Lyapunov-Based Deep Neural Networks for Adaptive Control of Stochastic Nonlinear Systems

TL;DR

This paper introduces Lyapunov-based deep neural networks that adaptively control stochastic nonlinear systems by compensating for both deterministic and non-deterministic uncertainties, ensuring stability and bounded tracking error.

Contribution

It extends Lyapunov-based DNN controllers to handle stochastic uncertainties, providing stability analysis and adaptive laws for unknown drift and diffusion terms.

Findings

Effective compensation for stochastic uncertainties demonstrated

Tracking error remains bounded in probability

Simulations confirm the method's efficacy

Abstract

Controlling nonlinear stochastic dynamical systems involves substantial challenges when the dynamics contain unknown and unstructured nonlinear state-dependent terms. For such complex systems, deep neural networks can serve as powerful black box approximators for the unknown drift and diffusion processes. Recent developments construct Lyapunov-based deep neural network (Lb-DNN) controllers to compensate for deterministic uncertainties using adaptive weight update laws derived from a Lyapunov-based analysis based on insights from the compositional structure of the DNN architecture. However, these Lb-DNN controllers do not account for non-deterministic uncertainties. This paper develops Lb-DNNs to adaptively compensate for both the drift and diffusion uncertainties of nonlinear stochastic dynamic systems. Through a Lyapunov-based stability analysis, a DNN-based approximation and corresponding DNN weight adaptation laws are constructed to eliminate the unknown state-dependent terms resulting from the nonlinear diffusion and drift processes. The tracking error is shown to be uniformly ultimately bounded in probability. Simulations are performed on a nonlinear stochastic dynamical system to show efficacy of the proposed method.