On-the-Fly Control of Unknown Systems: From Side Information to Performance Guarantees Through Reachability

In this article, we develop data-driven algorithms for reachability analysis and control of systems with a priori unknown nonlinear dynamics. The resulting algorithms not only are suitable for settings with real-time requirements but also provide provable performance guarantees. To this end, they merge noisy data from only a single finite-horizon trajectory and, if available, various forms of side information. Such side information may include knowledge of the regularity of the dynamics, algebraic constraints on the states, monotonicity, or decoupling in the dynamics between the states. Specifically, we develop two algorithms,
DaTaReach
and
DaTaControl
, to overapproximate the reachable set and design control signals for the system on the fly.
DaTaReach
constructs a differential inclusion that contains the unknown dynamics. Then, in a discrete-time setting, it overapproximates the reachable set through interval Taylor-based methods applied to systems with dynamics described as differential inclusions. We provide a bound on the time step size that ensures the correctness and termination of
DaTaReach
.
DaTaControl
enables convex-optimization-based control using the computed overapproximation and the receding-horizon control framework. Besides,
DaTaControl
achieves near-optimal control and is suitable for real-time control of such systems. We establish a bound on its suboptimality and the number of primitive operations it requires to compute control values. Then, we theoretically show that
DaTaControl
achieves tighter suboptimality bounds with an increasing amount of data and richer side information. Finally, experiments on a unicycle, quadrotor, and aircraft systems demonstrate the efficacy of both algorithms over existing approaches.