# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 3 Non-Adaptive Mixed Order Observer (NAMOO)  
Based on minimal representations, Luenberger observers [1] can be generalized for fractional order systems following the results in [3]. Consider system (6) with \((A,b,c)\) known and not necessarily in any canonical form. Define the estimate \(\hat{x}\) of the internal variable \(x\) by \[\left\{ \begin{array}{l}{{}_{0}D^{\alpha}\hat{x}=A\hat{x}+bu+F\left(y-\hat{y}\right) }\\ \hat{y}=c^{T}\hat{x}\\ \hat{x}\left(t=0,t,\alpha\right)=\hat{y}_{0}\end{array} \right. \tag{11}\]  
Note the FO is the same for the system (6) and the observer (11). It follows that \(e:=x-\hat{x}\) satisfies \(D^{\alpha}e=\left(A-Fc^{T}\right)e.\) Since it is a minimal representation, matrix \(F\) can be chosen such that the matrix \(\left(A-Fc^{T}\right)\) has arbitrary eigenvalues. Therefore, we get asymptotically \(e\to 0\) as \(t\rightarrow\infty\). We will build an alternative observer whose FO is not equal to the order of the original system. Moreover, it is minimal not only in the internal variable dimension but also in the use of the minimal numbers of parameters to describe the system, all the rest being freely chosen to simplify the design. Consider the state observer for system (6) given by  
\[\left\{ \begin{array}{l}{{}_{0}D^{\beta}\hat{x}=F\hat{x}+\hat{y}+\hat{u}}\\ \hat{y}=c^{T}\hat{x}\\ \hat{x}\left(t=0,t,\beta\right)=\hat{y}_{0}\end{array} \right. \tag{12}\]  
where \(\hat{u},\hat{y}\) are filtered versions of \(u,y\) given, in Laplace domain, by  
\[\left\{ \begin{array}{l}\hat{Y}=\left(s^{\beta}I-F\right)\left(s^{\alpha}I- F\right)^{-1}gY\\ \hat{U}=\left(s^{\beta}I-F\right)\left(s^{\alpha}I-F\right)^{-1}bU\\ \end{array} \right. \tag{13}\]  
with null IC for \(\hat{u},\hat{y}\). Consider system \(D^{\beta}x=Fx\) with \(F\) chosen such that \(x=0\) is asymptotically stable or more generally, globally attractive, then we have the following result stated in the next Theorem. Theorem 4.1: _Let us consider system defined in (6). Then we can state that system (12) is an asymptotic observer with speed of convergence regulated by the eigenvalues of \(F\). Besides, the observer is robust with respect to external perturbations, meaning that:_  
**(a)**: _If_ \(u,y\) _have additive bounded uncertainties, i.e.,_ \(u\left(t\right)+\Delta_{u}\left(t\right)\) _and_ \(y\left(t\right)+\Delta_{y}\left(t\right)\)_, the estimates of internal variables and the output error remain bounded, having also high frequency rejection._
**(b)**: _If parameter vector_ \(p\) _(i.e., the elements of_ \(A\) _and_ \(b\)_) is in fact_ \(p+\Delta_{p}\)_, where parameter disturbances_ \(n\Delta_{p},u\) _and_ \(y\) _are bounded functions, then_ \(x-\hat{x}\) _and_ \(y-\hat{y}\) _remain bounded._  
Proof: The solution of (12) is given by  
\[\hat{x}=\xi\left(t;\hat{y}_{0},F,\beta\right)+\left[\eta_{1}\ldots\eta_{2n} \right]p \tag{14}\]  
where we have used that the solution when \(\hat{y}_{0}\equiv 0\) is given, in Laplace domain, by \(\left(s^{\beta}I-F\right)^{-1}\left(s^{\beta}I-F\right)\left(s^{\alpha}I-F \right)^{-1}\left[gy+bu\right]=\left(s^{\alpha}I-F\right)^{-1}\left[gy+bu\right]\), the same solution for (7) when \(\hat{y}_{0}\equiv 0\). From (9) to (14), we obtain that, \(\hat{x}\left(t\right)-x\left(t\right)=\xi\left(t;\hat{y}_{0},F,\beta\right)- \xi\left(t;\hat{y}_{0},F,\alpha\right)\). Hence, \(\lim_{t\rightarrow\infty}\hat{x}\left(t\right)-x\left(t\right)=0\) and \(\lim_{t\rightarrow\infty}\hat{y}\left(t\right)-y\left(t\right)=\lim_{t \rightarrow\infty}c^{T}\hat{x}\left(t\right)-c^{T}x\left(t\right)=0\) accordingly to the choice of \(F\). If the input is now \(u\left(t\right)+\Delta_{u}\left(t\right)\) and the output is \(y\left(t\right)+\Delta_{y}\left(t\right)\), then (14) is modified to  
\[\hat{x}\left(t\right)=\xi\left(t;\hat{y}_{0},F,\beta\right)+\left[\eta_{1} \ldots\eta_{2n}\right]\left(t\right)p+\Delta\left(t\right) \tag{15}\]  
where \(\Delta(t)\) is the time domain expression of \(\left(s^{\alpha}I-F\right)^{-1}\left[g\Delta_{y}+b\Delta_{u}\right]\).