Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator

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