Online Learning Robust Control of Nonlinear Dynamical Systems

In this work we address the problem of the online robust control of nonlinear dynamical systems perturbed by disturbance. We study the problem of attenuation of the total cost over a duration $T$ in response to the disturbances. We consider the setting where the cost function (at a particular time) is a general continuous function and adversarial, the disturbance is adversarial and bounded at any point of time. Our goal is to design a controller that can learn and adapt to achieve a certain level of attenuation. We analyse two cases (i) when the system is known and (ii) when the system is unknown. We measure the performance of the controller by the deviation of the controller's cost for a sequence of cost functions with respect to an attenuation $\gamma$, $R^p_t$. We propose an online controller and present guarantees for the metric $R^p_t$ when the maximum possible attenuation is given by $\overline{\gamma}$, which is a system constant. We show that when the controller has preview of the cost functions and the disturbances for a short duration of time and the system is known $R^p_T(\gamma) = O(1)$ when $\gamma \geq \gamma_c$, where $\gamma_c = \mathcal{O}(\overline{\gamma})$. We then show that when the system is unknown the proposed controller with a preview of the cost functions and the disturbances for a short horizon achieves $R^p_T(\gamma) = \mathcal{O}(N) + \mathcal{O}(1) + \mathcal{O}((T-N)g(N))$, when $\gamma \geq \gamma_c$, where $g(N)$ is the accuracy of a given nonlinear estimator and $N$ is the duration of the initial estimation period. We also characterize the lower bound on the required prediction horizon for these guarantees to hold in terms of the system constants.