Linear Quadratic Regulator and Spectral Factorization for Discrete-Time Descriptor System

Abstract

This paper considers an infinite-horizon linear quadratic regulator problem and a related spectral factorization for constant discrete-time descriptor systems. We derive a descriptor Riccati equation (DRE) and the optimal feedback gain using a generalized Lyapunov equation. By applying a compact notation for descriptor transfer matrices, we also present a new spectral factorization algorithm that is valid even if the A-matrix is singular and/or the D-matrix is not of full column rank for the state-space model. Moreover, based on the solutions of (generalized) eigenvalue problems associated with the Hamiltonian equation, we develop a non-recursive method for obtaining the admissible solution of the DRE and the optimal feedback gain. A numerical result is also included.