On System Operators with Variation Bounding Properties

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.