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Lyapunov-based Adaptive Transformer (LyAT) for Control of Stochastic Nonlinear Systems

Saiedeh Akbari, Xuehui Shen, Wenqian Xue, Jordan C. Insinger, Warren E. Dixon

TL;DR

The paper introduces LyAT, a Lyapunov-based adaptive transformer for stochastic nonlinear control, which adapts drift and diffusion uncertainties in real time without offline training. By coupling an encoder–decoder transformer with a Lyapunov-derived weight update law, LyAT achieves probabilistic uniform ultimate boundedness of tracking and parameter estimation errors. The unified architecture compensates both drift and diffusion uncertainties using a single network, reducing computational burden while providing formal stability guarantees. Experimental validation on a quadrotor demonstrates rapid convergence to a bounded tracking error and smooth control actions under outdoor disturbances. This work advances real-time, stability-certified transformer-based control for stochastic systems with practical relevance to aerial robotics.

Abstract

This paper presents a novel Lyapunov-based Adaptive Transformer (LyAT) controller for stochastic nonlinear systems. While transformers have shown promise in various control applications due to sequential modeling through self-attention mechanisms, they have not been used within adaptive control architectures that provide stability guarantees. Existing transformer-based approaches for control rely on offline training with fixed weights, resulting in open-loop implementations that lack real-time adaptation capabilities and stability assurances. To address these limitations, a continuous LyAT controller is developed that adaptively estimates drift and diffusion uncertainties in stochastic dynamical systems without requiring offline pre-training. A key innovation is the analytically derived adaptation law constructed from a Lyapunov-based stability analysis, which enables real-time weight updates while guaranteeing probabilistic uniform ultimate boundedness of tracking and parameter estimation errors. Experimental validation on a quadrotor demonstrates the performance of the developed controller.

Lyapunov-based Adaptive Transformer (LyAT) for Control of Stochastic Nonlinear Systems

TL;DR

The paper introduces LyAT, a Lyapunov-based adaptive transformer for stochastic nonlinear control, which adapts drift and diffusion uncertainties in real time without offline training. By coupling an encoder–decoder transformer with a Lyapunov-derived weight update law, LyAT achieves probabilistic uniform ultimate boundedness of tracking and parameter estimation errors. The unified architecture compensates both drift and diffusion uncertainties using a single network, reducing computational burden while providing formal stability guarantees. Experimental validation on a quadrotor demonstrates rapid convergence to a bounded tracking error and smooth control actions under outdoor disturbances. This work advances real-time, stability-certified transformer-based control for stochastic systems with practical relevance to aerial robotics.

Abstract

This paper presents a novel Lyapunov-based Adaptive Transformer (LyAT) controller for stochastic nonlinear systems. While transformers have shown promise in various control applications due to sequential modeling through self-attention mechanisms, they have not been used within adaptive control architectures that provide stability guarantees. Existing transformer-based approaches for control rely on offline training with fixed weights, resulting in open-loop implementations that lack real-time adaptation capabilities and stability assurances. To address these limitations, a continuous LyAT controller is developed that adaptively estimates drift and diffusion uncertainties in stochastic dynamical systems without requiring offline pre-training. A key innovation is the analytically derived adaptation law constructed from a Lyapunov-based stability analysis, which enables real-time weight updates while guaranteeing probabilistic uniform ultimate boundedness of tracking and parameter estimation errors. Experimental validation on a quadrotor demonstrates the performance of the developed controller.

Paper Structure

This paper contains 18 sections, 3 theorems, 60 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. System Dynamics and Control Objective
  5. Transformer Neural Network Architecture
  6. Positional Encoding
  7. Attention Mechanism
  8. Encoder Block
  9. Decoder Block
  10. LyAT Output
  11. Weight Adaptation Law
  12. Control Design
  13. Stability Analysis
  14. Experiment
  15. Experimental Testbed
  16. ...and 3 more sections

Key Result

Lemma 1

(Akbari.Nino.ea2024) For the Ito process ${\tt z}\in\mathbb{R}^{n}$ and function ${\tt V}$, assume (A1) ${\tt V}$ is non-negative, ${\tt V}\left(0\right)=0$, and ${\tt V}\in\mathcal{C}^{2}$ over the open and connected set $Q_{m}\triangleq\left\{ {\tt z}:{\tt V}\left({\tt z}\right)<m\right\}$, where

Figures (6)

  • Figure 1: An overview of the encoder-decoder transformer architecture. The transformer architecture consists of $N$ encoder layers and $N$ decoder layers.
  • Figure 2: A modular schematic of each attention-based encoder and decoder in a transformer architecture. Across these blocks, there are three key attention mechanisms, illustrated as 1, 2, and 3: self-attention, masked self-attention, and cross-attention, respectively.
  • Figure 3: The Freefly Astro quadrotor (image courtesy of Freefly Systems, https://freeflysystems.com/astro).
  • Figure 4: Evolution of the RMS of the tracking error over 240 seconds.
  • Figure 5: Three-dimensional schematic of the trajectory tracking.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Lemma 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2
  • Theorem 1
  • Remark 5