Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator

TL;DR

This paper introduces a polynomial-time algorithm for robust adaptive control of the LQR problem, achieving sub-linear regret guarantees and exploring the relationship between regret and parameter estimation.

Contribution

It presents the first provably polynomial-time algorithm with high probability sub-linear regret guarantees for adaptive LQR control, and extends the approach to constrained demand forecasting.

Findings

Algorithm achieves sub-linear regret with high probability.

Lower bounds relate regret to exploration strategies.

Numerical experiments demonstrate effectiveness and flexibility.

Abstract

We consider adaptive control of the Linear Quadratic Regulator (LQR), where an unknown linear system is controlled subject to quadratic costs. Leveraging recent developments in the estimation of linear systems and in robust controller synthesis, we present the first provably polynomial time algorithm that provides high probability guarantees of sub-linear regret on this problem. We further study the interplay between regret minimization and parameter estimation by proving a lower bound on the expected regret in terms of the exploration schedule used by any algorithm. Finally, we conduct a numerical study comparing our robust adaptive algorithm to other methods from the adaptive LQR literature, and demonstrate the flexibility of our proposed method by extending it to a demand forecasting problem subject to state constraints.