On System Operators with Variation Bounding Properties

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.

💡 Research Summary

The paper investigates linear discrete‑time time‑invariant (LTI) system operators that preserve or reduce the number of sign changes (variations) of an input signal. The authors focus on two related properties: k‑variation bounding (VBk), which guarantees that an input with at most k sign changes yields an output with at most k sign changes, and k‑variation diminishing (VDk), which requires the VB property for all j ≤ k. When the order of sign changes is also preserved, the operator is called order‑preserving VDk (OVDk).

A central contribution is the use of k‑sign consistency (SSCk) and k‑sign regularity (SSRk)—concepts from total positivity theory—to obtain purely algebraic characterizations of VBk/VDk operators that are independent of matrix rank or dimension. The authors work with the multiplicative compound matrix X