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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 263045, 10 pages

http://dx.doi.org/10.1155/2014/263045

## Finite-Time Stability and Stabilization of Itô-Type Stochastic Singular Systems

^{1}School of Electrical Engineering and Automation, Qilu University of Technology, Jinan 250353, China^{2}College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266510, China

Received 14 October 2013; Accepted 19 November 2013; Published 14 January 2014

Academic Editor: Hui Zhang

Copyright © 2014 Zhiguo Yan and Weihai Zhang. 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 is concerned with the finite-time stability and stabilization problems for linear Itô stochastic singular systems. The condition of existence and uniqueness of solution to such class of systems are first given. Then the concept of finite-time stochastic stability is introduced, and a sufficient condition under which an Itô stochastic singular system is finite-time stochastic stable is derived. Moreover, the finite-time stabilization is investigated, and a sufficient condition for the existence of state feedback controller is presented in terms of matrix inequalities. In the sequel, an algorithm is given for solving the matrix inequalities arising from finite-time stochastic stability (stabilization). Finally, two examples are employed to illustrate our results.

#### 1. Introduction

Stochastic systems, especially for the systems governed by Itô-type stochastic differential equations, have received considerable attention due to its both theoretical and practical importance. Some results for this class of systems have been reported in the monographs and literatures, for example, stochastic stability and stabilization [1–3], linear/nonlinear stochastic control and filtering [4–7], and output tracking control for high-order stochastic nonlinear systems [8]. Meanwhile, singular systems (descriptor systems, implicit systems, generalized state-space systems, and differential-algebraic systems) have also attracted much attention of researchers and made a rapid progress. Many results have been achieved on different subjects related to such class of systems, for example, stability and impulsive elimination [9, 10], linear quadratic optimal control [11], andcontrol/filtering andcontrol [12–14]. Consequently, Itô stochastic singular systems have received attention in recent years. Reference [15] is concerned with the problems of stability of Itô singular stochastic systems with Markovian jumping. Reference [16] investigated thecontrol/filtering for a class of singular stochastic time-delay systems. To the best of our knowledge, most of the results on stability of Itô stochastic singular systems are concerned with Lyapunov asymptotic stability or exponential stability, which is defined over an infinite-time interval.

In many practical situations, however, we are interested in stability of the system over a fixed finite-time interval. Such kind of stability is called finite-time stability (FTS). The concept of FTS was first introduced in the Russian literature. Later, this concept appeared in the western control literatures. Roughly speaking, a system is said to be finite-time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. Compared with infinite-time stability, the FTS can be used in the problem of controlling the trajectory of a space vehicle from an initial point to a final point in a prescribed time interval and all those applications where large values of the states should be attained, for instance, in the presence of saturations. Much effort has been devoted to FTS for its stability analysis and stabilization, for instance, linear continuous-time systems [17], linear discrete-time systems [18], stochastic systems [19–22], singular systems/Markovian jumping singular systems [23, 24], and stochastic singular biological economic systems [25]. Nevertheless, the FTS in [17–25] only requires that the state trajectory does not exceed a given upper bound during a prespecified time interval. Recently, [26] gave a new “finite-time stochastic stability” for linear Itô stochastic systems, which quantifies the state trajectory of some complex practical systems over a finite-time interval in more detail. Roughly speaking, a stochastic Itô system is called finite-time stochastically stable if its state trajectories do not exceed an upper boundand are not less than a lower bound () in the mean square sense during a specific time interval.

In this paper, motivated by [26], we consider finite-time stability and stabilization problems for Itô stochastic singular systems. Because of the special structure of Itô stochastic singular systems, the problems considered are of more complexity than those in [26]. By using stochastic analysis technology, the stability criterion and some stabilizing conditions are obtained. The contributions of this paper lie in the following three aspects: the condition for the existence and uniqueness of solution to linear Itô stochastic singular systems is given. Our proof is different from that in [15], which may better reflect the essential characteristics of this class of systems. The definition of finite-time stochastic stability for linear Itô stochastic singular systems is given. By the generalized Itô formula and mathematical expectation properties, some new stability criteria and the conditions of existence for state feedback controller are obtained as well. A solving algorithm for the matrix inequalities arising from finite-time stochastic stability (stabilization) is given. By adjusting the parameters in this algorithm, the less conservative results can be attained.

