Approximate propagation of normal distributions for stochastic optimal control of nonsmooth systems

TL;DR

This paper introduces a method for approximating the propagation of mean and covariance in stochastic systems with discontinuous dynamics, enabling more efficient and accurate control under uncertainty without explicit switch detection.

Contribution

It presents an analytical approach for propagating distributions through nonsmooth ODEs, simplifying stochastic control of systems with discontinuities.

Findings

Enables direct integration of uncertain systems with discontinuities using standard schemes

Eliminates need for switch detection in stochastic control

Provides a structure-preserving linearization for piecewise smooth systems

Abstract

We present a method for the approximate propagation of mean and covariance of a probability distribution through ordinary differential equations (ODE) with discontinous right-hand side. For piecewise affine systems, a normalization of the propagated probability distribution at every time step allows us to analytically compute the expectation integrals of the mean and covariance dynamics while explicitly taking into account the discontinuity. This leads to a natural smoothing of the discontinuity such that for relevant levels of uncertainty the resulting ODE can be integrated directly with standard schemes and it is neither necessary to prespecify the switching sequence nor to use a switch detection method. We then show how this result can be employed in the more general case of piecewise smooth functions based on a structure preserving linearization scheme. The resulting dynamics can be straightforwardly used within standard formulations of stochastic optimal control problems with chance constraints.