Moment Propagation Through Carleman Linearization with Application to Probabilistic Safety Analysis

Abstract

We develop a method to approximate the moments of a discrete-time stochastic polynomial system. Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic polynomial system with finitely many states and transform it into an infinitedimensional system with linear deterministic dynamics, which describe the exact evolution of the moments of the original polynomial system. We then truncate this deterministic system to obtain a finite-dimensional linear system, and use it for moment approximation by iteratively propagating the moments along the finite-dimensional linear dynamics across time. We provide efficient online computation methods for this propagation scheme with several error bounds for the approximation. Our result also shows that precise values of certain moments can be obtained when the truncated system is sufficiently large. Furthermore, we investigate techniques to reduce the offline computation load using reduced Kronecker power. Based on the obtained approximate moments and their errors, we also provide probability bounds for the state to be outside of given hyperellipsoidal regions. Those bounds allow us to conduct probabilistic safety analysis online through convex optimization. We demonstrate our results on a logistic map with stochastic dynamics and a vehicle dynamics subject to stochastic disturbance.

Carleman Linearization of Nonlinear Systems and Its Finite-Section Approximations

“…Finite-Section Approximations. We review some notions related to the Carleman linearization and its finite section scheme [4,5,9,10,18,21,23,24]. For a given vector…”

“…The control system community has experienced several success stories through methods that are developed based on Carleman linearization ideas [2,3,10,12,19,23,24,16,17,25]. For instance, the author of [24] identified some connections between Carleman linearization and Lie series, and then utilized it to design optimal control laws for infinite-dimensional systems.…”

“…A common remedy to deal with the Carleman linearization is finite-section approximations, which is given by (2.13), where one truncates the infinite-dimensional linear system [9,18,23]. A fundamental question is whether the first block of the solution of the finite-section approximation converges to the solution of the original nonlinear dynamical system and what the convergence rate is.…”