## 4 Simulation Analysis  
While considering the linearized aircraft model of Equation (13), matrices \(A\) and \(B\) have been already given in Section 2.1, and the other matrices are listed, as follows:  
\[C=\left[ \begin{array}{rrrrrrr}0&0&0&0&0&1\\ 0&0&1&0&0&0&0\end{array} \right],D=\left[ \begin{array}{rrrrr}0&0\\ 0&0\end{array} \right]\]  
Figure 6: The stable inversion (SI) based robust autolanding controller (SIRAC) architecture. \[C_{m}=\left[ \begin{array}{cccccccc}0&0&0&0&0&0&1\\ 0&0&0&221&0&0&0\\ 0&0&1&0&0&0\\ 0&0&-0.04&-10.56&0&-21.64&0\\ 0&0&0&0&0&100&0\\ 0&0&0&0&100&0\\ 0&0&0&0&100&0\end{array} \right],D_{m}=\left[ \begin{array}{cccc}0&0\\ 0&0\\ 0&0\\ 0&0.98\\ 0&0\\ 0&0\end{array} \right]\]  
Additionally, \(z=\left[h,\dot{h},V,\dot{V},\theta,q\right]^{T},y=\left[l,V\right]^{T}\). To find the \(u_{d}\) and \(x_{d}\) in step 2 of SIRAC algorithm, we need to know \(y_{d}\) and its derivatives. According to Equations (10) and (11), in the glide slope,  
\[ \begin{array}{l}\dot{h}_{d}(t)=-\tan(3^{\circ})V_{d}\\ \ddot{h}_{d}(t)=0\\ \dddot{h}_{d}(t)=0\\ \dddot{V}_{d}(t)=0\end{array}  \tag{57}\]  
In the flare segment,  
\[ \begin{array}{l}\dot{h}_{d}(t)=-\frac{k_{1}V_{d}}{k_{2}}e^{-k_{2}V_{d}t}+ \frac{k_{1}V_{d}}{2k_{2}}e^{-2k_{2}V_{d}t}+k_{3}V_{d}\\ \ddot{h}_{d}(t)=k_{1}V_{d}^{2}e^{-k_{2}V_{d}t}-k_{1}V_{d}^{2}e^{-2k_{2}V_{d}t} \\ \dddot{h}_{d}(t)=-k_{1}k_{2}V_{d}^{3}e^{-k_{2}V_{d}t}+2k_{1}k_{2}V_{d}^{3}e^{- 2k_{2}V_{d}t}\\ V_{d}(t)=0\end{array}  \tag{58}\]  
where  
\[ \begin{array}{l}V_{d}=67.4(m/s)\\ k_{1}=0.0006,k_{2}=0.0056\\ k_{3}=-0.0031,k_{4}=1.7933\end{array} \]  
The proposed method is also compared with the method in [4] for the windshear effect, which developed a linear quadratic regulator (LQR) feedforward closed-loop controller for aircraft landing. The brief introduction is as follows. More details can be found in [4]. Defining a new variable \(\varepsilon\) as:  
\[\varepsilon=\int_{0}^{t_{f}}{(y-y_{d})dt} \tag{59}\]  
where \(t_{f}\) is the final time. While combining Equation (59) with the linearized aircraft model, it can be shown that  
\[\left[ \begin{array}{c}\dot{x}\\ \dot{\varepsilon}\end{array} \right]=\left[ \begin{array}{cc}A&0\\ C&0\end{array} \right]\left[ \begin{array}{c}x\\ \varepsilon\end{array} \right]+\left[ \begin{array}{c}B\\ 0\end{array} \right]u+\left[ \begin{array}{cc}E&0\\ 0&-I\end{array} \right]\left[ \begin{array}{c}d\\ y_{d}\end{array} \right] \tag{60}\]  
where \(d\) is the measurable disturbances. Using the assumption of constant steady-state values of \(y_{d}\), the Equation (60) can be written, as:  
\[\left[ \begin{array}{c}\dot{x}\\ y-y_{d}\end{array} \right]=G\left[ \begin{array}{c}x\\ u\end{array} \right]+H\left[ \begin{array}{c}d\\ y_{d}\end{array} \right] \tag{61}\]  
where  
\[G=\left[ \begin{array}{cc}A&B\\ C&0\end{array} \right],H=\left[ \begin{array}{cc}E&0\\ 0&-I\end{array} \right]\]  
While combining the optimal LQR with the integrator and feedforward controller, the form of the final control law is  
\[u=-F_{1}x-F_{2}\left[\int_{0}^{t_{f}}{(y-y_{d})dt}\right]-F_{3}\left[ \begin{array} []{c}d\\ y_{d}\end{array} \right] \tag{62}\]where  
\[F_{3}=\left[ \begin{array}{cc}F_{1}&I\end{array} \right]G^{-1}H\]  
It is evident that there are three terms in Equation (62). The first is the LQR, the second term is the integral control, and the third term is the feedforward of the disturbances and the desired trajectory. Figure 7 shows the block diagram of the LQR feedforward closed-loop controller. From the view of the deflection limits of the actuators and the time response, in the glide slope, the diagonal weighting matrices \(Q\) and \(R\) are chosen as:  
\[Q=diag\Big{(}10,10,1,10,1,10,10^{3},1,1\Big{)},R=diag\Big{(}1,10^{7}\Big{)}\]  
By solving the Riccati equation, \(F_{1}\), \(F_{2}\), and \(F_{3}\) are obtained, as:  
\[F_{1}=\left[ \begin{array}{cccc}-8.05&2.43&0.77&12.36&-123.32&0.56&2.38\\ -4.35&0.00&0.23&-1.45&-0.36&6.53&1.04\end{array} \right]\]  
\[F_{2}=\left[ \begin{array}{cc}0.99&0.48\\ 0.00&0.02\end{array} \right]\]  
\[F_{3}=\left[