Cover Page


Ligang Wu

Harbin Institute of Technology, China

Peng Shi

The University of Adelaide; and Victoria University, Australia

Xiaojie Su

Chongqing University, China

Wiley Logo

To Jingyan and Zhixin
L. Wu

To my family
P. Shi

To my family
X. Su

Series Preface

Electromechanical systems permeate the engineering and technology fields in aerospace, automotive, mechanical, biomedical, civil/structural, electrical, environmental, and industrial systems. The Wiley Book Series on dynamics and control of electromechanical systems will cover a broad range of engineering and technology within these fields. As demand increases for innovation in these areas, feedback control of these systems is becoming essential for increased productivity, precision operation, load mitigation, and safe operation. Furthermore, new applications in these areas require a reevaluation of existing control methodologies to meet evolving technological requirements, for example the distributed control of energy systems. The basics of distributed control systems are well documented in several textbooks, but the nuances of its use for future applications in the evolving area of energy system applications, such as wind turbines and wind farm operations, solar energy systems, smart grids, and the generation, storage and distribution of energy, require an amelioration of existing distributed control theory to specific energy system needs. The book series serves two main purposes: 1) a delineation and explication of theoretical advancements in electromechanical system dynamics and control, and 2) a presentation of application-driven technologies in evolving electromechanical systems.

This book series will embrace the full spectrum of dynamics and control of electromechanical systems from theoretical foundations to real-world applications. The level of the presentation should be accessible to senior undergraduate and first-year graduate students, and should prove especially well-suited as a self-study guide for practicing professionals in the fields of mechanical, aerospace, automotive, biomedical, and civil/structural engineering. The aim is to provide an interdisciplinary series, ranging from high-level undergraduate/graduate texts, explanation and dissemination of science and technology and good practice, through to important research that is immediately relevant to industrial development and practical applications. It is hoped that this new and unique perspective will be of perennial interest to students, scholars, and employees inthe engineering disciplines mentioned. Suggestions for new topics and authors for the series are always welcome.

This book, Sliding Mode Control of Uncertain Parameter-Switching Hybrid Systems, has the objective of providing a theoretical foundation as well as practical insights on the topic at hand. It is broken down into three parts: 1) sliding mode control (SMC) of Markovian jump singular systems, 2) SMC of switched state-delayed hybrid systems, and 3) SMC of switched stochastic hybrid systems. The book provides detailed derivations from first principles to allow the reader to thoroughly understand the particular topic. This is especially useful for Markovian jump singular systems with stochastic perturbations because a comprehensive knowledge of stochastic analysis is not required before understanding the material. Readers can simply dive into the material. It also provides several illustrative examples to bridge the gap between theory and practice. It is a welcome addition to the Wiley Electromechanical Systems Series because no other book is focused on the topic of SMC with a specific emphasis on uncertain parameter-switching hybrid systems.

Mark J. Balas
John L. Crassidis
Florian Holzapfel
Series Editors


Since the 1950s, sliding mode control (SMC) has been recognized as an effective robust control strategy for nonlinear systems and incompletely modeled systems. In the past two decades, SMC has been successfully applied to a wide variety of real world applications such as robot manipulators, aircraft, underwater vehicles, spacecraft, flexible space structures, electrical motors, power systems, and automotive engines. Basically, the idea of SMC is to utilize a discontinuous control to force the system state trajectories to some predefined sliding surfaces on which the system has desired properties such as stability, disturbance rejection capability, and tracking ability. Many important results have been reported for this kind of control strategy. However, when the controlled plants are uncertain parameter-switching hybrid systems including parameter-switching (Markovian jump or arbitrary switching), state-delay, stochastic perturbation, and singularly perturbed terms, the common SMC methodologies cannot meet the requirements.

It is known that the SMC of uncertain parameter-switching hybrid systems is much more complicated because sliding mode controllers must be designed so that not only is the sliding surface robustly reachable, but also the sliding mode dynamics can converge the system’s equilibrium automatically by choosing a suitable switching function. This book aims to present up-to-date research developments and novel methodologies on SMC of uncertain parameter-switching hybrid systems in a unified matrix inequality setting. The considered uncertain parameter-switching hybrid systems include Markovian switching hybrid systems, switched state-delayed hybrid systems, and switched stochastic hybrid systems. These new methodologies provide a framework for stability and performance analysis, SMC design, and state estimation for these classes of systems. Solutions to the design problems are presented in terms of linear matrix inequalities (LMIs). In this book, a large number of references are provided for researchers who wish to explore the area of SMC of uncertain parameter-switching hybrid systems, and the main contents of the book are also suitable for a one-semester graduate course.

In this book, we present new SMC methodologies for uncertain parameter-switching hybrid systems. The systems under consideration include Markovian jump systems, singular systems, switched hybrid systems, stochastic systems, and time-delay systems.

