Elsevier

Automatica

Volume 44, Issue 2, February 2008, Pages 534-542
Automatica

Brief paper
A new approach to quantized feedback control systems

https://doi.org/10.1016/j.automatica.2007.06.015Get rights and content

Abstract

This paper presents a new approach to the problems of analysis and synthesis for quantized feedback control systems. Both single- and multiple-input cases are considered, with complete results provided for stability and H performance analysis as well as controller synthesis for discrete-time state-feedback control systems with logarithmic quantizers. The most significant feature is the utilization of a quantization dependent Lyapunov function, leading to less conservative results, which is shown both theoretically and through numerical examples.

Introduction

Recent years have witnessed a growing interest in investigating effects of signal quantization on feedback control systems. This is mainly due to the widespread use in control systems of digital computers that employ finite-precision arithmetic. The emerging network based control (Antsaklis and Baillieul, 2004, Goodwin et al., 2004, Lian et al., 2003, Seiler and Sengupta, 2005, Yang et al., 2006, Yue et al., 2005, Zhivoglyadov and Middleton, 2003) where information between the controller and the plant is exchanged through a network medium with limited capacities has further strengthened the importance of the study on quantized feedback control. In these cases, the measurement and command signals are usually quantized before being communicated, and the number of quantization levels is closely related to the information flow between the components of the control system and thus to the capacity required to transmit the information. The classical control theory, which is based on the standard assumption that data transmission required by the system can be performed with infinite precision, may not be valid in the presence of signal quantization or capacity-limited feedback, and therefore there is a need for developing tools for analysis and design of quantized feedback systems.

The study on quantization can be traced back to as early as 1956, when Kalman investigated the effect of quantization in a sampled data control system and pointed out that if a stabilizing controller was quantized using a finite-alphabet quantizer, the feedback system would exhibit limit cycles and chaotic behavior (Kalman, 1956). Earlier work on quantized feedback control paid much attention to understanding and mitigation of quantization effects. While in recent studies, quantizers were usually regarded as information coders. The most fundamental question is how much information is required to be communicated by the quantizer in order to achieve a certain objective for the closed-loop system; many important results have been reported, see, for instance, Bicchi, Marigo, and Piccoli (2002), Brockett and Liberzon (2000), Fagnani (2004), Fagnani and Zampieri (2003), Feng and Loparo (1997), Ishii and Basar (2005), Ishii and Francis, 2002, Ishii and Francis, 2003, Liberzon (2003), Liu and Elia (2004), Sznaier and Sideris (1994) and the references therein.

Among these results, there are mainly two approaches for studying control problems with quantized feedback. The first approach considers memoryless feedback quantizers, which are usually called static quantization policies. Static policies (Delchamps, 1990, Elia and Mitter, 2001, Fagnani and Zampieri, 2003, Fu and Xie, 2005, Wong and Brockett, 1999) presume that data quantization at time k is dependent on the data at time k only, and leading to relatively simple structures for the coding/decoding schemes. The first mathematical treatment of control systems with uniform quantized feedback was given in Delchamps (1990), following which bounds on the number of quantization intervals needed to stabilize a linear system were proposed in Wong and Brockett (1999). The second approach regards a quantized feedback controller with an internal state, which allows the quantizer to be dynamic and time-varying. This approach is considered advantageous as it scales the quantization levels dynamically in order to increase the region of attraction and to attenuate the steady state limit cycle. Results on feedback control with dynamic quantizers can be found in Brockett and Liberzon (2000), Ling and Lemmon (2005), Nair and Evans, 2000, Nair and Evans, 2003, Tatikonda and Mitter (2004). As indicated in Fu and Xie (2005), however, most of the results on quantized feedback with dynamic quantizers are confined to the stabilization problem; control performance issues are not addressed, since there seems to lack a general approach/framework for extending to more complex cases.

