## 2 The Preliminary Background of the Proposed Control Method  
This section provides a description of the work's preliminary steps. The MIMO system's transfer function with n-input and n-output can be written as  
\[M(s)=\begin{bmatrix}m_{11}(s)e^{-\varphi_{11}(s)}&m_{12}(s)e^{-\varphi_{12}(s) }&\dots&m_{1n}(s)e^{-\varphi_{1n}(s)}\\ m_{21}(s)e^{-\varphi_{21}(s)}&m_{22}(s)e^{-\varphi_{22}(s)}&\dots&m_{21}(s)e^{- \varphi_{2n}(s)}\\ \dots&\dots&\dots&\dots\\ m_{n1}(s)e^{-\varphi_{n1}(s)}&m_{n2}(s)e^{-\varphi_{nn}(s)}&\dots&m_{nn}(s)e^{- \varphi_{nn}(s)}\end{bmatrix} \tag{1}\]  
Equation (1) makes clear how the process variables interact with one another. The loop interactions must be reduced in order to meet the design objectives. It has been stated in [29], interactions can be reduced using either a centralized or decentralized controller. However, given its advantages, the decentralized structure is favored. The independent controller is  
\[C(s)=\begin{bmatrix}C_{11}(s)&0&0&0\\.&C_{22}(s)&.&.\\ \dots&\dots&\dots&\dots\\ 0&0&0&C_{nn}(s)\end{bmatrix} \tag{2}\]  
Figure 1 shows the overall layout of the \(3\times 3\) MIMO system with a decentralized controller. First, second, and third loop effects are depicted in blue, green, and red, respectively. It is clear from the figure that loop interactions are a frequent occurrence in MIMO systems. The second and third loops' effects on the first loop are viewed as a disturbance that needs  
to be lessened in order to meet the design criteria. The second and third loops operate similarly. By creating decouplers, the loop interactions are reduced to a minimum.  
Figure 2 shows the MIMO system's schematics with a decoupler and controller. The decouplers basically reduce the interactions between the control loops.  
In addition, the control law is implemented to meet the appropriate operational standards. [PERSON] and [PERSON] [PERSON] and [PERSON] (2018) state that the decoupling matrix is  
\[D(s)=\left[\begin{array}{cccc}1&-\frac{m_{12}(s)}{m_{11}(s)}e^{(-\varphi_{ 12}-\varphi_{11})s}&\dots&-\frac{m_{1m}(s)}{m_{11}(s)}e^{(-\varphi_{1m}-\varphi _{11})s}\\ -\frac{m_{21}(s)}{m_{22}(s)}e^{(-\varphi_{21}-\varphi_{22})s}&1&\dots&-\frac{m_ {2m}(s)}{m_{22}(s)}e^{(-\varphi_{2m}-\varphi_{22})s}\\ \dots&\dots&\dots&\dots\\ -\frac{m_{21}(s)}{m_{2m}(s)}e^{(-\varphi_{21}-\varphi_{2m})s}&-\frac{m_{22}(s )}{m_{2m}(s)}e^{(-\varphi_{22}-\varphi_{2m})s}&\dots&1\end{array}\right] \tag{3}\]  
Consequently, the diagonal matrix \(J(s)\) can be calculated as  
\[J(s)=M(s)*C(s)=diag\{j_{11}(s),j_{22}(s),\dots,j_{mm}(s)\} \tag{4}\]  
For \(1=1,\ 2\), the decoupled elements \(j_{ii}\) need to be controlled.  
The higher-order system's complexity is shown in Equation (1). Additionally, the analysis gets more difficult as the system's rank rises. As a result, model reduction is preferred because controller design is challenging. The FOPDT model can be used to  
Figure 1: \(3\times 3\) MIMO system with a decentralized controller.  
Figure 2: Decentralized control.  
approximate the process dynamics though. As a result, the structure of the FOPDT model is determined using the proper model-reduction procedures. This model makes it simple to determine the gain, dead time, and time constant [31]. The scaled-down model is  
\[M_{ii}(s)=\frac{W_{ii}e^{-\mathcal{O}_{ii}(s)}}{\tau_{ii}s+1},j=1,2 \tag{5}\]  
The unknowns are identified using frequency-response fitting at two positions, \((0,\alpha_{cii})\) (\(0,\)\(cii\)), where \(\alpha_{cii}\) is the phase-crossover frequency.  
\[M_{ii}(0)=j_{ii}(0) \tag{6}\]  
\[|M_{ii}(j\alpha_{cii})|=|j_{ii}(j\alpha_{cii})| \tag{7}\]  
\[\angle\{M_{ii}(j\alpha_{cii})\}=\angle\{j_{ii}(j\alpha_{cii})\} \tag{8}\]  
The FOPDT specifications are provided by  
\[W_{ii}=\mathsf{s}j_{ii}(0) \tag{9}\]  
\[\tau_{ii}=\sqrt{\frac{W_{ii}^{2}-|j_{ii}(j\alpha_{cii})|^{2}}{|j_{ii}(j\alpha_ {cii})|^{2}a_{cii}^{2}}} \tag{10}\]  
\[\mathcal{O}_{ii}=\frac{\pi+tan^{-}(-\alpha_{cii}\tau_{ii})}{\alpha_{cii}\tau_ {ii}} \tag{11}\]