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, which measures how much deviation from the assumed linear approximation can be tolerated while maintaining a bounded $ell_2$-gain compared to the optimal control in hindsight. Some models cannot be stabilized even with perfect knowledge of their coefficients: the robustness is limited by the minimal distance between the assumed dynamics and the set of unstabilizable dynamics. Therefore it is necessary to assume a lower bound on this distance. Under this assumption, and with full observation of the $d$ dimensional state, we describe an efficient controller that attains $Omega(frac{1}{sqrt{d}})$ robustness together with an $ell_2$-gain whose dimension dependence is near optimal. We also give an inefficient algorithm that attains constant robustness independent of the dimension, with a finite but sub-optimal $ell_2$-gain.
تحميل البحث