Regulation of Discrete-Time Linear Input Delayed Systems Without Delay Knowledge

Abstract

Regulation of linear systems without using any knowledge of the delay in the input is an open problem that is known to be challenging. This is the case both in the continuous-time setting and in the discrete-time setting. This chapter presents a solution to this problem for a discrete-time linear system with an arbitrarily large bounded input delay. When the system has all its open loop poles at z = 1 or inside the unit circle, an adaptive feedback law is proposed to regulate the state and the input of the system to zero as time tends to infinity. The main features of the feedback law are its accommodation to unknown delay and its memorylessness . No knowledge of the delay, not even the knowledge of its variation with time and its upper bound, is needed in the implementation of the feedback law. Moreover, only the current state is used for feedback. These two features of the feedback law contribute to the simplicity of its implementation. The simplicity of our adaptive control design in turn entails a delicate regulation analysis. A new paradigm of regulation analysis is developed that does not follow any Lyapunov type analysis for time delay systems. Numerical study demonstrates the analyzed regulation results and further provides an indication of the robustness of our control scheme to exponentially unstable open loop poles.