On the Exponential Convergence of Input–Output Signals of Nonlinear Feedback Systems

This note studies the exponential convergence of input–output signals of discrete-time nonlinear systems composed of a feedback interconnection of a linear time-invariant system and a nonlinear uncertainty. Both the open-loop subsystems are allowed to be unbounded. Integral–quadratic-constraint-based conditions are proposed for these uncertain feedback systems, including the Lurye type, to exhibit the property that the endogenous input–output signals enjoy an exponential convergence rate for all initial conditions of the linear time-invariant subsystem. The conditions are established via a combination of tools, including integral–quadratic constraints, directed gap, and exponential weightings.