Elsevier

Automatica

Volume 44, Issue 5, May 2008, Pages 1385-1393
Automatica

Brief paper
H control for fast sampling discrete-time singularly perturbed systems

https://doi.org/10.1016/j.automatica.2007.10.010Get rights and content

Abstract

This paper is concerned with the H control problem via state feedback for fast sampling discrete-time singularly perturbed systems. A new H controller design method is given in terms of solutions to linear matrix inequalities (LMIs), which eliminates the regularity restrictions attached to the Riccati-based solution. A method for evaluating the upper bound of singular perturbation parameter ϵ with meeting a prescribed H performance bound requirement is also given. Furthermore, the results are extended to robust controller design for fast sampling discrete-time singularly perturbed systems with polytopic uncertainties. Numerical examples are given to illustrate the validity of the proposed methods.

Introduction

It is well known that the multiple time-scale systems otherwise known as singularly perturbed systems often raise serious numerical problems in the control engineering field. In order to avoid the difficulties linked with the stiffness of the equations involved in the design, they are usually modeled as singularly perturbed control systems with a small singular perturbation parameter ϵ, being exploited to determine the degree of separation between slow and fast parts of dynamical systems. In the past three decades, singularly perturbed systems have been intensively studied by many researchers, see Kafri and Abed (1996), Li and Li (1995), Li, Chiou, and Kung (1999), Li and Li (1992), Lim and Gajic (2000), Xu and Mizukami (1997), Naidu and Rao (1985) and Singh, Brown, and Naidu (1998) and a survey paper Naidu (2002). In the framework of singularly perturbed systems, a popular approach is the so-called reduction technique, which is a two-step design methodology. Firstly, through the separate stabilization of two lower-dimensional subsystems in two different time scales, a composite stabilizing controller is synthesized from the separate stabilizing controllers of the two subsystems, where the controller could be determined without the knowledge of the small singular perturbation parameter. Based on the reduction technique and Riccati equation approach, the H control problem for continuous-time singularly perturbed systems has been addressed in Fridman (1996), Lim and Gajic (2000), Pan and Basar, 1993, Pan and Basar, 1994, Shi and Dragan (1999) and Tan, Leung, and Tu (1998). The discrete-time case can be represented by three models: pure singularly perturbed discrete systems which are inherently discrete in nature, the slow sampling rate model, and the fast sampling rate model. The last two models are discretization of the singularly perturbed continuous-time systems. The fast sampling model is more practical in developing the stabilizing optimal control method (Kim, Kim, & Lim, 2002). Based on the two-step methodology and Riccati equation approach, H controller design methods are given, respectively, in Naidu, Charalambous, Moore, and Abdelrahma (1994) for slow sampling discrete-time singularly perturbed systems and in Datta and RaiChaudhuri (2002) for fast sampling ones. Moreover, a unified approach is provided in Singh, Brown, Naidu, and Heinen (2001) for H control for both continuous and discrete-time systems.

In general, most of these approaches for control synthesis are independent of ϵ for avoiding the ill-condition condition. Therefore, it is of great importance to find the ϵ-bound for the stability of the closed-loop systems. By considering critical stability criterion with a bialternate product, Ghosh, Sen, and Datta (1999) and Li et al. (1999) present systematic approaches to determine the exact stability bound of singularly perturbed discrete-time systems. Moreover, an algorithm to find the upper bound of singular perturbation parameter for D-stability is given in Hsiao, Pan, and Hwang (2000) and Hsiao, Hwang, and Pan (2003). However, by the authors’ knowledge, the approach for evaluating the upper bound of singular perturbation parameter with meeting H performance requirement has not been studied. In this paper, such approach will be presented.

In the past decade, linear matrix inequality (LMI) technique has been extensively exploited to solve control problems (Boyd, Ghaoui, Feron, & Balakrishnan, 1994). In contrast to the Riccati approach, the LMIs that arise in system and control theory can be formulated as convex optimization problems that are amenable to compute solution and can be solved effectively (Boyd et al., 1994). Another good feature of the LMI is their ability of adding constraints to the parametrical optimization problem provided they are themselves linear with respect to unknowns (Garcia, Daafouz, & Bernussou, 2002). In particular, for H synthesis, it has the merit of eliminating the regularity restrictions attached to the Riccati-based solution (Gahinet, Nemirovski, Laub, & Chilali, 1995). Motivated by the merits of the LMI formulations, some significant advances have been achieved for developing LMI-based approaches to the control synthesis for singularly perturbed systems. In particular, based on the LMI technique, state feedback H controller design methods are given in Fridman (2006) for continuous-time singularly perturbed systems with norm-bounded uncertainties. Garcia and Tarbouriech (2003) addresses the control design problem for linear singularly perturbed systems subject to bounded control. With pole-placement constraints, Lin and Li (2006) presents a sufficient condition for designing robust H dynamic output feedback controller. Most of the above-mentioned results for control synthesis are for continuous-time singularly perturbed systems. However, for the discrete-time case, there has been no LMI-based formulation for control synthesis.

