  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.
