Distributed Lebesgue Approximation Model for Distributed Continuous-Time Nonlinear Systems

Approximation models play a crucial role in model-based methods, as they enhance both accuracy and computational efficiency. This article studies distributed and asynchronous discretized models to approach continuous-time nonlinear systems. The considered continuous-time system consists of some distributed but physically coupled nonlinear subsystems that exchange information. We propose two Lebesgue approximation models (LAMs): 1) the unconditionally triggered LAM (CT-LAM) and 2) the CT-LAM. In both approaches, a specific LAM approximates an individual subsystem. The iteration of each LAM is triggered by either itself or its neighbors. The collection of different LAMs executing asynchronously together form the approximation of the overall distributed continuous-time system. The aperiodic nature of LAMs allows for a reduction in the number of iterations in the approximation process, particularly when the system has slow dynamics. The difference between the unconditionally and CT-LAMs is that the latter checks an "importance" condition, further reducing the computational effort in individual LAMs. Furthermore, the proposed LAMs are analyzed by constructing a distributed event-triggered system which is proved to have the same state trajectories as the LAMs with linear interpolation. Through this specific event-triggered system, we derive conditions on the quantization sizes in LAMs to ensure asymptotic stability of the LAMs, boundedness of the state errors, and prevention of Zeno behavior. Finally, simulations are carried out on a quarter-car suspension system to show the advantage and efficiency of the proposed approaches.