Standard versus strict Bounded Real Lemma with infinite-dimensional state space III: The dichotomous and bicausal cases

TL;DR

This paper extends the Bounded Real Lemma to infinite-dimensional discrete-time systems with dichotomy and bicausality, providing new conditions and proofs for non-analytic transfer functions.

Contribution

It introduces variations of the Bounded Real Lemma for dichotomous and bicausal systems, generalizing previous results to broader classes of infinite-dimensional systems.

Findings

Proves Bounded Real Lemma variations for dichotomous systems

Establishes Bounded Real Lemma for bicausal systems

Recovers known results for nonstationary systems with dichotomy

Abstract

This is the third installment in a series of papers concerning the Bounded Real Lemma for infinite-dimensional discrete-time linear input/state/output systems. In this setting, under appropriate conditions, the lemma characterizes when the transfer function associated with the system has contractive values on the unit circle, expressed in terms of a Linear Matrix Inequality, often referred to as the Kalman-Yakubovich-Popov (KYP) inequality. Whereas the first two installments focussed on causal systems with the transfer functions extending to an analytic function on the disk, in the present paper the system is still causal but the state operator is allowed to have nontrivial dichotomy (the unit circle is not contained in its spectrum), implying that the transfer function is analytic in a neighborhood of zero and on a neighborhood of the unit circle rather than on the unit disk. More generally, we consider bicausal systems, for which the transfer function need not be analytic in a neighborhood of zero. For both types of systems, by a variation on Willems' storage-function approach, we prove variations on the standard and strict Bounded Real Lemma. We also specialize the results to nonstationary discrete-time systems with a dichotomy, thereby recovering a Bounded Real Lemma due to Ben-Artzi--Gohberg-Kaashoek for such systems.