Unimodal self-oscillations and their sign-symmetry for discrete-time relay feedback systems with dead zone

TL;DR

This paper analyzes the conditions under which unimodal self-oscillations occur in discrete-time relay feedback systems with dead zones, revealing a sign-symmetry property and providing existence, bounds, and uniqueness criteria.

Contribution

It introduces a novel analytical framework based on total positivity to characterize unimodal self-oscillations and establishes their sign-symmetry and existence conditions.

Findings

Unimodal self-oscillations exist only if the loop gain's positive and negative values are balanced.

Conditions for the existence of unimodal self-oscillations are derived.

Bounds on the periods of self-oscillations are established.

Abstract

This paper characterizes self-oscillations in discrete-time linear time-invariant (LTI) relay feedback systems with nonnegative dead zone. Specifically, we aim to establish existence criteria for unimodal self-oscillations, defined as periodic solutions where the output exhibits a single-peaked period. Assuming that the linear part of system is stable, with a strictly monotonically decreasing impulse response on its infinite support, we propose a novel analytical framework based on the theory of total positivity to address this problem. We demonstrate that unimodal self-oscillations subject to mild variation-based constraints exist only if the number of positive and negative values of the system's loop gain coincides within a given strictly positive period, i.e., the self-oscillation is sign-symmetric. Building upon these findings, we derive conditions for the existence of such self-oscillations, establish tight bounds on their periods, and address the question of their uniqueness.