Core Concepts
A method is proposed to efficiently synthesize Control Lyapunov-Value Functions (CLVFs) for high-dimensional nonlinear systems by decomposing the system into low-dimensional subsystems, reconstructing the full-dimensional CLVF, and providing sufficient conditions for when the reconstruction is exact.
Abstract
The paper presents a method to efficiently synthesize Control Lyapunov-Value Functions (CLVFs) for high-dimensional nonlinear systems. The key ideas are:
System Decomposition: The original high-dimensional system is decomposed into low-dimensional subsystems with potentially shared states or controls.
Subsystem CLVF Computation: The CLVF is computed for each subsystem independently using standard techniques.
CLVF Reconstruction: The full-dimensional CLVF is reconstructed from the subsystem CLVFs. If there are no shared controls, the reconstruction is guaranteed to be exact. If there are shared controls, a sufficient condition is provided for when the reconstruction is exact.
Inexact Reconstruction: When the exact reconstruction is not possible, the subsystem CLVFs and their "admissible control sets" are used to generate a Lipschitz continuous CLF for the original system.
Optimal Controller Synthesis: A quadratic program is used to synthesize an optimal controller that guarantees exponential stabilization of the system.
The method is validated through several numerical examples, including 2D, 3D, and 10D systems, demonstrating significant computational efficiency compared to directly computing the CLVF for the high-dimensional system.