Linear quadratic regulation for linear time-varying systems with multiple input delays part I: discrete-time case

Although the problem of linear quadratic regulation has been well studied for discrete linear systems without delays or with only a single input delay in the past decades, the same problem for systems with multiple input delays remains to be solved. This paper aims to present a complete solution to the LQR problem for systems with multiple input delays. It shall be shown that the problem is dual to that of fixed-lag smoothing for a backward system without involving delays. Thus, the optimal LQR controller can be obtained by computing the gain matrices of the standard fixed-lag smoothing. An explicit solution for the LQR problem is given in terms of the solution of a Riccati difference equation.

1 Introduction

Systems with transportation delays exist in many engineering fields such as chemical processes and communication systems. In recent years, there have been a lot of interests in the study of continuous-time systems with input delays; see, e.g. [2], [6]. In the discrete-time context, the control problem for systems with input delays has also received some renewed interests due to the applications in network congestion control and networked control systems; see, e.g. [1], [7].