## 2 Materials and Methods
### IMC-Fractional P/PID Controller Design  
By using the controller structure as mentioned above, the multivariable processes become multi-loop systems. For each loop, a corresponding controller needs to be designed to meet the requirements of its closed loop responses. In this study, a new structure of a fractional PID controller is proposed for each loop, called \(1^{\sigma}\)PI\({}^{\lambda}\)D\({}^{\mu}\). In the case of the higher order process, Equation (19), a first order filter is also employed to improve its performances. Let the primary controller of each loop be as Equation (21):  
\[g_{ci}(s)=K_{ci}\frac{1}{s^{\sigma_{i}}}\bigg{(}1+\frac{1}{\tau_{li}s^{\lambda _{i}}}+\tau_{Di}s^{\mu_{i}}\bigg{)}F_{i}(s) \tag{21}\]  
where \(K_{ci},\tau_{li}\) and \(\tau_{Di}\) are proportional gain, integral time and derivative time respectively; \(\lambda_{i},\mu_{i}\) are fractional order of the integral and derivative terms; \(\sigma_{i}\) is the fractional order of the ideal integral and \(\sigma_{i}=1-\lambda_{i}\), in special case, when \(\lambda_{i}=1\) (integral term with integer order) then \(\sigma_{i}\) equals to zero; \(F_{i}(s)\) is the first order filter, \(F_{i}(s)=\frac{1}{\tau_{fi}s+1}\) where \(\tau_{fi}\) is a time constant. The IMC based PID procedure normally used for integer order processes is also addressed to design the proposed controller, \(1^{\sigma}\)PI\({}^{\lambda}\)D\({}^{\mu}\), for fractional order processes. Figure 2a,b show block diagrams of feedback control strategies including the classical feedback control and the internal model control as well [10; 11; 12; 13; 23]. Note that, in this case, the controlled process is a fractional order transfer function without delay time. According to the IMC theory, the process model is divided into two parts:  
\[\overline{q}_{io}(s)=p_{M}(s)p_{A}(s) \tag{22}\]  
where \(p_{A}(s)\) contains delay time terms and/or RHP zeros and \(p_{A}(0)=1\). According to Equations (19) and (20), \(p_{A}(s)=1\). Generally, the IMC controller is designed as  
\[\overline{g}_{ci}(s)={p_{M}}^{-1}(s)f_{i} \tag{23}\]  
The term \(f\) is called the IMC filter and normally has the form as follow:  
\[f_{i}=\frac{1}{(\tau_{ci}s+1)^{\tau_{i}}} \tag{24}\]  
where \(\tau_{ci}\) is an adjustable parameter which controls the tradeoff between the performance and robustness; \(r_{i}\) is relative order and to be selected large enough to make the IMC controller (semi-) proper. Substituting Equation (24) into Equation (23):  
\[\overline{g}_{ci}(s)={p_{M}}^{-1}(s)\frac{1}{(\tau_{ci}s+1)^{\tau_{i}}} \tag{25}\]  
Therefore, the ideal feedback controller for achieving the desired loop responses can be easily obtained by:  
\[g_{ci}(s)=\frac{\overline{g}_{ci}(s)}{1-\overline{q}_{io}(s)\overline{g}_{ci} (s)}=\frac{{p_{M}}^{-1}(s)}{(\tau_{ci}s+1)^{\tau_{i}}-{p_{A}(s)}}=\frac{{p_{M} }^{-1}(s)}{(\tau_{ci}s+1)^{\tau_{i}}-1} \tag{26}\]  
In this work, there are two cases of a process model to be considered:  
_Case 1_: The fractional first order system:  
\[\overline{q}_{io}(s)=\frac{K_{i}}{\tau_{i}s^{\alpha_{i}}+1}\ \ (0<\alpha_{i}<1) \tag{27}\]  
The proposed IMC filter structure  
\[f_{i}=\frac{1}{\tau_{ci}s+1} \tag{28}\]  
The ideal feedback controller is derived by:  
\[g_{ci}(s)=\frac{\tau_{i}s^{\alpha_{i}}+1}{K_{i}\tau_{ci}s}=\frac{\tau_{i}}{K_ {i}\tau_{ci}}\frac{1}{s^{1-\alpha_{i}}}\bigg{(}1+\frac{1}{\tau_{i}s^{\alpha_{ i}}}\bigg{)} \tag{29}\]  
Figure 2: The one degree of freedom feedback control diagram. (**a**) Classical feedback control (**b**) Internal model control.