# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 5 Fractional-Order Parameter Identifier  
The next identification scheme is an extension for fractional-order systems of the scheme presented in [27]. This identification scheme does not need a state measurement and only uses information about the input and output to construct the identifier. This scheme will be used to show that the fractional model proposed in the next section captures the behavior of the PK/PD model. Consider the commensurate fractional-order SISO system:  
\[ \begin{split}{}_{0}D_{t}^{a}x=& Ax+Bu,\\ z=& C^{T}x,\end{split}  \tag{30}\]  
where \(x\in R^{n}\), \(n\leq 3\), and only \(y,u\) are available for measurement. System (30) can be written as \[z=\frac{B(\lambda)}{A(\lambda)}u=C^{T}(\lambda I-A)^{-1}Bu, \tag{31}\]  
where \(\lambda=s^{\alpha}\) and \(\alpha\) is the commensurate order with \(0<\alpha<1\), and \(A(\lambda),B(\lambda)\) are in the forms  
\[A(\lambda)= \lambda^{n}+a_{n-1}\lambda^{n-1}+\cdots+a_{1}\lambda+a_{0}, \tag{32}\] \[B(\lambda)= b_{m}\lambda^{m}+b_{m-1}\lambda^{m-1}+\ldots b_{1}\lambda b_{0},\]  
where the constants \(a_{i}\) and \(b_{i}\) for \(i=0,1,2,\ldots,n-1\) are the system parameters, \(n\leq 2\) and \(m\leq 1\). Now, consider the linear parameterization of (31):  
\[z=W(\lambda)\theta^{*T}\psi, \tag{33}\]  
where \(\theta^{*T}\) is the parameter vector, \(\psi\) is the regressor that contains the filtered measurable signals, \(u,y\), and \(W(\lambda)\) is a proper stable transfer function. \[\theta^{*} = [b_{m},b_{m-1},\ldots,b_{0},a_{n-1},a_{n-2},\ldots,a_{0}]^{T}, \tag{34}\] \[z = \frac{1}{\Lambda(\lambda)}y=\frac{\lambda^{n}}{\Lambda(\lambda)}y,\] (35) \[\phi = \left[\frac{\beta_{n-1}^{T}(\lambda)}{\Lambda(\lambda)}u,-\frac{ \beta_{n-1}^{T}(\lambda)}{\Lambda(\lambda)}y\right],\] (36) \[\Lambda(\lambda) = \lambda^{n}+\eta_{n-1}\lambda^{n-1}+\eta_{n-2}\lambda^{n-2}+ \cdots+\eta_{0}\] (37) \[= \lambda^{n}+\eta^{T}\beta_{n-1}(\lambda). \tag{38}\]  
With the parameterization (33), the signals \(z\) and \(\psi\) can be generated only with the information of \(u\) and \(y\). Because \(\theta^{*}\) is a constant vector, we can write (33) in the form \(z=W(\lambda)L(\lambda)\theta^{*T}\phi\), where \(\phi=L^{-1}(\lambda)\psi\). \(L(s)\) is chosen so that \(L^{-1}(\lambda)\) is a proper stable transfer function, and \(W(\lambda)L(\lambda)\) is a proper SPR transfer function. A state-space representation of the parameterization (33) is given by  
\[{}_{0}D_{t}^{\alpha}\phi_{1}=\Lambda_{c}\phi_{1}+lu, \tag{39}\] \[A(\lambda)= \lambda^{n}+a_{n-1}\lambda^{n-1}+\cdots+a_{1}\lambda+a_{0}, \tag{32}\]0 \[A(\lambda)= \lambda^{n}+a_{n-1}\lambda^{n-1}+\cdots+a_{1}\lambda+a_{0}, \tag{32}\]1  
where  
\[\Lambda_{c}= \begin{bmatrix}-\eta_{n-1}&-\eta_{n-2}&\ldots&-\eta_{0}\\ 1&0&\ldots&0\\ \vdots&\ddots&&\vdots\\ 0&\ldots&1&0\end{bmatrix} ,\;l= \begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix} . \tag{40}\]  
Because \(\Lambda(\lambda)=det(\lambda I-\Lambda_{c})\) and \(\Lambda(\lambda)\) is stable, it follows that \(\Lambda_{c}\) is a stable matrix. The state-space model (39) has the same input-output response as (30) and (31), provided that all state initial condition are \(x_{0}=0\), \(\phi_{1}=\phi_{2}=0\). Now the identification scheme is formulated in the following theorem. **Theorem 4**. : _Consider that the system (30), the linear parameterization (33) and the matrices \(A_{c}\), \(B_{c}\) and \(C_{c}\) are associated with the state-space system, \(W(\lambda)L(\lambda)=C_{c}^{T}(\lambda I-A_{c})^{-1}B_{c}\), and the adaptive law_  
\[{}_{0}D_{t}^{\alpha}\dot{\theta}=\Gamma e\phi, \tag{41}\]_where \(\Gamma>0\) is the adaptation gain, and \(\hat{\theta}\) are the estimates \(\theta^{*}\) in (33), and the identification error is given by \(e=z-\hat{z}\) and \(\epsilon=C_{c}e\). Then, the identification error is \(\epsilon\to 0\) as \(t\to 0\)._  
Proof. : The dynamics of the identification error are given by  
\[{}_{0}D_{t}^{\alpha}e=A_{c}e-B_{c}\tilde{\theta}\phi, \tag{42}\]  
where the parameter error is defined as \(\tilde{\theta}=\hat{\theta}-\theta^{*}\).