The content of this book are divided into three parts. The first part is focused on SMC of Markovian jump singular systems. Some necessary and sufficient conditions are derived for the stochastic stability, stochastic admissibility, and optimal performances by developing new techniques for the considered Markovian jump singular systems. Then a set of new SMC methodologies are proposed, based on the analysis results. The main contents are as follows: Chapter 2 is concerned with the state estimation and SMC of singular Markovian switching systems; Chapter 3 studies the optimal SMC problem for singular Markovian switching systems with time delay; and Chapter 4 establishes the integral SMC method for singular Markovian switching stochastic systems.

In the second part, the problem of SMC of switched state-delayed hybrid systems is investigated. A unified approach of the piecewise Lyapunov function combining with the average dwell time technique is developed for analysis and synthesis of the considered systems. By this approach, some sufficient conditions are established for the stability and synthesis of the switched state-delayed hybrid system. More importantly, a set of SMC methodologies under a unique framework are proposed for the considered hybrid systems. The main contents of this part are as follows: Chapter 5 is devoted to the stability analysis and the stabilization problems for switched state-delayed hybrid systems; Chapter 6 investigates the optimal dynamic output feedback (DOF) control of switched state-delayed hybrid systems; and Chapters 7 and 8 study the SMC of continuous- and discrete-time switched state-delayed hybrid systems, respectively.

In the third part, the parallel theories and techniques developed in the second part are extended to deal with switched stochastic hybrid systems. The main contents include the following: Chapters 9 and 10 are concerned with the control of switched stochastic hybrid systems for continuous- and discrete-time cases, respectively; Chapter 11 studies the observer-based SMC of switched stochastic hybrid systems; and Chapter 12 focuses on the dissipativity-based SMC of switched stochastic hybrid systems.

This book is a research monograph whose intended audience is graduate and postgraduate students, academics, scientists and engineers who are working in the field.

Ligang Wu
Harbin, China

Peng Shi
Melbourne, Australia

Xiaojie Su
Chongqing, China
December 2013


There are numerous individuals without whose help this book would not have been completed. Special thanks go to Professor James Lam from The University of Hong Kong, Professor Daniel W. C. Ho from City University of Hong Kong, Professor Zidong Wang from Brunel University, Professor Wei Xing Zheng from University of Western Sydney, Professor Yugang Niu from East China University of Science and Technology and Professor Huijun Gao from Harbin Institute of Technology, for their valuable suggestions, constructive comments and support.

Next, our acknowledgements go to many colleagues who have offered support and encouragement throughout this research effort. In particular, we would like to acknowledge the contributions from Jianbin Qiu, Ming Liu, Guanghui Sun, and Hongli Dong. Thanks also go to our students, Rongni Yang, Xiuming Yao, Fanbiao Li, Xiaozhan Yang, Chunsong Han, Yongyang Xiong, and Huiyan Zhang, for their comments. The authors are especially grateful to their families for their encouragement and never-ending support when it was most required. Finally, we would like to thank the editors at Wiley for their professional and efficient handling of this project.

The writing of this book was supported in part by the National Natural Science Foundation of China (61174126, 61222301, 61134001, 61333012, 61174058), the Fok Ying Tung Education Foundation (141059), the Fundamental Research Funds for the Central Universities (HIT.BRETIV.201303), the Australian Research Council (DP140102180), the Engineering and Physical Sciences Research Council, UK (EP/F029195), the Fundamental Research Funds for the Central Universities (2013YJS021), the National Key Basic Research Program, China (2011CB710706, 2012CB215202), the 111 Project (B12018), and the Key Laboratory of Integrated Automation for the Process Industry, Northeast University.

Abbreviations and Notations



cone complementary linearization


common quadratic Lyapunov function


dynamic output feedback


linear matrix inequality


linear-quadratic regulator


linear time-invariant


multiple-input multiple-output


Markovian jump linear system


multiple Lyapunov function


single-input single-output


sliding mode control


static output feedback


switched quadratic Lyapunov functions


end of proof

end of remark

is defined as

belongs to

for all


field of complex numbers

field of real numbers

field of integral numbers

space of -dimensional real vectors

space of real matrices

set of -valued continuous functions on

mathematical expectation operator












rank of a matrix


trace of a matrix

minimum eigenvalue of a real symmetric matrix

maximum eigenvalue of a real symmetric matrix


block diagonal matrix with blocks

minimum singular value of a real symmetric matrix

maximum singular value of a real symmetric matrix

identity matrix with appropriate dimension

identity matrix


zero matrix with appropriate dimension

zero matrix of dimension

transpose of matrix

inverse of matrix

full row rank matrix satisfying



is real symmetric positive (negative) definite

is real symmetric positive (negative) semi-definite

space of square integrable functions


on (continuous case)

space of square summable infinite vector sequences


over (discrete case)

Euclidean vector norm

Euclidean matrix norm (spectral norm)

-norm: (continuous case)


-norm: (discrete case)

symmetric terms in a symmetric matrix