4.3 ANN modeling
4.3.5 Example: ANN modeling for snack food frying process control
network stability and to reduce noise fitting in modeling. Table 4.12 shows that incorporating weight-decay into the leave-one-out Levenberg–Marquardt BP training achieved better models which had higher R2 and lower validation MSE values. The decay constant λ was determined to see if it gave a better model generalization. The results indicated that these models had better out-put variation accounting and generalization. These models were evaluated as the best models with good output variation accounting and less noise fitting.
It was also interesting that incorporating weight decay made the training slightly more efficient vs. implementing the Levenberg–Marquardt algorithm alone.
for the equation of y1(t) and Θ2 is for the equation of y2(t): d represents the time lags between the process input (t) and the process output (t), that is, (t − d) = [u1(t − d1), u2(t − d2)]T in which d1 and d2 are the minimum time lags of the inputs related to the present outputs y1(t) and y2(t), respectively: (t) = [ε1(t), ε2(t)]T is the measurement noise vector in which ε1(t) is for the equation of y1(t) and ε2(t) is for the equation of y2(t) at time t, which is assumed to be two-dimensional Gaussian distributed with zero mean and constant variance.
Eq. (4.4) provided a more general description of this representation. In Eq. (4.47), the time lags are represented explicitly with the symbol d (5 s) because in the process any action from the actuators for controlling the temperature and the conveyor speed will take a period of time to take effect on the sensor readings for color and moisture content. These time lags were determined through field measurement and statistical analysis. They were 20 and 16 units (5 s), respectively, from inlet temperature, residence time to the process outputs, color and moisture content.
ANNs could be used to directly approximate the function f ( ) as described in Eq. (4.51) as follows
(4.52) This modeling approach has no network output feedback and takes the structure of a MFN. It can provide the one-step-ahead predictor, described in the next chapter, for internal model control (IMC) which is described in Chapter 6.
A one-hidden-layered feedforward neural network was trained with the BP algorithm to model the 2 × 2 system. Before training, the data were scaled as Eq. (3.1). Following the procedure of model identification for neural net-works, the smallest structure of the neural network process model was identified. From Table 4.13, it can be seen that although the FPE still decreased, 3 was chosen as the number of hidden nodes of the neural network process model because the test MSE had a minimum there. Table 4.14 shows that the test MSE and FPE had the minimum values at (2, 2, 2, 2) of the
Table 4.13 Results of the Determination of the Number of Hidden Nodes of the Neural Network Model for the Snack
Food Frying Process*
Number of
Hidden Nodes Training MSE Test MSE FPE
1 0.107641 0.132214 0.108852
2 0.031462 0.051293 0.032226
3 0.029928 0.050594 0.030994
4 0.029437 0.050692 0.030823
5 0.028460 0.052278 0.030130
* From Huang et al. (1998a). With permission.
u y
u
e
yˆ t( ) = fˆ y t 1( ( – ), y t 2( – ),…, y t p( – ), u t d( – –1), u t d( – –2),…, u t d( – –q), W )
model order (p1, p2, q1, q2). The order of the neural network model was determined as (2, 2, 2, 2). The resulting smallest structure of the neural network model was 8 × 3 × 2, that is, 8 inputs by 3 hidden nodes by 2 outputs, which is shown in Figure 4.9.
Table 4.14 Results of the Order Determination of the Neural Network Model for the Snack Food Frying Process*
Model Order Training MSE Test MSE FPE
(1, 1, 1, 1) 0.032226 0.055117 0.032648
(2, 2, 2, 2) 0.029928 0.050594 0.030655
(3, 3, 3, 3) 0.031292 0.052863 0.032407
(4, 4, 4, 4) 0.034187 0.054443 0.043841
* From Huang et al. (1998a). With permission.
Figure 4.9 Resulting structure of the neural network model for the snack food frying process.
y1(t)
y2(t) y1(t-1)
y2(t-1)
y1(t-2)
y2(t-2)
u1(t-17)
u2(t-17)
u1(t-18)
u2(t-18)
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