Generalized solutions of Riccati equalities and inequalities

TL;DR

This paper extends the analysis of Riccati equalities and inequalities for infinite-dimensional systems, allowing for weaker conditions and not requiring passivity, thereby broadening the scope of solutions in control theory.

Contribution

It introduces generalized solutions to Riccati equations and inequalities for infinite-dimensional systems without assuming passivity, building on prior work and relaxing key conditions.

Findings

Established new existence conditions for Riccati solutions.

Connected solutions to earlier results on the KYP inequality.

Broadened applicability to non-passive systems.

Abstract

The Riccati inequality and equality are studied for infinite dimensional linear discrete time stationary systems with respect to the scattering supply rate. The results obtained are an addition to and based on our earlier work on the Kalman-Yakubovich-Popov inequality in [6]. The main theorems are closely related to the results of Yu. M. Arlinski\u{\i} in [3]. The main difference is that we do not assume the original system to be a passive scattering system, and we allow the solutions of the Riccati inequality and equality to satisfy weaker conditions.