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I,1 ei (k Ϫ ni ϩ 2)] ϩ ydi(k ϩ 1)͖/bi (k) (k) ei (k) ϭ yi (k) Ϫ ydi(k) ri (k) ϭ ei (k) ϩ ␭i,niϪ1 ei (k Ϫ 1) ϩ . . ϩ ␭i,1 ei (k Ϫ ni ϩ 1) Filtered tracking error system 1ՅiՅm ri (k ϩ 1) ϭ Kvi rk(k) ϩ di(k) CEREBELLAR MODEL ARITHMETIC COMPUTERS _x d (k) – ΛT + x (k) _ Plant Kv _ [0 Λ T ] CMAC ^ f ( _x (k)) – – r (k) e (k) 159 + – – + Figure 6. Control of an unknown nonlinear system using CMAC neural network. The controller includes an inner feedback linearizing loop and an outer tracking loop.

13). ) ϭ [g1, g2, . , gm]T with gk (x) = Nn j n =1 ... , j n (x) f (x) = wT (x) + (16) where ⑀ is the function estimation error and ʈ⑀ʈ Յ ⑀N, with ⑀N a given bound. (12) BACKGROUND ON NONLINEAR DYNAMICAL SYSTEMS for some weights w. In fact, the weights can be shown to be the samples of the function components to be approximated at each of the knot points of the partition. Implementation Properties of CMAC Neural Networks The earliest use of CMACs in control applications was in the control of robot manipulators (22,26,40).

L. V. Fausett, Fundamentals of Neural Networks: Architectures, Algorithms, and Applications. Englewood Cliffs, NJ: Prentice Hall, 1994. 4. W. Elwasif, L. V. Fausett, and S. Harbaugh, Boltzmann machine generation of initial asset distributions. In S. K. Rogers and D. W. ), Proceedings, Applications and Science of Artificial Neural Networks, SPIE, Vol. 2492, 1995, pp. 331–340. 5. L. Gislen, C. Peterson, and B. , 4: 805–831, 1992. 6. R. S. S. thesis, Florida Institute of Technology, 1992. 7. D. H.

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