## 4 Proposed Adaptive Linear Quadratic Regulator with Fractional Integral (LQR-FI) Control Structure
### Mathematical Model and Constraints of the Proposed Controller Based on PSO Optimization  
As stated, in this paper, the control objective is simply to overcome the non-minimum phase behavior problems when the aircraft is pitching up. Therefore, a successful controller is designed based on optimal control theory in the case of actuator losses of effectiveness faults. A fractional-order integrator is used as the compensator to convert the system from a non-minimum phase to a minimum phase, and then the linear quadratic regulator is applied for the resultant minimum phase dynamics to obtain a closed-loop optimal control law. The LQR performance matrices \(\mathrm{Q}_{new}\), \(\mathrm{R}\) and the fractional integrator parameters, \(\lambda,\mathrm{K}_{I}\) are chosen as design parameter vectors, which can be determined by minimizing the objective function, which represents the errors between the required and the actual pitch angle utilizing PSO. The objective function is selected as an integral of the absolute magnitude of the error (IAE) performance index. The comparison between the fractional-order integrator and fractional-order derivative is obtained similarly by considering the fractional-order derivative instead of the fractional integral in the proposed structure as presented in Figure 2.  
The optimization problem is formulated mathematically as follows. \[\mathrm{Objective\ function}=\mathrm{minimize}(IAE)=\int_{0}^{T}|e(t)|dt\]  
Here, \(e(t)=r(t)-y(t)\) is subjected to:  
\[\mathrm{constraints}=\left\{ \begin{array}{c}\mathrm{Q}_{new_{lower}}<\mathrm{Q }_{new}<\mathrm{Q}_{new_{upper}}\\ \mathrm{R}_{lower}<\mathrm{R}\ <\mathrm{R}_{upper}\\ \mathrm{K}_{Imin}<\mathrm{K}_{I}<\mathrm{K}_{Imax}\\ \lambda_{min}<\lambda<\lambda_{max}\end{array} \right. \tag{12}\]The lower and upper bound for the design variables are set as follows. First, LQR performance matrices, Q\({}_{new}\) and R,  
\[ \begin{split}\text{Q}_{new,lower}&=\text{diag}[0.01,0.01,\ 0.01,\ 10,0.01,0\dots\ 0],\\ \text{Q}_{new,upper}&=\text{diag}[100,\ 100,\ 100,\ 1000,\ 100,\ 0 \dots 0]\end{split}  \tag{13}\]  
The Q\({}_{new}\) is selected in this way so that approximately 100 times as much effort is put to keep the state of pitch angle small. As known, large R means more control effort. \[\text{R}_{lower}=\text{diag}[10,10],\ \text{R}_{lower}=\text{diag}[100,100] \tag{14}\]  
Second, fractional-order integral parameters, K\({}_{I}\) and \(\lambda\),  
\[\text{upper and lower band}=\left\{ \begin{array}{rl}-150<\text{K}_{I}<150\\ \ \ \ \ 0.1<\lambda<0.9\\ \ \ \ 0.1<\mu<0.9\end{array} \right. \tag{15}\]  
In general, R and Q\({}_{new}\) matrices are known as the LQR controller performance matrices, which can be used as design parameter vectors to penalize the dynamic state variables and the control signals. In the case that is under consideration, Q\({}_{new}\) is penalizing the aircraft and pre-compensator states, whereas R is penalizing the elevator control signal. Still, the target is to penalize the state variables of aircraft, which represent pitch rate, true airspeed, angle-of-attack, pitch angle, and altitude. Nevertheless, the changes occurring in the fractional-order integrator states are not of concern. For this reason, the fractional-order integrator states are selected as equal to zeros in the Q\({}_{new}\) matrix, which means more attention is paid to keeping all aircraft states near zero as much as possible compared with the states of the fractional integrator compensator. Since the fractional-order integration parameters are simultaneously optimized with the state-feedback gain vector of K\({}_{lqr}\), it can ensure a good traceability performance while keeping all states near zero. Moreover, it can improve the convergence of the optimization algorithm to an optimal solution.