The remainder of this paper is organized as follows. The definition of finite-time stochastic stability of linear Itô stochastic singular systems and some preliminaries are presented in Section 2. A sufficient condition to verify finite-time stochastic stability is given in Section 3. Section 4 gives some sufficient conditions for finite-time stochastic stabilization and a solving algorithm for the matrix inequalities arising from finite-time stochastic stability (stabilization). Section 5 employs two examples to illustrate the results. Finally, concluding remarks are made in Section 6.

*Notation.* is transpose of a matrix or vector. () is positive definite (positive semidefinite) symmetric matrix. is identity matrix. is trace of a matrix. is the maximum (minimum) eigenvalue of a real symmetric matrix. is a probability space with natural filtration, andstands for the mathematical expectation operator with respect to the given probability measure:

#### 2. Preliminaries and Problem Statement

Consider an-dimensional Itô stochastic singular system onwith initial data;is the state vector;,,are-dimensional matrices with.is a scalar Brownian motion defined on the probability space. In order to guarantee the existence and uniqueness of solution for the system (1), the following lemma is given.

Lemma 1. *If there is a pair of nonsingular matrices,or,for the triplet such that (at least) one of the following conditions is satisfied:
**
whereis nilpotent matrix with nilpotent index, then (1) has a unique solution.*

*Proof. *Let , and , then under the conditions of (I), the system (1) is equivalent to
which are called the slow and fast subsystems, respectively.

Note that the slow subsystem (4) is nothing more than an Itô stochastic differential equation. Applying the existence and uniqueness theorem of stochastic differential equations [27], the solution of (4) exists and is unique.

We note that the fast subsystem (5) is actually an ordinary differential equation. Taking the Laplace transforms on both sides of (5) and letting, we have

From this equation, we obtain

Taking the inverse Laplace transform on both sides of (7) gives
which implies that (5) has a unique solution. So (1) has a unique solution.

When triplet satisfies the condition , the proof can be referred to [15].

The proof is different from that in [15], which may better reflect the essential characteristics of this class of systems, such as, impulse behaviors. It is obviously observed from the proof of Lemma 1 that the response of system (1) may contain impulse terms. For convenience, we introduce the following definition.

*Definition 2. *If the state response of an Itô stochastic singular system, starting from an arbitrary initial value, does not contain impulse terms, then the system is called impulse-free.

Referring to some results on impulse-free of singular systems in [28], the following result is obtained.

Proposition 3. *The following statements are equivalent under the conditions of Lemma 1:*(a)*system (1) is impulse-free;*(b) *in (2);*(c);
(d)* in (3).*

*Proof. *According to Definition 2 and the proof of Lemma 1, we can obtain the conclusion (a) *⇔* (b).

By (2),Obviously,if and only ifSo, we get (b) *⇔* (c).

By (3), it is easy to obtain thatif and only ifSo, (c) *⇔* (d).

Next, we extend the finite-time stochastic stability in [26] to Itô stochastic singular systems.

*Definition 4. *Given some positive scalars,,,, withand a positive definite matrix, system (1) is said to be finite-time stochastically stable with respect to, if
Definition 4 can be described as follows: system (1) is said to be finite-time stochastically stable if, given a bound on the initial condition and a fixed time interval, its state trajectories are required to remain in a certain domain of ellipsoidal shape in the mean square sense during this time interval.

A 2-dimension case of Definition 4 is illustrated by Figure 1. A point lies in the shaped area. The trajectory starting from point cannot escape the disc fromtoduring the time intervalin the mean square sense.

*Remark 5. *In [17–25], the finite-time stability only requires the state trajectory not to exceed a given upper bound. A 2-dimension case of this finite-time stability can be illustrated by Figure 2. Nevertheless, the current finite-time stability requires the state trajectory not only not to exceed a given upper bound but also not to be less than a given lower bound.

In the following, we give a proposition equivalent to Definition 4.

Proposition 6. *System (1) is finite-time stochastically stable with respect toif and only if
**
whereis the solution to
*

*Proof. *Letting , we easily obtain
Applying Itô’s formula to, we obtain

Under the conditions of Lemma 1, (14) has unique solution. So the proof is completed.

