# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 6 Application  
In this section we examine an application of the results previously presented to a control problem. In this application, we build a FOC for an IOS based on the design of an FOO (MO approach). This way of using FOO in classic IO problems have also been explored before in [28; 34] for adaptive and nonlinear systems. It has been observed that the FO of the controller/observer is a relevant optimization parameter for standard performance criteria. Besides, fault detection applications using our results can also be done based on the ideas contained in [35].  
Let us consider the IO LTIS defined as  
\[\left\{\begin{array}{l}\dot{x}=Ax+bu\\ y=c^{T}x\\ x(0)=x_{0}\end{array}\right. \tag{24}\]  
where \(x:[0,\infty)\rightarrow\mathbb{R}^{n}\), \(u:[0,\infty)\rightarrow\mathbb{R}\), \(y:[0,\infty)\rightarrow\mathbb{R}\) and matrices \(A,b,c\) of suited dimensions. The problem is, assuming that only \(y,u\) can be measured from the system, to find a control law \(u\) depending on \(y\) and some other known signals and parameters, such that \(x\) goes to zero asymptotically.  
It is well known that if the pair \((A,b)\) is completely controllable, there exists a vector \(k\) such that arbitrary eigenvalues can be assigned to the matrix \(H:=A-bk^{T}\). In particular, if \(x\) is available the controller \(u=-k^{T}x\) applied to system (24) drives \(x\) to zero aymtotical with speed determined by the eigenvalues of matrix \(H\). Since we are assuming that \(x\) is unknown, we will use an estimation \(\hat{x}\) for (24) obtained from the following FOO of order \(\beta\) whose design is based on (12) and (13) with \(\alpha=1\)  
\[\left\{\begin{array}{l}\vspace{0.1 cm}0D^{\beta}\hat{x}=F\hat{x}+\hat{y}+ \tilde{u}\\ \hat{y}=c^{T}\hat{x}\\ \hat{x}(t=0,t,\beta)=\hat{y}_{0}\end{array}\right. \tag{25}\]  
where \(\tilde{u},\hat{y}\) are filtered versions of \(u,y\) given, in Laplace domain, by  
\[\left\{\begin{array}{l}\tilde{Y}=\left(s^{\beta}I-F\right)\left(sI-F\right) ^{-1}gY\\ \tilde{U}=\left(s^{\beta}I-F\right)\left(sI-F\right)^{-1}bU\end{array}\right. \tag{26}\]  
with null IC for \(\tilde{u},\hat{y}\).  
Theorem 1 guarantees that \(e(t):=x(t)-\hat{x}(t)\) converges to zero asymptotically, for any pair of continuous functions \(u,y\), if the pair \((A,c)\) is observable. Then, using \(u=-k^{T}e(t)\), we have  
\[\hat{x}=Hx+bk^{T}e(t)\]  
Choosing \(H\) such that \(\left|arg\left(spec\left(H\right)\right)\right|>\frac{\pi}{2}\), the transfer matrix is \(\mathcal{L}^{1}\) (i.e., Lebesgue integrable). Since the convolution of an \(\mathcal{L}^{1}\) function with a function that converges to zero, converges to zero, it follows that \(x\to 0\) as \(t\rightarrow\infty\), with a speed regulated trough matrices \(H,F\). Thus, the resulting controller \(u=u\left(k,F,\beta,y\right)\) has an additional degree of freedom corresponding to the observer FO \(\beta\).  
Moreover, the control is robust meaning that if there exists an additive bounded measurement noise on the output, \(y+\Delta_{y}\), \(x\) remains bounded. Indeed, from the proof of Theorem 1, it follows that \(\|\hat{x}\|<C|\Delta_{y}|\) with \(C\) a constant number. Hence, \(\|x\|\leq C|\Delta_{y}|\)\(\|\int_{0}^{t}exp\left(-Ft\right)dt\|\) as \(t\rightarrow\infty\). Therefore, the choice of \(F\) also regulates the robustness of this control scheme with respect to bounded external perturbations on the system output.  
The adaptive observer (Theorem 2) can be applied in indirect adaptive control as was done in [28].