Perturbation-based Regret Analysis of Predictive Control in Linear Time Varying Systems

We study predictive control in a setting where the dynamics are time-varying and linear, and the costs are time-varying and well-conditioned. At each time step, the controller receives the exact predictions of costs, dynamics, and disturbances for the future k time steps. We show that when the prediction window k is sufficiently large, predictive control is input-to-state stable and achieves a dynamic regret of O(λ^kT), where λ < 1 is a positive constant. This is the first dynamic regret bound on the predictive control of linear time-varying systems. We also show a variation of predictive control obtains the first competitive bound for the control of linear time-varying systems: 1 + O(λ^k). Our results are derived using a novel proof framework based on a perturbation bound that characterizes how a small change to the system parameters impacts the optimal trajectory.