Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 7843940, 11 pages
http://dx.doi.org/10.1155/2016/7843940

Research Article

Robust Quadratic Stabilizability and Control of Uncertain Linear Discrete-Time Stochastic Systems with State Delay

Xiushan Jiang,¹ Xuemin Tian,¹ and Weihai Zhang²

¹College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266510, Shandong Province, China
²College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, Shandong Province, China

Received 20 May 2016; Revised 9 August 2016; Accepted 5 October 2016

Academic Editor: Abdellah Benzaouia

Copyright © 2016 Xiushan Jiang 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.

This paper mainly discusses the robust quadratic stability and stabilization of linear discrete-time stochastic systems with state delay and uncertain parameters. By means of the linear matrix inequality (LMI) method, a sufficient condition is, respectively, obtained for the stability and stabilizability of the considered system. Moreover, we design the robust state feedback controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust controllable. A sufficient condition for the existence of the desired robust controller is obtained. Finally, an example with simulations is given to verify the effectiveness of our theoretical results.

1. Introduction

It is well known that stability and stabilization are very important concepts in linear system theory. Due to a great number of applications of stochastic systems in the realistic world, the studies of stability and stabilization for stochastic systems attract lots of researchers’ attention in recent years; we refer the reader to the classic book [1] and the follow-up books [2, 3], together with references [4–11] and the references therein, which include robust stochastic stability [4], exponential stabilization [6], mean-square stability, and -stability and -stability [8]. The stabilization of various systems, including impulsive Markovian jump delay systems [4], stochastic singular systems [10, 12, 13], uncertain stochastic T-S fuzzy systems [14], and time-delay systems [6, 11, 15–17], has been studied extensively. control is one of the most important robust control approaches when the system is subject to the influence of external disturbance, which has been shown to be effective in attenuating the disturbance. The objective of standard control requires designing a controller to attenuate -gain from the external disturbance to controlled output below a given level ; see [18]. The study of control of general linear discrete-time stochastic systems with multiplicative noise seems to be first initiated by [19]. Then, stochastic control and its applications have been investigated extensively; see [14, 16, 20–24].

Because time-delay exists widely in practice and affects the system stability, there have been many works concerning the study in stability or control of stochastic systems [4, 6, 9, 11, 14–16, 22, 25]. Due to limitations of measurement technique and tools, it is not easy to construct exact mathematical models. Compared with the nominal stochastic systems without uncertain terms investigated in [2, 5, 24], our considered system allows the coefficient matrix to vary in a certain range.

Discrete-time stochastic difference systems have attracted a great deal of attention with the development of computer technology in recent years. In our viewpoint, there are at least two motivations to study discrete-time stochastic systems, Firstly, discrete-time stochastic systems are ideal mathematical models in practical modeling such as genetic regulatory networks [23]. Secondly, discrete-time stochastic systems provide a better approach to understand extensively continuous-time stochastic Itô systems [2, 3, 26]. Therefore, it is of significance to study the stabilization and control of discrete-time stochastic time-delay uncertain systems.

This paper will study quadratic stability, stabilization, and robust state feedback control for uncertain discrete-time stochastic systems with state delay. The parameter uncertainties are time varying and norm bounded. It can be found that, up to now, many criteria for testing quadratic stabilization and control have been given in terms of LMIs and algebraic Riccati equations by applying Lyapunov function approach. One of our main contributions is to study quadratic stability and stabilization via LMIs instead of algebraic Riccati equations which is hardly solved. What we have obtained extended the work of [15] about the quadratic stability and stabilization of deterministic uncertain systems. Another contribution is to solve the state feedback control and present a state feedback controller design.

The paper is organized as follows. In Section 2 we give some adequate preliminaries and useful definitions. In Section 3, sufficient conditions for quadratic stability and stabilization are given in terms of LMIs which is convenient to compute by the MATLAB LMI toolbox. Section 4 designs a state feedback controller. Two numerical examples with simulations are given in Section 5 to verify the efficiency of the proposed results. Finally, we end this paper in Section 6 with a brief conclusion.

