Research Article | Open Access

# Stochastic Stabilization of Itô Stochastic Systems with Markov Jumping and Linear Fractional Uncertainty

**Academic Editor:**Derong Liu

#### Abstract

For a class of Itô stochastic linear systems with the Markov jumping and linear fractional uncertainty, the stochastic stabilization problem is investigated via state feedback and dynamic output feedback, respectively. In order to guarantee the stochastic stability of such uncertain systems, state feedback and dynamic output control law are, respectively, designed by using multiple Lyapunov function technique and LMI approach. Finally, two numerical examples are presented to illustrate our results.

#### 1. Introduction

Modeled by a set of linear systems with the transitions among the linear systems governed by the Markov chain, the Markov jumping linear system (MJLS) can characterize and model different types of systems [1], for instance, fault-tolerant systems, target tracking systems, manufactory processes, networked control systems, multiagent systems, and so on. In the past years, many important results on MJLS have been addressed in the literature, such as, the stability analysis and control design which were discussed in [2–8]. Besides the aforementioned theoretical studies, MJLS also found applications in practical systems, such as networked direct current motor systems [9].

The interest of this topic lies in the fact that many practical systems, with random abrupt changes, sudden environmental changes, and so on, can be modeled by Itô stochastic linear systems with Markov jumping and configurable uncertainty [10]. Therefore, it is very important to investigate robustly stochastic stabilization problem for the aforementioned Itô stochastic MJLS with linear fractional uncertainties. Recently, some results for Itô stochastic MJLS have been addressed in [11–18].

In this paper, motivated by the recent work [19], we consider robustly stochastic stabilization problem for Markov jumping Itô stochastic linear system subject to linear fractional uncertainty. An effective LMI-based approach for state feedback and dynamic output feedback controller is designed such that the closed-loop system is robust stable in the mean square sense, respectively.

The reminder of this paper is organized as follows: the problem statement is described in Section 2, while in Section 3 the stochastic stability for Itô stochastic MJLS with linear fractional uncertainty is addressed. In Section 4, the robustly stochastic stabilization of Itô stochastic MJLS with linear fractional uncertainty is investigated via state feedback and dynamical output feedback, respectively. Two numerical examples are presented in Section 5 to illustrate the obtained results. Finally, some conclusions are drawn in Section 6.

#### 2. Problem Formulation

Consider the following unforced Itô stochastic MJLS with linear fractional uncertainties: where is a Markov process taking values in a finite space and is the state vector. is a standard Wiener process that is supposed to be independent of the Markov process . denotes the initial state of system at , and stands for the initial mode in the Markov process at . and , are system matrices with compatible dimensions that depend on and , and where , , , and are known real constant matrices with appropriate dimensions. is an unknown real matrix function with Lebesgue-measurable elements, satisfying with being the identity matrix.

The evolution of the Markov process is governed by the following probability transitions: where is the transition rate from mode to mode at time instant when , and is the little- notation defined by .

For the stochastic stability, we adopt the following definition. For more details, please refer to [18] and the references therein.

*Definition 1. *The Itô stochastic MJLS in (1) with all modes and all is said to achieve stochastic stability if there exists a finite positive constant such that the following inequality holds for any initial condition :
where is the expectation conditioning on the initial value of .

#### 3. Stochastic Stability

Before proceeding further, we recall the following lemma, which will be used in the proof of the stochastic stability of the Itô stochastic MJLS in (1).

Lemma 2 (see [20]). *Let be given constant matrices with appropriate dimensions and norm-bounded time-varying uncertain matrix with . For any , one has
*

Lemma 3 (see [19]). *Let the matrix G satisfy ; define the following set:
**
Then, the set can be rewritten as
*

Theorem 4. *The Itô stochastic uncertain MJLS in (1) is stochastically stable if there exists a set of symmetric and positive-definite matrices and a set of positive scalars such that the following sets of coupled inequalities hold for each and all admissible linear fractional uncertainties:
**
where represents the blocks that are induced by symmetry and
*

*Proof. *Consider the following Lyapunov-like function:
where denotes the positive symmetric matrix. The infinitesimal generator can be considered as a derivative of the Lyapunov function along the trajectory of the Markov process in point at time . We need to derive the infinitesimal generator of first of all. According to the definition of , we have
where is a small positive number. Setting and applying the law of total probability and the probability transition of the Markov process yield

According to the Itô stochastic differential coefficient theorem [18], we have
Note that and , then the infinitesimal generator becomes
where .

