On the scalability and convergence of simultaneous parameter identification and synchronization of dynamical systems

TL;DR

This paper introduces a new adaptation law for the synchronization of dynamical systems that overcomes scalability issues and guarantees parameter convergence, supported by Lyapunov stability proofs and numerical simulations.

Contribution

The paper proposes a novel adaptation law that improves scalability and ensures parameter convergence in the synchronization of dynamical systems.

Findings

The new method outperforms Chen's method in complex parametric models.

The adaptation law guarantees convergence of parameters.

Numerical simulations validate the effectiveness of the proposed approach.

Abstract

The synchronization of dynamical systems is a method that allows two systems to have identical state trajectories, appart from an error converging to zero. This method consists in an appropriate unidirectional coupling from one system (drive) to the other (response). This requires that the response system shares the same dynamical model with the drive. For the cases where the drive is unknown, Chen proposed in 2002 a method to adapt the response system such that synchronization is achieved, provided that (1) the response dynamical model is linear with a vector of parameters, and (2) there is a parameter vector that makes both system dynamics identical. However, this method has two limitations: first, it does not scale well for complex parametric models (e.g., if the number of parameters is greater than the state dimension), and second, the model parameters are not guaranteed to converge, namely as the synchronization error approaches zero. This paper presents an adaptation law addressing these two limitations. Stability and convergence proofs, using Lyapunov's second method, support the proposed adaptation law. Finally, numerical simulations illustrate the advantages of the proposed method, namely showing cases where the Chen's method fail, while the proposed one does not.