## Robust Control, Optimization, and Applications to Markovian Jumping Systems

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# Stochastic Finite-Time Performance Analysis of Continuous-Time Systems with Random Abrupt Changes

**Academic Editor:**Zhengguang Wu

#### Abstract

The problem of control performance analysis of continuous-time systems with random abrupt changes is concerned in this paper. By employing an augmented multiple mode-dependent Lyapunov-Krasovskii functional and using some integral inequalities, new sufficient conditions are obtained relating to finite-time bounded and an performance index. The finite-time control performance problem is solved and desired controller is given to ensure the system trajectory stays within a prescribed bound during a given time interval. At last, two numerical examples are provided to show that our results are less conservative than the existing ones.

#### 1. Introduction

It is well known that Markovian jump systems were introduced when the physical models are always subject to random changes, which can be also regarded as a special class of hybrid systems because of the structures are subject to random abrupt changes [1]. In the recent years, there are a lot of people towards to Markovian jump systems for its widely applications, for example, target tracking, robotics, manufacturing systems, aircraft control, and power systems [2–4]. Markovian jump systems are regarded as a special class of stochastic systems which switches from one to another at different time in the finite operation modes. Many important topics have been studied for Markovian jumping systems such as stability, control synthesis, stabilization, and filter design [5–7].

On the other hand, time delay is very common in practical dynamical systems, for example, networked control systems, chemical processes, communication systems, and so on [8–20]. Therefore, during the past two decades, various research topics have been considered for Markovian jump systems with time-varying delays [8–14]. It worth pointing out that when time delay is small enough in linear Markovian jump systems, the delay-dependent criteria are always less conservative than delay-independent ones. Over the past few years, for Markovian jump systems, many important topics related to delay-dependent have been extensively studied [14, 15].

Generally speaking, finite-time stability is investigated to address these transient performances of control systems in finite-time interval. Up to now, the concept of finite-time stability has been revisited with different systems, and many important results are obtained for finite-time stability and finite-time boundedness [21–26]. However, to the best of authors' knowledge, the stochastic finite-time control for Markovian jump systems has not been fully studied. There is some room for next investigation due to the fact that analysis methods in existing references seem still conservative.

The major contribution of this paper is that we introduce a newly Lyapunov-Krasovskii functional for Markovian jump system. Some sufficient conditions are obtained to ensure the finite-time stability and bounded of the closed-loop Markovian jump systems. Compared with traditional methods of MJSs, it is shown the less conservative results can be obtained and the desired control performance is obtained by employing mode-dependent Lyapunov functional instead of mode-independent Lyapunov functional. The finite-time bounded criterion can be dealt with in the terms of LMIs. Finally, the effectiveness of the developed techniques is also illustrated by two numerical examples.

#### 2. Preliminaries

Given the probability space , where , , and represent the sample space, the algebra of events, and the probability measure defined on , respectively, the following Markovian jump systems over the probability space are considered: where represents the state vector of Markovian jump system, is the control input, denotes the controlled output and , , where is initial condition. denotes the disturbance input which satisfies

Firstly, taking value on the finite set , let the random form process be the stochastic process with transition rate matrix , and let the transition probabilities also be denoted as follows: where and , , for , denotes the mode in time to time with mode , for each mode , . denotes the time-varying delay, which satisfies where is the given upper bound of time-varying delays and is the given upper bound of . All the matrices are known matrices with the appropriate dimension.

In this paper, the objective is to design a state feedback controller as follows: where is the controller gains to be designed.

*Definition 1. *System (1) is said to be finite-time bounded with respect to , if condition (2) and the following inequality hold:
where and .

*Definition 2 (see [8]). *Considering system (1) with the stochastic Lyapunov function , we get the weak infinitesimal operator as follows:

*Definition 3. *Given a constant scalar and for all admissible given in condition (2), if the Markovian jump system (1) is finite-time stochastic bounded and controller outputs satisfy condition (7) with attenuation ,
The Markovian jump system (1) is called the finite-time stochastic bounded with a disturbance attenuation .

