Reachability of Nonlinear Systems with Unknown Dynamics

Abstract

Determining the reachable set for a given nonlinear control system is crucial for system control and planning. However, computing such a set is impossible if the system's dynamics are not fully known. This paper is motivated by a scenario where a system suffers an adverse event mid-operation, resulting in a substantial change to the system's dynamics, rendering them largely unknown. Our objective is to conservatively approximate the system's reachable set solely from its local dynamics at a single point and the bounds on the rate of change of its dynamics. We translate this knowledge about the system dynamics into an ordinary differential inclusion. We then derive a conservative approximation of the velocities available to the system at every system state. An inclusion using this approximation can be interpreted as a control system; the trajectories of the derived control system are guaranteed to be trajectories of the unknown system. To illustrate the practical implementation and consequences of our work, we apply our algorithm to a simplified model of an unmanned aerial vehicle.Notice of Previous Publication. This manuscript substantially improves the work of [1]. Theory has been generalized to include a class of non-invertible matrices and improved to provide a larger set of reachable states. All lemmas, corollaries, and Theorems 1, 2, and 4 are entirely novel. Theorem 3 has been slightly modified from existing theorems in [1] given our new results.

Online Guaranteed Reachable Set Approximation for Systems with Changed Dynamics and Control Authority

“…for > 1. In case any of the constituent systems possess a decoupled structure, simplifications of the form of ( 15) can be made in (18) or (19). 1) Numerical Example: We consider the following system:…”

“…While other methods have been proposed to compute reachable sets under system impairment, due to their computational complexity, these have either used reduced order models, or have been limited to offline applications [17]. In more extreme cases of system impairments, such as those were very little is known about the system's present capabilities, more conservative methods for computing reachable sets exist [18]. Here, we present a general algorithm that yields guaranteed inner and outer approximations, given limited knowledge of the failure modes as expressed by a bound on the trajectory deviation growth dynamics.…”

“…Here, we present a general algorithm that yields guaranteed inner and outer approximations, given limited knowledge of the failure modes as expressed by a bound on the trajectory deviation growth dynamics. We leverage the fact that the offnominal system dynamics are related to the nominal system's dynamics, allowing us to leverage reachable sets computed for the nominal system, unlike in [18]. Given a sufficiently tight deviation growth bound, our approach can be applied online to high dimensional systems with no additional computational cost for the growth in system dimension.…”