# Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System
## 4 Fractional-Order LQR Method for State Feedback Control  
### Design Method for Fractional-Order LQR  
In this section, a fractional-order LQR is designed for the fractional-order vibratory system described in the previous section. The control input is assumed to be given as follows:  
\[u(t)=-\mathbf{F}\mathbf{x}(t), \tag{17}\]  
where \(\mathbf{F}\) designates the feedback-gain matrix.  
First, the fractional-order state equation is forced to transform into an integer-order state equation. Setting the closed-loop system matrix as \(\mathbf{A}_{cl}=\mathbf{A}-\mathbf{B}\mathbf{F}\), Equation (17) is substituted into Equation (15), and the following relation is obtained:  
\[D^{\frac{1}{2}}\mathbf{x}(t)=\mathbf{A}_{cl}\mathbf{x}(t). \tag{18}\]  
The operator \(D^{\frac{1}{2}}\) is applied to both sides of Equation (18) once, and then the following equation is obtained [29]:  
\[D\mathbf{x}(t)=\mathbf{A}_{cl}^{2}\mathbf{x}(t). \tag{19}\]  
Second, a quadratic cost function \(J\) is considered for the controlled object modeled by Equation (15):  
\[J=\int_{0}^{\infty}\Big{[}\mathbf{x}(t)^{T}\mathbf{Q}\mathbf{x}(t)+u(t)^{T}\mathbf{R}u(t) \Big{]}dt, \tag{20}\]  
where the weighting matrices \(\mathbf{Q}\in\mathbb{R}^{n\times n}\) and \(R\in\mathbb{R}^{r\times r}\) are positive semi-definite and positive definite, respectively, and the pair of matrices \((\mathbf{Q}^{\frac{1}{2}},\mathbf{A})\) is observable. Moreover, Equation (17) is substituted into Equation (20) in order to obtain the following:  
\[J=\int_{0}^{\infty}\mathbf{x}(t)^{T}\Big{[}\mathbf{Q}+\mathbf{F}^{T}\mathbf{R}\mathbf{F}\Big{]}\bm {x}(t)dt. \tag{21}\]  
Modifying Equation (21) using both Equation (19) and the Lyapunov equation yields the following equations:  
\[J=\mathbf{x}(0)^{T}\mathbf{P}\mathbf{x}(0), \tag{22}\]  
\[\mathbf{P}\mathbf{A}_{cl}^{2}+\Big{(}\mathbf{A}_{cl}^{2}\Big{)}^{T}\mathbf{P}=-\mathbf{Q}-\mathbf{F}^{ T}\mathbf{R}\mathbf{F}, \tag{23}\]  
where \(\mathbf{P}\) is a positive-definite symmetric solution that satisfies the algebraic Riccati equation. Here, using [PERSON]'s method for undetermined multipliers, matrices \(\mathbf{P}\) and \(\mathbf{F}_{r}\) which minimize Equation (22), can be obtained under the constraint condition of Equation (23). If Equation (23) is assumed to be expressed as \(\mathbf{G}(n\times n)=\big{\{}g_{ij}\big{\}}=\mathbf{0}\), then the number of undetermined multipliers is required to be \(n^{2}\). Therefore, the undetermined multipliers are arranged in the form of \(\mathbf{K}(n\times n)=\big{\{}k_{ij}\big{\}}\). As a result, the Lagrangian function can be expressed as follows [29]:  
\[L=J+\sum_{i=1}^{n}\sum_{j=1}^{n}k_{ij}g_{ij}=J+tr\Big{[}\mathbf{K}^{T}\mathbf{G}\Big{]} =\mathbf{x}(0)^{T}\mathbf{P}\mathbf{x}(0)+tr\bigg{[}\mathbf{K}^{T}\bigg{\{}\mathbf{P}\mathbf{A}_{cl}^{ 2}+\Big{(}\mathbf{A}_{cl}^{2}\Big{)}^{T}\mathbf{P}+\mathbf{Q}+\mathbf{F}^{T}\mathbf{R}\mathbf{F}\bigg{\}} \bigg{]}, \tag{24}\]  
where \(tr\) designates the trace of the matrix.  
Third, the Lagrangian function \(L\) is differentiated with respect to the matrices \(\mathbf{F}\), \(\mathbf{P}_{r}\) and \(\mathbf{K}\), respectively, in order to obtain each extreme value. Finally, the equations to obtain the matrices \(\mathbf{F}\), \(\mathbf{P}\), and \(\mathbf{K}\) are derived.  
However, it has been clarified that the feedback-gain matrix \(\mathbf{F}\) cannot be obtained based on the above-mentioned method reported in a previous study [29]. This is because the equation resulting from differentiation directly with respect to \(\mathbf{F}\) becomes too complicated to be appropriate for obtaining the optimal feedback-gain matrix \(\mathbf{F}_{opt}\).