# Fractional-Order Closed-Loop Model Reference Adaptive Control for Anesthesia
## 4 Fractional-Order Closed-Loop Model Reference Adaptive Control  
One recent approach of adaptive control that improves transient behavior by using a closed-loop architecture for reference models, named closed-loop model reference adaptive control (CMRAC), has been proposed [79]. In this approach, the focus is to have an adaptive system with output feedback. It has been shown that such closed-loop reference models can lead to a separation principle-based adaptive controller which is simpler to implement than classical ones based on observers or filtered signals. The simplification comes with the use of the reference model states in the construction of the regressor and not the classic approach where the regressor is constructed from filtered plant inputs and outputs. Next, the generalizations of this scheme (Figure 3), denoted the fractional-order closed-loop model reference adaptive control (FOCMRAC), are presented. A preliminary version of this extension was presented in reference [80]. The commensurate fractional-order system, which is a state-space representation of the models (13-15) is given by  
\[ \begin{split}{}_{0}D_{t}^{\mathrm{z}}x(t)=& Ax(t)-B \Lambda u(t)\\ z(t)=& C^{T}x(t)\end{split}  \tag{16}\]  
Figure 3: FOCMRAC (Fractional-Order Closed-Loop Model Reference Adaptive Control) scheme. where \(x\in R^{n}\), \(n\leq 3\), \(u\in R\), and \(z\in R\). \(A\) and \(\Lambda\) are unknown, \(B\) y C are known, and only \(z\) is available for measurement. The control objective is to design a control signal, \(u\), such that \(x\) follows the state, \(x_{m}\), of the reference model given by  
\[ \begin{split} 0D_{t}^{\alpha}x_{m}(t)=& A_{m}x_{m}(t)- Br(t)-L(z-z_{m})\\ z_{m}(t)=& C^{T}x_{m}(t)\end{split}  \tag{17}\]  
where \(r\in R\) is the reference signal, and \(L\) is a feedback gain to be designed. The following assumptions are made:  
* The product, \(C^{T}B\), is full rank. * The pair, \(A_{m},C^{T}\), is observable. * The system in (16) is at the minimum phase. * \(\Theta^{*}\in R^{nx1}\) exists such that \(A+B\Lambda\Theta^{*T}=A_{m}\) and \(K^{*}\in R\) such that \(\Lambda K^{*T}=I\) (_matching conditions_). * \(\Lambda\) is diagonal with positive elements. * The uncertain matching parameter, \(\Theta^{*}\), and the input uncertainty matrix, \(\Lambda\), have _a priori_ upper bounds. \[\bar{\theta}^{*}\triangleq sup\|\Theta^{*}\|\;\;\text{and}\;\;\bar{\lambda} \triangleq sup\|\Lambda\|.\] (18)  
Next, some results needed for the design and analysis of the control scheme are presented. The proofs can be found in reference [79]. Lemma 4: _For the SISO case, in system (16), satisfying the suppositions 1-3, a \(L_{s}\) exists, such that_  
\[C^{T}(sI-A_{m}-L_{s}C^{T})^{-1}B=\frac{a}{s+\rho} \tag{19}\]  
_where \(\rho>0\) is an arbitrary parameter and \(a=C^{T}B\)._  
Lemma 5: _If \(L_{s}\) is chosen as (19) and \(M\triangleq C^{T}B\), the SISO transfer function, \(M^{T}C^{T}(sI-A_{m}-L_{s}C^{T})^{-1}B\), is SPR. Therefore, \(P=P^{T}>0\) and \(Q_{s}=Q_{s}^{T}>0\) exist, such that_  
\[ \begin{split}(A_{m}+L_{s}C^{T})^{T}P+P(A_{m}+L_{s}C^{T})=& -Q_{s}\\ PB=& CM.\end{split}  \tag{20}\]  
Lemma 6: _Choosing \(L=L_{s}-\rho BM^{T}\) where \(L_{s}\) is defined by (19) andis arbitrary, \(\rho>0\), the transfer function \(M^{T}C^{T}(sI-A_{m}-LC^{T})^{-1}B\) is SPR and satisfies_  
\[ \begin{split}(A_{m}+LC^{T})^{T}P+P(A_{m}+LC^{T})=&- Q\\ Q\triangleq& Q_{s}+2\rho CM^{T}C^{T}\end{split}  \tag{21}\]  
_where \(P\) and \(Q_{s}\) are defined in (20) and \(M=C^{T}B\)._  
Assuming that \(L\) is chosen using the above lemmas, in the following theorem, the adaptive scheme is formulated. Theorem 3: _Consider the fractional-order system given by (17) that satisfies assumptions 1-6 and the closed-loop model reference given by (17) and the control law_  
\[u(t)=\Theta^{T}(t)x_{m}+K^{T}(t)r(t) \tag{22}\]with adaptive laws_  
\[ \begin{split}{}_{0}D_{t}^{a}\Theta&=-\Gamma x_{m}e_{y}^ {T}\\ {}_{0}D_{t}^{a}K&=-\Gamma re_{y}^{T}\end{split}  \tag{23}\]  
_where \(e_{y}=C^{T}e\), \(\Gamma>0\).