# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 2 Linear Systems Preliminaries  
Along this paper we consider a generic fractional order differential operator (GFODO) denoted as \({}_{t=0}D^{\alpha}\) or \({}_{0}D^{\alpha}\), which satisfies the property that for any continuous function \(f\), its Laplace transform holds that  
\[\mathcal{L}\left[{}_{t=0}D^{\alpha}f\right]\left(s\right)=s^{\alpha}F\left(s \right)+F_{IC}\left(t=0,s,\alpha\right),\quad\forall\alpha>0, \tag{1}\]  
where \(F(s)\) denotes the Laplace transform (with lower limit \(t=0\)) of the time function \(f(t)\). \(F_{IC}\left(t=0,s,\alpha\right)\) is the Laplace transform of function \(f_{IC}\left(t=0,t,\alpha\right)\) and depends on the function \(f\), the order \(\alpha\), (\(t=0\) is the lower limit of the Laplace Transform) and the type of fractional derivative. Examples are Caputo (e.g., \(F_{IC}\left(t=0,s,\alpha\right)=f\left(0\right)s^{\alpha-1}\) for \(\alpha<1\)), Riemann-Liouville (\(F_{IC}\left(t=0,s,\alpha\right)=\left\{D^{\alpha-1}f\right\}\left(0\right)\) for \(\alpha<1\)), Grunwald-Letnikov (\(F_{IC}\left(t=0,s,\alpha\right)=0\) for \(\alpha<1\)) and Miller-Ross fractional derivative ([16], Section 2.8) among others. For the fractional integral, there is a commonly chosen definition, which in Laplace domain takes the form,  
\[\mathcal{L}\left[{}_{t=0}I^{\alpha}f\right]\left(s\right)=s^{-\alpha}F\left(s \right)\quad\forall\alpha>0, \tag{2}\]  
and in time domain, it is given by  
\[{}_{0}I^{\alpha}f(t):=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{\alpha-1 }f(\tau)d\tau. \tag{3}\]  
where \(\Gamma(\cdot)\) is the Gamma function [16]. Fractional order linear systems (FOLS) can be represented in Laplace domain by taking Laplace transform to the input/output pair \(\left(u(t),y(t)\right)\) to obtain \(\left(U\left(s\right),Y\left(s\right)\right):=\mathcal{L}\left\{\left(u,y \right)\right\}\left(s\right)\). Assuming null IC conditions, a linear relationship is obtained expressed in terms of the complex variable \(s\) of the form \(Y(s)=H(s,U(s))=H(s)U(s)\), where \(H\) is called the Transfer Function (TF) (Transfer Matrix in the vector case). In particular, we are interested in TF of the type  
\[H\left(s\right)=\frac{\sum_{i=0}^{m}b_{i}s^{\beta_{i}}}{\sum_{i=0}^{n}a_{i}s^{ \alpha_{i}}}, \tag{4}\]  
where \(\alpha_{0}<\ldots<\alpha_{n}\); \(\beta_{0}<\ldots<\beta_{m};a_{0}\ldots,a_{n}\) and \(b_{0},\ldots,b_{n}\) are numbers in \(\mathbb{R}\). It follows that  
\[Y(s)=\frac{b_{n}s^{\beta_{n}}+\cdots+b_{0}}{a_{n}s^{\alpha_{n}}+\cdots+b_{0}}U (s)+\frac{\sum_{i=1}^{m}b_{i}U_{ICi}\left(0,s,\beta_{i}\right)-\sum_{i=1}^{n}a _{i}Y_{ICi}\left(0,s,\alpha_{i}\right)}{a_{n}s^{\alpha_{n}}+\cdots+a_{0}}, \tag{5}\]  
where \(\mathcal{L}[a_{i}D^{\alpha_{i}}y]=a_{i}s^{\alpha_{i}}Y(s)+a_{i}Y_{ICi}\left(0,s,\alpha_{i}\right)\) and \(\mathcal{L}[b_{i}D^{\beta_{i}}u]=b_{i}s^{\beta_{i}}U(s)+b_{i}U_{ICi}(0,s,\beta _{i})\). In view of (5), the terms \(Y\), \(U\) in time domain can be called initial conditions (IC), since they uniquely define the output \(y\) for each input \(u\) (see discussion in Section 5.4). More precisely, the IC must be all what is undetermined in \(U\), \(Y\). For instance, if \(Y=s^{\alpha-1}\{y\left(0\right)\}\), the IC must be \(y(0)\).This procedure can be easily generalized for the vector case. From this, we obtain some elementary properties for generic fractional order linear systems (GFOLS):  
* **P1:** The superposition of responses to a linear combination of inputs holds when the IC are null. * **P2:** The solution to (5) is the sum of a zero state solution (first term of (5)) plus a zero input solution (second term of (5)). Since we are interested in observing internal variables (state or pseudo state variables), we look for a suited mathematical model of internal variables for (4). In the commensurate case \(\alpha_{i}=l_{i}\alpha\) and \(\beta_{i}=m_{i}\beta\) for some integers numbers \(l_{i},m_{i}\) and making the identification \(s^{\alpha}=s\), some standard results for IOLS also hold for FOLS.