Closed-Loop Finite-Time Analysis of Suboptimal Online Control

Suboptimal methods in optimal control arise due to a limited computational budget, unknown system dynamics, or a short prediction window among other reasons. In this work, we study the transient closed-loop performance of such methods by providing finite-time suboptimality gap guarantees. We consider the control of discrete-time, nonlinear time-varying dynamical systems and establish sufficient conditions for such guarantees. These allow the control design to distribute a limited computational budget over a time horizon and estimate the on-the-go loss in performance due to suboptimality. We study exponential incremental input-to-state stabilizing policies and show that for nonlinear systems, under some mild conditions, this property is directly implied by exponential stability without further assumptions on global smoothness. The analysis is showcased on a suboptimal model predictive control use case.