On reachability sets for optimal feedback controllers: Monitoring the approach of a region of attraction

Abstract

Sums-of-Squares optimization represents an important tool for the direct computation of a local Lyapunov function for a nonlinear dynamic system. Specifically, it can certify a sub-levelset of the cost-to-go from an optimal feedback controller like the Linear Quadratic Regulator (LQR), geometrically an ellipsoid in the state space, as Region of Attraction (ROA) of the closed-loop system. More complex robotic tasks however require switching control to first take the system into the ROA before invoking the LQR stabilizer. In this paper, we propose computationally efficient measures of the ROA distance based on quadratic and conic optimization to effectively supervise such a trajectory as it approaches the ROA. As a one-dimensional condensate of the multi-dimensional state trajectory, monitoring the ROA distance evolution allows us to early detect deviations, e.g. due to input saturation or time delay, in order to quickly take corrective action such as replanning. Importantly, computing the ROA distance adds only a small overhead on top of the ROA calculation itself and can be done concurrently.