Construction of Periodic Counterexamples to the Discrete-Time Kalman Conjecture

TL;DR

This paper develops a method to construct periodic counterexamples for the discrete-time Kalman conjecture by designing destabilizing nonlinearities in Lurye systems, challenging existing stability bounds.

Contribution

It introduces a novel procedure to explicitly construct destabilizing nonlinearities, providing less conservative bounds for stability analysis in discrete-time Lurye systems.

Findings

Constructed nonlinearities induce non-trivial periodic cycles.

Demonstrated examples show the effectiveness of the construction.

Provides tighter bounds for absolute stability analysis.

Abstract

This paper considers the Lurye system of a discrete-time, linear time-invariant plant in negative feedback with a nonlinearity. Both monotone and slope-restricted nonlinearities are considered. The main result is a procedure to construct destabilizing nonlinearities for the Lurye system. If the plant satisfies a certain phase condition then a monotone nonlinearity can be constructed so that the Lurye system has a non-trivial periodic cycle. Several examples are provided to demonstrate the construction. This represents a contribution for absolute stability analysis since the constructed nonlinearity provides a less conservative upper bound than existing bounds in the literature.