# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 4 Adaptive Mixed Order Observer (AMOO)  
In this section we consider that vector \(p\) (i.e., the elements of \(A\) and \(b\)) in system (7) is unknown. Thus, instead of \(p\) we will use an estimation of it, \(\hat{p}=\hat{p}\left(t\right)\), for the proposed observer scheme in Section 3. Using expression (14), we write  
\[\hat{x}\left(t\right)=\xi\left(t;\hat{\psi}_{0},F,\beta\right)+\left[\eta_{1} \ldots\eta_{2n}\right]\left(t\right)\hat{p}\left(t\right)\,, \tag{16}\]  
where \(\eta_{i}\) are calculated using only the input and output information as in (7) or (8), and \(c\) is known from the canonical representation chosen for the system (6). We will use the information vector defined by \(W:=\left[\eta_{1}\ldots\eta_{2n}\right]c\) and it will be assumed the Caputo derivative along this section. We will state a previous result to assure the stability of the AMOO, given in the following Lemma. **Lemma 1**. : _The solutions to the Caputo system \({}_{0}D^{\alpha}x=Ax\), with \(x\left(0\right)\in\mathbb{R}^{n}\), have bounded quadratic \(\gamma-\)fractional integral, if \(0<\gamma\leq 2\alpha\) and \(\left|arg\left(spec\left(A\right)\right)\right|>\frac{\alpha\gamma}{2}\)._  
Proof. : Let \(\xi\) be a solution for \(x\left(0\right)=x_{0}\). Since the solution is continuous at \(\left[0,T\right]\) for every \(T>0\)[16], provided that \(u\) is continuous, only the behavior of \(\xi^{2}\) as \(t\rightarrow\infty\) is decisive to determine if its fractional integral is bounded. In Laplace domain, we can write \(\hat{\xi}\left(s\right)=s^{\alpha-1}(s^{\alpha}-A)^{-1}x_{0}=:s^{\alpha-1}H_{ \alpha}\left(s\right)\). We have that \(H_{\alpha}\) in time domain is an \(L^{1}\) function in each component (see e.g., [23], or [24]). Therefore, as \(t\rightarrow\infty\), \(\xi\) has a behavior in Laplace domain given by \(\lim_{s\to 0}s^{\alpha-1}H_{\alpha}\left(s\right)\). Since the relative degree of each component of \(H_{\alpha}\) is non-negative, this limit is \(s^{\alpha-1}x_{0}\) which in time domain is proportional to \(t^{-\alpha}x_{0}\). Therefore, the condition \(\gamma\leq 2\alpha\) to have bounded fractional integral of \(\xi^{2}\) is thus easily obtained. **Theorem 2**. : _Consider the observer (16) for a stable system (6) such that the parameter estimation is performed as_  
\[{}_{t_{0}}D^{\gamma}\hat{p}=-\kappa\left(\hat{y}-y\right)W, \tag{17}\]  
_where \(t_{0}\geq 0\), \(\kappa>0\), \(\gamma\leq 2\min\{\alpha,\beta\}\) or \(\gamma\leq 2\alpha\) and \(\hat{x}_{0}=0\). Then,_  
1. _All the variables of the system remain bounded,_ \(\left(\hat{y}-y\right)\) _has bounded_ \(\gamma\)_-integral and its RMS value converges to zero. These claims remain true if a bounded additive perturbation of quadratic bounded_ \(\gamma-\)_integral is added on the input and/or output of the system (_6_)._
2. _If the spectral measure of_ \(u\) _is not concentrated on_ \(k<2n\) _points, then_ \(\hat{p}\) _converges to the true parameters of the plant_ \(p\) _and_ \(\hat{x}\) _converges to_ \(x\) _as_ \(t\rightarrow\infty\)_._  
Proof. :
1. From (9) to (16), it follows that \[\hat{x}\left(t\right)-x\left(t\right)=\xi\left(t;\hat{\psi}_{0},F,\beta\right)- \xi\left(t;\psi_{0},F,\alpha\right)-\left[\eta_{1}\ldots\eta_{2n}\right] \left(t\right)\left(\hat{p}\left(t\right)-p\right).\] (18)Hence,  
\[c^{T}\left(\hat{x}\left(t\right)-x\left(t\right)\right) =c^{T}\left(\xi\left(t;\hat{\psi}_{0},F,\beta\right)-\xi\left(t; \psi_{0},F,\alpha\right)\right)+c^{T}\left[\eta_{1}\ldots\eta_{2n}\right]\left( t\right)\left(\hat{p}\left(t\right)-p\right)\] \[e\left(t\right) :=\hat{y}\left(t\right)-y\left(t\right)=\
u\left(t\right)+W^{T} \left(t\right)\tilde{p}\left(t\right) \tag{19}\]  
where \(e\) is the output error and \(\tilde{p}=\hat{p}\left(t\right)-p\) is the parameter error. It follows that \(\
u^{2}:=\left[c^{T}\left(\xi\left(t;\hat{\psi}_{0},F,\beta\right)-\xi\left(t ;\psi_{0},F,\alpha\right)\right)\right]^{2}\) has bounded \(\gamma-\)integral.