# Fractional-Order LQR and State Observer for a Fractional-Order Vibratory System
## 4 Fractional-Order LQR Method for State Feedback Control
### Iteration-Based Method for Obtaining Optimal Feedback Gains  
Therefore, in order to overcome this difficulty, the approach must be altered to an iteration-based method in order to obtain the optimal feedback-gain matrix \(F_{opt}\). To do so, \(\mathbf{A}_{cl}^{2}\) in Equation (24) is expressed as follows:  
\[\mathbf{A}_{cl}^{2}=(\mathbf{A}-\mathbf{B}\mathbf{F})^{2}=(\mathbf{A}-\mathbf{B}\mathbf{F}_{a})(\mathbf{A}-\mathbf{ B}\mathbf{F}_{b}). \tag{25}\]  
Here, \(\mathbf{F}_{a}\) is properly determined so that all the real parts of the eigenvalues of \(\mathbf{A}_{cl}^{2}\) are negative, and \(\mathbf{F}_{b}\) is treated as a variable. By setting \(\mathbf{A}_{cl,a}=\mathbf{A}-\mathbf{B}\mathbf{F}_{a}\), Equation (24) can be rewritten using Equation (25) as follows:  
\[L=\mathbf{x}(0)^{T}\mathbf{P}\mathbf{x}(0)+tr\Big{[}\mathbf{K}^{T}\Big{\{}\mathbf{P}\mathbf{A}_{cl,a}( \mathbf{A}-\mathbf{B}\mathbf{F}_{b})+\big{[}\mathbf{A}_{cl,a}(\mathbf{A}-\mathbf{B}\mathbf{F}_{b})\big{]}^ {T}\mathbf{P}+\mathbf{Q}+\mathbf{F}_{b}^{T}\mathbf{R}\mathbf{F}_{b}\Big{\}}\Big{]}. \tag{26}\]  
As a result, differentiating the above equation with respect to \(\mathbf{F}_{b}\), \(\mathbf{P}\), and \(\mathbf{K}\), respectively, produces the following relationships [30]:  
\[\frac{\partial L}{\partial\mathbf{P}}=\mathbf{K}(\mathbf{A}-\mathbf{B}\mathbf{F}_{b})^{T}\mathbf{A}_{ cl,a}^{T}+\mathbf{A}_{cl,a}(\mathbf{A}-\mathbf{B}\mathbf{F}_{b})\mathbf{K}+\mathbf{x}(0)\mathbf{x}(0)^{T}= \mathbf{0}, \tag{27}\]  
\[\frac{\partial L}{\partial\mathbf{F}_{b}}=-(\mathbf{P}\mathbf{A}_{cl,a}\mathbf{B})^{T}\mathbf{K}- (\mathbf{A}_{cl,a}\mathbf{B})^{T}\mathbf{P}\mathbf{K}^{T}+\mathbf{R}^{T}\mathbf{F}_{b}\mathbf{K}+\mathbf{R} \mathbf{F}_{b}\mathbf{K}^{T}=\mathbf{0}, \tag{28}\]  
\[\frac{\partial L}{\partial\mathbf{K}}=\mathbf{P}\mathbf{A}_{cl,a}(\mathbf{A}-\mathbf{B}\mathbf{F}_{b} )+(\mathbf{A}-\mathbf{B}\mathbf{F}_{b})^{T}\mathbf{A}_{cl,a}^{T}\mathbf{P}+\mathbf{Q}+\mathbf{F}_{b}^{T} \mathbf{R}\mathbf{F}_{b}=\mathbf{0}. \tag{29}\]  
From Equations (27) and (28), the following equation is obtained because the matrices \(\mathbf{R}\), \(\mathbf{P}\), and \(\mathbf{K}\) are symmetric:  
\[\mathbf{F}_{b}=\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{A}_{cl,a}^{T}\mathbf{P}. \tag{30}\]  
Next, Equations (29) and (30) generate the following algebraic Riccati equation:  
\[\mathbf{P}\mathbf{A}_{cl,a}\mathbf{A}+\big{(}\mathbf{A}_{cl,a}\mathbf{A}\big{)}^{T}\mathbf{P}-\mathbf{P} \mathbf{A}_{cl,a}\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{T}\mathbf{A}_{cl,a}^{T}\mathbf{P}+\mathbf{Q}=\mathbf{0}. \tag{31}\]  
In the end, an iterative method to obtain the optimal feedback-gain matrix \(\mathbf{F}_{opt}\) can be explained as follows. Here, \(\mathbf{P}\) can be obtained from Equation (31), and \(\mathbf{F}_{b}\) can be determined by substituting the obtained \(\mathbf{P}\) into Equation (30). Next, the determined \(\mathbf{F}_{b}\) is treated as a new \(\mathbf{F}_{a}\), and Equations (25)-(31) are repeatedly calculated until the feedback-gain matrix converges to \(\mathbf{F}_{b}=\mathbf{F}_{a}\). Eventually, the converged feedback-gain matrix can be considered to be the optimal feedback-gain matrix \(\mathbf{F}_{opt}\) to minimize the quadratic cost function \(\mathbf{J}\).