The Descriptor Discrete–time Riccati Equation: Numerical Solution and Applications∗

We investigate the numerical solution of a descriptor type discrete–time Riccati equation and give its main applications in several key problems in robust control formulated under very general hypotheses: rank compression (squaring down) of improper or polynomial systems with L∞–norm preservation, (J, J ′)–spectral and (J, J ′)–lossless factorizations of a completely general discrete–time system having any type of singularity, including arbitrary normal rank, poles and zeroes at infinity, at zero, or on the unit circle. Necessary and sufficient existence conditions together with computable formulas are given for the stabilizing and antistabilizing solutions of the descriptor Riccati equation in terms of an associated matrix pencil, without imposing any unnecessary restrictive assumptions on the matrix coefficients.