Adaptive identifier for uncertain complex-valued discrete-time nonlinear systems based on recurrent neural networks

Recently, the study of dynamic systems and signals in the frequency domain motivates the emergence of new tools. In particular, electrophysiological and communications signals in the complex domain can be analyzed but hardly, they can be modeled. This problem promotes an attractive field of researching in system theory. As a consequence, adaptive algorithms like neural networks are interesting tools to deal with the identification problem of this kind of systems. In this study, a new learning process for recurrent neural network applied on complex-valued discrete-time nonlinear systems is proposed. The Lyapunov stability framework is applied to obtain the corresponding learning laws by means of the so-called Lyapunov control functions. The region where the identification error converges is defined by the power of uncertainties and perturbations that affects the nonlinear discrete-time complex system. This zone is obtained as an alternative result of the same Lyapunov analysis. An off-line training algorithm is derived in order to reduce the size of the convergence zone. The training is executed using a set of some off-line measurements coming from the uncertain system. Numerical results are developed to prove the efficiency of the methodology proposed in this study. A first example is oriented to identify the dynamics of a nonlinear discrete time complex-valued system and the second one to model the dynamics of an electrophysiological signal separated in magnitude and phase.