Uncertainty Quantification in Hybrid Dynamical Systems

TL;DR

This paper introduces a novel, fast uncertainty quantification method for hybrid dynamical systems, extending polynomial chaos with wavelet expansions and transport theory to handle discontinuities and resets.

Contribution

It develops a new UQ approach combining wavelet-based polynomial chaos and transport theory for hybrid systems, addressing limitations of existing methods.

Findings

The wavelet-based Wiener-Haar expansion effectively captures discontinuities.

The transport theory approach enables uncertainty propagation through reset conditions.

Demonstrated effectiveness on example hybrid systems.

Abstract

Uncertainty quantification (UQ) techniques are frequently used to ascertain output variability in systems with parametric uncertainty. Traditional algorithms for UQ are either system-agnostic and slow (such as Monte Carlo) or fast with stringent assumptions on smoothness (such as polynomial chaos and Quasi-Monte Carlo). In this work, we develop a fast UQ approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems. To capture discontinuities, we use a wavelet-based Wiener-Haar expansion. We develop a boundary layer approach to propagate uncertainty through separable reset conditions. We also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems. Here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics. The solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems. The above methods are demonstrated on example problems.