By Kronecker’s product theory, (14) can be rewritten as wheredenotes the vector formed by stacking the rows ofinto one long vector; that is, andrepresents the Kronecker product of two matrices.

*Remark 7. *Proposition 6 is actually to solve a set of ordinary differential equations and avoids solving a stochastic differential equation (1), which provides an easier method to test finite-time stochastic stability of system (1).

*Remark 8. *If, then system (1) becomes normal Itô stochastic systems and Proposition 6 reduces to Proposition 1 in [26].

Based on Definition 4, we define the finite-time stochastic stabilization as follows.

*Definition 9. *The following Itô stochastic singular controlled system
is said to be finite-time stochastically stabilizable, if there exists a state feedback control law, such that
is finite-time stochastically stable.

Before proceeding further, we give some lemmas which will be used in the next sections.

Lemma 10 (Gronwall inequality). *Letting be a nonnegative function such that
**
for some constants,, then one has
*

Lemma 11 (modified Gronwall inequality [26]). *Letting be a nonnegative function such that
**
for some constants,, then one has
*

Lemma 12 (see [24]). *(i) Assume that, there exist two nonsingular matricesandsuch thathas the decomposition as
**(ii) Define, obviously,,. Ifsatisfies
**
thenwithandsatisfying (24) if and only if
**
with. In addition, whenis nonsingular, one hasand. Furthermore, satisfying (25) can be parameterized as
**
where, , andis an arbitrary parameter matrix.**(iii) Ifis a nonsingular matrix,andare two symmetric positive definite matrices,andsatisfy (25),is a diagonal matrix from (27), and the following equality holds:
**
Then the symmetric positive definite matrixis a solution of (28).*

#### 3. Finite-Time Stochastic Stability

In this section, we provide an impulse-free and finite-time stochastic stability condition for system (1).

Theorem 13. *Under the conditions of Lemma 1, if there exist positive matrices, nonsingular matrix, and two scalarandsatisfying
**
then the system (1) is impulse-free and finite-time stochastically stable with respect to.*

*Proof. *We split the proof of Theorem 13 into three steps as follows.*Step 1*. We prove system (1) to be impulse-free. By condition (33), we obtain

Take nonsingularandsuch thathas the decomposition as
Denote
From (36), (37), and (29), it is easy to obtain
Substitute (37) and (38) into (35), then it becomes
where,.

From (39),, which implies that is nonsingular, by Proposition 3, system (1) is impulse-free.*Step 2*..

Construct a stochastic quadratic function as
wheresatisfies (29)–(33).

Applying generalized Itô formula [1, 15] foralong the trajectory of system (1) and considering condition (29), we have
which leads to

By condition (33), it is easy to see that

Integrating both sides of (43) fromtowithand then taking the expectation, it yields that

By Lemma 10, we obtain

According to condition (30), it follows that

From (46), we easily obtain

By condition (31), it is obvious that*Step 3*..

By (34) and (41), we obtain

Integrating both sides of (48) fromtowithand then taking the expectation, it yields that

By Lemma 11, we conclude that

According to condition (30), it follows that

From (32), (51), we obtain

So, the proof is completed.

Theorem 13 provides a criterion for finite-time stochastic stability of system (1). To design finite-time controller conveniently, the following corollary is given.

Corollary 14. *Under the conditions of Lemma 1, if there exist positive matrices, nonsingular matrix, and two scalars,satisfying
**
then the system (1) is impulse-free and finite-time stochastically stable with respect to .*

*Proof. *Premultiply and postmultiply (53) by the matricesand. Premultiply and postmultiply (57) and (58) byand. Let, by (29)–(33), this proof completes.

*Remark 15. *If, then Corollary 14 reduces to Theorem 1 in [26].

#### 4. Finite-Time Stochastic Stabilization

In this section, we aim to design a finite-time stabilizing controller for system (18). To this aim, the following result is obtained.

Theorem 16. *Under the conditions of Lemma 1, if there exist positive matrices, nonsingular matrix, and two scalars,satisfying (53)–(56) and matrix inequalities
**
then system (18) is impulse-free and finite-time stochastically stabilizable with respect to . In addition, the feedback controller gain can then be given by.*

*Proof. *If the state feedback controller is taken into account, then the state equation of system (18) becomes
Therefore, we can replacebyin Corollary 14. As a result, condition (57) and (58) turn to

By setting, (62) and (63) become (59) and (60), respectively. This completes the proof of Theorem 16.

