# An Analytical Design of Simplified Decoupling Smith Predictors for Multivariable Processes
## 2 Materials and Methods  
### Simplified Decoupling Smith Predictor Structure  
Consider an \(n\)-input and \(n\)-output open-loop multivariable process with the general transfer function matrix for stable, square, and multidelays, which is represented by the following matrix:  
\[\mathbf{G}(s)=\left[ \begin{array}{cccc}g_{11}(s)&g_{12}(s)&\ldots&g_{1n}(s)\\ g_{21}(s)&g_{22}(s)&\ldots&g_{2n}(s)\\ \vdots&\vdots&\vdots&\vdots\\ g_{n1}(s)&g_{n2}(s)&\ldots&g_{m}(s)\end{array} \right] \tag{1}\]  
where \(g_{ij}(s)=g_{ij0}(s)e^{-\theta_{ij}s}\), \(i,j=1,2,\ldots,n\), of which \(g_{ij0}(s)\) denotes the physically proper, stable, and delay-free transfer function. \(\theta_{ij}\) represents the time delay. The multivariable Smith predictor control scheme is shown in Figure 1, where \(\mathbf{G}(s)\) and \(\mathbf{\hat{G}}(s)\) are the process and its model, respectively. \(\mathbf{\hat{G}}_{0}(s)\) is also the process model but all the delays have been eliminated. \(\mathbf{G}_{c}(s)\) is the primary controller of the following form:  
\[\mathbf{G}_{c}(s)=\left[ \begin{array}{cccc}g_{c1}(s)&0&\ldots&0\\ 0&g_{c2}(s)&\ldots&0\\ \vdots&\vdots&\vdots&\vdots\\ 0&0&\ldots&g_{\alpha n}(s)\end{array} \right]. \tag{2}\]  
The closed-loop transfer function matrix from set-points to outputs becomes  
\[\mathbf{T}(s)=\mathbf{G}(s)\mathbf{G}_{c}(s)\big{(}\mathbf{I}+\big{(}\mathbf{ \hat{G}}_{o}(s)-\mathbf{\hat{G}}(s)\big{)}\mathbf{G}_{c}(s)+\mathbf{G}(s) \mathbf{G}_{c}(s)\big{)}^{-1} \tag{3}\]  
When the model is perfect, \(\mathbf{\hat{G}}(s)=\mathbf{G}(s)\), and Equation (3) is rewritten as  
\[\mathbf{T}(s)=\mathbf{\hat{G}}(s)\mathbf{G}_{c}(s)\big{(}\mathbf{I}+\mathbf{ \hat{G}}_{o}(s)\mathbf{G}_{c}(s)\big{)}^{-1} \tag{4}\]  
The time delay terms have been removed from the denominators of the closed-loop system transfer matrix. Similarly, there are no delay terms in the transfer function matrix from disturbances to outputs, and when the model is perfect, (\(\mathbf{\hat{G}}(s)=\mathbf{G}(s)\)). Therefore, the primary controller \(\mathbf{G}_{c}(s)\) can be designed for the delay-free part \(\mathbf{\hat{G}}_{0}(s)\) and it is the main attractiveness of this control scheme. Equation (4) can be rewritten as follows:  
\[\mathbf{T}(s)=\mathbf{\hat{G}}(s)\mathbf{\hat{G}}_{0}{}^{-1}(s)\mathbf{T}_{0} (s) \tag{5}\]  
where \(\mathbf{T}_{o}(s)=\mathbf{\hat{G}}_{o}(s)\mathbf{G}_{c}(s)\big{(}\mathbf{I}+ \mathbf{\hat{G}}_{o}(s)\mathbf{G}_{c}(s)\big{)}^{-1}\) is the closed-loop transfer matrix corresponding to the designed controller of the delay-free part. Figure 1: Block diagram of multivariable Smith predictor (SP) control. It can be seen that the actual system performances could not be guaranteed because of the \(\mathbf{\hat{G}}(s)\mathbf{\hat{G}}_{o}{}^{-1}(s)\) part in Equation (5), regardless of how good the performance of \(\mathbf{T}_{0}(s)\) is. In special cases where all delay terms of each row of the transfer function matrix \(\mathbf{\hat{G}}(s)\) are identical, \(\mathbf{\hat{G}}(s)\) can be rewritten by \(diag\big{[}e^{-\theta_{ss}}\big{]}\mathbf{\hat{G}}_{0}(s)\). Consequently, \(\mathbf{T}(s)=diag\big{\{}e^{-\theta_{ss}}\big{\}}\mathbf{T}_{0}(s)\) and the system outputs are the delayed outputs of \(\mathbf{T}_{0}(s)\); that is, to some extent, the system performance and robustness still meet requirements. In most cases, however, this desired property is not preserved. In order to overcome this problem and improve the performance of the multivariable Smith predictor control system, a decoupling Smith predictor (DSP) system was proposed (Figure 2), where \(\mathbf{D}(s)\) is a decoupling matrix for the system transfer function matrix \(\mathbf{G}(s)\), \(\mathbf{Q}(s)\) is the decoupled process, and \(\mathbf{Q}(s)=\mathbf{G}(s)\mathbf{D}(s)\). \(\mathbf{Q}_{0}(s)\) is the same as \(\mathbf{Q}(s)\) except that all the delays are removed. It is obvious that the \(\mathbf{Q}(s)\) and \(\mathbf{Q}_{0}(s)\) will be diagonal matrices due to the properties of decoupling techniques.