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Conditions for Global Asymptotic Stability (GAS) of a nonlinear relaxation process realized by a Recurrent Neural Network (RNN) are provided. Existence. convergence, and robustness of such a process are analyzed. This is undertaken based upon the Contraction Mapping Theorein (CMT) and the corresponding Fixed Point Iteration (FPI). Upper bounds for such a process are shown to be the conditions of convergence for a commonly analyzed RNN with a linear state dependence.

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Abstract-A novel data-driven gradient descent (GD) adaptive controller, for discrete-time single-input and single output (SISO) systems, is presented. The controller operates as the least mean squares (LMS) algorithm, applied to a nonlinear system with feedback. Normalization of the learning rate parameter provides robustness of the overall system to modeling errors and environment nonstationarity. Convergence analysis reveals that the controller forces tracking error to zero, with bounded control signal and the controller weights. The experiments on the benchmark systems support the analysis.