Elsevier

Automatica

Volume 79, May 2017, Pages 27-34
Automatica

Brief paper
Sliding mode control for singular stochastic Markovian jump systems with uncertainties

https://doi.org/10.1016/j.automatica.2017.01.002Get rights and content

Abstract

This paper considers sliding mode control design for singular stochastic Markovian jump systems with uncertainties. A suitable integral sliding function is proposed and the resulting sliding mode dynamics is an uncertain singular stochastic Markovian jump system. A set of new sufficient conditions is developed which not only guarantees the stochastic admissibility of the sliding mode dynamics, but also determines all the parameter matrices in the integral sliding function. Then, a sliding mode control law is synthesized such that reachability of the specified sliding surface can be ensured. Finally, three examples are given to demonstrate the effectiveness of the results.

Introduction

Markovian jump systems (MJSs) have the advantage of better representing physical systems with random changes in both structure and parameters. Much recent attention has been paid to the investigation of these systems (Fang and Loparo, 2002, Xiong and Lam, 2006, Yue and Han, 2005). Singular systems have extensive applications in fields related to electrical circuits and power systems (Lewis, 1986, Yang et al., 2006). When singular systems experience abrupt changes in their structure, it is natural to model them as singular Markovian jump systems (SMJSs) (Boukas, 2008, Huang and Mao, 2010). In practice, these systems are often corrupted by noise, for example Brownian motion. Therefore it is of significance to study singular stochastic Markovian jump systems (SSMJSs).

Sliding mode control (SMC) has been recognized as an effective strategy for control of systems with uncertainties and nonlinearity (Hung et al., 1993, Ma and Boukas, 2009). The sliding mode dynamics is a reduced-order system and completely insensitive to matched uncertainties (Edwards and Spurgeon, 1998, Utkin et al., 1999). Sliding mode methods can also be applied to systems in the presence of mismatched uncertainties (Yan, Spurgeon, & Edwards, 2005). To obtain similar levels of robustness from a classical linear state feedback controller, high gain is required (Young, Utkin, & Özgüner, 1999) which can be limiting in terms of controller saturation and practical application. A novel augmented sliding mode observer is presented for the augmented system of MJSs and is utilized to eliminate the effects of sensor faults and disturbances (Li, Gao, Shi, & Zhao, 2014). Sliding mode methods are successfully applied to uncertain time-delay systems (Alwi and Edwards, 2008, Fridman et al., 2003, Yan et al., 2013), interconnected systems (Yan, Spurgeon, & Edwards, 2010), stochastic systems (Niu et al., 2007, Shi et al., 2006), SMJSs (Wu and Daniel, 2010, Wu et al., 2012, Wu and Zheng, 2009). When a linear sliding function is used, the dimension of the resulting sliding motion will be reduced and the regular form typically used for sliding mode control design (Edwards & Spurgeon, 1998) is necessary in order to solve the corresponding existence problem. When considering singular systems, this regular form is available only if the column vector of the input matrix B is a linear representation of that of the derivative matrix E. In comparison, the integral-type sliding function introduces a compensator whose dimension is equal to the dimension of the input vector and the resulting sliding motion is of full order. In this case the regular form typically adopted for sliding mode controller design is not required and the integral-type sliding function (Wu and Daniel, 2010, Wu et al., 2012, Wu and Zheng, 2009) is suitable for any singular system. In Wu et al. (2012) and Wu and Zheng (2009), parameter matrices Gi (G) in the sliding function need to be designed in advance. If the selection of these parameter matrices is not appropriate, additional conservatism will be introduced into the stability analysis of the resulting sliding mode dynamics. In order to decrease the conservatism, these parameter matrices need be redesigned but no constructive design approach is given. In Wu and Daniel (2010), although a method of how to design all the parameter matrices in the sliding function is given, a particular constraint must be satisfied so that EBi for system matrices E and Bi must have full column rank.

