Numerical conditioning of delta-domain Lyapunov and Riccati equations

Fundamental issues related to numerical conditioning of the discrete-time Lyapunov and Riccati equations, given in so-called delta-domain forms, are addressed. Having observed that forward shift operator techniques for solving these equations become ill-conditioned for a sufficiently small sampling period, the author shows that numerical robustness and reliability of computations can be significantly improved by utilising the delta-operator representations of the origin equations. Relative condition numbers of these equations are defined to evaluate their conditioning. Results from numerical experiments dealing with reachability and observability Gramians as well as suboptimal control in H_∞ are reported that confirm the claim that the delta-domain formulations are much better-conditioned than their counterpart versions based on the forward shift operator