For the problem of feedback control with static quantizers, Elia and Mitter (2001) considered the problem of quadratic stabilization for discrete-time single-input–single-output (SISO) linear time-invariant systems; it is proven that for a quadratically stabilizable system, the quantizer needs to be logarithmic. Following this work, very recently, Fu and Xie (2005) revisited the sector bound approach to quantized feedback control, gave a comprehensive study on feedback control systems with logarithmic quantizers, and presented complete results for both SISO and multiple-input–multiple-output (MIMO) linear discrete-time systems. Moreover, not only stabilization but also control performance issues, including guaranteed cost control and H control, have been treated in a unified framework. It is worth noting that the basic idea behind these two papers is the notion of quadratic stability. By quadratic stability, we refer to the utilization of a fixed Lyapunov function (say, V(x)=xTPx with P being a constant positive definite matrix) for checking the stability of a set of systems over an uncertainty region. Though this constant Lyapunov function greatly facilitates the analysis and synthesis of quantized feedback control systems, the results obtained within this quadratic framework can be conservative due to the fact that the adopted Lyapunov function is independent of quantization errors. Our main objective is to propose a new general framework based on quantization dependent Lyapunov functions and to obtain less conservative results for quantized feedback systems.

In this paper, we present a new approach to the problem of analysis and synthesis for quantized feedback control systems. As in Fu and Xie (2005), we first consider the single-input case in Section 2. By introducing a quantization dependent Lyapunov function, new stability and stabilization results for the closed-loop quantized state-feedback control systems are obtained. It is shown both theoretically and through a numerical example that these results are generally less conservative than existing ones in the quadratic framework. This quantization dependence idea is then further extended to H performance analysis (Section 3) and multiple-input case (Section 4). The most significant feature of the approach presented here is the utilization of quantization dependent Lyapunov functions, leading to less conservative results than existing ones employing a fixed Lyapunov function in the quadratic framework.

Notation: The notation used throughout the paper is fairly standard. The superscript “T” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space (R stands for R1) and the notation P>0(0) means that P is real symmetric and positive definite (semi-definite). In symmetric block matrices, we use an asterisk (*) to represent a term that is induced by symmetry and diag{} stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. The space of square summable infinite sequence is denoted by l2[0,), and for ω={ω(k)}l2[0,), its norm is given by ω2=t=0|ω(k)|2.

Section snippets

Stability analysis

Consider the following discrete-time system:x(k+1)=Ax(k)+Bu(k).Here x(k)Rn is the state vector and u(k)R is the control input. A and B are system matrices. It is assumed that A is unstable and the pair (A,B) is stabilizable. We consider the quantized state-feedback in the following form:u(k)=f(v(k)),v(k)=Kx(k),where K is the state-feedback gain and f(·) is a quantizer which is assumed to be symmetric, that is, f(-v)=-f(v). In this paper, we are interested in the logarithmic static and

Extension to H performance analysis

In this section, we extend the quantization dependence idea to H performance analysis of quantized feedback control systems. Consider the following discrete-time system:x(k+1)=Ax(k)+Bu(k)+Ew(k),z(k)=Cx(k)+Du(k)+Fw(k).Here x(k),u(k) are similar to those in (1); z(k)Rq is the controlled output; w(k)Rl is the disturbance input which satisfies w{w(k)}l2[0,). (A,B,C,D,E,F) are system matrices with appropriate dimensions. We still consider the quantized state-feedback in the form of (2) and (3)

Extension to multiple-input case

All the above sections are concerned with the single-input case. In this section, we will generalize the quantization dependent approach presented above to the multiple-input case. We still consider the discrete-time system in (1), but with u(k)Rp instead of u(k)R. In this case, the quantized state-feedback becomesu(k)=f(v(k))=[f1(v1(k))f2(v2(k))fp(vp(k))]T,v(k)=Kx(k),where fi(·) is the quantizer for the ith component of the control input v(k). It is assumed that the quantizer fi(·) is

Conclusions

The problem of quantized state-feedback control has been investigated through a quantization dependent approach. New stability and stabilization conditions have been proposed for linear state-feedback discrete-time systems with logarithmic quantizers, which are shown both theoretically and through numerical examples to be less conservative than the existing results in the quadratic framework. These conditions are obtained by introducing new quantization dependent Lyapunov functions, which is

Acknowledgment

The authors thank the associate editor and anonymous reviewers for their valuable comments and suggestions that have helped them in improving the paper.