This paper considers the design of a SMC for a class of uncertain SSMJSs. Key questions to be addressed are stated as follows:

  • Q1.

    How to design a suitable sliding function such that conditions developed for the stochastic admissibility of the resulting sliding mode dynamics can determine all the parameter matrices in the sliding function complementing existing design methods?

  • Q2.

    How to analyze and synthesize a SMC law so that the proposed approach can effectively reject the effect of Markovian switching on the desired dynamic performance of uncertain SSMJSs?

Section snippets

System representation and preliminaries

Consider a nonlinear SSMJS described as follows: Edx(t)=[(A(rt)+ΔA(rt,t))x(t)+B(rt)(u(t)+f(x(t),rt))]dt+D(rt)x(t)dϖ(t) where x(t)Rn is the state vector, u(t)Rm is the control input and ϖ(t) is a one-dimensional Brownian motion satisfying E{dϖ(t)}=0 and E{dϖ2(t)}=dt, E{} denotes the mathematical expectation of the stochastic process or vector. The matrix ERn×n may be singular. It is assumed that rank(E)=rn. Matrices A(rt),B(rt) and D(rt) are known and real with appropriate dimensions where B

SMC synthesis

In this section, a sliding surface is designed and the corresponding sliding motion is analyzed. Then sliding mode controllers are synthesized such that the closed-loop system has the desired performance.

For the system (1), consider the following integral sliding function: s(t)=BiTP̄iEx(t)0tBiTP̄i(Ai+BiKi)x(θ)dθ where P̄iRn×n and KiRm×n are real matrices to be designed with BiTP̄iBi being nonsingular. It should be noted that due to the assumption that Bi is full column rank, the

SMC with H performance

In this section, a set of sufficient conditions will be developed under which the sliding mode dynamics of the considered system is guaranteed to be stochastically admissible with H performance.

Consider the following SSMJS in the presence of external disturbance: Edx(t)=[(Ai+ΔAi(t))x(t)+Bi(u(t)+fi(x))+Fiw(t)]dt+Dix(t)dϖ(t)z(t)=C1ix(t)+C2iw(t) where w(t)Rp is the disturbance input which belongs to L2p[0,); z(t)Rq is the controlled output; Fi,C1i and C2i are constant matrices with appropriate

Numerical examples

Example 1

Consider a SSMJS (24) with two modes and parameters as follows: Mode  1:A1=[1.511.21.31.61.10.60.80.8],B1=[1.00.50.2],M1=[0.100.1],D1=[0.10000.20000],F1=[0.100],N1=[0.20.10.1],C11=[0.100.2],C21=0Mode  2:A2=[0.50.60.71.22.40.40.60.21.5],B2=[0.81.00.3],M2=[0.200.1],D2=[0.30000.10000],F2=[0.100]T,N2=[0.30.20.1],C12=[0.200.3],C22=0. The other parameters for models 1 and 2 are given as follows E=[100010000],Π=[0.50.50.40.4],f1(x)=f2(x)=0.3sin(t)x1,γ=1,F1(t)=F2(t)=0.2sin(t). By Theorem 3 and

Conclusions

In this paper, SMC laws have been designed for a class of SSMJSs in the presence of uncertainties. A suitable integral-type sliding surface is designed such that the resulting sliding mode dynamics is stochastically admissible. Investigation of stochastic MJSs is considered as well using sliding mode techniques such that the corresponding closed-loop systems are stochastically stable. The conditions developed are easily testable and can be regarded as complementary to the existing results

Qingling Zhang received B.S. and M.S. degrees in Mathematics Department, and Ph.D. degree in Automatic Control Department from Northeastern University, Shenyang, China, in 1982, 1986 and 1995, respectively. He finished his two-year Postdoctoral work in Automatic Control Department of Northwestern Polytechnical University, Xian, China, in 1997. Since then he has been a Professor and serve College of Science at Northeastern University as dean from 1997 to 2006. He was also a member of the

