A Revisited Tsypkin Criterion for Discrete-Time Nonlinear Lur’e Systems with Monotonic Sector-Restrictions

Abstract

This paper revisits a well-known Tsypkin criterion for stability analysis of discrete-time nonlinear Lur’e systems. When nonlinearities are monotonic and sector restricted by [0, Δ ̄ ] , where Δ ̄ is positive definite, it is shown by Kapila and Haddad that the system is absolutely stable if a function G 0(z)= Δ ̄ -1+{I+(1−z -1) K +}G(z) is strictly positive real, where K + is nonnegative diagonal and G( z) represents a transfer function of the linear part of the Lur’e system with invertible or identically zero G(0). This paper extends this criterion when Δ ̄ is positive diagonal, by choosing a new Lyapunov function to obtain an LMI criterion. From a frequency-domain interpretation of this LMI criterion, another sufficient criterion is generated which establishes that the system is absolutely stable if a function G 0(z)= Δ ̄ -1+{I+(1−z -1) K ++(1−z) K -}G(z) is strictly positive real, where K + and K - are nonnegative diagonal and orthogonal to each other.