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Recently it has been shown that CMACs can be constructed to approximate nonlinear function’s with arbitrary accuracy. Consider the partition ȏi, 1 Յ i Յ n, given earlier. Then the following theorem can be proved (33). Theorem 4 The function estimate g(x) defined in Eq. (6) uniformly approximates any C1-continuous function f(x): Rn Ǟ Rm on ⍀ ʚ Rn. ) on ⍀, the maximum partition size ͳ can be chosen such that f (x) − g(x) ≤ (9) where δ≤ (10) mL 157 hand side of Eqs. (7) and (8). In the implementation of CMAC neural networks, it is customary to employ the following submappings (18,22,26): R:X ⇒M Q:M ⇒I (14) :I×M ⇒A where R(x) is the receptive field function described in Eq.

The closed-loop implementation is as shown in Fig. 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.

N − 1 (19)   · yd (k + n) Feedback Linearizing Controller. The tracking problem above can be solved using a feedback linearizing controller if the complete dynamics in Eq. (20) are known. In this implementation the system is first expressed in terms of the filtered error system and the filter gains selected to make the error dynamics Hurwitz (Table 1). +n m−1 +2 (k + 1) = xn 1 +n 2 +···+n m−1 +3 · xn (k + 1) = f m (x(k)) + bm um (k) + dm (k) Tracking error Filtered tracking error Control input 1ՅiՅn ui (k) ϭ ͕Ϫfi (x(k)) Ϫ Kvi ri (k) Ϫ [␭i,niϪ1 ei(k) ϩ ␭i,niϪ2 ei (k Ϫ 1) ϩ .

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

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