Synthesizing Control Lyapunov-Value Functions for High-Dimensional Systems Using System Decomposition and Admissible Control Sets

Abstract

Control Lyapunov functions (CLFs) play a vital role in modern control applications, but finding them remains a problem. Recently, the control Lyapunov-value function (CLVF) and robust CLVF have been proposed as solutions for nonlinear time-invariant systems with bounded control and disturbance. However, the CLVF suffers from the ''curse of dimensionality,'' which hinders its application to practical high-dimensional systems. In this paper, we propose a method to decompose systems of a particular coupled nonlinear structure, in order to solve for the CLVF in each low-dimensional subsystem. We then reconstruct the full-dimensional CLVF and provide sufficient conditions for when this reconstruction is exact. Moreover, a point-wise optimal controller can be obtained using a quadratic program. We also show that when the exact reconstruction is impossible, the subsystems' CLVFs and their ``admissible control sets'' can be used to generate a Lipschitz continuous CLF. We provide several numerical examples to validate the theory and show computational efficiency.