Asymptotic Behaviour of Gradient Learning Algorithms in Neural Network Models for the Identification of Nonlinear Systems

Abstract

This paper deals with studying the asymptotical properties of multilayer neural networks models used for the adaptive identification of wide class of nonlinearly parameterized systems in stochastic environment. To adjust the neural network's weights, the standard online gradient type learning algorithms are employed. The learning set is assumed to be infinite but bounded. The Lyapunov-like tool is utilized to analyze the ultimate behaviour of learning processes in the presence of stochastic input variables. New sufficient conditions guaranteeing the global convergence of these algorithms in the stochastic frameworks are derived. The main their feature is that they need no a penalty term to achieve the boundedness of weight sequence. To demonstrate asymptotic behaviour of the learning algorithms and support the theoretical studies, some simulation examples are also given.

Some Features of Neural Networks as Nonlinearly Parameterized Models of Unknown Systems Using an Online Learning Algorithm

“…The feature (12) gives that in the presence of nonlinear g there exist, at least, two different see [35]. Therefore, the set of * s θ will be not one-point if g is nonlinear.…”

“…To overcome these difficulties, the penalty term to an error function has been introduced in [33]. Recently we however established in [35] that the global convergence of these algorithms with probability 1 can be achieved without any additional term, at least, in the case when the activation function of the network output layer is linear.…”

“…Novelty of this paper which extends the basic ideas of [35] to the case where the activation function of the output layer is nonlinear, consists in establishing sufficient conditions under which the gradient algorithm for learning neural networks will globally converge in the sense almost sure for the case when the learning rate can be constant. The proposed approach to deriving these convergence results is based on utilizing the Lyapunov methodology [36].…”