Uncertainty propagation for nonlinear dynamics: A polynomial optimization approach

TL;DR

This paper introduces a polynomial optimization method using Lyapunov-like functions and convex optimization to rigorously propagate uncertainty in nonlinear dynamical systems, providing convergent bounds on expected quantities even with limited initial data.

Contribution

The paper develops a novel approach combining polynomial optimization and semidefinite programming to compute rigorous bounds on uncertainty propagation in nonlinear systems without approximations.

Findings

Bounds are tight and converge to the true values in compact sets.

Method successfully applied to van der Pol oscillator and Lorenz system.

Provides a systematic way to handle uncertainty with limited initial statistics.

Abstract

We use Lyapunov-like functions and convex optimization to propagate uncertainty in the initial condition of nonlinear systems governed by ordinary differential equations. We consider the full nonlinear dynamics without approximation, producing rigorous bounds on the expected future value of a quantity of interest even when only limited statistics of the initial condition (e.g., mean and variance) are known. For dynamical systems evolving in compact sets, the best upper (lower) bound coincides with the largest (smallest) expectation among all initial state distributions consistent with the known statistics. For systems governed by polynomial equations and polynomial quantities of interest, one-sided estimates on the optimal bounds can be computed using tools from polynomial optimization and semidefinite programming. Moreover, these numerical bounds provably converge to the optimal ones in the compact case. We illustrate the approach on a van der Pol oscillator and on the Lorenz system in the chaotic regime.