Using Lemmas 2 and 3, we have
where , .

Hence , where
According to the Schur complement, , implies
Therefore .

By the generalized Dynkin formula, we have
The last inequality implies
Furthermore, the condition in (8) indicates , so
holds for any . Letting go to infinity, then, we know that

According to Definition 1, the Itô stochastic uncertain MJLS in (1) is stochastically stable. This ends the proof.

#### 4. Stochastic Stabilization

##### 4.1. State Feedback Stochastic Stabilization

In this subsection, we discuss how to design the robust state feedback control law for the following Itô stochastic uncertain MJLS: where , are system matrices with compatible dimensions that depend on and , and where , , , and are known real constant matrices with appropriate dimensions. is an unknown real matrix function with Lebesgue-measurable elements, satisfying with being the identity matrix. The robust state feedback control law to be designed is where is a gain with appropriate dimension to be determined for each mode . Notice that the gain of the above controller is model dependent, which requires the knowledge of the model at time to choose the appropriate gain among . Thus when the model switches from mode , which uses the gain , to mode, the controller gains must be switched instantaneously to the gain to guarantee the desired performances.

Theorem 5. *Suppose that there exists a set of symmetric and positive-definite matrices , a set of matrices , and a set of positive scalars and such that the following sets of coupled inequalities hold for each and all admissible linear fractional uncertainties. Then the controller (24) with gain can stabilize stochastically the Itô stochastic uncertain MJLS in (22)
**
where represents the blocks that are induced by symmetry and
*

*Proof. *With the use of the state feedback control law in (24) for system (22), the closed-loop system becomes

In order to guarantee the stochastic stability of system (27), using the stochastically stable condition in Theorem 4, we know that the following inequality should be satisfied for each :
Let , and before and after multiplying the matrix inequality (28) by , then, we have

Applying expressions (2) and (23) of the linear fractional uncertainties, the above inequality can be described as

Again in view of Lemmas 2 and 3, the following two matrix inequalities are obvious:
where
Setting
Then can be rewritten as follows:
Let , and by means of the matrix inequalities (31) and (34), the closed-loop system (27) will be stochastically stable if the following matrix inequality holds:
After using the Schur complement, we can get the LMIs (25), which end the proof of Theorem 5.

##### 4.2. Dynamic Output Feedback Stochastic Stabilization

In Section 4.1, the design method of state feedback controller had been given under that the state vector and the mode are available at each time . However, some of the state variables are not measurable by their construction or the lack of appropriate sensors to give information. Alternatively we can use dynamic output feedback controller that uses the system’s measurement to design and to compute the control gains in this subsection.

In this subsection, we discuss how to design the dynamic output feedback control law for the following Itô stochastic uncertain MJLS: where , are system matrices with compatible dimensions that depend on and , and where , , , and are known real constant matrices with appropriate dimensions. is an unknown real matrix function with Lebesgue-measurable elements, satisfying with being the identity matrix. The dynamic output feedback control law is to be described by the following structure: where is the controller state and , , and are the controller gains to be determined.

Theorem 6. *If there exists a set of matrices , , , and and a set of positive scalars , , and such that the following sets of inequalities (39)–(41) hold for each and all admissible linear fractional uncertainties, then the dynamic output feedback controller (38) with control gain (43) can stabilize stochastically the Itô stochastic uncertain MJLS in (36)*

where represents the blocks that are induced by symmetry and

The dynamical output feedback controller gain is given by

*Proof. *Combining the dynamic system (36) with controller dynamics (38), we can obtain the following extended dynamics:
where and , are system matrices with compatible dimensions that depend on and , and

Applying Theorem 4, the dynamics (44) is stochastically stable if there exists a set of symmetric and positive-definite matrices such that the following inequalities hold for every :

Let , , and , and using Lemma 3, we have
In view of (47) and Lemma 2, we have

Based on the matrix inequality (48), the condition (46) will be satisfied if the following holds:

It is obvious that inequality (49) is nonlinear in the design parameters , , , and , . In order to transform inequality (49) into an LMI, define , , , and as follows:
where , are symmetric and positive-define matrices.

Before and after multiply inequality (49) by and , respectively, we have