For convenience, the notations in this paper are quite standard such as the following: we let and represent the set of all real -dimensional vectors and real matrices. For symmetric matrices and , (resp., stands for the idea that the matrix is positive semidefinite (resp., positive definite). denotes the identity matrix of appropriate dimensions and denotes the matrix transpose of . represents the Euclidean norm or spectral norm of the vector . , especially, , , and , represents the set of integers between and (inclusive). In symmetric block matrices, the symbol “” is used as an ellipsis for terms induced by symmetry. is the expectation operator.

2. Preliminaries

Consider a class of uncertain linear discrete-time stochastic systems with state delay described by

where is the system state and is the control input, and are independent white noise process satisfying the following assumptions:

), , where is a Kronecker function defined by for while for .() are defined on the filtered probability space with . In addition, is an increasing sequence of -algebras with .

, , , , , are known real constant matrices with compatible dimensions. , , , , , are norm bounded and time-varying uncertain parameter which are assumed to have the following form:

where , , , , , , are constant matrices and is the uncertain matrix satisfying

For the purpose of simplicity, throughout this paper, we write system (1) in the following form:

where , , , , are bounded uncertain system matrices with

Below, we define robust quadratic stability and robust quadratic stabilizability for the uncertain time-delay discrete-time system (1), which generalize Definition 1 of [15] to stochastic systems.

Definition 1. Uncertain discrete time-delay system (1) is said to be robustly quadratically stable, if there exist matrices , and a scalar such that, for all admissible uncertain terms and given initial condition for , the unforced system of (1) (with ) satisfies

for with and

Definition 2. Uncertain discrete time-delay system (1) is said to be robustly quadratically stabilizable if there exists a matrix such that closed-loop system (1) with , that is,

is robustly quadratically stable for given for .

3. Robust Quadratic Stabilization

In this section, a sufficient condition about robust quadratic stability and robust quadratic stabilization will be presented via LMIs, respectively. First, we cite the following lemma which is essential in proving our main results.

Lemma 3 (see [27]). Suppose that , satisfies (2), and then for any real matrices , , and of suitable dimensions we have

if and only if (iff), for some ,

Theorem 4. Consider uncertain discrete-time stochastic delay system (1) with . This system is robustly quadratically stable if there exist positive definite matrices , such that the following LMI holds.

where

Proof. From Definition 1, taking a Lyapunov function as in the form of (7), if uncertain discrete time-delay stochastic system (1) is quadratically stable, then, for all admissible uncertainties of (1), there exist matrices , and a scalar such that associated with unforced system (8) satisfies (6). In view of the assumption (), it is easy to compute

where , , , and are given in (5) and is shown as

By Definition 1, system (1) with is robustly quadratically stable, only if

which is equivalent to

Note that can be rewritten as

By Schur’s complement, it is easy to derive that is equivalent to

where

Then, using the same way as in (16)–(19) yields

The above inequality can be rewritten as

where

Because is a symmetric matrix, applying Lemma 3, (21) holds iff the following inequality holds:

where

Take

and then by substituting (25) into (23), for , we get

where , , are shown in (12).
Using the same method as in (16)–(20), (11)-(12) follow immediately from the above inequality.

Theorem 5. System (1) is robustly quadratically stabilizable if there exist positive matrices , , and a scalar with such that the following LMI holds.

where

Moreover, a quadratically stabilizing state feedback controller is given by

Proof. By Definition 2, using the same way as in the proof of Theorem 4, the following inequality which has a similar form to (11)-(12) can be obtained by taking

where

In order to eliminate the nonlinear quadratic terms

pre- and postmultiplying

on both sides of (30) and considering , (27)-(28) can be obtained easily. This theorem is proved.

Remark 6. Compared with the results about quadratic stability and quadratic stabilizability of deterministic systems given in [14], our two theorems not only extend the results of [14] to stochastic systems, but also provide the corresponding LMI criteria which can be easily tested by MATLAB LMI toolbox.