Lemma 4 (see [27]). *Let : have positive values in an open subset of . Then, the reciprocally convex combination of over satisfies
*

Lemma 5. *For any constant matrix with , scalars , vector function , such that the integrals in the following are well-defined; then,
*

#### 3. Finite-Time Performance Analysis

The issue of stability analysis of Markovian jump system (1) subject to is given firstly. Therefore, the finite-time stability is obtained in this section.

Theorem 6. *System (1) is called the finite-time bounded with respect to , if there exist matrices
**
scalars , , , , , and , such that and the inequalities hold as follows:
**
where
*

*Proof. *Firstly, a novel process is defined in this paper as follows:
Then, the following Lyapunov-Krasovskii functional is considered:
where
where .

Letting represent the time , that is, , one has
Noting for and , one has
It follows from (15) and (28) that
By employing Lemma 4, we can obtain that
It follows from (30) and (31) that
Consider
Moreover, the following two zero equalities with any symmetric matrices and are considered:
With the above two zero equalities (34) and (35), an upper bound of is
From (19), (20), and (36), one can obtain
Now, is obtained as follows:
By using Lemma 5, one has
Together with (38) and (39), it implies that
From (26)–(40), we can eventually obtain
where
It follows from (45) that
Multiplying the above inequality by yields that
Integrating the inequality from to , we have
Denoting , , , , , , , , and yields that
For scalars and , (46) turns out to be
To illustrate the bounded, (26) takes the following form:
From inequalities (46)–(48), one has
Finally, inequalities (24) and (49) guarantee that

Therefore, the Markovian jump system (1) is finite-time stochastic bounded with respect to .

*Remark 7. *It should be noted that and may, respectively, get the different upper bound due to the fact that condition (6) holds. However, and always lead to conservativeness for and in [14–18], and this case can be improved with employing the different Lyapunov-Krasovskii functional (26).

*Remark 8. *It should be pointed out that, in Theorem 6, the novelty of the Lyapunov functional (26) lies in the following: (i) triple-integral terms and and four-integral term are introduced and (ii) the distinct Lyapunov matrices are chosen for different system modes .

For the condition , the Markovian jump system given in this paper is followed by
where

Theorem 9. *System (53) is finite-time stochastic bounded with respect to with a disturbance attenuation, if there exist matrices
**
scalars , , , , and , such that for all , (15)–(20) and the following inequalities hold:**
where
*

*Proof. *Considering the Lyapunov-Krasovskii functional in Theorem 6 and from Schur's Lemma, it turns out to be

Thanks to (54), we have
Multiplying the (58) by , (58) can be written as
Under the condition of zero initial and , one has
Using the Dynkin formula, it results that
Finally, it is easy to obtains that
Therefore, the Markovian jump system (53) is finite-time stochastic bounded with an performance .

#### 4. Finite-Time Control

Theorem 10. *System (53) is finite-time stochastic bounded with respects to with an disturbance attenuation, if there exists matrices
**
scalars , , , , and , such that for all , (15)–(20) and the following inequalities hold:**
where
**
and the state feedback gain matrices considered in this paper could be designed as follows:
*

*Proof. *This proof can be completed in view of Theorem 9 with and .

#### 5. Illustrative Example

*Example 1. *Considering the following example with parameters
the transition probabilities matrix is given as follows:

Given the different upper bounds of and , the results of the maximum upper bound of decay rates and maximum values of for different time delays are obtained in Tables 1 and 2, respectively. This example indicates fully that the method proposed in the paper plays an important role in reducing conservatism. It can be also seen that our results in this paper show significant improvement over the results obtained in [11, 12]. This clearly shows that our results have less conservatism in the above two cases.

*Example 2. *Consider the Markovian jump system (1) where