Elsevier

Automatica

Volume 49, Issue 2, February 2013, Pages 457-462
Automatica

Brief paper
Stabilization bound of singularly perturbed systems subject to actuator saturation

https://doi.org/10.1016/j.automatica.2012.11.004Get rights and content

Abstract

This paper considers the stabilization bound problem for singularly perturbed systems (SPSs) subject to actuator saturation. A state feedback stabilization controller design method is proposed and a basin of attraction depending on the singular perturbation parameter is constructed, which facilitates the formulation of the convex optimization problem for maximizing the basin of attraction of SPSs. Finally, examples are given to show the advantages and effectiveness of the obtained results.

Introduction

Singularly perturbed systems (SPSs), with a small singular perturbation parameter ε determining the degree of separation between the “slow” and “fast” modes of the systems, have been one of the major research subjects of control science due to their widespread applications. The stability bound problem for SPSs, which is referred to as the problem of determining the stability bound ε0 such that the system is stable for all ε(0,ε0) or (0,ε0], is a fundamental problem and has attracted much attention (Abed, 1985, Cao and Schwartz, 2004, Feng, 1988, Saydy, 1996, Sen and Datta, 1993). Frequency- and time-domain methods were proposed in Cao and Schwartz (2004) and Feng (1988) to provide the largest stability bound for SPSs. The stabilization bound problem aiming at designing controllers to enlarge the stability bound of SPSs has also been considered (Chiou et al., 1999, Li and Li, 1992, Liu et al., 1997).

Actuator saturation is a common phenomenon in practical systems and thus intensive research efforts have been devoted to control systems subject to actuator saturation. The problem of global/semi-global stabilization is one of the most interesting topics and has been discussed in great depth (Cao et al., 2002, Hu et al., 2006, Lin and Saberi, 1993). Since global stabilization cannot be achieved for open-loop unstable systems in the presence of actuator saturation, local results have to be developed. In this context, a key issue is to estimate the domain of stability for the closed-loop system (estimation of the basin of attraction). Most of the results on this topic are based on characterizing the basin of attraction by Lyapunov functions, by which the design parameters can be incorporated into optimization problems to maximize the basin of attraction for the closed-loop systems (Cao et al., 2002, Hu et al., 2006). However, the associated Lyapunov function for SPSs is usually ε-dependent, which leads to difficulties in generalizing the approaches for normal systems to SPSs.

Recently, the problems of analysis and synthesis for SPSs subject to actuator saturation have attracted more attention. Applying the routine methods for normal systems to SPSs usually leads to ill-conditioned numerical problems (Kokotovic, Khalil, & O’Reilly, 1986). The conventional approaches to avoiding the ill-conditioned problem are based on decomposing the original SPSs into fast and slow subsystems. Liu (2001) proposed a controller design method for SPSs subject to actuator saturation under the assumption that the fast dynamics is stable. Garcia and Tarbouriech (2003) designed a composite stabilizing controller and estimated the basin of attraction of SPSs by solving a convex optimization problem. In Xin et al., 2010, Xin et al., 2008, some methods to estimate the basin of attraction of SPSs were proposed by introducing the so-called reduced-order adjoint systems. These methods are all based on the decomposition of the original systems, which leads to difficulties for analyzing stability bound of the systems. An alternative approach that is independent of system decomposition was proposed in Lizarraga, Tarbouriech, and Garcia (2005) to avoid the possible ill-conditioned numerical problems. However, the proposed results did not consider the stability bound either. All in all, the problems of stabilization bound and optimization of the basin of attraction of SPSs subject to actuator saturation are still open.

This paper will consider the stabilization bound problem for SPSs subject to actuator saturation. The objective is to propose a state feedback controller design method to achieve a given stabilization bound of the closed-loop system. First, by a Lyapunov function that gives full consideration of the singular perturbation structure, a state feedback controller is designed such that the closed-loop system is asymptotically stable and a basin of attraction is constructed. Then, an optimization problem is formulated to enlarge the basin of attraction of the closed-loop system. Finally, two examples are given to show the effectiveness of the obtained results. The main contributions of the paper are as follows: (1) the proposed method implicitly employs the singular perturbation structure of the SPSs rather than depends on decomposing the original systems into reduced-order subsystems, which provides convenience for stability bound analysis and synthesis of SPSs subject to actuator saturation; (2) a given stabilization bound is one of the design objectives; (3) a novel basin of attraction is constructed, which facilitates the formulation of a well-conditioned convex optimization problem for maximizing the basin of attraction of SPSs.