Remark 7. From these two theorems, we also can get the result about quadratic stability with the given decay rate. Take the function

and then, substituting (34) into (8), we obtain the following new system:

where

So system (1) is quadratically stabilizable with decay rate if system (35) is quadratically stabilizable.

4. State Feedback Control

In this section we consider the state feedback discrete-time control problem for the following uncertain linear stochastic system with state delay:

where and are called the controlled output and external disturbance, respectively. In addition, the effect of the disturbance on the controlled output is described by a perturbation operator , which maps any finite energy disturbance signal into the corresponding finite energy output signal of the closed-loop system. The size of this linear operator, that is, , measures the influence of the disturbances in the worst case. We denote by the set of all nonanticipative square summable -valued stochastic processes

-norm of is defined by

Firstly, for system (37), we define the perturbed operator and its norm as follows.

Definition 8. The perturbed operator of system (37), , is defined as

with its norm

Next, we present the definition about stochastic robust control.

Definition 9. For a certain level , is the control of the system (37), if

(i)system (37) is internally stabilizable when ;(ii)the norm of the perturbed operator of system (37) satisfies for all external disturbance .

Besides, if exists, then system (37) is called controllable in the disturbance attenuation. Furthermore, it is called strongly robust controllable if .

Theorem 10. Consider system (37). For the given and some with and if there exist , , and satisfying the following LMI

where

then system (37) is robustly controllable with a control law .

Proof. By Theorem 5, when disturbance , it is easy to test that system (37) is internally stabilizable with . Now we only need to show . By Definition 1, choose the Lyapunov function with and to be determined, and then

So in the case of , , we have

where

Obviously, it is easy to get that if . Then, we need to eliminate the uncertainties. Using the same method as in the proof of Theorem 4, we know that, for some , a sufficient condition for can be got from the following matrix inequality.

where

Then, by pre- and postmultiplying

on both sides of (47), we have

For some constant with , Theorem 10 is concluded; that is, an control of system (37) is obtained by solving LMIs (42)-(43). This completes the proof.

5. Simulation Example

In this section, we consider two simple examples with simulations to illustrate the effectiveness of the proposed approach.

Example 11. Consider discrete-time stochastic system (1) with the following parameters:

Using LMI toolbox to solve (11)-(12) in Theorem 4, we find out that which means that there is no feasible solution and indicates that system (1) with is unstable. Figure 1 verifies the result. By solving LMI (27), a group of feasible solutions with are shown as and

By Theorem 5, the system is mean-square stabilizable which is verified by Figure 2. A robust stabilizing controller is given by

Figure 1: State trajectories of the autonomous system.

Figure 2: State trajectories of the closed-loop system.

Example 12. Consider system (42) with the following parameters:

For perturbed system (42), we take the external disturbance as and the certain level as . In addition, according to Lemma 3, an appropriate is given as . Then, by the result of Theorem 10, using LMI toolbox to solve (43) and (47), we find that , which means we have got a group of feasible solutions with

The simulation results of state trajectories and controlled output trajectories of system (42) are given in Figures 3 and 4 with the controller

This further verifies the effectiveness of Theorem 10.

Figure 3: State trajectories of the closed-loop system.

Figure 4: Controlled output trajectories of the closed-loop system.

6. Conclusion

In this paper, we have studied the robust quadratic stability, quadratic stabilization, and robust state feedback control of discrete-time stochastic systems with state delay and uncertain parameters. Based on LMI technique, a sufficient condition about quadratic stability and quadratic stabilization of our considered system is, respectively, given. Moreover, an state feedback controller is obtained by solving two LMIs. Finally, we supply two simulation examples to show the validity of the proposed results. It is expected to solve the output feedback control and filtering in our forthcoming work.

Competing Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61573227 and no. 61273160), Natural Science Foundation of Shandong Province of China (no. 2013ZRE28089), the Research Fund for the Taishan Scholar Project of Shandong Province of China, the Postgraduate Innovation Funds of China University of Petroleum (East China) (no. YCX2015051), and the SDUST Research Fund (no. 2015TDJH105).

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