Feedback Optimization of Nonlinear Strict-Feedback Systems

Feedback optimization aims at regulating the output of a dynamical system to a value that minimizes a cost function. This problem is beyond the reach of the traditional output regulation theory, because the desired value is generally unknown and the reference signal evolves according to a gradient flow using the system's real-time output. This paper complements the output regulation theory with the nonlinear small-gain theory to address this challenge. Specifically, the authors assume that the cost function is strongly convex and the nonlinear dynamical system is in lower triangular form and is subject to parametric uncertainties and a class of external disturbances. An internal model is used to compensate for the effects of the disturbances while the cyclic small-gain theorem is invoked to address the coupling between the reference signal, the compensators, and the physical system. The proposed solution can guarantee the boundedness of the closed-loop signals and regulate the output of the system towards the desired minimizer in a global sense. Two numerical examples illustrate the effectiveness of the proposed method.