A Framework for Adaptive Stabilisation of Nonlinear Stochastic Systems

TL;DR

This paper introduces an adaptive control framework for nonlinear stochastic systems with parameter uncertainty, providing probabilistic stability guarantees using a learning-based approach.

Contribution

It proposes a certainty equivalence adaptive control strategy with stability bounds, extending to global stability under fully informative state spaces.

Findings

Stability bounds are derived for the closed-loop system.

High probability stability guarantees are achieved under global stabilisability.

The approach handles linearly parameterised uncertainties in nonlinear stochastic systems.

Abstract

We consider the adaptive control problem for discrete-time, nonlinear stochastic systems with linearly parameterised uncertainty. Assuming access to a parameterised family of controllers that can stabilise the system in a bounded set within an informative region of the state space when the parameter is well-chosen, we propose a certainty equivalence learning-based adaptive control strategy, and subsequently derive stability bounds on the closed-loop system that hold for some probabilities. We then show that if the entire state space is informative, and the family of controllers is globally stabilising with appropriately chosen parameters, high probability stability guarantees can be derived.