)𝑅1/2(𝑖))}𝑐21+max𝑖{𝜆max(𝑄2(𝑖))}𝑑2} for all 𝑖. This completes the proof of the theorem.

By Theorem 3.1, Corollary 3.2, and Theorem 3.3, we have the following corollary.

Corollary 3.4. There exists a state feedback controller 𝑢=𝐾(𝑟𝑡)𝑥(𝑡) with 𝐾(𝑟𝑡)=𝐿𝑇(𝑟𝑡)𝑃𝑇(𝑟𝑡),𝑟𝑡=𝑖 such that the closed-loop stochastic singular system with Markovian jumps (2.9) with 𝑤(𝑡)=0 is SSFTS with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡)), if there exist a scalar 𝛼0, a set of positive matrices {𝑋(𝑖),𝑖} with 𝑋(𝑖)𝑛×𝑛, and a set of matrices {𝑌(𝑖),𝑖} with 𝑌(𝑖)𝑛×(𝑛𝑟𝑖), two sets of positive scalars {𝜎𝑖,𝑖} and {𝜖𝑖,𝑖}, for all 𝑟𝑡=𝑖 such that (3.22b), (3.23a) and Φ11(𝑖)𝑃(𝑖)𝐿(𝑖)Φ14(𝑖)𝑈𝑖𝑅11(𝑖)000𝑅21(𝑖)00𝜖𝑖𝐼0𝑊𝑖<0(3.36) holds, where Φ11(𝑖)=𝑃(𝑖)𝐴𝑇(𝑖)+𝐿(𝑖)𝐵𝑇(𝑖)+(𝑃(𝑖)𝐴𝑇(𝑖)+𝐿(𝑖)𝐵𝑇(𝑖))𝑇+𝜖𝑖𝐹(𝑖)𝐹𝑇(𝑖)[(𝑁𝑖1)𝜋𝑖𝛼]𝑃(𝑖)𝐸𝑇(𝑖), Φ14(𝑖)=𝑃(𝑖)𝐸𝑇1(𝑖)+𝐿(𝑖)𝐸𝑇2(𝑖). Furthermore, the other matrical variables are the same as Theorem 3.3, and a guaranteed cost bound for the stochastic singular system can be chosen as 𝜓0𝑒=max𝛼𝑇𝜆max𝑅1/2(𝑖)𝑀𝑇(𝑖)𝑋1(𝑖)𝑀(𝑖)𝑅1/2𝑐(𝑖)21.,𝑖(3.37)

Remark 3.5. It is easy to check that condition (3.1d) and (3.22b) can be guaranteed by imposing the conditions, respectively, 𝜂1𝐼<𝑅1/2(𝑖)𝑀1(𝑖)𝑋(𝑖)𝑀𝑇(𝑖)𝑅1/2𝜂(𝑖)<𝐼,(3.38a)3𝐼<𝑄2(𝑖)<𝜂2𝑒𝐼,(3.38b)𝛼𝑇𝑐22𝑑2𝜂2𝑐1𝑐1𝜂1𝜂>0,(3.38c)1𝐼<𝑅1/2(𝑖)𝑀1(𝑖)𝑋(𝑖)𝑀𝑇(𝑖)𝑅1/2(𝑒𝑖)<𝐼,(3.39a)𝛼𝑇𝑐22𝑐1𝑐1𝜂1>0.(3.39b) In addition, conditions (3.23b) and (3.36) are not strict LMIs; however, once we fix parameter 𝛼, conditions (3.23b) and (3.36) can be turned into LMIs-based feasibility problem.

Remark 3.6. From the above discussion, one can see that the feasibility of conditions stated in Theorem 3.3 and Corollary 3.4 can be turned into the following LMIs-based feasibility problem with a fixed parameter 𝛼, respectively, min𝑐22𝑋(𝑖),𝑌(𝑖),𝐿(𝑖),𝑄2(𝑖),𝜖𝑖,𝜎𝑖,𝜂1,𝜂2,𝜂3s.t.(3.23a),(3.23b),and(3.38a)-(3.38c)(3.40)min𝑐22𝑋(𝑖),𝑌(𝑖),𝐿(𝑖),𝜖𝑖,𝜎𝑖,𝜂1s.t.(3.23a),(3.36),and(3.39a)-(3.39b).(3.41)

Remark 3.7. If 𝛼=0 is a solution of feasibility problem (3.41), then the closed-loop stochastic singular system with Markovian jumps (2.9) with 𝑤(𝑡)=0 is SSFTS with respect to (𝑐1,𝑐2,𝑇,𝑅(𝑟𝑡)) and is also stochastically stable.

