# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 2 Linear Systems Preliminaries
In particular, we can consider a completely controllable and observable system (i.e., (4) cannot be further simplified), for which we choose, without loss of generality (since, by applying Laplace transform, the resulting system will be input-output equivalent to (4)), the following canonical representation for \(x\in\mathbb{R}^{n}\) and \(y,u\in\mathbb{R}\) and the matrices \(A,b,c\) having suited dimensions:  
\[\left\{ \begin{array}{l}{}_{0}D^{\alpha}x=Ax+bu\\ y=c^{\top}x\\ x_{IC}\left(t=0,t,\alpha\right)=\psi_{0},\end{array} \right. \tag{6}\]  
where \(\alpha>0\), \(x_{IC}\left(t=0,t,\alpha\right)\) is the IC (a generalized time function or the Laplace transform inverse of \(X_{IC}\left(t=0,s,\alpha\right)\) obtained according to (1)) and  
\[A= \begin{bmatrix}-a_{1}&1&0&\ldots&0\\ \vdots&0&\ddots&&\vdots\\ \vdots&\vdots&&\ddots&\vdots\\ -a_{n-1}&0&\ldots&0&1\\ -a_{n}&0&\ldots&\ldots&0\end{bmatrix} ,b= \begin{bmatrix}b_{1}\\ \vdots\\ \vdots\\ \vdots\\ b_{n}\end{bmatrix} ,c= \begin{bmatrix}1\\ 0\\ \vdots\\ \vdots\\ 0\end{bmatrix} .\]  
Note that we use the minimal number of parameters necessary to describe the system and, since the system is completely controllable and observable, the minimal dimension for the internal variables corresponding to the number of components of \(x\). Any other minimal representation \(x^{\prime}\) of the system is related to (6) through a linear transformation \(x^{\prime}=Tx\) with T nonsingular. Consider the following matrices  
\[F=\left[ \begin{array}{ccccc}-f_{1}&1&0&\ldots&0\\ \vdots&0&\ddots&&\vdots\\ \vdots&\vdots&&\ddots&\vdots\\ -f_{n-1}&0&\ldots&0&1\\ -f_{n}&0&\ldots&\ldots&0\end{array} \right],g= \begin{bmatrix}g_{1}\\ \vdots\\ \vdots\\ \vdots\\ g_{n}\end{bmatrix} ,\]  
where the elements of \(F\) are chosen such that \(x=0\) is globally attractive for the system \(D^{\alpha}x=Fx\), i.e., any of its solutions converges to zero as time goes to infinity. We can write \(Ax=(F+A-F)x=Fx+(A-F)x\) and, choosing \(g_{i}=a_{i}-f_{i}\) together with the structures of \(A\) and \(F\), it follows that \((A-F)x=gc^{T}x=gy\). Thus, we can write \(Ax=Fx+gy\) and therefore system (6) can be re-expressed as  
\[\left\{ \begin{array}{l}{}_{0}D^{\alpha}x=Fx+gy+bu\\ y=c^{T}x\\ x_{IC}\left(t=0,t,\alpha\right)=\psi_{0}\end{array} \right. \tag{7}\]  
By linearity, the solution to (7) can be expressed as the sum of the response to a null input with nonzero IC (zero input response) plus the response when \(\psi_{0}\equiv 0\) (null IC) to a nonzero input (zero state response). The former is just the solution to \(D^{\alpha}x=Fx\) when \(X\left(t=0,t,\alpha\right)=\psi_{0}\); we denote it by \(\zeta\left(t;x_{0},\alpha\right)\). For the latter, we observe that superposition can be used since the IC are null. To this aim, we define for \(i=1,\ldots,n\) the following system  
\[\left\{ \begin{array}{l}{}_{0}D^{\alpha}\eta_{i}=F\eta_{i}+e_{i}y\\ {}_{0}D^{\alpha}\eta_{i+n}=F\eta_{i+n}+e_{i}u\\ \eta_{IC_{i}}(t=0,t,\alpha)\equiv\eta_{IC_{i+n}}(t=0,t,\alpha)\equiv 0, \end{array} \right. \tag{8}\]  
where \(e_{i}\) is the \(i-\)element of the canonical base of \(\mathbb{R}^{n}\). Hence, the response to \(gy+bu=\sum_{i=1}^{n}g_{i}e_{i}y+h_{i}e_{i}u\) is \(\sum_{i=1}^{n}g_{i}\eta_{i}+h_{i}\eta_{i+1}\). Therefore, for \(t>0\)  
\[x(t)=\zeta(t;\psi_{0},F,\alpha)+[\eta_{1}\ldots\eta_{2n}](t)p, \tag{9}\]  
where \(p=(g_{1},\ldots,g_{n},b_{1},\ldots,b_{n})^{T}\) is a vector of parameters. To end this section, we recall that for Caputo or Riemann-Liouville FOS, if \(|arg\left(spec\left(F\right)\right)|>\frac{\alpha\pi}{2}\), then \(x=0\) is globally attractive for the system \({}_{0}D^{\alpha}x=Fx\), where \(spec\left(F\right)\) is the set of eigenvalues of \(F\)[11]. It follows that if \(F\) is such that \(x=0\) is globally attractive for \({}_{0}D^{\alpha}x=Fx\), then \(x=0\) it is globally attractive for \({}_{0}D^{\beta}x=Fx\), where \(0<\beta<\alpha\).