# On the Use of FOPID Controllers for Maintenance Phase of General Anesthesia
## 1 Introduction  
Many closed-loop control algorithms for total intravenous anesthesia (TIVA) have been developed in the last several years [1; 2] because of the benefits that an automatic control system can provide in this context. In fact, an effective control system can be a valuable tool for the anesthesiologist, who supervises the overall system, because it has the potential to limit errors caused by distraction and to reduce possible drug overdosing and underdosing, thereby increasing the safety of the patient. Although in TIVA, analgesia and neuromuscular blockade should also be provided, the main task generally requires controlling the Depth of Hyponsis (DoH) through the hypnotic drug propofol. A feedback control system can be implemented by measuring the DoH by means of a processed electroencephalogram (pEEG) mechanism. Among these techniques, the most common one is the Bispectral Index Scale (BIS), which provides a measure of the DoH ranging from 100 (fully awake patient) to 0 (flat EEG). The overall anesthesia process consists of three phases. At the beginning, there is the induction phase, where the BIS has to be driven to the set-point value of 50 in a time interval of less than 5 min and without an excessive undershoot that might cause problems for the patient. Usually, this is achieved through the administration of a bolus of propofol, where the amount of drug depends on the physical characteristics of the patient (thus, a feedforward action is applied from a control system point of view). Once the BIS has settled to the set-point value, the maintenance phase starts. During maintenance, the control task is to keep the BIS in the range from 40 to 60 despite the surgical activity and the presence of noxiousstimuli (excessively low values can result in dangerous hypotension or burst suppression episodes, while excessively high values can result in traumatic patient awareness). Finally, in the emergence phase, drug administration is stopped, and the patient wakes up. The maintenance phase can last for hours and is critical in anesthesia, which implies that a control system should be designed by mainly considering the disturbance rejection task. Further, the controller should also be robust to intra- and inter-patient variability, since satisfactory clinical outcomes should be guaranteed for all patients. The nonlinearity of the pharmacokinetic/pharmacodynamic (PK/PD) model [3] makes the control design challenging, and for this reason, many approaches have been proposed in the literature. Proportional-Integral-Derivative (PID) controllers have been widely investigated, as they are the most widespread controllers in industry, and, indeed, they have also been proven to be very effective in anesthesia [3; 4; 5; 6; 7; 8; 9]. Some variations of the standard PID algorithm have also been devised, including the event-based PID controller, which has been shown to be able to better mimic the behavior of the anesthesiologist [9]. In the context of PID control, a great interest is in the use of fractional-order PID (FOPID) controllers, as they provide additional flexibility in the controller design, which, in principle, should result in a better performance. Indeed, a FOPID controller has five parameters to tune: in addition to the three classic parameters (proportional gain, integral time constant and derivative time constant), the integral and derivative orders can also be freely chosen as real numbers, and this implies, for example, that a loop-shaping design can be improved. The use of fractional control in anesthesia has already been considered. In particular, a CRONE controller was first applied in [10] for DoH control, where the _WAV_CNS system was used instead of the BIS, resulting in a high-order controller. The CRONE approach was also used in [11] for the design of a fault-tolerant control system consisting of a PID controller with a fractional-order filter.