Robust Online Control with Model Misspecification

TL;DR

This paper develops an efficient control algorithm for unknown nonlinear systems approximated linearly, achieving robustness that surpasses traditional small gain bounds and is nearly optimal in dimension dependence.

Contribution

It introduces a computationally efficient controller that guarantees robustness polynomial in system dimension, overcoming exponential time limitations of prior methods.

Findings

Achieves robustness polynomial in system dimension

Provides an efficient control algorithm for nonlinear systems

Attains near-optimal dimension dependence in $ ext{ell}_2$-gain

Abstract

We study online control of an unknown nonlinear dynamical system that is approximated by a time-invariant linear system with model misspecification. Our study focuses on robustness, a measure of how much deviation from the assumed linear approximation can be tolerated by a controller while maintaining finite $\ell_2$-gain. A basic methodology to analyze robustness is via the small gain theorem. However, as an implication of recent lower bounds on adaptive control, this method can only yield robustness that is exponentially small in the dimension of the system and its parametric uncertainty. The work of Cusumano and Poolla shows that much better robustness can be obtained, but the control algorithm is inefficient, taking exponential time in the worst case. In this paper we investigate whether there exists an efficient algorithm with provable robustness beyond the small gain theorem. We demonstrate that for a fully actuated system, this is indeed attainable. We give an efficient controller that can tolerate robustness that is polynomial in the dimension and independent of the parametric uncertainty; furthermore, the controller obtains an $\ell_2$-gain whose dimension dependence is near optimal.