Global Asymptotic Stability Analysis for Autonomous Optimization

This article studies the feedback optimization problem for nonlinear systems. In particular, we assume that the controlled plant admits a global asymptotic stable equilibrium corresponding to each fixed reference input and consider a modified gradient-flow optimizer augmented with a nonlinear perturbation function. With the output map of the controlled plant satisfying a linear growth condition, this article proves the existence of a perturbation function such that the resulting feedback optimization system is globally asymptotically stable at the desired equilibrium. The proof is based on the seamless integration of tools from the singular perturbation theory, input-to-state stability, and the nonlinear small-gain theorem. The possibility of extending the proposed approach to other optimizers is also discussed. Three numerical examples are employed to verify the effectiveness of the proposed method.