Page 216 - Neural Network Modeling and Identification of Dynamical Systems

6.2 SEMIEMPIRICAL MODELING OF LONGITUDINAL SHORT-PERIOD MOTION FOR A MANEUVERABLE AIRCRAFT 207

FIGURE 6.7 Accuracy of the estimated dependencies C L (α,q,δ e ) and C m (α,q,δ e ) based on the ANN model testing results
(point mode, identification, and testing with polyharmonic control signal). The values of the outputs for system (6.5)and the
ANN model are shown by a solid line and a dashed line, respectively.

. When the training of the s m,l , i.e.,
k,j
example, C L α and C m α
semiempirical model is completed, it is possi- l
ble to extract from this model the ANN mod- ∂a k m = s m,l · ∂n j , s m,l = ∂a m ,
k
ules that represent the approximations for the n l k,j p j k,j n l
j (j,l)∈IC i j
functions C L and C m . Then, we can estimate the
l
derivatives of functions C L and C m with respect where n is the weighted input of the jth neuron
j
to α, q,and δ e . We can do this by computing the of the lth layer; IC i is the set of pairs of indices
derivatives of the corresponding outputs of the j,l defining the number j of aneuroninan lth
ANN modules concerning their inputs. We can layer that has a connection with the ith input p i .
m,l
perform the computation of these derivatives by In this case, the sensitivities s k,j are computed
an algorithm similar to the forward propaga- during execution of the forward propagation al-
l
gorithm, and the derivatives (∂n /∂p j ) are equal
tion, originally designed to calculate the deriva- j
to the weights of the corresponding input con-
tives of the network outputs with respect to its
nections (for a neuron with a weighted summa-
weights and biases.
tion as an input mapping).
Using the chain rule for differentiation, we For example, application of this algorithm
can express the derivative of the output a m for
k gives the following values of the derivatives C L α
the kth neuron of the mth (output) layer with re- and C m α , corresponding to the point mode (q =
spect to the input p i in terms of the sensitivities 0) and the balancing values of the deflection an-