ISSUE 181, article 4


Kibern. vyčisl. teh., 2015, Issue 181, pp.

Zhiteckii L.S., Nikolaienko S.A., Solovchuk K.Yu.

International Research and Training Center for Information Technologies and Systems of the National Academy of Science of Ukraine and Ministry of Education and Sciences of Ukraine, Kiev, Ukraine


Introduction. The paper deals with studying the asymptotical properties of the standard discrete-time gradient online learning algorithm in the two-layer neural network model of the uncertain nonlinear system to be identified. Also, the design of the discrete-time adaptive closed-loop system containing the linear multivariable memoryless plant with possibly singular but unknown matrix gain in the presence of unmeasurable bounded disturbances having the unknown bounds are addressed in this paper. It is assumed that the learning process in the neural network model is implemented in the stochastic environment whereas the adaptation of the plant model in the control system is based on the non-stochastic description of the external environment.
The purpose of the paper is to establish the global convergence conditions of the gradient online learning algorithm in the neural network model by utilizing the probabilistic asymptotic analysis and to derive the convergent adaptive control algorithm guaranteeing the boundedness of the signals in the closed-loop system which contains the multivariable memoryless plant with an arbitrary matrix gain in the presence of unmeasurable disturbances whose bounds are unknown.
Results. The Lyapunov function approach as the suitable tool for analyzing the asymptotic behavior both of the gradient learning algorithm in the neural network identification systems and of the adaptive gradient algorithm in the certain closed-loop control systems is utilized. Within this approach, the two groups of global sufficient conditions guaranteeing the convergence of the online gradient learning algorithm in neural network model with probability 1 are obtained. The first group of these conditions defines the requirements under which this algorithm will be convergent almost sure with a constant learning rate. Such an asymptotic property holds in the ideal case where the nonlinearity to be identified can exactly be described by a neural network model. The second group of convergence conditions shows that this property can also be achieved in non-ideal case. It turns out that adding a penalty term to the current error function is indeed not necessary to guarantee this property. It is established that in a worst case where the matrix gain of multivariable plant is unknown and may be singular, and the bounds on the arbitrary unmeasurable disturbances remain unknown, the convergence of the gradient adaptation algorithm and the boundedness of all signals in the adaptive closed-loop system can be ensured.
Conclusions. In order to guarantee the global convergence of the online learning algorithm in the neural network identification system with probability 1, the certain conditions should be satisfied. Also the boundedness of all signals in the closed-loop adaptive control system containing the multivariable memoryless plant whose matrix gain is unknown and possibly singular can be achieved even if the bounds on the unmeasurable disturbances are unknown.

Keywords: neural network, gradient learning algorithm, convergence, multivariable memoryless plant, adaptive control algorithm, boundedness of the signals.

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Received 06.07.2015