4. Numerical Examples

Example 4.1. Consider a two-mode Markovian jumping singular system (2.1) with

Mode. ,𝐸(1)=100010000,𝐴(1)=2.611130110,𝐵(1)=012110001𝐹(1)=0.50000.11010,𝐸1,𝐸(1)=0.0300.20.010.200.300.12(1)=0.0600.020.010.100.0400.5,𝐸3000,(1)=0.010.01,𝐺(1)=0.1(4.1)

Mode. ,𝐸(2)=100010000,𝐴(2)=201001011,𝐵(2)=0.50.10110010𝐹(2)=0.10000.10000,𝐸1,𝐸(2)=0.0200.20.010.200.100.52(2)=0.0400.010.010.100.300.1,𝐸300,(2)=0.040.010.3,𝐺(2)=0.1(4.2) and 𝑑=2,Δ(𝑖)=diag{𝑟1(𝑖),𝑟2(𝑖),𝑟3(𝑖)}, where 𝑟𝑗(𝑖) satisfies |𝑟𝑗(𝑖)|1 for all 𝑖=1,2 and 𝑗=1,2,3.
The switching between the two modes is described by the transition rate matrix Γ=𝜋11𝜋12𝜋21𝜋22. The lower and upper bounds parameters of 𝜋𝑖𝑗 for all 𝑖,𝑗 are given in Table 1.
Then, we choose 𝑅1(1)=𝑅1(2)=𝑅2(1)=𝑅2(2)=𝑅(1)=𝑅(2)=𝐼3, 𝑇=1.5, 𝑐1=1, 𝛼=2. Using the LMI control toolbox of Matlab, we can obtain from Theorem 3.3 that the optimal value 𝑐2=20.6686, 𝜓0=426.2786, and ,,,𝜂𝑋(1)=0.09460.024400.02440.07480000.5271,𝑋(2)=0.05440.002500.00250.08060000.5271𝑌(1)=0.01810.00250.1558,𝑌(2)=0.11820.03290.3341𝐿(1)=0.10410.93670.80870.97460.93100.05960.03830.00230.4082,𝐿(2)=0.37320.09710.04190.82390.98220.05790.13110.98170.04391=0.0541,𝜂2=0.6948,𝜂3𝜖=0.2301,1=0.1571,𝜖2=0.5278,𝜎1𝜎=0.1110,2=0.0809,𝑄2(1)=0.6920,𝑄2(2)=0.6875.(4.3)
Then, we can obtain the following state feedback controller gains: 𝐾(1)=2.410213.80740.24557.313310.06320.01469.68212.44552.6195,𝐾(2)=8.194410.31440.39258.692811.25342.93860.52150.78750.1313.(4.4)
Furthermore, let 𝑅1(1)=𝑅1(2)=𝑅2(1)=𝑅2(2)=𝑅(1)=𝑅(2)=𝐼3, 𝑇=1.5, 𝑐1=1; by Theorem 3.3, the optimal bound with minimum value of 𝑐22 relies on the parameter 𝛼. We can find feasible solution when 0.37𝛼12.92. Figure 1 shows the optimal value with different value of 𝛼. When 𝛼=1.4, it yields the optimal value 𝑐2=18.3686 and 𝜓0=337.0518. Then, by using the program fminsearch in the optimization toolbox of Matlab starting at 𝛼=1.4, the locally convergent solution can be derived as 𝐾(1)=2.969817.32530.40079.035512.36140.017811.80342.38183.3381,𝐾(2)=10.429012.91861.040516.397119.42814.67412.95054.33930.3515,(4.5) with 𝛼=1.4217 and the optimal value 𝑐2=18.3341 and 𝜓0=336.0016.

tab1
Table 1: Partially known rate parameters.
431751.fig.001
Figure 1: The local optimal bound of 𝑐2.

Remark 4.2. From the above example and Remark 3.6, condition (3.23b) in Theorem 3.3 is not strict in LMI form; however, one can find the parameter 𝛼 by an unconstrained nonlinear optimization approach, which a locally convergent solution can be obtained by using the program fminsearch in the optimization toolbox of Matlab.

