# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 1 Introduction  
In medical practice, the application of general anesthesia plays a significant role in a patient's well-being, through the administration of a combination of drugs to provide adequate hypnosis (unconsciousness and amnesia to avoid traumatic recalls), paralysis, or muscle relaxation (to attain immobility, an absence of reflexes, and proper operating conditions), and analgesia (pain relief). This process is accomplished by an anesthesiologist who must continuously observe and adjust the rates and overall amounts of anesthetic agents delivered to the patient, preserving the stability of the autonomic, cardiovascular, respiratory, and thermoregulatory systems [1]. Even though been a subject of intense research in previous decades, the process of anesthesia is a complicated process and is still not well understood, resulting in a challenging control problem [2; 3; 4]. Moreover, in drug delivery systems, the controller has to tackle issues, such as inter- and intra-patient variability, multivariable characteristics, variable time delays, dynamics dependent on the hypnotic agent, model variability, and stability issues [5; 6; 7]. The current state of the art in the understanding of consciousness and the mechanisms of anesthetic-induced loss of consciousness is limited. Consciousness is very subjective and ethereal, so it is still very difficult to model. At present, the models available are the mean field models of drug action [5], which describe the phenomena presented in electroencephalograms (EEG) associated with different brain states [7]. There have been many attempts to automatize this process, aimed at the application of closed-loop control to drug delivery systems to assist anesthesiologists to improve the safety of the patient by avoiding excessive over-dosages and under-dosages in their patients, minimizing side effects and the risk of awareness during anesthesia [8]. Optimizing the delivery of anesthetics could lead to personalized healthcare, where the individual patient characteristics are taken into account for optimal and flexible drug administration. The first step towards an automated anesthesia process is to derive a mathematical model that adequately describes the system (or the experimental data). For control purposes, we need avoid very complex models with many parameters that cannot be determined or estimated independently mainly due to the lack of measurements and adequate sensors. Commonly, the mathematical model used to study the depth of anesthesia is a PK/PD (pharmacokinetic/pharmacodynamic) model, with a third-order linear PK model and a PD model consisting of a first-order linear transfer function (which represents the time-lag between the drug infusion and the observed response) and a static nonlinearity, i.e., a Wiener structure [6]. Despite its plausibility being accepted by biomedical and control community, being a physiologically based empirical model, it presents many difficulties for controller designs, namely due to its large number of uncertain parameters resulting from significant variability among different individuals, time delay, lack of state measurements, and nonlinearity. The limited amount of real-time data and the poor excitation properties of the input signals constitute further challenges, making the identification of an online individualized model very difficult. Many attempts to simplify this model have been made, including PK/PD model structures with some fixed parameters [9; 10], first-order plus time-delay models with an output nonlinearity [11; 12], piecewise linear models [13], and low-complexity, control-oriented models [14; 11]. One notable different approach was the recent introduction of fractional PK models [15; 16]. In reference [17], the authors presented the relationship between the diffusion process and fractional-order models.