This paper will be concerned with the H controller design problem for fast sampling discrete-time singularly perturbed systems. By constructing ϵ-dependent Lyapunov function, an H controller design method is given in terms of solutions to a set of ϵ-independent LMIs, which can avoid the ill-conditioned numerical problem in LMIs, and eliminates the regularity restrictions attached to the Riccati-based solutions in Vu and Sawan (1993) and Datta and RaiChaudhuri (2002). Moreover, an approach is presented to estimate the upper bound of ϵ of a singularly perturbed control system subject to H performance bound constraint. The paper is organized as follows. Section 2 presents system description and problem statement. In Section 3, an H controller design method is given based on ϵ-independent LMIs, and a method of evaluating the upper bound of singular perturbation parameter ϵ with meeting a prescribed H performance bound requirement is also presented. Section 4 extends the results to systems with polytopic uncertainties. The effectiveness of the proposed methods is illustrated by two numerical examples in Section 5. Finally, conclusions are given in Section 6.

Notation:() is used for the blocks induced by symmetry. The superscript T stands for matrix transposition and the notation MT denotes the transpose of the inverse matrix of M. L2 is the Lebesgue space consisting of all discrete-time vector-valued functions that are square-summable over [0,). z2 denotes the L2-norm of a vector-valued function z.

Section snippets

System description and problem statement

Consider the following fast sampling discrete-time singularly perturbed system: [x1(k+1)x2(k+1)]=[In1+ϵA11ϵA12A21A22][x1(k)x2(k)]+[ϵB11B12]w(k)+[ϵB21B22]u(k)z(k)=[C11C12][x1(k)x2(k)]+D11w(k)+D12u(k) and x1(k)Rn1, x2(k)Rn2, are state vectors, w(k)Rm1 is disturbance input, u(k)Rm2 is the control input, z(k)Rp is the output to be controlled. The positive singular perturbation parameter is denoted by ϵ.

Remark 1

The model (1) was investigated in Datta and RaiChaudhuri (2002), Li and Li (1995), Li et al.

H control

In this section, an LMI-based H controller design method is presented. Moreover, a sufficient condition is derived for evaluating the upper bound ϵ of ϵ subject to a prescribed H performance bound constraint.

Robust H control synthesis

In this section, the results obtained in Section 3 are extended to robust H control synthesis.

Consider the following uncertain fast sampling discrete-time singularly perturbed system [x1(k+1)x2(k+1)]=[In1+ϵA11ϵA12A21A22][x1(k)x2(k)]+[ϵ1112]w(k)+[ϵ2122]u(k)z(k)=[C11C12][x1(k)x2(k)]+D11w(k)+D12u(k) and x1(k)Rn1, x2(k)Rn2 are state vectors, w(k)Rm1 is disturbance input, u(k)Rm2 is the control input, z(k)Rp is the controlled output. The positive singular perturbation parameter is denoted

Examples

In this section, two numerical examples are given to illustrate the effectiveness of the proposed methods, where Example 1 presents a comparison between Theorem 4 and the method in Vu and Sawan (1993), and the robust H controller design using Theorem 9 is illustrated by Example 2.

Example 12

Consider the following nuclear reactor mood (Abdelrahman, Naidu, Charalambous, & Moore, 1998), ẋ1=λx1+λx2ẋ2=βΛx1βΛx2+ρΛ where x1 and x2 are the normalized precursors’ concentration and neutron density,

Conclusion

In this paper, the problem of H control synthesis for fast sampling discrete-time singularly perturbed systems has been studied. The main contribution is as follows. A new design method is given in terms of solutions to a set of LMIs, which eliminates the regularity restrictions attached to the Riccati-based solutions. Moreover, a technique for evaluating the upper bound of singular perturbation parameter with meeting a prescribed H performance bound requirement is given. Furthermore, the

Acknowledgments

This work was supported in part by Program for New Century Excellent Talents in University (NCET-04-0283), the Funds for Creative Research Groups of China (No. 60521003), Program for Changjiang Scholars and Innovative Research Team in University (No. IRT0421), the State Key Program of National Natural Science of China (Grant No. 60534010), the Funds of National Science of China (Grant No. 60674021) and the Funds of Ph.D. program of MOE, China (Grant No. 20060145019).

Jiuxiang Dong received the B.S. degree in mathematics and applied mathematics and the M.S. degree in applied mathematics from Liaoning Normal University, China, in 2001 and 2004, respectively. He is currently pursuing the Ph.D. degree at Northeastern University, China. His research interests include fuzzy control, robust control and fault-tolerant control.

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    Guang-Hong Yang received the B.S. and M.S. degrees in mathematics from Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, nonfragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), and an Associate Editor of the Conference Editorial Board of the IEEE Control Systems Society.

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