Balancing Exploration for Online Receding Horizon Learning Control with Provable Regret Guarantees

We address the problem of simultaneously learning and control in an online receding horizon control setting. We consider the control of an unknown linear dynamical system with general cost functions and affine constraints on the control input. Our goal is to develop an online learning algorithm that minimizes the dynamic regret, which is defined as the difference between the cumulative cost incurred by the algorithm and that of the best policy with full knowledge of the system, cost functions and state and that satisfies the control input constraints. We propose a novel approach to explore in an online receding horizon setting. The key challenge is to ensure that the control generated by the receding horizon controller is persistently exciting. Our approach is to apply a perturbation to the control input generated by the receding horizon controller that balances the exploration-exploitation trade-off. By exploring the conditions under which sub-linear regret is guaranteed, We show that the proposed controller's performance is upper bounded by $\tilde{\mathcal{O}}(T^{3/4})$ for both regret and cumulative constraint violation when the controller has preview of the cost functions for the interval that doubles in size from one interval to the next.Comment: The proof of the method in this paper is incorrect. The correct method, which is different from this paper is given in "online learning for predictive control with provable regret guarantees": arXiv:2111.1504