H∞ control of discrete-time uncertain linear systems with quantized feedback

Abstract

This paper studies the problem of $H_{\infty }$ control for uncertain linear discrete-time systems with quantized state feedback. Consider that the uncertain parameters are supposed to reside in a polytope. The system state is quantized by a logarithmic static and time-invariant quantizer. Via giving a new control law and using parameter dependent Lyapunov function approach, new results on the quantized $H_{\infty }$ state feedback control are expressed in terms of linear matrix inequalities (LMIs). A numerical example is introduced to illustrate the effectiveness and applicability of the proposed methodology.

Introduction

In recent years, quantization in feedback control systems has attracted a growing interest [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. This is mainly due to the wide application of digital computers in control systems and the rapid development of network based control. Comparing with classical control theory, quantized feedback control is a common source of errors, which may degrade the system performance as described by Kalman [1], 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. Consequently, a lot of works have focused on understanding and mitigating the quantization effects in the early. While in recent studies, a general practice is to treat the quantizers as information coders. Among these results, there are mainly two approaches for studying control problem with quantized feedback. The first approach handles static quantizers such as uniform and logarithmic quantizers [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], while the second approach considers the dynamic quantizers which scales the quantization levels dynamically in order to improve the steady-state performance [13], [14].

For the problem of quantized feedback control, many important achievements have been obtained. Elia [2] had proven that a logarithmic quantizer is needed for stabilization of discrete-time single-input–single-output (SISO) linear time-invariant systems. Fu [3] gave a comprehensive study on feedback control systems with logarithmic quantizers by the sector bound approach. Both stabilization and $H_{\infty }$ performance issues have been considered. Following this work, Gao [7] noticed that the constant Lyapunov function is conservative for quantized feedback problem and proposed a new general framework based on quantization dependent Lyapunov functions. Recently, Zhou [8] revisited the absolute stability approach, and gave a less conservative result. For some new results about quantized feedback control, see [15], [16], [17]. Considerable attention, however, have been paid toward the study of $H_{\infty }$ control for linear systems [[18], [19], [20], [21], [22], [23], [24]], nonlinear systems [[25], [26], [27], [28]]. For polytopic uncertain systems, refs. [19], [20], [21], [22], [23] focus on the improved problem of the bounded real lemma (BRL) for the polytopic uncertainty systems, that is, how to find a less conservative LMI-based method of designing $H_{\infty }$ controller. It was noted that the basic idea behind these papers is based on constant feedback matrix K. However, the constant feedback matrix is independent of polytopic uncertainty parameters, the results obtained with constant feedback matrix are conservative when extended to polytopic uncertainty system with quantized feedback. Our main objective is to propose a new parameter dependent control law and to obtain less conservative results for polytopic uncertainty quantized feedback systems.

In this paper, we present a new way to deal with the quantized feedback problem for polytopic uncertainty systems, that is, change constant feedback matrix K into parameter dependent $K\left(\eta \right)$ by Lagrange׳s interpolation. Obviously the result obtained by parameter dependent feedback matrix is less conservative than the ones by constant feedback matrix for polytopic uncertainty systems with quantized feedback. Finally, we will illustrate the effectiveness and reduced conservatism of our main results by a numerical example.

Notations: The symbol ⁎ induces a symmetric structure in LMIs. Generally, for a square matrix $A$, $A^T$ denote its transpose and $\mathit{He}\left\{A\right\}$ denotes $(A+A^T)$. Matrices are assumed to have compatible dimensions.