Stochastic Relative Degree and Path-wise Control of Nonlinear Stochastic Systems

TL;DR

This paper introduces a stochastic relative degree concept and a normal form transformation for nonlinear stochastic systems, enabling the design of both idealistic and practical path-wise control strategies with proven convergence and tracking capabilities.

Contribution

It develops a novel stochastic normal form and hybrid control architecture for nonlinear stochastic systems, bridging idealistic and practical control approaches.

Findings

Hybrid control achieves idealistic performance as compensation period approaches zero

Normal form transformation simplifies nonlinear stochastic dynamics for control design

Practical control effectively tracks outputs in numerical simulations

Abstract

We address the path-wise control of systems described by a set of nonlinear stochastic differential equations. For this class of systems, we introduce a notion of stochastic relative degree and a change of coordinates which transforms the dynamics to a stochastic normal form. The normal form is instrumental for the design of a state-feedback control which linearises and makes the dynamics deterministic. We observe that this control is idealistic, i.e. it is not practically implementable because it employs a feedback of the Brownian motion (which is never available) to cancel the noise. Using the idealistic control as a starting point, we introduce a hybrid control architecture which achieves \emph{practical} path-wise control. This hybrid controller uses measurements of the state to perform periodic compensations for the noise contribution to the dynamics. We prove that the hybrid controller retrieves the idealistic performances in the limit as the compensating period approaches zero. We address the problem of asymptotic output tracking, solving it in the idealistic and in the practical framework. We finally validate the theory by means of a numerical example.