A new Tsypkin criterion for discrete-time Lur´e systems with monotonic sector-restrictions

In this paper we revisit a well-known Tsypkin criterion for absolute stability analysis of discrete-time nonlinear Lur´e systems. When nonlinearities are monotonic and sector-restricted by [0,Δ̅] and G(z) represents a transfer function of the linear part of the Lur´e system, it is shown that the system is absolutely stable if a function G₀(z) = Δ̅^-1 + (1 + (1 - z^-1)M)G(z) is strictly positive real, where M is nonnegative diagonal and G(0) is invertible or identically zero. We extend this criterion by choosing a new Lyapunov function, to obtain a new criterion that the system is absolutely stable if a function G₀(z) = Δ̅^-1 + {1 + (1 - z^-1)M₁ + (1 - z)M₂}G(z) is strictly positive real, where M₁ and M₂ are are nonnegative diagonal and no condition on G(0) is assumed.