# An Analytical Design of Simplified Decoupling Smith Predictors for Multivariable Processes
## 2 Materials and Methods
### IMC-PI/PID Approach for Controller Design  
The multivariable Smith predictor is based on the multiple single-loop Smith predictor control design. Let the primary controller be as in Equation (2), and then, each diagonal controller has the standard form of the PID controller in series with the FOLF:  
\[g_{ci}(s)=K_{c}\bigg{(}1+\frac{1}{\tau_{iI\!S}}+\tau_{D\!i}s\bigg{)}\frac{1}{ \tau_{Fi}s+1}. \tag{21}\]  
The \(g_{ci}(s)\) was designed with respect to the delay-free part \(q_{i0}(s)\) of \(q_{i\!i}(s)\) such that the closed-loop system formed by \(g_{ci}(s)\) and \(q_{i0}(s)\) had the desired performance. Figure 5 shows block diagrams of feedback control strategies, including the classical feedback control and the internal model control as well [24]. Note that, in this case, the controlled plant is a delay-free transfer function.  
According to the IMC theory, the process models are factored into two parts:  
\[q_{i0}(s)=p_{M}(s)p_{A}(s) \tag{22}\]  
Figure 5: The one-degree-of-freedom feedback control diagram: (**a**) classical feedback control; (**b**) internal model control.  
where \(p_{A}(s)\) contains delay time terms and/or right half plane (RHP)zeroes and \(p_{A}(0)=1\). According to Equations (18)-(20), \(p_{A}(s)=1\).  
Generally, the IMC controller is designed as  
\[\widetilde{g}_{ci}(s)={p_{M}}^{-1}(s)f_{i}. \tag{23}\]  
The term \(f\) is called the IMC filter and normally has the following form:  
\[f_{i}=\frac{\sum\limits_{j=1}^{m}{\alpha_{ij}s^{j}}+1}{{\left({\lambda_{i}s+1} \right)^{r}}}. \tag{24}\]  
There are some criteria used to choose the appropriate value of \(\lambda_{i}\) which can be found in the available literature. For example, [PERSON] (2003) suggested choosing \(\lambda\) to be equal to the time delay of the process. However, in this work, \(\lambda_{i}\) was an adjustable parameter which controlled the tradeoff between performance and robustness; \(r\) was the relative order and had to be large enough to make the IMC controller (semi)proper; and \(\alpha_{ij}\) was chosen to cancel the \(j\)th unstable poles in the process model.  
\[\widetilde{g}_{ci}(s)={p_{M}}^{-1}(s)\frac{\sum\limits_{j=1}^{m}{\alpha_{ij}s^ {j}}+1}{{\left({\lambda_{i}s+1}\right)^{r}}}. \tag{25}\]  
Therefore, the ideal feedback controller for achieving the desired loop responses can be easily obtained by  
\[g_{ci}(s)=\frac{\widetilde{g}_{ci}(s)}{1-\widetilde{q}_{i0}(s)\widetilde{g}_{ ci}(s)}=\frac{{p_{M}}^{-1}(s){\left({\sum\
olimits_{j=1}^{m}{\alpha_{ij}s^ {j}}+1}\right)}}{{\left({\lambda_{i}s+1}\right)^{r}}-{p_{A}(s){\left({\sum \
olimits_{j=1}^{m}{\alpha_{ij}s^{j}}+1}\right)}}}. \tag{26}\]  
The resulting controller given by Equation (26) normally does not have the standard PI/PID controller form despite it being physically realizable. To take advantage of an available commercial PID controller, it must be converted into the equivalent PID type. In this case, it is necessary to utilize some approximation techniques. In this paper, Maclaurin expansion was used to approximate a PID controller for some typical industrial processes such as the transfer functions (18-20).  
Since the feedback controller has the offset-free integral term, \(g_{ci}(s)\) can be rewritten as  
\[g_{ci}(s)=s^{-1}p_{i}(s). \tag{27}\]  
The rational approximation form of Equation (26) can be found by expanding \(g_{ci}(s)\) in a Maclaurin series [1; 2; 5]:  
\[g_{ci}(s)=\frac{1}{s}\bigg{[}p_{i}(0)+p_{i}^{\prime}(0)s+\frac{1}{2!}p_{i}^{ \prime\prime}\left(0\right)s^{2}+\frac{1}{3!}p^{\prime\prime\prime}\left(0 \right)s^{3}+\ldots\bigg{]}. \tag{28}\]