# Mixed Order Fractional Observers for Minimal Realizations of Linear Time-Invariant Systems
## 6 Application
### Simulation Example  
In what follows, a simulation example is presented, corresponding to the application presented above. The plant to be observed and controlled has the form indicates in (24) with  
\[A=\left[ \begin{array}{cc}4&1\\ -3&0\end{array} \right]\qquad b=\left[ \begin{array}{cc}3\\ -6\end{array} \right]\qquad c=\left[ \begin{array}{cc}1&0\end{array} \right]^{T }\qquad x_{0}=\left[ \begin{array}{cc}5&3\end{array} \right]^{T} \tag{27}\]  
The plant is unstable, controllable, observable and in observable canonical form. Thus, an observer with structure given by (13) and (25) is implemented, using  
\[F=\left[ \begin{array}{cc}-25&1\\ -150&0\end{array} \right], \tag{28}\]  
which makes the observer asymptotically stable with poles in \(-10\) and \(-15\). The observed state \(\hat{x}\) given by (25) and (26) with parameters given by (28) is used to construct a control signal given by  
\[u=-k^{T}\hat{x}+Nr \tag{29}\]  
where the parameter \(k\) is chosen as \(k=\left[ \begin{array}{cc}17&7\end{array} \right]^{T}\), \(N\) is the correction factor given by \(N=\left(c\left(A-bk^{T}\right)^{-1}\right)^{-1}\) and \(r\) is the reference signal. The choice of \(k\) makes that the closed-loop roots of \(A-bk^{T}\) are located at \(-2\) and \(-3\). Using the control law (29) and the observer (25) and (26), the system was simulated using a step of magnitude \(5\) as reference signal. Different values of the FO \(\beta\) were explored for the observer (25) and (26) in order to check its influence in the control scheme. Figure 1 shows the evolution of the resulting control error \(e_{c}\left(t\right)=r\left(t\right)-y\left(t\right)\) whereas Figure 2 shows the evolution of the norm of the estimation error \(e\left(t\right)=x\left(t\right)-\hat{x}\left(t\right)\), for different values of \(\beta\in\left[0.7;1.3\right]\). Figure 1: Evolution of the control error for a step reference of magnitude \(5\), using different values of the FO \(\beta\) for the observer. It can be seen from Figure 1 that the control error converges asymptotically to zero for all the FOO used in the estimation stage. The main differences are appreciated in the transient, where the largest overshoots correspond to those cases where the FO are far from 1. Also, the convergence time is higher for these cases. Observing Figure 2 it can be seen that the five FOO work as expected, since the estimation errors are converging to zero, which means that the observers correctly estimate the real states. Again, in this case, the convergence time is higher as the FO \(\beta\) is farther from 1, which is the order of the plant to be controlled. To add more information on these results, the following indexes were calculated for the simulation, corresponding to the integral of the squared control error (\(ISE_{c}\)) and the integral of the squared norm of the estimation error (\(ISE_{e}\)). \[ISE_{c}=\int\limits_{0}^{T}e_{c}^{2}\left(\tau\right)d\tau \tag{30}\]  
\[ISE_{e}=\int\limits_{0}^{T}\|e\|^{2}\left(\tau\right)d\tau \tag{31}\]  
Table 1 presents the corresponding values of indexes (30) and (31) for these simulations. It can be seen that in fact, the best estimation is achieved with the IOO (\(\beta=1\)), since it has the smallest \(ISE_{e}\). However, since the observer is used to implement the control signal, it can be observed that the behavior of the control error is better for the case when \(\beta=0.9\), since it has the smallest \(ISE_{c}\). However, differences are not so important as in the case of the \(ISE_{e}\). \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\beta=0.7\) & \(\beta=0.9\) & \(\beta=1\) & \(\beta=1.1\) & \(\beta=1.3\) \\ \hline \(ISE_{c}\) & \(1.74\times 10^{3}\) & \(1.57\times 10^{3}\) & \(1.61\times 10^{3}\) & \(1.73\times 10^{3}\) & \(2.76\times 10^{3}\) \\ \(ISE_{e}\) & \(10.66\) & \(3.1693\) & \(3.36\times 10^{-3}\) & \(3.20\) & \(9.91\) \\ \hline \end{tabular}
\end{table} 
Table 1: Performance indexes (30) and (31) for the ideal case, using a step reference of magnitude 5.