# Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System
## 7 Illustrative Examples
### Comparison between Proposed Method and Sierociuk and Vinagre Method  
Moreover, the control results for the fractional-order LQR using the feedback gains obtained using the proposed method and the fractional-order LQR using the feedback gains obtained with the Sierociuk and Vinagre method [19] are compared. Using the same weighting matrices, \(\mathbf{Q}\) and \(\mathbf{R}\) in Equation (60), the feedback gains are calculated with the Sierociuk and Vinagre method as follows:  
\[\mathbf{F}_{SV}=\big{[}\begin{array}{cccc}0.5394&14.7824&13.0640&5.1116&\end{array} \big{]}. \tag{69}\]  
The poles of the closed-loop system-matrix \(A-\mathbf{B}F_{SV}\) are as follows:  
\[\textit{Poles}=\left\{\begin{array}{c}-1.5649\pm 1.5733i\\ -0.9908\pm 0.9774i\end{array}\right.. \tag{70}\]  
These poles are located in region 3 in Figure 1. As a result, the controlled states with the feedback gains in Equation (61) and the controlled states with the feedback gains in Equation (69) can be compared, as shown in Figure 9. The results with control are obtained with output feedback control using a fractional-order LQR.  
From Figure 9, it is clear that both feedback gains exhibit good control effects. In particular, in the vibration waveforms controlled using the feedback gains obtained with the Sierociuk and Vinagre method, all of the states are confirmed to be overdamped. However, it is obvious that the convergence of \(x_{1}\) (displacement) to the target value is considerably slow.  
Next, the design of the weighting matrices \(\mathbf{Q}\) and \(\mathbf{R}\) in the quadratic cost function / is modified as follows:  
\[\mathbf{Q}=\left[\begin{array}{cccc}10&0&0&0\\ 0&10&0&0\\ 0&0&10&0\\ 0&0&0&10\end{array}\right],\ \mathbf{R}=1. \tag{71}\]  
In this case, the weights in \(\mathbf{Q}\) are placed on not only the integer-order derivative states, but also the fractional-order derivative states. Using weighting matrices \(\mathbf{Q}\) and \(\mathbf{R}\), the feedback gain matrix derived using the proposed method is as follows:  
\[\mathbf{F}=\big{[}\begin{array}{cccc}0.5394&-2.3077&3.7258&-0.9831&\end{array} \big{]}. \tag{72}\]  
Figure 8: Results for states with/without fractional-order LQR control (with control: blue lines, without control: red broken lines).  
The poles of the closed-loop system matrix \(\mathbf{A}-\mathbf{B}\)F are as follows:  
\[\begin{split}\mathit{Poles}=\left\{\begin{array}{l}-1.0806\pm 1.5848i\\ -0.5891\pm 1.4985i\end{array}\right..\end{split} \tag{73}\]  
These poles are located in region 2 in Figure 1. On the other hand, the feedback gain matrix obtained by the Sierociuk and Vinagre method is as follows:  
\[\begin{split}\mathbf{F}_{SV}=\left[\begin{array}{cc}0.5394&16.2648 &15.1137&6.3425&\end{array}\right].\end{split} \tag{74}\]  
The poles of the closed-loop system matrix \(\mathbf{A}-\mathbf{B}\mathbf{F}_{SV}\) are as follows:  
\[\begin{split}\mathit{Poles}=\left\{\begin{array}{l}-2.3380\pm 0.3989i \\ -0.8332\pm 1.0007i\end{array}\right..\end{split} \tag{75}\]  
These poles are located in region 3 in Figure 1. The states controlled with the above-mentioned feedback-gain matrices are compared in Figure 10. The results with control are obtained with output feedback control using a fractional-order LQR.  
Figure 9: Results for states with fractional-order LQR control (proposed method: blue lines, Sierociuk and Vinagre method (S–V method): red broken lines).  
Figure 10 demonstrates that the results of the states are oscillations damped by the feedback-gain matrix \(\mathbf{F}\) and are overdamped by the feedback-gain matrix \(\mathbf{F}_{SV}\). Furthermore, the convergence property of \(x_{1}\) in Figure 10 is similar to that in Figure 9. However, in the proposed method, the vibration suppression effect becomes worse when the weights in \(\mathbf{Q}\) are placed on the fractional-order derivative states. On the other hand, in the Sierociuk and Vinagre method, the weights on the fractional-order derivative states in \(\mathbf{Q}\) can play an important role for better vibration control effects.