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Extra resources for 39.Neural Networks

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In the early approaches learning in CMAC was first accomplished off-line. The CMAC was presented with training samples, and the corresponding weights were updated until the network could reconstruct the unknown function with reasonable accuracy over the domain of interest. In these works the CMAC weight update rules were similar to the least mean CEREBELLAR MODEL ARITHMETIC COMPUTERS squares (LMS) algorithm. This way they ensured convergence of CMAC learning to some local minima. The convergence properties of CMAC were also studied by Wong and Sideris (40).

9. (25) M(q)q¨ + Vm (q, q)q ˙ + G(q) + F (q) ˙ + τd = τ (28) Table 2 Tracking error Filtered error Filtered tracking error dynamics Control input Closed-loop dynamics e ϭ x Ϫ xd eiϩ1 ϵ y (i) (t) Ϫ y d(i)(t), i ϭ 1, 2, . . , n Ϫ 1 r ϭ ⌳Te, where ⌳ ϭ [⌳ 1] ϭ [␭1 ␭2 . . ␭nϪ11] T s nϪ1 ϩ ␭nϪ1 s nϪ2 ϩ . . ϩ ␭1 is Hurwitz. r˙ ϭ f (x) ϩ g(x)u ϩ d ϩ Yd ͸ nϪ1 ␭i eiϩ1 where Yd ϵ Ϫy (n) d ϩ iϭ1 1 [Ϫf (x) Ϫ yd Ϫ ⌳r] Uϭ g(x) r˙ ϭ ⌳r ϩ d CEREBELLAR MODEL ARITHMETIC COMPUTERS where the tracking error is defined as e(t) ϵ q(t) Ϫ qd(t), M is a constant diagonal matrix approximation of the inertia matrix, and Kv, Kp are constant diagonal matrices of the derivative and proportional gains.

Nn) ϫ one-dimensional space I. The map ⌫ is now defined by I and M. Specifically the 2n nonzero values of R(x) are placed into the matrix ⌫(x) at the locations specified by I(x). This ordering of the indices uniquely determines w and ⌫ in Eq. (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.

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

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