On the exponential convergence of input-output signals of nonlinear feedback systems
We show that the integral-constraint-based robust feedback stability theorem for certain Lurye systems exhibits the property that the endogenous input-output signals enjoy an exponential convergence rate for all initial conditions of the linear time-invariant subsystem. More generally, we provide conditions under which a feedback interconnection of possibly open-loop unbounded subsystems to admit such an exponential convergence property, using perturbation analysis and a combination of tools including integral quadratic constraints, directed gap measure, and exponential weightings. As an application, we apply the result to first-order convex optimisation methods. In particular, by making use of the Zames-Falb multipliers, we state conditions for these methods to converge exponentially when applied to strongly convex functions with Lipschitz gradients.
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