Online convex optimization for constrained control of nonlinear systems

Abstract

This paper proposes a modular approach that combines the online convex optimization framework and reference governors to solve a constrained control problem featuring time-varying and a priori unknown cost functions. Compared to existing results, the proposed framework is uniquely applicable to nonlinear dynamical systems subject to state and input constraints. Furthermore, our method is general in the sense that we do not limit our analysis to a specific choice of online convex optimization algorithm or reference governor. We show that the dynamic regret of the proposed framework is bounded linearly in both the dynamic regret and the path length of the chosen online convex optimization algorithm, even though the online convex optimization algorithm does not account for the underlying dynamics. We prove that a linear bound with respect to the online convex optimization algorithm's dynamic regret is optimal, i.e., cannot be improved upon. Furthermore, for a standard class of online convex optimization algorithms, our proposed framework attains a bound on its dynamic regret that is linear only in the variation of the cost functions, which is known to be an optimal bound. Finally, we demonstrate implementation and flexibility of the proposed framework by comparing different combinations of online convex optimization algorithms and reference governors to control a nonlinear chemical reactor in a numerical experiment.