Lyapunov-Based Kolmogorov-Arnold Network (KAN) Adaptive Control

TL;DR

This paper introduces Lyapunov-based Kolmogorov-Arnold Networks (Lb-KAN), a novel adaptive control method that offers interpretable, real-time learning for uncertain nonlinear systems, outperforming existing deep neural network approaches.

Contribution

It develops the first Lyapunov-based KAN adaptive control framework that provides interpretability through functional decomposition and ensures stability and convergence in real-time.

Findings

Reduces function approximation error by over 20%.

Provides interpretable control architecture with visualizable decomposition.

Achieves stable, real-time adaptation with formal error bounds.

Abstract

Recent advancements in Lyapunov-based Deep Neural Networks (Lb-DNNs) have demonstrated improved performance over shallow NNs and traditional adaptive control for nonlinear systems with uncertain dynamics. Existing Lb-DNNs rely on multi-layer perceptrons (MLPs), which lack interpretable insights. As a first step towards embedding interpretable insights in the control architecture, this paper develops the first Lyapunov-based Kolmogorov-Arnold Networks (Lb-KAN) adaptive control method for uncertain nonlinear systems. Unlike MLPs with deep-layer matrix multiplications, KANs provide interpretable insights by direct functional decomposition. In this framework, KANs are employed to approximate uncertain dynamics and embedded into the control law, enabling visualizable functional decomposition. The analytical update laws are constructed from a Lyapunov-based analysis for real-time learning without prior data in a KAN architecture. The analysis uses the distinct KAN approximation theorem to formally bound the reconstruction error and its effect on the performance. The update law is derived by incorporating the KAN's learnable parameters into a Jacobian matrix, enabling stable, analytical, real-time adaptation and ensuring asymptotic convergence of tracking errors. Moreover, the Lb-KAN provides a foundation for interpretability characteristics by achieving visualizable functional decomposition. Simulation results demonstrate that the Lb-KAN controller reduces the function approximation error by 20.2% and 18.0% over the baseline Lb-LSTM and Lb-DNN methods, respectively.