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The synchronization of dynamical systems is a method that allows two systemsto 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 sharesthe same dynamical model with the drive. For the cases where the drive isunknown, Chen proposed in 2002 a method to adapt the response system such thatsynchronization is achieved, provided that (1) the response dynamical model islinear with a vector of parameters, and (2) there is a parameter vector thatmakes both system dynamics identical. However, this method has two limitations:first, it does not scale well for complex parametric models (e.g., if thenumber of parameters is greater than the state dimension), and second, themodel parameters are not guaranteed to converge, namely as the synchronizationerror approaches zero. This paper presents an adaptation law addressing thesetwo limitations. Stability and convergence proofs, using Lyapunov's secondmethod, support the proposed adaptation law. Finally, numerical simulationsillustrate the advantages of the proposed method, namely showing cases wherethe Chen's method fail, while the proposed one does not.

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