Notation

The superscript T stands for matrix transposition and the notation MT denotes the transpose of the inverse matrix of M. denotes the block induced by symmetry. For a matrix M, M(i) denotes the ith row of M.

Section snippets

Problem formulation

Consider the following system E(ε)ẋ(t)=Ax(t)+Bsat(u(t)), where x=[x1x2]Rn is the state, x1Rn1,x2Rn2, uRm is the control input, E(ε)=[In100εIn2]Rn×n,ARn×n,and BRn×m are constant matrices. sat() is a componentwise saturation map RmRm defined as: sat(ui(t))=sign(ui(t))min{1,|ui(t)|},i=1,2,,m.

The following state feedback controller is considered to stabilize system (1), u(t)=K(ε)x(t).

Then, we have the following closed-loop system E(ε)ẋ(t)=Ax(t)+Bsat(K(ε)x(t)).

The problem under

Main results

In this section, a state feedback controller is designed and then a convex optimization problem is formulated to enlarge the basin of attraction of the closed-loop system.

Examples

This section will illustrate various features of the proposed methods and show their advantages over the existing results.

Example 1

To show the advantages of the proposed methods over the existing results, consider system (1) with E(ε)=[100ε],A=[5111],B=[01].

Using the method in Lizarraga et al. (2005), we have the following controller u=[16.34610.3371]x.

Based on the results of Lizarraga et al. (2005), the system is stabilized by (34) if the perturbation parameter ε is small enough. The obtained basin

Conclusion

In this paper, we have considered the stabilization bound problem for singularly perturbed systems subject to actuator saturation. We first proposed a state feedback stabilization controller design method and constructed an ε-dependent basin of attraction, by which a convex optimization algorithm was formulated to maximize the basin of attraction of the closed-loop system. The results presented in this paper generalize the existing ones for normal systems. Finally, examples were given to

Chunyu Yang received his B.S. degree in Applied Mathematics and his Ph.D. degree in Control Theory and Control Engineering from Northeastern University, China in 2002 and 2009, respectively. He is currently an Associate Professor of China University of Mining and Technology, Xuzhou, China. His research interests include descriptor systems and robust control.

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    Chunyu Yang received his B.S. degree in Applied Mathematics and his Ph.D. degree in Control Theory and Control Engineering from Northeastern University, China in 2002 and 2009, respectively. He is currently an Associate Professor of China University of Mining and Technology, Xuzhou, China. His research interests include descriptor systems and robust control.

    Jing Sun received her Ph.D. degree from the University of Southern California in 1989, and her B.S. and M.S. degrees from University of Science and Technology of China in 1982 and 1984 respectively. From 1989 to 1993, she was an Assistant Professor in the Electrical and Computer Engineering Department, Wayne State University. She joined the Ford Research Laboratory in 1993 where she worked in the Powertrain Control Systems Department. She joined the faculty of the College of Engineering at the University of Michigan in 2003, where she is now a Professor in the Department of Naval Architecture and Marine Engineering and Department of Electrical Engineering and Computer Science. Her research interests include system and control theory and its applications to marine and automotive propulsion systems.

    Xiaoping Ma received his B.S., M.S. and Ph.D. degrees from the School of Information and Electrical Engineering of China University of Mining and Technology, Xuzhou, China in 1982, 1989, 2001, respectively. Now he is a Professor of China University of Mining and Technology, Xuzhou, China. His research interests include process control, networked control system and fault detection.

    This work was supported by the National Natural Science Foundation of China (60904009, 60904079, 61020106003, 61074029) and the National Basic Research Program of China (2009CB320601). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Fen Wu under the direction of Editor Roberto Tempo.

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