On the basis of Theorem 16, the following theorem gives a sufficient condition for designing a finite-time stabilizing controller of (18), which is easy to solve.

Theorem 17. *Under the conditions of Lemma 1, if there exist positive matrices,, matrices,, and scalars,,,, satisfying the following matrix inequalities:
**
then there exists a controller such that close-loop system of system (18) is impulse-free and finite-time stochastically stable with respect to , where , is nonsingular. In addition, the feedback controller is .*

*Proof. *By Lemma 12,satisfying (53) in Theorem 16 can be parameterized as
and (54) holds when, where,.

Substitutinginto (59) and (60), by Schur Complement, (53) and (65) are obtained, respectively.

Since
it is easy to check that conditions (55)-(56) are guaranteed by (66)-(68). This completes the proof.

*Remark 18. *It is important to notice that once we have fixed the values forand, the feasibility of the conditions stated in Theorem 17 can be turned into the following LMIs based feasibility problem. The algorithm of how to chooseandfor Theorem 17 is given in the following.

*Algorithm 19. *Consider the following steps.*Step 1*. Given,,,,, and. *Step 2*. Take a series of () by a step sizeand a series ofby a step size. *Step 3*. Set, take a. *Step 4*. Set, take a. *Step 5*. If () makes (53)–(68) have feasible solutions, then store () intoand, go to Step 5; otherwise go to Step 6. *Step 6*. If, thenand take, go to Step 5. Otherwise, go to Step 7. *Step 7*. Stop. If, then we cannot findmaking (53)–(68) have feasible solution; otherwise, there existsmaking (53)–(68) have feasible solution.

*Remark 20. *By Algorithm 19, we can obtain a region surrounded byand, if it exists, which is used to selectandfor appropriate conditions.

*Remark 21. *Ifobtained from LMIs (53)–(68) is singular, then we can adjustandsuch thatis non-singular.

#### 5. Examples

In this section, we will present two examples to illustrate the obtained results.

*Example 1. *Consider the Itô stochastic singular system (1) with

For system (1), there exists a pair of nonsingular matrices
such that
which satisfy Lemma 1, so system (1) has a unique solution and is also impulse-free. We find that system (1) is equivalent to the following system:
Based on this,.

By Proposition 3, we solve (16), where

By a simple calculation, we obtain . It is easy to obtain that,, so (1) is finite-time stochastically stable with respect to. Figure 3 depicts the evolution ofof system (1).

*Example 2. *Consider the Itô stochastic singular system (18) with

By Lemma 12, we obtain,and. Apply Algorithm 19 to Theorem 17, a region surrounded byand, which is illustrated in Figure 4.

Selecting , and solving (53)–(68), we obtain

Hence, the feedback gain matrix is given by
Under the following state feedback controller
the closed-loop system of (18) is impulse-free and finite-time stochastically stable with respect to .

Figure 5 depicts the evolution of of system (18). Figure 6 gives the evolution of .

#### 6. Conclusion

In this paper, we have dealt with finite-time stability and stabilization problems for linear Itô stochastic singular systems and also established a condition of the existence and uniqueness of solution of linear Itô stochastic singular systems. A new sufficient condition has been provided to guarantee that the linear Itô stochastic singular system is impulse-free and finite-time stochastic stable. Based on the obtained result, we have also derived the corresponding stabilization criteria. Moreover, the finite-time stochastic stabilization has been studied via state feedback, and some new sufficient conditions have been given. Two examples are presented to illustrate the effectiveness of the proposed results. In addition, we can refer to [29–31] and extend the results of this paper to Markovian jump systems, networked systems, and linear parameter varying systems.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of the paper.

#### Acknowledgments

This work is supported by the Starting Research Foundation of Qilu University of Technology (Grant no. 12045501), NSF of Shandong Province (Grant no. ZR2013FM022), NSF of China (Grant no. 61174078), the Research Fund for the Taishan Scholar Project of Shandong Province of China and the SDUST Research Fund (Grant no. 2011KYTD105), and the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant no. LAPS13018).

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