On System Operators with Variation Bounding Properties

TL;DR

This paper investigates the properties of linear discrete-time system operators that bound the number of sign changes in inputs and outputs, providing algebraic characterizations and computational insights relevant to control systems.

Contribution

It introduces a tractable algebraic characterization of variation bounding properties using $k$-sign consistency, independent of rank and dimension, and applies this to bounding sign changes in impulse responses.

Findings

Provides an algebraic characterization of variation bounding operators

Establishes computational tractability of the property

Applies results to control problems involving sign change bounds

Abstract

The property of linear discrete-time time-invariant system operators mapping inputs with at most $k-1$ sign changes to outputs with at most $k-1$ sign changes is investigated. We show that this property is tractable via the notion of $k$-sign consistency in case of the observability/controllability operator, which as such can also be used as a sufficient condition for the Hankel operator. Our results complement the mathematical literature by providing an algebraic characterization, independent of rank and dimension for variation bounding and diminishing matrices as well as by discussing their computational tractability. Based on these, we conduct our studies of variation bounding system operators beyond existing studies on order-preserving $k$-variation diminishment. Our findings are applied to the open problem of bounding the number of sign changes in a system's impulse response, which appears, e.g., when bounding the number of over- and undershoots in a step response or the number of bangs in bounded optimal control problems.