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  • Cited by (0)

    Qingling Zhang received B.S. and M.S. degrees in Mathematics Department, and Ph.D. degree in Automatic Control Department from Northeastern University, Shenyang, China, in 1982, 1986 and 1995, respectively. He finished his two-year Postdoctoral work in Automatic Control Department of Northwestern Polytechnical University, Xian, China, in 1997. Since then he has been a Professor and serve College of Science at Northeastern University as dean from 1997 to 2006. He was also a member of the University Teaching Advisory Committee of National Ministry of Education, and now he is vice chairman of the Chinese Biomathematics Association, member of technical committee on control theory of the Chinese Association of Automation, member of the Chinese Association of Mathematics and Chairman of Mathematics Association of Liaoning Province. He has published 16 books and more than 600 papers about control theory and applications. Prof. Zhang received 14 prizes from central and local governments for his research. He has also received the Gelden Scholarship from Australia in 2000. During these periods, he visited Hong Kong University, Seoul University, Alberta University, Lakehead University, Sydney University, Western Australia University, Windsor University, Hong Kong Polytechnic University and Kent University as a Research Associate, Research Fellow, Senior Research Fellow and Visiting Professor, respectively.

    Li Li received the B.Sc. and M.Sc. degrees in mathematics from the Liaoning Normal University, Dalian, China, in 1999 and 2005, respectively, and the Ph.D. degree in the system complexity theory from Northeastern University, Shenyang, China, in 2015. She is a Lecturer at Bohai University, Jinzhou, China. Her current research interests include robust control, stochastic systems, singular systems, and fuzzy modeling and control.

    Xing-Gang Yan received the B.Sc. degree of Applied Mathematics from Shaanxi Normal University, in 1985, the M.Sc. degree of Control and Optimisation from Qufu Normal University in 1991, and the Ph.D. degree of Control Engineering from Northeastern University, P. R. China in 1997. Currently, he is appointed as a Senior Lecturer at the University of Kent, United Kingdom. He was a Lecturer in Qingdao University, P. R. China from 1991 to 1994. He worked as a Research Fellow or Research Associate in the Northwestern Polytechnical University, China, the University of Hong Kong, China, Nanyang Technological University, Singapore and the University of Leicester, United Kingdom. He is the Editor-In-Chief of the International Journal of Engineering Research and Science & Technology. He was a lead guest editor for the special issue “Advanced Control of Complex Dynamical Systems with Applications” in Mathematical Problems in Engineering. He serves as a member of the Editorial Board for several engineering journals. He has published three books, a few book chapters and over 140 referred papers. His research interests include sliding mode control, decentralized control, fault detection and isolation, nonlinear control and time delay systems with applications.

    Sarah K. Spurgeon OBE, FREng, FInstMC, FIET, FIMA is Professor of Control Engineering and Head of Department of Electronic and Electrical Engineering at University College London in the UK. She is President of the Institute of Measurement and Control. Sarah Spurgeon’s research interests are in the area of systems modeling and analysis, robust control and estimation in which areas she has published over 270 refereed research papers. She was awarded the Honeywell International Medal for ‘distinguished contribution as a control and measurement technologist to developing the theory of control’ in 2010 and an IEEE Millennium Medal in 2000. She is currently a member of the Council of the International Federation of Automatic Control (IFAC) and a member of the General Assembly of the European Control Association.

    This work was supported by the National Natural Science Foundation of P. R. China under grant No. 61673099,61603055, 61304054 and 61273008, respectively and the Royal Academy of Engineering of United Kingdom under grant No. 12/13RECI027. The initial version of this paper was completed during the period from Oct. 19, 2013 to Jan. 18, 2014 when the first author, Professor Qingling Zhang, visited the University of Kent, funded by the Royal Academy of Engineering of the United Kingdom under grant No. 12/13RECI027. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Fabrizio Dabbene under the direction of Editor Richard Middleton.

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