Competitive-Ratio and Regret-Optimal Control with General Weights

Motivated by the online policy design approaches in learning theory, new controller design paradigms such as competitive-ratio and regret-optimal control have been recently proposed as alternatives to the classical ${\mathcal{H}_2}$ and ${\mathcal{H}_\infty }$ controllers. These metrics respectively consider the performances against a clairvoyant controller, which has access to future disturbances, in terms of ratio and difference. Even though the prior works on regret-optimal control provide its exact solution, in the competitive-ratio setting the solution is only provided for the suboptimal problem. In this work, we give the first exact solution to the optimal competitive-ratio control problem and present an explicit construction of the optimal competitive-ratio controller. A key technique that underpins our explicit solution is a reduction of the competitive-ratio control problem to the Nehari extension problem (similar to the regret-optimal control setting). The resulting optimal competitive-ratio controller is given by an explicit state space and the state-feedback law that is inherited from the ${\mathcal{H}_2}$ controller. Inspired by this explicit solution, we generalize regret-optimal control to have weight functions on the state, input, and noise sequences and show that competitive-ratio control is an instance of this general framework. The utilization of weight functions allow penalization of particular sequences, but still enjoying the explicit and optimal solution for the regret-optimal control problem.