# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 3 Anesthesia
### Proposed Model  
Under a process where a considerable amount of uncertainty exists between individuals (inter-patient variability) and within the same individual (intra-patient variability), to have a valid deterministic model is challenging (in practice, it is almost impossible or too complex to be useful). So, if we could have a model for a single patient, this model is only valid for a period because of the changes in the physiological variables during surgery, which imply a change in the parameters of the model, hence the need to update the model online. Therefore, it would be convenient to have a generic model that has the ability of capture a wide range of dynamics (in this case, a range characterizing inter-patient variability) and adapted online to individualize the model for each patient.  
Based on these facts, simple FOMs were considered. Three commensurate fractional-order models to represent the input-output behavior of the Wiener system (10-12) are proposed as follows:  
\[G_{1}(\lambda)=\frac{b_{0}}{\lambda+a_{0}}, \tag{13}\]  
\[G_{2}(\lambda)=\frac{b_{1}\lambda+b_{0}}{\lambda^{2}+a_{1}\lambda+a_{0}}, \tag{14}\]  
\[G_{3}(\lambda)=\frac{b_{2}\lambda^{2}+b_{1}\lambda+b_{0}}{\lambda^{3}+a_{2} \lambda^{2}+a_{1}\lambda+a_{0}}, \tag{15}\]  
where \(\lambda=s^{\alpha}\), \(s\) is the complex variable, and \(\alpha\) is the commensurate order, with \(0<\alpha<1\), \(a_{0}\), \(a_{1}\), \(a_{2}\) and \(b_{0}\), \(b_{1}\), \(b_{2}\) as the model parameters.  
These control-oriented models should be evaluated based on the performance in the closed-loop, rather than from their prediction capabilities, because the latter is not the objective of the model, but ideally, a good prediction capability is also desired, as well as, not instead of, a closed-loop performance.  
The proposed fractional models have a generic structure, and it can be shown that the input-output response of the patient models given by the PK/PD model can be captured by the fractional models proposed, given that the S-shape response of the patient model is part of all the possible responses of the proposed fractional models.  
These models we can see them as phenomenological models (or some kind black-box model), namely, models that can capture the input-output dynamics of the patient model, but with the disadvantage of the loss of physical meaning of the model parameters.  
It was shown in reference[74] that a set of parameters exists (depending on the respective structure) with which the fractional-order model can capture the response of a particular patient's model. However, with the use of adaptive control, we aim to control a large set of patients with one controller, so that we do not need to identify a specific set of parameters for the FOM for a given patient--we only need a model structure capable of capture the overall response.  
It is known that the particular response of a PK/PD model has an S-shape response, and reference [75; 76] showed that simple structures like those proposed can capture this type of response.  
Also, it is worth noting that one interesting application of the fractional operators and fractional differential equations is that of the study of the representation of an integer-order system by a fractional-order system. Fractional systems make it possible to carry out an efficient reduction of high-order integer-order systems, so we can represent those systems by a fractional system characterized by lower fractional order (compared with the high-order of the integer system) and a more simple structure [77; 78].