# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 1 Introduction
Also, fractional-order pharmacokinetic models [15; 16; 18; 19] that represent the experimental data in a more precise way thanks to the \(t^{-\alpha}\) decay of the fractional operators, were introduced, and with this, a new line of investigation on the area of drug delivery systems has opened. The problem of control of drug administration in anesthesia has been studied since the 1950s [20]. It is clear that the control of anesthesia has many challenges, such as multivariate features [21], different dynamics that are dependent on the drug and place of administration [22; 23], stability problems [24], and performance of the control algorithm [25; 26]. Given the complex nature and uncertainty of the process, it is not surprising that reliable models for the control of drug administration are not available. The level of system uncertainty of fixed and robust gain controllers can unnecessarily sacrifice system performance, while adaptive controls can tolerate much higher levels of uncertainty and improve performance [27]. The control of anesthesia has been an active subject of research in past decades, and many control schemes have been developed, for example, fixed-gain controllers, mostly PID, were considered in references [25; 28; 29; 30; 31; 32]. Adaptive controllers were developed in [33; 34; 35]. To deal with delays in the system, predictive controllers were used in reference [36; 37; 38]. Also, sliding mode control has been used in [39] and control based on neural networks in [40; 41]. There have been a few published studies using a fractional control [42; 43; 2]. The implementation of such controllers is based on the assumption that the state measurements are available and that we have a clear and measurable output with little noise influence. However, in reality, the measured output may be noisy, and the system measurements do not produce this information directly. Instead, the state information needs to be inferred from the available output measurements. All of these challenges bring us the need to use estimation techniques that can estimate the state of each patient and adjust them based on the dynamics of each patient to deal with the system constraints [44]. This paper considers the problem of designing a fractional-order closed-loop model reference adaptive control (FOCMRAC) for anesthesia. To this end, three simple fractional-order model (FOMs) are proposed, which are shown to be able to follow the anesthesia dynamics of the PK/PD model. Then, based on these models, a simple FOCMRAC is designed. The stability of the closed-loop system is established using Lyapunov analysis. The main contributions of this paper are the proposal of the FOCMRAC scheme for fractional-order systems and its application to the control of anesthesia. Also, according to the best knowledge of the authors, the application of fractional-order controllers in control of anesthesia is very limited, so a novel and direct approach is proposed using fractional calculus tools. Section 2 presents the mathematical tools used for the main results. Section 3 introduces the principles, the PK/PD model, and the challenges of the anesthesia process and presents the proposedFOMs. Section 4 gives the design of the FOCRMAC using the Lyapunov analysis. Section 5 gives the design of an identification scheme for fractional-order systems. Section 5 shows the simulation results using 30 virtual patients, and Section 6 presents the conclusions.