Page 57 - Neural Network Modeling and Identification of Dynamical Systems

2.1 ARTIFICIAL NEURAL NETWORK STRUCTURES 45

FIGURE 2.13 Examples of a structural organization for feedforward dynamic neural networks. (A) NARX (Nonlinear
AutoRegression with eXternal inputs). (B) LDDN (Layered Digital Dynamic Network). D in aresource(input) data; D out are
output data (results); L (0) is input layer; L (1) is hidden layer; L (2) is output layer for NARX network and hidden layer for
LDDN; L (3) is output layer for LDDN; TDL (m) , TDL (m) , TDL (n1) , TDL (n2) are tapped delay lines of order m, m, n1,and n2
1 2 1 1
respectively.

FIGURE 2.15 General structure of the operational (hid-
(0)
den and output) ANN layers: s is the ith neuron of the
i
pth ANN layer; W(L (p) ) is a matrix of synaptic weights for
connection entering the neurons of the L (p) layer.

0
inputs; and a ∈ R is the value of the ith input.
i
For each ith neuron of an lth layer we denote
l
the following: n is the weighted sum of neu-
i
l
ron inputs; ϕ : R → R is the activation function;
i
l
and a ∈ R is the output of an activation function
i
FIGURE 2.14 ANN input layer as a data structure. (the neuron state). Outputs a of activation func-
L
(0) i
(A) One-dimensional array. (B) Two-dimensional array. s ,
i tions of Lth layer neurons are called the network
(0)
s are numeric or character variables.
ij outputs. Also, W ∈ R n w is the total vector of net-
l
work parameters, which consists of biases b ∈ R
i
Neural Network (LFNN). We introduce the fol- and connection weights w l i,j ∈ R. Thus, the lay-
lowing notation: L ∈ N is the total number of ered feedforward neural network is a parametric
l
layers; S ∈ N is the number of neurons within function family, mapping the network inputs a 0
L
0
the lth layer; S ∈ N is the number of network and parameters W to the outputs a according