# On the Use of FOPID Controllers for Maintenance Phase of General Anesthesia
## 2 Materials and Methods
### FOPID Controller  
The transfer function of the FOPID controller is  
\[C(s)=K_{p}\Bigg{(}1+\frac{1}{T_{i}s^{\lambda}}+\frac{T_{d}s^{\mu}}{\frac{T_{d}}{N ^{\mu}}s^{\mu}+1}\Bigg{)}, \tag{8}\]  
where \(\lambda\) and \(\mu\) are the fractional-order terms of the integral and derivative actions, respectively. These new parameters introduce, compared to the PID controller, additional design flexibility, which can be exploited to improve the robustness/performance trade-off. Looking at the control scheme shown in Figure 3, it is possible to see that a back-calculation anti-windup technique has been implemented in this case, as well [28]. The advantage of the back-calculation approach is that it relies on the integrator's own dynamics, thereby making it suitable for PID and FOPID alike. To implement a fractional-order controller, it is necessary to approximate the non-integer-order part with a transfer function that consists of a series of zeros and poles of integer order. In this context, the CRONE approximation [29] was used with 8 pairs of poles and zeros and within a range of frequencies from \(10^{-4}\) to 10 (rad/s). Two different methods were employed to tune the FOPID controller. In the first case, the FOPID controller was tuned by applying the same methodology employed for the PID controller, namely, by minimizing the worst-case IAE for the set of 13 patients. The second method exploits the isodamping property. This approach consists of flattening the phase at the gain crossover frequency to guarantee zero phase margin variations with respect to changes in the process dc gain [23]. In this context, a nominal process model is needed to design the controller. The demographic values of the thirteenth patient in the previously mentioned database were used as a reference. Moreover, the model nonlinearity, i.e., the Hill function, was linearized around a BIS value of 50. Thus, the process can be written as:  
\[P(s)=K\frac{\left(\frac{s}{z_{1}}+1\right)\left(\frac{s}{z_{2}}+1\right)}{ \left(\frac{s}{p_{1}}+1\right)\left(\frac{s}{p_{2}}+1\right)\left(\frac{s}{p_{ 3}}+1\right)\left(\frac{s}{p_{4}}+1\right)}, \tag{9}\]  
where \(z_{1}\), \(z_{2}\) and \(p_{1}\), \(p_{2}\), \(p_{3}\) are the zeroes and poles that come from the PK model, \(p_{4}\) is the pole deriving from the PD model, and \(K\) is the product of the dc gains of the PK model, the PD model and the slope of the Hill function at \(BIS=50\). More precisely, the linearized Hill function is  
\[K_{Hill}=E_{max}\Bigg{[}\frac{\gamma Ce_{ref}^{\gamma-1}Ce_{50}^{\gamma}}{ \left(Ce_{ref}^{\gamma}+Ce_{50}^{\gamma}\right)^{2}}\Bigg{]}, \tag{10}\]  
where \(Ce_{ref}\) is the drug concentration value in the patient that produces a BIS equal to 50, and it is calculated as follows:  
\[Ce_{ref}=\Bigg{[}\frac{Ce_{50}^{\gamma}(E_{0}-50)}{E_{max}+50-E_{0}}\Bigg{]}^ {1/\gamma}. \tag{11}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Controller** & \(\mathbf{K_{p}}\) & \(\mathbf{T_{i}}\) & \(\mathbf{T_{d}}\) & \(\mathbf{T_{r}}\) & \(\mathbf{\Lambda}\) & \(\mathbf{\mu}\) & **IAE** \\ \hline PID & 0.0458 & 156.4 & 22.34 & 84.44 & - & - & 1685.2 \\ FOPIDIAE & 0.0393 & 59.56 & 22.94 & 89.96 & 0.8467 & 0.7978 & 1531.3 \\ FOPID ISO1 & 0.2053 & 207.1 & 13.01 & 51.90 & 0.1136 & 1.005 & 1695.9 \\ FOPID ISO2 & 0.0552 & 373.9 & 30.34 & 106.5 & 0.9994 & 1.2084 & 2354.1 \\ \hline \end{tabular}
\end{table} 
Table 1: Tuning parameters for each controller. Note that for the isodamping scenarios (ISO), the value of \(T_{t}\) is computed as \(\sqrt{T_{i}T_{d}}\). The worst-case IAE achieved on the 13 patients is shown in the IAE column. The values of \(E_{0}\), \(E_{max}\), \(Ce_{50}\) and \(\gamma\) were taken from [30].