Example 4.3. Consider a two-mode stochastic singular system (2.1) with 𝑤(𝑡)=0 and 𝐴(1)=2.611130110,𝐴(2)=101001011.(4.6) In addition, the transition rate matrix and the other matrices parameters are the same as Example 4.1.
Then, let 𝑅1(1)=𝑅1(2)=𝑅2(1)=𝑅2(2)=𝑅(1)=𝑅(2)=𝐼3,𝑇=1.5,𝑐1=1. By Corollary 3.4, the optimal bound with minimum value of 𝑐22 relies on the parameter 𝛼. We can find feasible solution when 0𝛼13.37. Thus the above system is stochastically stable, and when 𝛼=0, it yields the optimal value 𝑐2=2.7682,𝜓0=7.6608, and the following optimized state feedback controller gains 𝐾(1)=0.36337.26050.12853.75676.35170.00464.23290.62842.0607,𝐾(2)=3.78407.51890.00974.13619.56272.64840.00890.01980.0046.(4.7)

5. Conclusions

In this paper, we deal with the problem of stochastic finite-time guaranteed cost control of Markovian jumping singular systems with uncertain transition probabilities, parametric uncertainties, and time-varying norm-bounded disturbance. Sufficient conditions on stochastic singular finite-time guaranteed cost control are obtained for the class of stochastic singular systems. Designed algorithms for the state feedback controller are provided to guarantee that the underlying stochastic singular system is stochastic singular finite-time guaranteed cost control in terms of restricted linear matrix equalities with a fixed parameter. Numerical examples are also presented to illustrate the validity of the proposed results.

Acknowledgments

The authors would like to thank the reviewers and the editors for their very helpful comments and suggestions which have improved the presentation of the paper. The paper was supported by the National Natural Science Foundation of P.R. China under Grant 60874006, Doctoral Foundation of Henan University of Technology under Grant 2009BS048, by the Natural Science Foundation of Henan Province of China under Grant 102300410118, Foundation of Henan Educational Committee under Grant 2011A120003, and Foundation of Henan University of Technology under Grant 09XJC011.

