# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 2 Preliminaries  
In this section, the mathematical tools employed to develop the main results are briefly summarized. The notion of fractional-order integral of order \(\alpha>0\in\mathbb{R}\) is a natural consequence of [PERSON]'s formula for repeated integrals, which reduces the computation of the primitive corresponding to the \(n\)-fold integral of a function \(f(t)\) to a simple convolution [45]. The most common definition of a fractional-order integral is the Riemman-Liouville integral. **Definition 1**. : _The Riemann-Liouville integral is defined as_  
\[{}_{t_{0}}D_{t}^{-\alpha}f(t)=_{t_{0}}I_{t}^{\alpha}f(t)=\frac{1}{\Gamma( \alpha)}\int_{t_{0}}^{t}\frac{f(\tau)}{(t-\tau)^{1-\alpha}}d\tau, \tag{1}\]  
_where \(\alpha\in\mathbb{R}_{+}\) and_  
\[\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}e^{-t}dt. \tag{2}\]  
_is the Euler's gamma function._  
Among the existing definitions of the fractional derivative, there are three main definitions: the Grunwald-Letnikov, Riemman-Liouville and Caputo [46]. A review of the existing definitions of the fractional operator can be consulted in [47]. One of the advantages of Caputo derivative, which is a modification of the Riemann-Liouville definition, is that the initial conditions for fractional differential equations take the same form as those for integer-order differentiation which has a well-understood physical meaning. This definition is the one used in this paper. **Definition 2**. : _The Caputo derivative is defined as_  
\[{}_{t_{0}}D_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{t_{0}}^{t}\frac{f ^{(n)}(\tau)}{(t-\tau)^{\alpha-n+1}}d\tau, \tag{3}\]  
_where \(n-1<\alpha<n\in\mathbb{Z}_{+}\)._  
The following property is established using the of Riemann-Liouville integral and Caputo derivative:  
**Property 1**. : _Newton-Leibniz formula generalization [48]_  
\[{}_{t_{0}}I_{t_{0}}^{\alpha}D_{t}^{\alpha}f(x)=f(t)-f(t_{0}). \tag{4}\]  
All modeling applications of the fractional calculus give rise to the notion of fractional-order models, namely, models described by an integrodifferential equation involving fractional order derivatives. Based on the Caputo derivative, a fractional-order system can be described as  
\[{}_{t_{0}}D_{t}^{\alpha}x(t)=f(x,t), \tag{5}\]  
with initial conditions \(x(t_{0})\), where \(\alpha\in(0,1)\), \(f:\Omega\times\mathbb{R}_{+}\to\mathbb{R}^{n}\) is piecewise continuous in \(t\) and locally Lipschitz in \(x\) on \(\Omega\times\mathbb{R}_{+}\), and \(\Omega\in\mathbb{R}^{n}\) is a domain that contains the origin, \(x=0\). **Definition 3**. : _[_49_]_ _The constant \(x_{0}\) is an equilibrium point of the Caputo fractional dynamical system (5), if and only if \(f(t,x_{0})=0\)._When \(\alpha\in(0,1)\), the fractional-order system (5) has the same equilibrium points as the integer-order system \(\dot{x}=f(t,x)\)[50]. In nonlinear systems, the Lyapunov direct method provides an important tool to analyze the stability of a system without explicitly solving the differential equations. The following method generalizes the idea for fractional-order systems which shows that a system is stable if there is a Lyapunov function for the system. **Theorem 1**. : _(Fractional-order extension of the Lyapunov direct method [50]) Let \(x=0\) be an equilibrium point for the fractional-order system (5) and \(\mathbb{D}\subset\mathbb{R}^{n}\) be a domain containing the origin. Let \(V(t,x(t)):[0,\infty)\times\mathbb{D}\to\mathbb{R}\) be a continuously differentiable function and locally Lipschitz with respect to \(x\), such that_  
\[\gamma_{1}\parallel x\parallel\leq V(t,x(t))\leq\gamma_{2}\parallel x \parallel^{ab} \tag{6}\]  
\[{}_{t_{0}}D_{t}^{\alpha}V(t,x(t))\leq-\gamma_{3}\parallel x\parallel^{ab} \tag{7}\]  
_where \(t\geq 0\), \(x\in\mathbb{D}\), \(\alpha\in(0,1)\), \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}\), a and \(b\) are arbitrary positive constants. Then, \(x=0\) is Mittag-Leffler stable.