Analysis and Synthesis of Low-Gain Integral Controllers for Nonlinear Systems with Application to Feedback-Based Optimization

Relaxed conditions are given for stability of a feedback system consisting of an exponentially stable multi-input multi-output nonlinear plant and an integral controller. Roughly speaking, it is shown that if the composition of the plant equilibrium input-output map and the integral feedback gain is infinitesimally contracting, then the closed-loop system is exponentially stable if the integral gain is sufficiently low. The main result is applied to analyze stability of an optimal frequency regulation scheme for AC power systems. We demonstrate how the key condition can be checked computationally via semidefinite programming, and how gain matrices can be synthesized for nonlinear systems to achieve guaranteed performance bounds. Finally, we apply the results to show closed-loop stability of recently developed feedback-based schemes which optimize the steady-state behaviour of a dynamic system.