# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 2 Preliminaries
If the assumptions hold globally on \(\mathbb{R}^{n}\), then \(x=0\) is globally Mittag-Leffler stable._  
Similar to integer-order systems, the idea of this fractional-order extension of the Lyapunov theorem is that the stability condition is derived by constructing a positive definite function, \(V\), and calculating the fractional derivative of the \(V\) function. Next is a version of Theorem 1 that is useful for the analysis of adaptive systems, especially when the \({}_{t_{0}}D_{t}^{\alpha}V(x,t)\) negative semidefinite is present. **Theorem 2**. : _[_51_]_ _Let \(x=0\) be an equilibrium point for the non-autonomous, fractional-order system (5). Let us assume that a continuous function exists, \(V(x,t)\), such that_  
* \(V(x,t)\) _is positive definite._
* \({}_{t_{0}}D_{t}^{\alpha}(x,t)\)_, with_ \(\alpha\in(0,1]\)_, is negative semidefinite._  
_Then, the origin of the system (5) is Lyapunov stable. Furthermore, if \(V(x(t),t)\) is decrescent, then the origin of system (5) is Lyapunov uniformly stable._  
One of the most used Lyapunov candidate functions to analyze the stability of integer-order system is the quadratic functions. However, in fractional cases, the use of these functions is not immediate, since evaluating the fractional derivative of the Lyapunov candidate function, in general, involves the evaluation of infinite sums, which include higher order integrals and derivatives of the fractional states of the system, which is not an easy task [52]. Next, some inequalities that facilitate the use of quadratic Lyapunov candidate functions in the analysis of stability of fractional-order system using [PERSON]'s direct method are presented. **Lemma 1**. : _[_51_]_ _Let \(x(t)\in\mathbb{R}^{n}\) be a vector of differentiable functions. Then, for any time instant \(t\geq t_{0}\), the following relationship holds:_  
\[\frac{1}{2}\,{}_{t_{0}}D_{t}^{\alpha}(x^{T}(t)Px(t))\leq x^{T}(t)P_{t_{0}}D_{t }^{\alpha}x(t) \tag{8}\]  
_where \(P\in\mathbb{R}^{nxn}\)  
[PERSON]'s lemma is a well-known and useful tool to deduce the asymptotic stability of integer-order nonlinear uncertain and time-varying systems (like adaptive systems). Due to some different properties between fractional-order integrals and integer-order integrals, it is not easy (but is possible) to use [PERSON]'s lemma in fractional-order systems. In this paper, we will use the extension of this lemma for fractional-order systems that was presented in reference [43]. Lemma 3: _(Extension of [PERSON]'s lemma [43]) Let \(f:\mathbb{R}\to\mathbb{R}_{+}\) be a function that is uniformly continuous, and \({}_{t_{0}}I_{t}^{a}\) is given by the Riemann-Liouville integral with \(\alpha\in(0,1)\)._  
_If \(\lim\limits_{t\to\infty}{}_{t_{0}}I_{t}^{a}f(t)\) exists and is finite, then \(f(t)\to 0\), as \(t\to\infty\)._  
This extension of the [PERSON] lemma is very useful for concluding the convergence of the error in fractional-order adaptive systems. Other important concepts are the positive realness (PR) and strictly positive realness (SPR). According to [53; 54], these concepts for integer-order systems are valid for fractional-order commensurate systems with \(\alpha\in(0,1)\). Definition 4: The \(m\times m\) transfer function matrix \(G(s)\) is called strictly positive real (**SPR**) if  
1. All elements of \(G(s)\) are analytic in \(\mathcal{R}\geq 0\). 2. \(G(s)\) is real for real \(s\). 3. \(G(s)+G^{\mathcal{T}*}(s)\) for \(\mathcal{R}\geq 0\) and a finite \(s\). A more extense review on fractional-order Lyapunov theory can be consulted in references [55; 56].