## 2 Materials and Methods
### Simplified Decoupling Smith Predictor Structure based on FO (F-SDSP) for TITO Processes  
In this work, the controller structure called simplified decoupling Smith predictor (SDSP) [15] is used to deal with the main issues of multivariable processes including interactions between process variables and multiple delay times. The whole structure of the controller is shown in Figure 1.  
In the Figure 1, **G**(\(s\)), **D**(\(s\)), **Q**(\(s\)) are the transfer function matrix, the decoupling matrix and the decoupled matrix of a TITO process, respectively. Due to the properties of the simplified decoupling, they have the forms as following:  
\[\textbf{G}(s)=\left[ \begin{array}{cc}g_{11}(s)&g_{12}(s)\\ g_{21}(s)&g_{22}(s)\end{array} \right]\text{; }\textbf{D}(s)=\left[ \begin{array}{cc}1&d_{12}(s)\\ d_{21}(s)&1\end{array} \right]\text{; }\textbf{Q}(s)=\left[ \begin{array}{cc}q_{11}(s)&0\\ 0&q_{22}(s)\end{array} \right] \tag{9}\]  
where the elements of **D**(\(s\)) and **Q**(\(s\)) can be calculated as shown in (10)-(12) [15]:  
\[d_{12}(s)=-\frac{g_{12}(s)}{g_{11}(s)}\text{; }d_{21}(s)=-\frac{g_{21}(s)}{g_{22}(s)} \tag{10}\]  
\[q_{11}(s)=g_{11}(s)-\frac{g_{12}(s)g_{21}(s)}{g_{22}(s)} \tag{11}\]  
\[q_{22}(s)=g_{22}(s)-\frac{g_{12}(s)g_{21}(s)}{g_{11}(s)} \tag{12}\]  
Various methods were proposed to approximate the diagonal elements of the decoupled matrix (\(q_{11}\), \(q_{22}\)) to some standard forms [15; 16; 18; 20; 21]. However, all of them only deal with integer order transfer functions. In this work, a fractional order transfer function (FOTF) is employed to be the equivalent transfer function of Equations (11) and (12). The general form of the FOTF is as follows:  
\[g_{m}(s)=\frac{Ke^{-\theta s}}{\tau_{2}s^{\alpha_{2}}+\tau_{1}s^{\alpha_{1}}+ 1}\ \ (0<\alpha_{1}\leq 1<\alpha_{2}<2) \tag{13}\]  
where \(\tau_{1},\tau_{2}\) are time constants; \(K\) is a gain; \(\alpha_{2},\alpha_{1}\) are fractional orders, assuming \(0<\alpha_{1}\leq 1<\alpha_{2}<2\); \(\theta\) is a delay time. In special case, when \(\tau_{2}=0\), Equation (13) will become a simpler form:  
\[g_{m}(s)=\frac{Ke^{-\theta s}}{\tau s^{\alpha}+1}\ \ (0<\alpha<1) \tag{14}\]  
In order to facilitate the approximation procedure, the parameter \(\theta\) is a priori value that could be determined by the unit step response of the original model. Therefore, the number of tuning parameters will be written in the vector form as follow:  
\[\textbf{x}=\left[ \begin{array}{cccc}K&\tau_{2}&\tau_{1}&\alpha_{2}&\alpha_ {1}\end{array} \right] \tag{15}\]  
Figure 1: The simplified decoupling Smith predictor (SDSP) structure with approximated fractional order processes (F-SDSP). The different constraints of these parameters are given in (16):  
\[\left\{ \begin{array}{l}K_{\min}<K<K_{\max}\\ 0<\tau_{1}<\tau_{\max}\\ 0\leq\tau_{2}<\tau_{\max}\\ 0<\alpha_{1}\leq 1\\ 1<\alpha_{2}<2\end{array} \right. \tag{16}\]  
where \(K_{\min},K_{\max},\tau_{\max}\) are determined based on the open-loop unit step responses of the original model. The approximation technique using the PSO algorithm proposed by [PERSON] et al. [15] is addressed to find out the parameters in (15). Note that, from the condition (16), only \(\tau_{2}\) can be equal to zero; and if happened, the reduced model becomes the simpler form as (14). Therefore, the approximated transfer function normally has one of the following forms:  
\[\overline{q}_{i}(s)=\frac{K_{i}e^{-\theta_{i}s}}{\tau_{2i}s^{\alpha_{2i}}+\tau _{1i}s^{\alpha_{1i}}+1}\ \ (0<\alpha_{1i}\leq 1<\alpha_{2i}<2,\ i=1\div 2) \tag{17}\]  
\[\overline{q}_{i}(s)=\frac{K_{i}e^{-\theta_{i}s}}{\tau_{i}s^{\alpha_{i}}+1}\ \ (0< \alpha_{i}<1,\ i=1\div 2) \tag{18}\]  
where \(\overline{q}_{i}(s)\) is the equivalent transfer function of \(q_{ii}\) in Equations (11) and (12). It is obvious that the delays still exist in the diagonal elements of the decoupled matrix and that make sluggish responses in the outputs [15; 22].