# Identification of Nonlinear Systems Using Neural Networks by Andrzej Janczak

By Andrzej Janczak

This monograph systematically provides the prevailing id tools of nonlinear platforms utilizing the block-oriented strategy It surveys a variety of recognized methods to the identity of Wiener and Hammerstein structures that are acceptable to either neural community and polynomial types. The ebook supplies a comparative research in their gradient approximation accuracy, computational complexity, and convergence premiums and moreover offers a few new and unique tools about the version parameter adjusting with gradient-based concepts. "Identification of Nonlinear platforms utilizing Neural Networks and Polynomal types" comes in handy for researchers, engineers and graduate scholars in nonlinear platforms and neural community thought.

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**Additional resources for Identification of Nonlinear Systems Using Neural Networks and Polynomial Models: A Block-Oriented Approach**

**Example text**

39) xj (n) . t. its parameters vj1 , vj0 , v1j , j = 1, . . 40) ∂zj (n) ∂v (1) ∂ϕ zj (n) j1 = ∂ˆ g y(n), v ∂ϕ zj (n) ∂zj (n) (2) = v1j ϕ zj (n) , (1) ∂z (n) ∂ϕ zj (n) j ∂vj0 ∂ˆ g y(n), v (2) ∂v1j = ϕ zj (n) , ∂ˆ g y(n), v (2) ∂v10 = 1. t. the weights wj1 , wj0 , w1j , j = 1, . . 45) ∂xj (n) ∂w(1) ∂ϕ xj (n) j0 ∂ fˆ sˆ(n),w (2) ∂w1j = ϕ xj (n) , ∂ fˆ sˆ(n),w (2) ∂w10 = 1. 47) 42 2 Neural network Wiener models In spite of the fact that the series-parallel model is of the feedforward type, its training with the backpropagation (BPS) method is quite complex.

In general, SISO neural network Hammerstein models can be represented by a multilayer perceptron model of the nonlinear element and a linear node with two tapped delay lines used as a model of the linear system [73]. Both the series-parallel and parallel models can be considered. They have a similar architecture, the only diﬀerence is the feedback connection in the parallel model [100]. For the series-parallel model, the gradient of its output with respect to model parameters can be obtained using the computationally eﬀective backpropagation algorithm.

A ˆna , ˆb1 , . . , ˆbnb are the parameters of the linear dynamic model, (1) (1) (2) (2) w = [w10 . . wM1 w10 . . w1M ]T is the parameter (weight) vector of the (2) (2) (1) (1) nonlinear element model, and v = [v10 . . vM1 v10 . . v1M ]T is the parameter vector of the inverse nonlinear element model. Note that gˆ(·) does not denote the inverse of fˆ(·) but describes the inverse nonlinear element. The architecture of the parallel model (Fig. 5), which does not contain any inverse model, is even simpler in comparison with that of the series-parallel one.