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

Saiedeh Akbari, Cristian F. Nino, Omkar Sudhir Patil, Warren E. Dixon

TL;DR

The paper addresses trajectory tracking for nonlinear stochastic systems with unknown, unstructured drift and diffusion terms. It introduces three Lyapunov-based deep neural networks (Lb-DNNs) to separately approximate drift and diffusion components and employs Lyapunov-driven weight updates to guarantee stability. The main result shows the tracking error is uniformly ultimately bounded in probability (UUB-p) with an explicit escape risk, even when the diffusion noise does not vanish at the origin. Simulations on a five-dimensional stochastic system demonstrate robust tracking and resilience to changes in noise mean and covariance, highlighting practical applicability of the approach.

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.

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

TL;DR

The paper addresses trajectory tracking for nonlinear stochastic systems with unknown, unstructured drift and diffusion terms. It introduces three Lyapunov-based deep neural networks (Lb-DNNs) to separately approximate drift and diffusion components and employs Lyapunov-driven weight updates to guarantee stability. The main result shows the tracking error is uniformly ultimately bounded in probability (UUB-p) with an explicit escape risk, even when the diffusion noise does not vanish at the origin. Simulations on a five-dimensional stochastic system demonstrate robust tracking and resilience to changes in noise mean and covariance, highlighting practical applicability of the approach.

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.
Paper Structure (11 sections, 2 theorems, 59 equations, 3 figures)

This paper contains 11 sections, 2 theorems, 59 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Deep Neural Network Model
  4. Dynamics and Control Objective
  5. Control Design
  6. Deep Neural Network Architecture
  7. Adaptive Update Law
  8. Controller and Closed-Loop Error System
  9. Stability Analysis
  10. Simulation
  11. Conclusion

Key Result

Lemma 1

For the Ito process ${\tt z}\in\mathbb{R}^{n}$ and function ${\tt V}$, assume

Figures (3)

  • Figure 1: For a UUB-p system, if the states are initialized within the set $\mathcal{S}$, they remain inside the set $\mathcal{D}$ with probability $1-\vartheta$ and eventually exponentially converge to the set $\mathcal{D}$, staying within the bounded set (blue trajectory). However, there is an escape risk of $\vartheta$, meaning the trajectories can potentially become unbounded (red trajectory). Additionally, $\lambda$ is the radius of an arbitrary level set, whose size corresponds to either the minimum size of $\mathcal{B}$ or the maximum size of $\left\{ z:V_{L}<m\right\}$.
  • Figure 2: Performance of the tracking error over time for the developed Lb-DNN controller.
  • Figure 3: Performance of the RMS of the tracking error with respect to changes in mean and covariance of the stochastic noise for the developed Lb-DNN controller.

Theorems & Definitions (3)

  • Definition 1
  • Lemma 1
  • Theorem 1