Safe Control of Partially-Observed Linear Time-Varying Systems with Minimal Worst-Case Dynamic Regret

We present safe control of partially-observed linear time-varying systems in the presence of unknown and unpredictable process and measurement noise. We introduce a control algorithm that minimizes dynamic regret, i.e., that minimizes the suboptimality against an optimal clairvoyant controller that knows the unpredictable future a priori. Specifically, our algorithm minimizes the worst-case dynamic regret among all possible noise realizations given a worst-case total noise magnitude. To this end, the control algorithm accounts for three key challenges: safety constraints; partially-observed time-varying systems; and unpredictable process and measurement noise. We are motivated by the future of autonomy where robots will autonomously perform complex tasks despite unknown and unpredictable disturbances leveraging their on-board control and sensing capabilities. To synthesize our minimal-regret controller, we formulate a constrained semi-definite program based on a System Level Synthesis approach for partially-observed time-varying systems. We validate our algorithm in simulated scenarios, including trajectory tracking scenarios of a hovering quadrotor collecting GPS and IMU measurements. Our algorithm is observed to have better performance than either or both the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ controllers, demonstrating a Best of Both Worlds performance.