# Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System
## 7 Illustrative Examples  
### Fractional-Order LQR Control for Viscoelastic Damper System  
In this section, the viscoelastic damper described in Section 3 is set to be the controlled object. Moreover, the feedback gains of the fractional-order LQR are obtained using the method explained in Section 4, and the states are estimated with the fractional-order state observer described in Section 5. In addition, each state is simulated using the numerical calculation method introduced in Section 6. Another derivation method for the feedback gains of the fractional-order LQR, which is different from the method proposed herein, was proposed by [PERSON] and [PERSON] [19]. Accordingly, vibration waveforms for the states controlled by the LQR using the feedback gains obtained with the proposed method are compared with those obtained using the [PERSON] method. The parameters for the viscoelastic damper system are set as \(\zeta=0.1\) and \(\omega_{n}=3.0\) rad/s. The observable is chosen to be the displacement. Therefore, the output matrix \(\mathbf{C}\) in Equation (35) is as follows:  
\[\mathbf{C}=\big{[} \begin{array}{cccc}1&0&0&0&\\ \end{array} \big{]}. \tag{56}\]  
First, the stability of the viscoelastic damper system expressed by Equation (15) is investigated. The eigenvalues of the system matrix \(\mathbf{A}\) are given as:  
\[\lambda=1.2263\pm 1.3098i,\ -1.2263\pm 1.1366i. \tag{57}\]  
Here, the stability of the system is determined using Equation (7):  
\[\frac{2}{\pi}|\text{arg}(\lambda_{i})|=0.5210,\ 0.5210,\ 1.5241,\ 1.5241>0.5. \tag{58}\]  
As a result of this inequality, this system can be said to be asymptotically stable. Next, the controllability and the observability of the viscoelastic damper system are examined. The ranks of the matrices \(\mathbf{N}_{c}\) and \(\mathbf{N}_{o}\) in Equations (9) and (10) can be calculated as follows:  
\[rank(\mathbf{N}_{c})=4,\ rank(\mathbf{N}_{o})=4. \tag{59}\]  
The pair of matrices \((\mathbf{A},\mathbf{B})\) for the viscoelastic damper system is controllable, and the pair of matrices \((\mathbf{C},\mathbf{A})\) for the viscoelastic damper system is observable. Based on these results, the system can be said to be controllable and observable. Therefore, a state feedback controller and a state observer can be designed for this system. If the weighting matrices \(\mathbf{Q}\) and \(\mathbf{R}\) in the quadratic cost function \(J\) are designed as follows:  
\[\mathbf{Q}=\left[ \begin{array}{cccc}10&0&0&0\\ 0&0&0&0\\ 0&0&10&0\\ 0&0&0&0\end{array} \right],\ \mathbf{R}=1, \tag{60}\]  
then the optimal feedback-gain matrix \(\mathbf{F}_{opt}\) is obtained as follows:  
\[\mathbf{F}_{opt}=\big{[} \begin{array}{cccc}0.5394&-0.5520&3.3285&-0.0098\\ \end{array} \big{]}. \tag{61}\]  
In this case, the weights in \(\mathbf{Q}\) are placed only on the integer-order derivative states. With these feedback gains, the eigenvalues for the closed-loop system-matrix \(\mathbf{A}_{cl}=\mathbf{A}-\mathbf{B}\mathbf{F}\) are shown below:  
\[\lambda_{cl}=0.8475\pm 1.5897i,\ -0.8426\pm 1.4931i. \tag{62}\]  
From Equation (62), since the poles of the closed-loop system are located in region 2 in Figure 1, the system is asymptotically stable. Next, a fractional-order state observer is designed using Equation (36) by the pole-assignment method. Furthermore, generally, when a state observer and a regulator are used simultaneously, the state observer must be designed to be able to estimate the states of thesystem faster than the regulator makes the states converge to zeros. Here, the eigenvalues of the matrix \(\mathbf{A}_{so}=\mathbf{A}-\mathbf{H}\mathbf{C}\) are designed as follows:  
\[\lambda_{so}=-10.0,\ -9.0,\ -8.0,\ -7.0. \tag{63}\]  
The stability of these eigenvalues can be determined using Equation (7) as:  
\[\frac{2}{\pi}|\text{arg}(\lambda_{cl\_i})|=0.6882,\ 0.6882,\ 1.3271,\ 1.3271>0.5, \tag{64}\]  
\[\frac{2}{\pi}|\text{arg}(\lambda_{so\_i})|=2,\ 2,\ 2,\ 2>0.5. \tag{65}\]  
The closed-loop system \(\mathbf{A}_{cl}\) and the state observer \(\mathbf{A}_{so}\) are asymptotically stable.