# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 6 Simulations
### Inter-Patient Robustness  
The inter-patient variability denotes the variation in the mathematical models among individuals. Every patient has a specific model.  
The simulations were performed using 30 virtual patients applied to four different control schemes. The FOMRAC presented in reference [43] and the proposed FOCMRAC given by Theorem 3 were tested.  
In the case of the FOCMRAC scheme, three controllers were implemented, each one based on the models (13-15) respectively, and applied to the 30 virtual patients.  
Figure 9 shows the response of the 30 virtual patients with the four control schemes. We can observe that all schemes met the control objective-taking all of the patients to \(BIS=50\). It can be seen that the controllers based on the second-order (14) and third-order models (15) had better performances than the controllers based on the first-order structure (13).  
Figure 10 shows the control input of the four implemented controllers and it a similar behavior can be seen between the controllers.  
In Figure 11, the tracking errors are shown. The errors of the FOMRAC scheme caused more oscillations in the induction phase and a higher convergence time than in the FOCMRAC schemes.  
Figure 9: BIS output of the 30 virtual patients with the FOMRAC and FOCMRAC schemes.  
In Figures 12-15 are shown the evolution of the controller parameters.  
Figure 11: Tracking errors of the FOMRAC and FOCMRAC schemes with the 30 virtual patients.  
Figure 10: Control input of the FOMRAC and FOCMRAC schemes with the 30 virtual patients.  
Figure 12: Controller parameters of the FOMRAC scheme using the 1 st-order structure.  
Figure 13: Controller parameters of the FCOMRAC scheme using the 1 st-order structure.  
To evaluate controller performance, performance measures were calculated during the maintenance phase of anesthesia, in accordance with [31].  
Figure 14: Controller parameters of the FOCMRAC scheme using the 2 nd-order structure.  
Figure 15: Controller parameters of the FOCMRAC scheme using the 3 rd-order structure.  
The performance error (PE) was calculated as the difference between the actual and the target values:  
\[PE=100\left(\frac{BIS_{measured}-BIS_{target}}{BIS_{target}}\right). \tag{50}\]  
The bias or median performance error (MDPE) described whether the measured values were either above or below the target values and thus represented the direction (under-shoot or over-shoot) of the PE:  
\[MDPE=median(PE). \tag{51}\]  
The inaccuracy or median absolute performance error (MDAPE) described the size of the errors:  
\[MDAPE=median(|PE|). \tag{52}\]  
The Wobble measured the intra-individual variability in the PE:  
\[Wobble=median(|PE-MDPE|). \tag{53}\]  
The global score (GS) was calculated according to the following equation:  
\[GS=\frac{MDAPE+Wobble}{\%\text{ }time\text{ }BIS\text{ }value\text{ }between\text{ }40\text{ }and\text{ }60}. \tag{54}\]  
The overall control performances of the FOMRAC scheme [43] and the proposed FOCRMAC scheme are shown in Table 3.  
From Table 3, we can observe that the three FOCRMAC controllers led to an overall improved performance compared to the FOMRAC scheme. Moreover, the performance also improved depending on the model used, for example, the MDPE, MDAPE, Wobble, and GS indices decreased with an increase in the order of the controller, thus improving performance.