The Koopman Expectation: An Operator Theoretic Method for Efficient Analysis and Optimization of Uncertain Hybrid Dynamical Systems

TL;DR

This paper introduces the Koopman Expectation, an operator theoretic method that efficiently computes expectations in uncertain hybrid dynamical systems, enabling faster analysis and optimization without explicit Koopman operator representation.

Contribution

The paper presents the Koopman Expectation, a novel approach that accelerates expectation calculations in complex systems without explicit operator representation, improving efficiency in uncertainty analysis.

Findings

Achieves 1700x speedup over Monte Carlo methods

Applicable to discrete, continuous, and hybrid systems with non-Gaussian uncertainties

Demonstrates significant accuracy improvements in probabilistic computations

Abstract

For dynamical systems involving decision making, the success of the system greatly depends on its ability to make good decisions with incomplete and uncertain information. By leveraging the Koopman operator and its adjoint property, we introduce the Koopman Expectation, an efficient method for computing expectations as propagated through a dynamical system. Unlike other Koopman operator-based approaches in the literature, this is possible without an explicit representation of the Koopman operator. Furthermore, the efficiencies enabled by the Koopman Expectation are leveraged for optimization under uncertainty when expected losses and constraints are considered. We show how the Koopman Expectation is applicable to discrete, continuous, and hybrid non-linear systems driven by process noise with non-Gaussian initial condition and parametric uncertainties. We finish by demonstrating a 1700x acceleration for calculating probabilistic quantities of a hybrid dynamical system over the naive Monte Carlo approach with many orders of magnitudes improvement in accuracy.