Suboptimality analysis of receding horizon quadratic control with unknown linear systems and its applications in learning-based control

This work analyzes how the trade-off between the modeling error, the terminal value function error, and the prediction horizon affects the performance of a nominal receding-horizon linear quadratic (LQ) controller. By developing a novel perturbation result of the Riccati difference equation, a novel performance upper bound is obtained and suggests that for many cases, the prediction horizon can be either 1 or $+\infty$ to improve the control performance, depending on the relative difference between the modeling error and the terminal value function error. The result also shows that when an infinite horizon is desired, a finite prediction horizon that is larger than the controllability index can be sufficient for achieving a near-optimal performance, revealing a close relation between the prediction horizon and controllability. The obtained suboptimality performance upper bound is applied to provide novel sample complexity and regret guarantees for nominal receding-horizon LQ controllers in a learning-based setting. We show that an adaptive prediction horizon that increases as a logarithmic function of time is beneficial for regret minimization.