Huijun Gao was born in Heilongjiang Province, China, in 1976. He received the M.S. degree in Electrical Engineering from Shenyang University of Technology, Shengyang, China, in 2001, and the Ph.D. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, China, in 2005. He was a Research Associate in the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, from November 2003 to August 2004. From October 2005 to September 2007, he carried out his

References (30)

  • D.F. Delchamps

    Stabilizing a linear system with quantized state feedback

    IEEE Transactions on Automatic Control

    (1990)
  • L. El Ghaoui et al.

    A cone complementarity linearization algorithm for static output-feedback and related problems

    IEEE Transactions on Automatic Control

    (1997)
  • N. Elia et al.

    Stabilization of linear systems with limited information

    IEEE Transactions on Automatic Control

    (2001)
  • F. Fagnani

    Chaotic quantized feedback stabilizers: the scalar case

    Communications and Information Systems

    (2004)
  • F. Fagnani et al.

    Stability analysis and synthesis for scalar linear systems with a quantized feedback

    IEEE Transactions on Automatic Control

    (2003)
  • Cited by (365)

    View all citing articles on Scopus

    Huijun Gao was born in Heilongjiang Province, China, in 1976. He received the M.S. degree in Electrical Engineering from Shenyang University of Technology, Shengyang, China, in 2001, and the Ph.D. degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, China, in 2005. He was a Research Associate in the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, from November 2003 to August 2004. From October 2005 to September 2007, he carried out his postdoctoral research in the Department of Electrical and Computer Engineering, University of Alberta, Canada, supported by an Alberta Ingenuity Fellowship and an Honorary Izaak Walton Killam Memorial Postdoctoral Fellowship. He joined Harbin Institute of Technology, in November 2004, where he is currently a Professor.

    Dr. Gao's research interests include network based control, robust control/filter theory, model reduction, time-delay systems, multidimensional systems and their engineering applications. He is an Associate Editor of IEEE Transactions on Systems, Man and Cybernetics Part B: Cybernetics, Journal of Intelligent and Robotics Systems, International Journal of Innovative Computing, Information and Control, and serves on the Editorial Board of International Journal of Systems Science, Journal of the Franklin Institute and Nonlinear Dynamics and Systems Theory. He was the recipient of the University of Alberta Dorothy J. Killam Memorial Postdoctoral Fellow Prize in 2005, and was a co-recipient of the Outstanding Science and Technology Development Awards, from the Ministry of Machine-Building Industry of China, and from the Liaoning Provincial Government of China, both in 2002. He was an Outstanding Reviewer for Automatica in 2007, and an Appreciated Reviewer for IEEE Transactions on Signal Processing in 2006. He received the National Excellent Doctoral Dissertation Award in 2007 from the Ministry of Education of China.

    Tongwen Chen received the B.Eng. degree in Automation and Instrumentation from Tsinghua University (Beijing) in 1984, and the M.A.Sc. and Ph.D. degrees in Electrical Engineering from the University of Toronto in 1988 and 1991, respectively. From 1991 to 1997, he was an Assistant/Associate Professor in the Department of Electrical and Computer Engineering at the University of Calgary, Canada. Since 1997, he has been with the Department of Electrical and Computer Engineering at the University of Alberta, Edmonton, Canada, and is presently a Professor. He held visiting positions at the Hong Kong University of Science and Technology, Tsinghua University, Kumamoto University, and Harbin Institute of Technology Shenzhen Graduate School.

    His research interests include computer and network based control systems, process control, multirate digital signal processing, and their applications to industrial problems. He co-authored with B.A. Francis the book Optimal Sampled-Data Control Systems (Springer, 1995). Dr. Chen received a McCalla Professorship for 2000–2001 and a Killam Professorship for 2006–2007, both from the University of Alberta, and a Fellowship from the Japan Society for the Promotion of Science for 2004. He was elected an IEEE Fellow in 2005. He has served as an Associate Editor for several international journals, including IEEE Transactions on Automatic Control, Automatica, Systems and Control Letters, and Journal of Control Science and Engineering. He is a registered Professional Engineer in Alberta, Canada.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Minyue Fu under the direction of Editor Roberto Tempo. This work was partially supported by an Alberta Ingenuity Fellowship, Natural Sciences and Engineering Research Council of Canada, National Natural Science Foundation of China (60528007, 60504008), Program for New Century Excellent Talents in University, China, and an Honorary Izaak Walton Killam Memorial Postdoctoral Fellowship.

    View full text