## 4 Case Study and Discussion
### Case Study  
In this section, representative data-driven methods for IFGD modeling used in previous research works are compared with the proposed approach. The methods are BP network, which is typically used as a benchmark in the IFGD modeling field (e.g., [6; 14]), Hammerstein model in [30] and RNN in [31]. As for the RNN model, the back propagation through time (BPTT) algorithm proposed in [32] is used as the training algorithm; the stochastic gradient descent is employed for the BP network and the TCN with mini-batch size setting as 16, and the learning rate starts from 0.1 and is divided by 10 when the error plateaus. Training terminates after 200 epochs or once there was no improvement on the validation set for over six consecutive epochs. We use grid search to determine a good combination of hyperparameters for BP network and RNN (with the searching scope for hidden layer number [1,2,3,4], hidden units [5; 20]) over the validation set. The determination of the TCN structure is realized by adjusting the depth of the network n and kernel size \(k\), such that the receptive field covers sufficient input information for identification. To prevent the randomness of result brought about by the initial parameters in network, for each structure we run 20 trials with different initial weights, and the best-preforming one on the validation set is chosen and presented below. Hammerstein is a non-linear identifier with a linear dynamic and static nonlinearity cascading together, and extended least square [33; 34] can be used for parameter estimation. Based on the performance on the validation set, the final model structure for above models is summarized in Table 2. It is noteworthy to mention that dilation factor \(d\), kernel size \(k\), layer number \(i\) and residue block number \(n\) are several most important parameters in TCN. Among them, exponential dilation technique is employed to expand the receptive field through layers, and we have \(d=2^{i}\) for \(i\)th layer in the network. By adjusting \(k\), \(i\) and \(n\) it was ensured that TCN has \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Season**} & \multicolumn{5}{c}{**Statistical Index**} \\ \cline{2-6}  & **Minimum** & **Maximum** & **Mean** & **SD\({}^{\text{1}}\)** & **Median** \\ \hline Spring & 5.504/5.362 & 40.35/5.613 & 18.84/5.491 & 5.62/0.0675 & 18.84/5.499 \\ Summer & 5.635/5.563 & 37.14/5.823 & 20.61/5.681 & 7.041/0.0582 & 20.54/5.672 \\ Autumn & 5.007/5.446 & 50.53/5.699 & 15.53/5.632 & 5.323/0.0437 & 16.92/5.641 \\ Winter & 5.43/5.526 & 38.22/5.778 & 19.11/5.641 & 7.095/0.0534 & 19.17/5.623 \\ \hline \hline \end{tabular} \({}^{\text{1}}\) SD denotes standard deviation.  
\end{table} 
Table 1: Statistical characteristics of measured slurry pH in four different seasons. Figure 5: The photograph of desulfurization tower in the IFGD process. enough receptive field that is required for our IFGD identification task, and the result is also included in Table 2 below. As for the BEMD-TCN approach, BEMD should be first performed to decompose the original IFGD process into several subprocesses, each of which is made up of corresponding IMF in two channels. In our case study, bivariate signal is made up of limestone feed flow and slurry pH and decomposed into six subprocesses (i.e., six groups of IMFs), due to space limitations and results for spring are presented in Figure 6 below. Since the decomposed IMFs have less complex frequency components, lower amplitude and are relatively stationary, hence, resulting subprocesses are more readily to identify. Likewise, IMF pairs in other seasons can be decomposed by BEMD and are in turn used to constitute subprocesses. Having decomposed process in each season into several parts, the next step is to use the TCN to identify each of them, and the final identification result is yielded by integrating them together. Keeping the same IMF number as that in spring, process in each season is partitioned into six IMFs and a residue.