# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 5 Discussion
### Unknown Initial Time  
When \(\alpha\
eq 1\), the schemes proposed in Sections 3 and 4, implicitly assumes that the initial time of the observed system is known and the same for the observer system. This is an unrealistic assumption which is found in all the revised papers (see e.g., [3]), with the exception of those that use diffusive fractional systems. This consideration does not apply to the adaptive law (17) which can be applied at any instant without changing Theorem 2. We show that the framework of Section 2 allows to deal with this case in a consistent way. Let \(t_{0}<0\) the unknown initial time of the observed system and \(t=0\) the initial time of the observing process. Consider the system given by  
\[t_{0}D^{\alpha}x=Ax+Bu=:\bar{A}(x,u), \tag{22}\]  
where the derivative is generic in the sense of (1). In particular, by applying Laplace transform for the time variable \(t^{\prime}:=t-t_{0}\), it holds that \([\,{}_{t_{0}}D^{\alpha}x](s^{\prime})=\hat{X}(t_{0},s^{\prime},\alpha)+s^{ \prime\alpha}\hat{x}\). Applying \(\alpha-\)fractional integral from \(t^{\prime}=0\), we obtain  
\[s^{\prime-\alpha}[\,{}_{t_{0}}D^{\alpha}x](s^{\prime}) =s^{\prime-\alpha}\hat{X}(t_{0},s^{\prime},\alpha)+\hat{x}(s^{ \prime})\] \[=:\hat{\psi}_{0}(t_{0},s^{\prime},\alpha)+\hat{x}.\]  
Hence, the \(\alpha-\)fractional integral of (22) yields in the time variable \(t\)  
\[x(t) =\psi_{0}(t_{0},t,\alpha)+\frac{1}{\Gamma(\alpha)}\int_{t_{0}}^{0 }(t-\tau)^{\alpha-1}\bar{A}d\tau+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^ {\alpha-1}\bar{A}d\tau\] \[=:\psi_{1}(0,t,\alpha)+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t- \tau)^{\alpha-1}\bar{A}d\tau\]  
Applying Laplace Transform for the time variable \(t\), we obtain  
\[x(s) =\hat{\psi}_{1}(0,s,\alpha)+s^{-\alpha}[A\hat{x}+B\hat{u}]\] \[s^{\alpha}x =s^{\alpha}\hat{\psi}_{1}(0,s,\alpha)+A\hat{x}+B\hat{u}\] \[=:\hat{\psi}(0,s,\alpha)+A\hat{x}+B\hat{u}.\]  
Therefore, by defining \([\bar{D}^{\alpha}x](s):=s^{\alpha}x-\hat{\psi}(0,s,\alpha)\), we have that the system \(\bar{D}^{\alpha}x=Ax+Bu\) is equivalent to (22) for \(t\geq 0\), in the sense that both have the same solution for \(t\geq 0\). In particular, the term \(\psi(0,t,\alpha)\) can be identified as the IC term. Note that, in general \(\,{}_{0}\bar{D}^{\alpha}\
eq\,{}_{0}D^{\alpha}\), since both have different IC terms. Theorems 1 and 2 still hold, but the care must be put in finding \(F\) verifying Equation (10) of Section 2. Another important fact is that if the observed system started at \(t=t_{0}\) from _rest_, meaning that any kind of IC becomes identically zero, i.e., \(\psi_{0}\equiv 0\) for any chosen derivative, then the FOS is unequivocally defined, meaning that there is not ambiguity in the choice of the derivative, since any of them yields the same IC term (\(\psi\)) for any \(t\geq t_{0}\). Only in this way, we can talk of a FOS, in the sense of [30]. On the other hand, if the system started at non-null IC, it has sense talk of a Caputo or RL or any other generic FOS. We have the following specifications to the above procedure, for a Caputo FOS. \[\psi_{0}(t_{0},s^{\prime},\alpha) =s^{\prime-\alpha}s^{\prime\alpha-1}x(t_{0})=s^{\prime-1}x(t_{0})\] \[\psi_{0}(t_{0},t,\alpha) =x(t_{0}),\quad\forall t>t_{0},\] \[s^{\prime-\alpha}[\,{}_{t_{0}}D^{\alpha}x](s^{\prime}) =s^{\prime-\alpha}\hat{X}(t_{0},s^{\prime},\alpha)+\hat{x}(s^{ \prime})\]0 \[s^{\prime-\alpha}[\,{}_{t_{0}}D^{\alpha}x](s^{\prime}) =s^{\prime-\alpha}\hat{X}(t_{0},s^{\prime},\alpha)+\hat{x}(s^{ \prime})\]1  
In particular, the resulting \(\bar{D}^{\alpha}\) is not a Caputo one. This is what some authors refer as that Caputo derivative does not take account of the past [31]. To the best of our knowledge, it seems there is no a fractional derivative operator yielding \(\bar{D}^{\alpha}=D^{\alpha}\) (the diffusive approach is not realizing a known fractional derivative operator).