Steady-State Cascade Operators and their Role in Linear Control, Estimation, and Model Reduction Problems

TL;DR

This paper provides a comprehensive analysis of steady-state cascade operators in linear systems, introducing new design methods for control and estimation, and exploring extensions to nonlinear systems.

Contribution

It systematically categorizes cascade operators, revealing new recursive and low-gain design frameworks for control and observation of cascaded systems.

Findings

New recursive control design methods

Low-gain observation frameworks

Extensions to nonlinear systems

Abstract

Certain linear matrix operators arise naturally in systems analysis and design problems involving cascade interconnections of linear time-invariant systems, including problems of stabilization, estimation, and model order reduction. We conduct here a comprehensive study of these operators and their relevant system-theoretic properties. The general theory is leveraged to delineate both known and new design methodologies for control and observation of cascades, and to characterize structural properties of reduced models. Several entirely new designs arise from this systematic categorization, including new recursive and low-gain design frameworks for observation of cascaded systems. The benefits of the results beyond the linear time-invariant setting are demonstrated through preliminary extensions for nonlinear systems, with an outlook towards the development of a similarly comprehensive nonlinear theory.