Standard versus Bounded Real Lemma with infinite-dimensional state space II: The storage function approach

TL;DR

This paper extends the Bounded Real Lemma to infinite-dimensional systems using the storage function approach, providing new proofs and insights into the KYP inequality in this complex setting.

Contribution

It adapts Willems' storage-function method to infinite-dimensional systems, offering alternative proofs and deeper understanding of the Bounded Real Lemma and KYP inequality.

Findings

Reproves key results using storage functions in infinite dimensions

Connects storage functions with solutions to the generalized KYP inequality

Provides a framework for analyzing infinite-dimensional control systems

Abstract

For discrete-time causal linear input/state/output systems, the Bounded Real Lemma explains (under suitable hypotheses) the contractivity of the values of the transfer function over the unit disk for such a system in terms of the existence of a positive-definite solution of a certain Linear Matrix Inequality (the Kalman-Yakubovich-Popov (KYP) inequality). Recent work has extended this result to the setting of infinite-dimensional state space and associated non-rationality of the transfer function, where at least in some cases unbounded solutions of the generalized KYP-inequality are required. This paper is the second installment in a series of papers on the Bounded Real Lemma and the KYP inequality. We adapt Willems' storage-function approach to the infinite-dimensional linear setting, and in this way reprove various results presented in the first installment, where they were obtained as applications of infinite-dimensional State-Space-Similarity theorems, rather than via explicit computation of storage functions.