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J. Werbos, Back propagation: Past and future, Proc. 1988 Int. Conf. , Vol. 1, pp. 1343–1353, 1989. 8. Y. Iiguni, H. Sakai, and H. Tokumaru, A nonlinear regulator design in the presence of system uncertainties using multilayer neural networks, IEEE Trans. , 2: 410–417, 1991. 9. K. S. Narendra and K. Parthasarathy, Gradient methods for the optimization of dynamical systems containing neural networks, IEEE Trans. , 2: 252–262, 1991. 10. T. Yabuta and T. Yamada, Neural network controller characteristics with regard to adaptive control, IEEE Trans.

21) with Ͱ, ͱ Ͼ 0 design parameters. Then for small enough outer-loop gains Kvi (as specified in the proof), the filtered ˆ (k) are Unitracking error r(k) and the weight estimates w formly Ultimately Bounded (UUB). Further, the filtered tracking error can be made arbitrarily small by proper selection of the feedback gains Kvi. Remark 2 explains how the CMACs overcome one of the serious difficulties in the implementation of fully connected NNs. ) is the vector of the activation function of each node.

The weight-update laws for guaranteed stability, tracking performance, and robustness issues are discussed. BACKGROUND ON CMAC NEURAL NETWORKS Structure of CMAC Neural Networks The cerebellar model arithmetic computer (CMAC) is a perceptronlike associative memory that performs nonlinear function mapping over a region of the function space. This highly structured nonfully connected neural network model was established by J. Albus (22,26) based on a model of the human memory and neuromuscular control system.

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39.Neural Networks by John G. Webster (Editor)

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