A Lyapunov-Lurye Functional Parametrization of Discrete-Time Zames-Falb Multipliers

We consider the absolute stability of discrete-time Lurye systems with SISO/MIMO (non-repeated SISO) nonlinearities that are sector bounded and slope restricted. For this class of systems, we present a parametrization of Lyapunov-Lurye functional (LLF) that is the time-domain equivalence to finite impulse response (FIR) Zames-Falb multipliers. As searches over FIR Zames-Falb multipliers provide the best-known results in the literature, the parametrization here provides the best-known Lyapunov function for absolute stability. A motivation of this alternative is making it easy to analyze the system in the time domain, especially when the frequency domain expression of the system is not straightforward. In this letter, we show the equivalence between the proposed LLF and FIR multipliers theoretically and numerically.

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