Pseudo-predictor feedback stabilization of linear systems with time-varying input delays

This paper is concerned with stabilization of (time-varying) linear systems with a single time-varying input delay by using the predictor based delay compensation approach. Differently from the traditional predictor feedback which uses the open-loop system dynamics to predict the future state and will result in an infinite dimensional controller, we propose in this paper a pseudo-predictor feedback (PPF) approach which uses the (artificial) closed-loop system dynamics to predict the future state and the resulting controller is finite dimensional and is thus easy to implement. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system under the PPF are obtained in terms of the stability of a class of integral delay operators (systems). Moreover, it is shown that the PPF can compensate arbitrarily large yet bounded input delays provided the open-loop (time-varying linear) system is only polynomially unstable and the feedback gain is well designed. Comparison of the proposed PPF approach with the existing results are well explored. Numerical examples demonstrate the effectiveness of the proposed approaches.

1 Introduction

Since (large) time delay can be a source of performance degradation and even instability of control systems [7], control of time-delay systems has attracted much attention for several decades and various problems that were initially solved for delay-free systems have been investigated in the time-delay setting (see, e.g., [4], [7], [13], [15] and the references cited therein). Stability analysis and stabilization of time-delay systems are two fundamental problems that are important in the other analysis and synthesis problems for time-delay systems. One of the most efficient method for handling asymptotic stability analysis and stabilization of time-delay systems is the Lyapunov-Krasovskii functional based method (see, e.g., [3], [4] and [17]). The basic idea is to find a positive-definite functional such that its time-derivative along the trajectories of the time-delay system is negative-definite. The results obtained by this kind of methods for stability analysis can be easily recast into linear matrix inequalities and can also be easily adopted to the stabilizing controllers design. However, only sufficient conditions can be obtained by this approach.