Exploiting nonlinear invariants and path constraints to achieve tighter reachable set enclosures using differential inequalities

Abstract

This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative by exploiting state constraints that are known to hold for all trajectories of interest (e.g., path constraints that describe feasible trajectories in the context of dynamic optimization, or constraints that explicitly describe invariant sets containing all system trajectories). However, effective bounding algorithms of this type are currently only available for problems with linear constraints. Moreover, the theoretical results underlying these algorithms do not apply to constraints that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear constraints. This article contributes a new differential inequalities theorem that permits the use of a very general class of nonlinear state constraints. Moreover, a new algorithm is presented for efficiently exploiting nonlinear constraints to achieve tighter bounds. The proposed approach is shown to produce very sharp bounds for two challenging case studies.