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The aim of this study was to design an adaptive control strategy based on recurrent neural networks (RNNs). This neural network was designed to obtain a non-parametric approximation (identification) of discrete-time uncertain nonlinear systems. A discrete-time Lyapunov candidate function was proposed to prove the convergence of the identification error. The adaptation laws to adjust the free parameters in the RNN were obtained in the same stability analysis. The control scheme used the states of the identifier, and it was developed fulfilling the necessary conditions to establish a behavior comparable with a quasi-sliding mode regime. This controller does not use the regular form of the switching function that commonly appears in the sliding mode control designs. The Lyapunov candidate function to design the controller and the identifier simultaneously requires the existence of positive definite solutions of two different matrix inequalities. As consequence, a class of separation principle was proven when the RNN-based identifier and the controller were designed by the same analysis. Simulations results were designed to show the behavior of the proposed controller solving the tracking problem for the trajectories of a direct current (DC) motor. The performance of the proposed controller was compared with the solution obtained when a classical proportional derivative controller and an adaptive first-order sliding mode controller assuming poor knowledge of the plant. In both cases, the proposed controller showed superior performance when the relation between the tracking error convergence and the energy used to reach it was evaluated. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This study introduces a method for designing stable learning laws of Long Short-Term Memory (LSTM) networks working as a non-parametric identifier of nonlinear systems with uncertain models. The strategy applies the concept of stability for discrete-time systems in the sense of Lyapunov to prove that origin is a practical stable equilibrium point for the identification error. The laws consider a general class of sigmoidal functions placed at the different gates of a LSTM structure (long and short memory). The design of the learning laws uses a matrix inequality framework to obtain the rate gains associated with the evolution of the weights. Numerical results show the designed learning laws for the non-parametric identifier based on a LSTM approximation tested on two classes of nonlinear systems: the first one describes the ozone-based degradation of organic contaminants, and the second one represents the dynamics of a Van Der Poll oscillator. The LSTM identifier is compared against a classical Lyapunov-based recurrent neural network. This comparison demonstrates how the proposed algorithm approximates the trajectories of both systems with a smaller mean squared error, which serves as an indicator of the benefits obtained with these new learning laws. [ABSTRACT FROM AUTHOR]

This study aims to develop stable learning laws (in the sense of Lyapunov) for a general class of multilayer recurrent neural network (RNN) non-parametric identifier. The application of control Lyapunov functions in the discrete-time domain ensures the ultimate boundedness for the identification error enforcing it to reach a boundary set related to the power of parametric uncertainties and the modeling error. This work presents a general algorithm to design RNNs with n layers: one input layer, n - 2 hidden layers, and one output layer. Numerical simulations support the theoretical results showing the advantages of increasing the number of layers in an RNN structure. A first example identifies the unknown states of the Van Der Pol oscillator. Then, a second example demonstrates the behavior of an RNN in a third-order system with high nonlinear dynamics describing an ozonation system of a single contaminant. For both cases, the application of the developed learning laws succeeds in estimating the uncertain dynamics. Moreover, the numerical simulations show how the identification error decreases as the number of layers increases. [ABSTRACT FROM AUTHOR]