Standard versus Strict Bounded Real Lemma with infinite-dimensional state space I: The State-Space-Similarity Approach

TL;DR

This paper unifies various results on the bounded real lemma for infinite-dimensional systems using a novel state-space-similarity approach, providing a comprehensive theoretical framework.

Contribution

It introduces a new state-space-similarity theorem that unifies and extends existing results on the bounded real lemma for infinite-dimensional systems.

Findings

Unified theorems for bounded real lemma in infinite dimensions

Reconciliation of bounded/unbounded solution conditions

Framework applicable to various system stability and controllability settings

Abstract

The Bounded Real Lemma, i.e., the state-space linear matrix inequality characterization (referred to as Kalman-Yakubovich-Popov or KYP inequality) of when an input/state/output linear system satisfies a dissipation inequality, has recently been studied for infinite-dimensional discrete-time systems in a number of different settings: with or without stability assumptions, with or without controllability/observability assumptions, with or without strict inequalities. In these various settings, sometimes unbounded solutions of the KYP inequality are required while in other instances bounded solutions suffice. In a series of reports we show how these diverse results can be reconciled and unified. This first instalment focusses on the state-space-similarity approach to the bounded real lemma. We shall show how these results can be seen as corollaries of a new State-Space-Similarity theorem for infinite-dimensional linear systems.