Linear Quadratic Regulation for Discrete-time Systems with Multiple Delays in Single Input Channel

Abstract

This paper is concerned with the linear quadratic regulation (LQR) problem for linear discrete-time systems with multiple delays in a single input channel. Although the LQR problem for discrete-time systems with single delay in each of the multiple input channels has been studied in existing literature, the problem to be addressed in this paper is known to be very difficult and has not been well investigated. In this paper, we address the LQR problem for systems with multiple delays in a single input channel by first establishing a duality between the LQR problem and a smoothing problem for an associated stochastic backward system. An analytical solution to the LQR control is then derived by solving the smoothing problem and is given in terms of the solutions of Riccati difference equations of the same dimension as the plant (ignoring the delays). The in finite horizon LQR problem is also considered in this paper and the convergence and stability analysis of the LQR controller is provided.