# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 4 Adaptive Mixed Order Observer (AMOO)
Indeed, from the hypothesis on \(\gamma\), the choice of \(F\) and Lemma 1, \(c^{T}\xi\left(t;\hat{x}_{0},F,\beta\right)\) and \(c^{T}\xi\left(t;x_{0},F,\alpha\right)\) have bounded quadratic \(\gamma-\)integral. By recalling that \(||x+y||^{2}\leq 2||x||^{2}+2||y||^{2}\), we obtain that \(\
u\) has bounded quadratic \(\gamma-\)integral by (say) a constant number \(C\). Without loss of generality, we assume \(\kappa=1\) (otherwise, we always can redefine \(W\) as \(\kappa W\)). Let \(V:=\tilde{p}^{T}\tilde{p}+C-1/2I^{\gamma}\
u^{2}\). Note that \(V\) is non-negative, since \(\
u\) has bounded quadratic \(\gamma-\)integral. By ([25], Theorem 3) and continuity of the solutions, \(\left[\iota_{0}D^{\gamma}\tilde{p}^{T}\tilde{p}\right]\left(t\right)\leq 2 \tilde{p}^{T}\left(t\right)\iota_{0}D^{\gamma}\tilde{p}\left(t\right)\)[26; 27]. Using that \(\iota_{0}D^{\gamma}\tilde{p}=\iota_{0}D^{\gamma}\hat{p}\), since \(p\) is constant, together with (17) and (19), we obtain  
\[D^{\gamma}V \leq-2\left(\tilde{p}^{T}W\right)^{2}-2\tilde{p}^{T}W\
u-1/2\
u^ {2}\] \[=-2\left(\tilde{p}^{T}W+1/2\
u\right)^{2}\leq 0. \tag{20}\]  
By \(\gamma-\)integration of (20), we obtain \(V\left(t\right)\leq V\left(0\right)\). That is, \(\tilde{p}^{T}\tilde{p}+C-1/2I^{\gamma}\
u^{2}\) is bounded. Since \(C>1/2I^{\gamma}\), it follows that \(\hat{p}\) is bounded. Using the BIBO stability property of system (8) due to the choice of \(F\), it follows that \(W\) is bounded whenever \(y,u\) are bounded. Then, \(e\) is bounded. By \(\gamma-\)integration of (20), we also conclude that \(e^{2}=\left(\tilde{p}^{T}W+1/2\
u\right)^{2}\) has bounded \(\gamma\)-integral. Thus, \(e_{RMS}\) converges to zero ([28], Proposition 1). Using expression (15) and redefining \(\
u\), we have also proved that if a bounded perturbation \(\Delta\) with bounded quadratic \(\gamma-\)fractional integral, is added to the model input and/or output, all the above claims remain true. If the spectral measure of \(u\) is not concentrated on \(k<2n\) points, \(W=\eta_{1}\) is a persistently exciting (PE) function ([28], Property 11). Using this and the fact that \(\
u\) converges to zero, it follows that \(\tilde{p}\) converges to zero ([24], Section 3) i.e., \(\hat{p}\) converges to \(p\). Since \(u,y\) are bounded, \(\eta_{i}\) is bounded for \(i=1,\ldots,2n\). Therefore, using (18), \(x-\hat{x}\) converges to zero. For additive bounded perturbations on the parameters, we can redefine \(\
u\) which is now just a bounded function. According to ([24], Section 3), \(\left(\hat{p},e\right)\) remains bounded, when \(W\) is a PE function. Moreover, if the perturbation vanishes as \(t\) goes to \(\infty\), then \(\left(\hat{p},e\right)\) converges to zero. **Remark 2**: _A benefit of the adaptive scheme (17) when \(\gamma<1\) is shown by the following reasoning. From the facts that (a) if \(\
u^{2}\) has bounded \(\gamma_{1}-\)integral, then it has bounded \(\gamma_{2}-\)integral for any \(0<\gamma_{2}\leq\gamma_{1}\) (the proof is by contradiction and using \(I^{\gamma_{1}}\
u^{2}=I^{\gamma_{1}-\gamma_{2}}I^{\gamma_{2}}\
u^{2}\)) and that (b) \(t^{-\gamma}\) has bounded \(\gamma-\)integral but unbounded integer integral when \(\gamma<1\), it follows that the set of non-destabilizing perturbations for the adaptive scheme is strictly enlarged for \(\gamma<1\). The following heuristic rule can be stated as the lesser the value of \(\gamma\) the more robust is the response._