Uncertainty Quantification in Hybrid Dynamical Systems

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.

💡 Research Summary

The paper addresses the challenge of quantifying uncertainty in hybrid dynamical systems—systems that combine continuous dynamics with discrete mode switches and state resets. Traditional uncertainty quantification (UQ) tools such as Monte‑Carlo (MC) and Quasi‑Monte‑Carlo (QMC) are either too slow or suffer from the curse of dimensionality, while polynomial chaos (PC) methods assume smooth dynamics and therefore break down when discontinuities are present.

To overcome these limitations, the authors extend the generalized polynomial chaos (gPC) framework to hybrid systems. They first rewrite a two‑mode hybrid system using indicator functions 1_R₁(x) and 1_R₂(x) that select the appropriate vector field depending on the state. Expanding the state x(t,λ) in an orthogonal polynomial basis H_α(λ) yields coefficient ODEs that involve integrals over time‑dependent regions R₁(t) and R₂(t). Because the integration domains change with the state, a naïve implementation would be computationally expensive.

The paper therefore introduces two complementary techniques. The first is a Wiener‑Haar wavelet expansion. By expressing the uncertain parameters through their cumulative distribution function u(λ) and using Haar wavelets ψ_{j,k}(u), the expansion coefficients become piecewise‑constant on dyadic intervals. This property allows the integrals in the coefficient equations to be evaluated analytically on each interval, dramatically reducing computational cost and preserving accuracy near switching boundaries where probability densities can be highly non‑smooth.

The second technique tackles state resets, which are common in hybrid models (e.g., a bouncing ball’s velocity reversal). The authors embed the reset map into a thin “boundary layer” of width ε around the guard surface. Inside the layer the dynamics are modified to smoothly interpolate between pre‑ and post‑reset states, while outside the layer the original dynamics are retained. As ε → 0 the modified system converges to the original hybrid system, yet the resulting ODEs are continuous and amenable to the gPC expansion.

Beyond the PC‑based approaches, the authors develop a transport‑theory formulation. They write a conservation law for the joint probability density ρ(x,λ,t) and project it onto the Haar basis. The resulting coefficients satisfy a hyperbolic system of partial differential equations. By integrating along characteristic curves, the method propagates uncertainty without resorting to high‑dimensional quadrature, making it especially suitable for systems with many uncertain parameters or complex switching logic (e.g., Zeno phenomena).

The methodology is demonstrated on two benchmark problems. The first is a switching oscillator where the forcing term changes sign depending on the sign of the state. Using the Haar‑based expansion, the authors accurately capture the mean and variance of both position and velocity, even when the switching induces sharp probability jumps. The second example is a bouncing ball with uncertain gravitational acceleration and a restitution coefficient γ<1. The boundary‑layer approach reproduces the mean trajectory and variance observed in large‑scale Monte‑Carlo simulations, while requiring orders of magnitude fewer samples.

Performance comparisons show that the hybrid PC methods achieve the same statistical accuracy as MC/QMC with 10–100× fewer evaluations. The wavelet expansion mitigates Gibbs‑type oscillations near discontinuities, and the boundary‑layer treatment handles resets without introducing artificial stiffness. The transport‑theory approach further scales well with the number of uncertain parameters, maintaining stability and accuracy.

In conclusion, the paper contributes two novel tools for UQ in hybrid dynamical systems: (1) a Wiener‑Haar wavelet‑based polynomial chaos expansion that handles discontinuous probability distributions, and (2) a transport‑theory based characteristic method that solves the resulting hyperbolic PDEs efficiently. Both techniques extend the applicability of fast UQ methods to systems with mode switching, state resets, and even Zeno behavior. The authors suggest future work on more complex guard conditions, adaptive boundary‑layer sizing, and integration with real‑time control and optimization frameworks.