References

  1. L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. F. L. Lewis, “A survey of linear singular systems,” Circuits, Systems, and Signal Processing, vol. 5, no. 1, pp. 3–36, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. M. S. Mahmoud, F. M. Al-Sunni, and Y. Shi, “Dissipativity results for linear singular time-delay systems,” International Journal of Innovative Computing, Information and Control, vol. 4, no. 11, pp. 2833–2846, 2008. View at Google Scholar · View at Scopus
  4. J. Y. Ishihara and M. H. Terra, “On the Lyapunov theorem for singular systems,” IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1926–1930, 2002. View at Publisher · View at Google Scholar · View at Scopus
  5. I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “H control for descriptor systems: a matrix inequalities approach,” Automatica, vol. 33, no. 4, pp. 669–673, 1997. View at Publisher · View at Google Scholar · View at Scopus
  6. Y. Xia, P. Shi, G. Liu, and D. Rees, “Robust mixed H/H2 state-feedback control for continuous-time descriptor systems with parameter uncertainties,” Circuits, Systems, and Signal Processing, vol. 24, no. 4, pp. 431–443, 2005. View at Publisher · View at Google Scholar · View at Scopus
  7. L. Zhang, B. Huang, and J. Lam, “LMI synthesis of H2 and mixed H2/H controllers for singular systems,” IEEE Transactions on Circuits and Systems II, vol. 50, no. 9, pp. 615–626, 2003. View at Publisher · View at Google Scholar · View at Scopus
  8. G. Zhang, Y. Xia, and P. Shi, “New bounded real lemma for discrete-time singular systems,” Automatica, vol. 44, no. 3, pp. 886–890, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. X. Sun and Q. Zhang, “Delay-dependent robust stabilization for a class of uncertain singular delay systems,” International Journal of Innovative Computing, Information and Control, vol. 5, no. 5, pp. 1231–1242, 2009. View at Google Scholar · View at Scopus
  10. Z. G. Wu and W. N. Zhou, “Delay-dependent robust stabilization for uncertain singular systems with state delay,” ICIC Express Letters, vol. 1, no. 2, pp. 169–176, 2007. View at Google Scholar · View at Zentralblatt MATH
  11. E. K. Boukas, Control of Singular Systems with Random Abrupt Changes, Communications and Control Engineering Series, Springer, Berlin, Germany, 2008.
  12. S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, vol. 332 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2006.
  13. X. Mao, “Stability of stochastic differential equations with Markovian switching,” Stochastic Processes and Their Applications, vol. 79, no. 1, pp. 45–67, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. K. C. Yao, “Reliable robust output feedback stabilizing computer control of decentralized stochastic singularly-perturbed systems,” ICIC Express Letters, vol. 1, no. 1, pp. 1–7, 2007. View at Publisher · View at Google Scholar
  15. Y. Xia, E. K. Boukas, P. Shi, and J. Zhang, “Stability and stabilization of continuous-time singular hybrid systems,” Automatica, vol. 45, no. 6, pp. 1504–1509, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. C. E. de Souza, “Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 836–841, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. P. Shi, Y. Xia, G. P. Liu, and D. Rees, “On designing of sliding-mode control for stochastic jump systems,” IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 97–103, 2006. View at Publisher · View at Google Scholar · View at Scopus
  18. L. Wu, P. Shi, and H. Gao, “State estimation and sliding-mode control of Markovian jump singular systems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, Article ID 5406129, pp. 1213–1219, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. S. K. Nguang, W. Assawinchaichote, and P. Shi, “Robust H-infinity control design for fuzzy singularly perturbed systems with Markovian jumps: an LMI approach,” IET Control Theory and Applications, vol. 1, no. 4, pp. 893–908, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. G. Wang, Q. Zhang, C. Bian, and V. Sreeram, “H Control for discrete-time singularly perturbed systems with distributional properties,” International Journal of Innovative Computing, Information and Control, vol. 6, no. 4, pp. 1781–1792, 2010. View at Google Scholar · View at Scopus
  21. S. Xu, T. Chen, and J. Lam, “Robust H filtering for uncertain Markovian jump systems with mode-dependent time delays,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 900–907, 2003. View at Publisher · View at Google Scholar · View at Scopus
  22. K. C. Yao and C. H. Hsu, “Robust optimal stabilizing observer-based control design of decentralized stochastic singularly-perturbed computer controlled systems with multiple time-varying delays,” International Journal of Innovative Computing, Information and Control, vol. 5, no. 2, pp. 467–478, 2009. View at Google Scholar · View at Scopus
  23. J. Qiu and K. Lu, “New robust passive stability criteria for uncertain singularly Markov jump systems with time delays,” ICIC Express Letters, vol. 3, no. 3, pp. 651–656, 2009. View at Google Scholar · View at Scopus
  24. R. Amjadifard and M. T. H. Beheshti, “Robust disturbance attenuation of a class of nonlinear singularly perturbed systems,” International Journal of Innovative Computing, Information and Control, vol. 6, no. 3, pp. 913–920, 2010. View at Google Scholar · View at Scopus
  25. H. J. Kushner, “Finite time stochastic stability and the analysis of tracking systems,” IEEE Transactions on Automatic Control, vol. 11, pp. 219–227, 1966. View at Google Scholar
  26. L. Weiss and E. F. Infante, “Finite time stability under perturbing forces and on product spaces,” IEEE Transactions on Automatic Control, vol. 12, pp. 54–59, 1967. View at Google Scholar · View at Zentralblatt MATH
  27. F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459–1463, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. F. Amato, M. Ariola, and P. Dorato, “Finite-time stabilization via dynamic output feedback,” Automatica, vol. 42, no. 2, pp. 337–342, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  29. W. Zhang and X. An, “Finite-time control of linear stochastic systems,” International Journal of Innovative Computing, Information and Control, vol. 4, no. 3, pp. 689–696, 2008. View at Google Scholar · View at Scopus
  30. D. Y. Xin and Y. G. Liu, “Finite-time stability analysis and control design of nonlinear systems,” Journal of Shandong Universituy, vol. 37, pp. 24–30, 2007. View at Google Scholar
  31. G. Garcia, S. Tarbouriech, and J. Bernussou, “Finite-time stabilization of linear time-varying continuous systems,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 364–369, 2009. View at Publisher · View at Google Scholar · View at Scopus
  32. R. Ambrosino, F. Calabrese, C. Cosentino, and G. de Tommasi, “Sufficient conditions for finite-time stability of impulsive dynamical systems,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 861–865, 2009. View at Publisher · View at Google Scholar · View at Scopus
  33. F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems with jumps,” Automatica, vol. 45, no. 5, pp. 1354–1358, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  34. F. Amato, M. Ariola, and C. Cosentino, “Finite-time control of discrete-time linear systems: analysis and design conditions,” Automatica, vol. 46, no. 5, pp. 919–924, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  35. S. He and F. Liu, “Stochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 2631–2638, 2011. View at Publisher · View at Google Scholar
  36. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, 1994.