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Construction of a Destabilizing Nonlinearity for Discrete-Time Uncertain Lurye Systems

This letter considers the instability of a Lurye system consisting of an uncertain, discrete-time, linear time-invariant plant in feedback with a slope-restricted nonlinearity. There is a large literature on analyzing the stability of such systems. This includes various conditions for proving stability of the Lurye system, including the Circle criterion and the use of O’Shea-Zames-Falb multipliers. In many cases, these conditions are sufficient but not necessary to prove stability. In contrast, there is also some work to construct specific nonlinearities that demonstrate the instability of the Lurye system (with the nominal plant dynamics). This letter considers a more general case where the plant has dynamic uncertainty. The goal is to construct both an instance of the uncertain model and a corresponding nonlinearity that combined make the Lurye system unstable. A limit cycle oscillation is also computed to verify the instability. A simple example is provided to demonstrate the results.