Abstract and Applied Analysis

Volume 2014, Article ID 198616, 11 pages

http://dx.doi.org/10.1155/2014/198616

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

School of Information and Electrical Engineering, Panzhihua University, Panzhihua, Sichuan 617000, China

Received 10 December 2013; Accepted 21 January 2014; Published 3 March 2014

Academic Editor: Zhengguang Wu

Copyright © 2014 Bing Wang. 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 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
and corresponding transition rate matrix is

Assuming that , , , and , by suing LMI toolbox, Theorem 10 provides the following controller gains:

#### 6. Conclusions

We have presented the problems of finite-time stochastic performance analysis of continuous-time systems with random abrupt changes in this paper. By using a different Lyapunov-Krasovskii functional, several sufficient conditions are provided to ensure the Markovian jump system is finite-time stochastic bounded. The controller gains can be dealt with by LMIs toolbox and optimization techniques. At last, two numerical examples are proposed to illustrate the effective and advantage of the developed theories.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Basic Research Program of China (2010CB732501) and the Fund of Sichuan Provincial Key Laboratory of signal and information processing, Xihua University (SZJJ2009-002, SGXZD0101-10-1).

#### References

- N. N. Krasovskiĭ and È. A. Lidskiĭ, “Analytical design of controllers in systems with random attributes,”
*Automation and Remote Control*, vol. 22, pp. 1021–1025, 1961. View at Google Scholar · View at MathSciNet - J. Cheng, H. Zhu, S. Zhong, and G. Li, “Novel delay-dependent robust stability criteria for neutral systems with mixed time-varying delays and nonlinear perturbations,”
*Applied Mathematics and Computation*, vol. 219, no. 14, pp. 7741–7753, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. Cheng, H. Zhu, S. Zhong, Y. Zhang, and Y. Zeng, “Improved delay-dependent stability criteria for continuous system with two additive time-varying delay components,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, no. 1, pp. 210–215, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - H. Shen, S. Xu, J. Lu, and J. Zhou, “Passivity-based control for uncertain stochastic jumping systems with mode-dependent round-trip time delays,”
*Journal of the Franklin Institute*, vol. 349, no. 5, pp. 1665–1680, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Shen, S. Xu, J. Zhou, and J. Lu, “Fuzzy ${H}_{\infty}$ filtering for nonlinear Markovian jump neutral systems,”
*International Journal of Systems Science*, vol. 42, no. 5, pp. 767–780, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Zhang and L. Yu, “Exponential state estimation for Markovian jumping neural networks with time-varying discrete and distributed delays,”
*Neural Networks*, vol. 35, pp. 103–111, 2012. View at Publisher · View at Google Scholar - D. Zhang and L. Yu, “Passivity analysis for stochastic Markovian switching genetic regulatory networks with time-varying delays,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 8, pp. 2985–2992, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Zhang, L. Yu, Q. Wang, and C. Ong, “Estimator Design for discrete-time switched neural networks with asynchronous switching and time-varying delay,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 23, no. 5, pp. 827–834, 2012. View at Publisher · View at Google Scholar - Z. Wu, P. Shi, H. Su, and J. Chu, “Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data,”
*IEEE Transactions on Cybernetics*, vol. 43, no. 6, pp. 1796–1806, 2013. View at Google Scholar - Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Passivity analysis for discrete-time stochastic markovian jump neural networks with mixed time delays,”
*IEEE Transactions on Neural Networks*, vol. 22, no. 10, pp. 1566–1575, 2011. View at Publisher · View at Google Scholar · View at Scopus - H. Huang, G. Feng, and X. Chen, “Stability and stabilization of Markovian jump systems with time delay via new Lyapunov functionals,”
*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 59, no. 10, pp. 2413–2421, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Gao, Z. Fei, J. Lam, and B. Du, “Further results on exponential estimates of Markovian jump systems with mode-dependent time-varying delays,”
*IEEE Transactions on Automatic Control*, vol. 56, no. 1, pp. 223–229, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Wu, P. Shi, H. Su, and J. Chu, “Asynchronous ${{l}_{2}-l}_{\infty}$ ltering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities,”
*Automatica*, vol. 50, no. 1, pp. 180–186, 2014. View at Publisher · View at Google Scholar - S. Hu, D. Yue, and J. Liu, “${H}_{\infty}$ filtering for networked systems with partly known distribution transmission delays,”
*Information Sciences. An International Journal*, vol. 194, pp. 270–282, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-H. Lien, “Robust observer-based control of systems with state perturbations via LMI approach,”
*IIEEE Transactions on Automatic Control*, vol. 49, no. 8, pp. 1365–1370, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - P. Shi, E.-K. Boukas, and R. K. Agarwal, “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 11, pp. 2139–2144, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang, J. Lam, and X. Liu, “Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances,”
*IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 51, no. 5, pp. 262–268, 2004. View at Publisher · View at Google Scholar · View at Scopus - Y. Wang, C. Wang, and Z. Zuo, “Controller synthesis for Markovian jump systems with incomplete knowledge of transition probabilities and actuator saturation,”
*Journal of the Franklin Institute*, vol. 348, no. 9, pp. 2417–2429, 2011. View at Publisher · View at Google Scholar · View at Scopus - Q. Zhu, X. Yang, and H. Wang, “Stochastically asymptotic stability of delayed recurrent neural networks with both Markovian jump parameters and nonlinear disturbances,”
*Journal of the Franklin Institute*, vol. 347, no. 8, pp. 1489–1510, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. M. Kwon, J. Park, S. Lee, and E. Cha, “Analysis on delay-dependent stability for neural networks with time-varying delays,”
*Neurocomputing*, vol. 103, pp. 114–120, 2013. View at Publisher · View at Google Scholar - F. Amato, R. Ambrosino, C. Cosentino, and G. De Tommasi, “Input-output finite time stabilization of linear systems,”
*Automatica*, vol. 46, no. 9, pp. 1558–1562, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cheng, H. Zhu, S. Zhong, Y. Zeng, and X. Dong, “Finite-time ${H}_{\infty}$ control for a class of Markovian jump systems with mode-dependent time-varying delays via new Lyapunov functionals,”
*ISA Transactions*, vol. 52, pp. 768–774, 2013. View at Google Scholar - J. Cheng, H. Zhu, S. Zhong, Y. Zhang, and Y. Li, “Finite-time ${H}_{\infty}$ control for a class of discrete-time Markov jump systems with partly unknown time-varying transition probabilities subject to average dwell time switching,”
*International Journal of Systems Science*, 2013. View at Publisher · View at Google Scholar - J. Cheng, G. Li, H. Zhu, S. Zhong, and Y. Zeng, “Finite-time ${H}_{\infty}$ control for a class of Markovian jump systems with mode-dependent time-varying delay,”
*Advances in Difference Equations*, vol. 2013, article 214, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. He and F. Liu, “Stochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities,”
*Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems*, vol. 35, no. 6, pp. 2631–2638, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. N. ElBsat and E. E. Yaz, “Robust and resilient finite-time bounded control of discrete-time uncertain nonlinear systems,”
*Automatica*, vol. 49, no. 7, pp. 2292–2296, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - P. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,”
*Automatica*, vol. 47, no. 1, pp. 235–238, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet