Zames–Falb Multipliers for Convergence Rate: Motivating example and convex searches

The bound of convergence rates for discrete-time Lur’e systems has recently attracted much attention. The contributions of this technical note are twofold. Firstly, we show an example where the asymptotic convergence rate is slower than the linear case. This example can be seen as a counterexample of the Kalman conjecture extended to the analysis of convergence rates and motivates a rigorous analysis. Secondly, we develop a result to bound the convergence rate as a convex search over the suitable subclass of noncausal FIR Zames-Falb multipliers. The convex search is demonstrated with several numerical examples.