Lyapunov-Based Deep Residual Neural Network (ResNet) Adaptive Control

TL;DR

The paper tackles real-time adaptive control for uncertain nonlinear systems by extending Lyapunov-based weight adaptation to deep residual networks (ResNets). It introduces a recursive, Taylor-series-approximation framework to derive per-layer weight updates and proves asymptotic tracking via a nonsmooth Lyapunov analysis, with a control law that combines a ResNet feedforward, a sliding term, and a tracking term. The key contributions include the first Lyapunov-derived adaptation laws for an arbitrary-depth ResNet in adaptive control, a rigorous stability guarantee, and a 64% performance improvement over a fully-connected DNN-based controller demonstrated through extensive Monte Carlo simulations. The work shows that shortcut connections in ResNets mitigate vanishing gradients, enabling faster, more reliable online adaptation and improved function approximation, with potential extensions to LSTM-enhanced architectures and composite adaptive strategies.

Abstract

Deep Neural Network (DNN)-based controllers have emerged as a tool to compensate for unstructured uncertainties in nonlinear dynamical systems. A recent breakthrough in the adaptive control literature provides a Lyapunov-based approach to derive weight adaptation laws for each layer of a fully-connected feedforward DNN-based adaptive controller. However, deriving weight adaptation laws from a Lyapunov-based analysis remains an open problem for deep residual neural networks (ResNets). This paper provides the first result on Lyapunov-derived weight adaptation for a ResNet-based adaptive controller. A nonsmooth Lyapunov-based analysis is provided to guarantee asymptotic tracking error convergence. Comparative Monte Carlo simulations are provided to demonstrate the performance of the developed ResNet-based adaptive controller. The ResNet-based adaptive controller shows a 64% improvement in the tracking and function approximation performance, in comparison to a fully-connected DNN-based adaptive controller.

Figures (5)

• Figure 1: Illustration of the ResNet architecture in (\ref{['eq:ResNet1_arch']}). The ResNet is shown at the top of the figure and is composed of building blocks that involve a shortcut connection across a fully-connected DNN component. The fully-connected DNN component for the $p^{th}$ building block (bottom) is denoted by $\Phi_{p}^{\theta_{p}}$ for all $p\in\{1,\ldots,m\}$, where the input and the vector of weights of $\Phi_{p}$ are denoted by $\eta_{p}$ and $\theta_{p}$, respectively. Then the output of the $p^{th}$ building block after considering the shortcut connection is represented by $\eta_{p+1}=\eta_{p}+\Phi_{p}^{\theta_{p}}(\eta_{p})$ for all $p\in\{1,\ldots,m-1\}$, and the output of the ResNet is $\eta_{m}+\Phi_{m}^{\theta_{m}}\left(\eta_{m}\right)$.
• Figure 2: Plots of the tracking error norm $\left\Vert e\right\Vert$ and function approximation error norm $\left\Vert \tilde{f}\right\Vert$ with ResNet and fully-connected DNN-based adaptive controller.
• Figure 3: Plot of the weight estimates of the ResNet and fully-connected DNN. There are a total of 2,000 individual weights in each architecture. For better visualization, 10 arbitrarily selected weights are shown. The fully-connected DNN weights adapt slowly due to the problem of vanishing gradients. However, the ResNet weights are able to adapt faster since the ResNet does not have vanishing gradients.
• Figure 4: Comparative plots of the tracking error norm $\left\Vert e\right\Vert$ and the function approximation error norm $\left\Vert \tilde{f}\right\Vert$ with the ResNet and a shallow NN with 10 neurons.
• Figure 5: Plots demonstrating the tracking and function approximation errors with the ResNet-based controller, using sigma modification in the adaptation law.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Theorem 1
• Remark 3
• Remark 4