By Edgar N. Sanchez

The target of this paintings is to provide fresh advances within the thought of neural keep an eye on for discrete-time nonlinear platforms with a number of inputs and a number of outputs. the consequences that seem in each one bankruptcy contain rigorous mathematical analyses, in response to the Lyapunov strategy, that warrantly its houses; moreover, for every bankruptcy, simulation effects are integrated to make sure the profitable functionality of the corresponding proposed schemes. on the way to entire the therapy of those schemes, the ultimate bankruptcy provides experimental effects regarding their software to a electrical 3 part induction motor, which express the applicability of such designs. The proposed schemes will be hired for various functions past those offered during this publication. The publication offers strategies for the output trajectory monitoring challenge of unknown nonlinear structures in accordance with 4 schemes. For the 1st one, a right away layout procedure is taken into account: the well-known backstepping strategy, below the belief of entire sate dimension; the second considers an oblique technique, solved with the block keep an eye on and the sliding mode thoughts, less than an analogous assumption. For the 3rd scheme, the backstepping procedure is reconsidering together with a neural observer, and eventually the block keep an eye on and the sliding mode recommendations are used back too, with a neural observer. all of the proposed schemes are constructed in discrete-time. For either pointed out keep watch over tools in addition to for the neural observer, the online education of the respective neural networks is played via Kalman Filtering.

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**Sample text**

22) as x2d (k) = B1−1 (x1 (k))[−f1 (x1 (k)) + Kz1 (k)]. It is desired that x2 (k) = x2d (k). In this way, it is deﬁned as a second new error vector z2 (k) = x2 (k) − x2d (k). Then z2 (k + 1) = f3 (x1 (k)) + B2 (k)u(k), with f3 (x1 (k)) = f2 (x2 (k)) − B1−1 (x1 (k + 1))[−f1 (x1 (k + 1)) + Kz1 (k + 1)]. Let us select the manifold for the sliding mode as SD (k) = z2 (k). To design a control law, a discrete-time sliding mode version is implemented as u(k) = ueq (k) u (k) u0 ueq eq (k) if if ueq (k) ≤ u0 , ueq (k) > u0 , where ueq (k) = −B2−1 (k)f3 (x1 (k)) is calculated from SD (k) = 0 and u0 is the control resources that bound the control.

It is desired that x2 (k) = x2d (k). In this way, it is deﬁned as a second new error vector z2 (k) = x2 (k) − x2d (k). Then z2 (k + 1) = f3 (x1 (k)) + B2 (k)u(k), with f3 (x1 (k)) = f2 (x2 (k)) − B1−1 (x1 (k + 1))[−f1 (x1 (k + 1)) + Kz1 (k + 1)]. Let us select the manifold for the sliding mode as SD (k) = z2 (k). To design a control law, a discrete-time sliding mode version is implemented as u(k) = ueq (k) u (k) u0 ueq eq (k) if if ueq (k) ≤ u0 , ueq (k) > u0 , where ueq (k) = −B2−1 (k)f3 (x1 (k)) is calculated from SD (k) = 0 and u0 is the control resources that bound the control.

ILi } is a collection of nonordered subsets of {1, 2, . . , n + m}, n is the state dimension, m is the number of external inputs, wi (i = 1, 2, . . 8). 1) by the following discrete-time RHONN series–parallel representation [5]: xi (k + 1) = wi∗ zi (x(k), u(k)) + zi , i = 1, . . 3) where xi is the ith plant state, zi is a bounded approximation error, which can be reduced by increasing the number of the adjustable weights [5]. Assume that there exists an ideal weights vector wi∗ such that zi can be minimized on a compact set Ωzi ⊂ Li .