Continuous-Time Enclosures for Uncertain Implicit Differential Equations

The computation of enclosures for the reachable set of uncertain dynamic systems is a crucial component in a wide variety of applications, from global and robust dynamic optimization to safety verification and fault detection. Even though many systems in engineering are best modeled as implicit differential equations (IDEs) and differential algebraic equations (DAEs), methods for the construction of enclosures for these are not as well developed as they are for ordinary differential equations (ODEs). In this paper, we propose a continuous-time approach for the guaranteed over approximations of the reachable set for quasilinear IDEs. This approach builds on novel high-order inclusion techniques for the solution set of algebraic equations and state-of-the-art techniques for bounding the solution of nonlinear ODEs. We show how this approach can be used to bound the reachable set of uncertain semi-explicit DAEs by bounding the underlying IDEs. We demonstrate this approach on two case studies, a double pendulum where it proves superior with delayed break-down times compared to other methods, and anaerobic digestion of microalgae which has nine differential and two algebraic states.