Simulation results

The evaluation function that is used for testing the neural networks is a root mean squared percentage (RMSP) error index [57], which is defined as:

where, N is the number of sampling data pairs; X^"1 is the measured (actual) value of biomass at sampling time t; Xt is the corresponding estimated value predicted by the neural softsensors.

The RMSP error between the network output and the measured output of test data set was used to evaluate the merit of the network. In this study, extensive test simulations were carried out. For each network structure, 150 networks were trained; the one that produced the smallest RMSP error for the test data sets was retained. The selection was finally narrowed to two choices: six hidden neurons and 12 neurons. A representative set of error distributions is shown in Figure 4.4 for various combinations of delays and the number of hidden neurons. As shown in the Figure, 12-hidden-neuron networks frequently out-performed the six-hidden neuron networks with the exception of the "0/4/1" structure (zero input delay, four output feedback delays and one activation feedback delay). The testing RMSP errors for these two network structures were very close, and were smaller than other types of network structures. The network with six hidden neurons was therefore chosen for the on-line biomass estimation because of the small prediction error and small size of the network.

0/0/0 0/1/1 0/2/1 0/3/1 0/4/1 1/1/1 1/2/1 1/3/1 1/4/1

Fig. 4.4. Estimation root mean squared percentage error on testing data sets for neural networks with different combinations of delays. '0/4/1' indicates that input delay is zero, the number of output feedback delays are 1, 2, 3, and 4, the number of activation feedback delay is 1.

0/0/0 0/1/1 0/2/1 0/3/1 0/4/1 1/1/1 1/2/1 1/3/1 1/4/1

Number of delays

Fig. 4.4. Estimation root mean squared percentage error on testing data sets for neural networks with different combinations of delays. '0/4/1' indicates that input delay is zero, the number of output feedback delays are 1, 2, 3, and 4, the number of activation feedback delay is 1.

One of the simulation results of biomass prediction is plotted in Figure 4.5. The feed rate profile was saw-wave (see Figure 4.3(b)). The softsensor provided a good prediction of the growth of biomass with high fidelity. The prediction error showed oscillations occurring at the initial phase. This happened because the input delay was set to be zero. Previous inputs were not incorporated into the network, only the current inputs were presented for prediction. However, with the activation feedback and the output (estimated biomass concentration) feedback, the network just took a few iteration steps to settle down and then was able to move along the right track. From the prediction error in Figure 4.5, one can also see the prediction offset is small. The maximum percentage error of prediction is less than 3%.

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Soft sensor's output Actual values

Fig. 4.5. Simulation result of softsensor using six hidden neuron network for a fed-batch fermentation process.

Fig. 4.5. Simulation result of softsensor using six hidden neuron network for a fed-batch fermentation process.

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