# A Study of Emergency Aircraft Control During Landing
## 6 Control System
### Preparation of Simulink Models
#### 6.2.1 Flight Path Angle Control System  
The flight path angle control system (Figure 1) uses feedback from measurements of equivalent airspeed \(u_{EAS}\), angle of attack \(\alpha\), pitch rate q, pitch angle \(\theta\), and flight path angle \(\gamma\), to determine the control signal for collective thrust levers deflection. The control system consists of two discrete LQR controllers, an integrator I, a control signal limiter, and an 'anti-windup' component preventing the control signal from rising when the limit of the thrust lever position is reached.  
The developed system takes into account the change in aircraft dynamics related to the landing gear position, realising a linear transition between the determined settings of the LQRI controller according to Equation (17). The purpose of this solution is to reduce overshoot and eliminate the control signal disturbance caused by controller gain matrix switching during landing gear extension or retraction. The control limit used in the model has constant limits [\(-0.3\); \(0.7\)]. In reality, however, these values are constantly changing, and such changes are mainly dependent on the flight altitude.  
\[\mathrm{K_{lqri}}=(1-\mathrm{lg})\cdot\mathrm{K_{lon_{g0}}}+\mathrm{lg}\cdot \mathrm{K_{lon_{g1}}} \tag{17}\]  
The flight path angle controller settings were computed using the _lqrd()_ function implementing the discrete control law (18) that minimises the discrete cost function (19) based on the continuous linear model and the assumed Q and R weight matrices.  
\[\mathrm{u_{k}}=-\mathrm{K_{d}}\mathrm{x_{k}} \tag{18}\]  
\[\mathrm{J}=\sum_{k=0}^{\infty}\left(\mathrm{x_{k}^{T}Qx_{k}+u_{k}^{T}Ru_{k}}\right) \tag{19}\]  
The process of determining the controller settings started with the formulation of state and input matrices for the model with an integrating component. The applied output matrix \(\mathrm{C_{lon}}=[0,-1,0,1]\) is responsible for calculating the flight path angle \(\gamma\) from the state variables \(\alpha\) and \(\theta\) according to relation (20).  
\[\Delta\gamma=\theta-\alpha \tag{20}\]  
Subsequently, the weight matrices Q and R were determined. To select the weight values, the diagonal form of the Q matrix was adopted, in which the last element of the last row denotes the weight of the \(\gamma\) error integral. In general, high weight values were assigned to the state variables to be tracked by the control system. As a result of _lqrd()_execution, the gain vector K\({}_{\text{d}}\) was obtained. Its first four elements formed the K\({}_{\text{lqri}}\) gain vector for state feedback, and the last element was the gain of integral term F\({}_{\text{lqri}}\). The values of initial weight matrices and the longitudinal controller settings are presented in Table A4a in Appendix C.  
Of all the coefficients in the Q matrix, increasing the weight of \(\gamma\), the error integral had the greatest effect on reducing the settling time, while leaving the values of u and q weights low. Faster control was achieved at the expense of an overshoot of a few percent, and in future, this can be eliminated by increasing the weights of \(\alpha\) and \(\theta\). When using the same weight matrices to determine the controller settings for the aircraft model with extended landing gear, the resulting step responses were almost identical.  
A verification of controller settings conducted on the prepared control system Simulink model revealed that, regardless of the weights, the control signal limitation had a dominant effect on the obtained settling time.  
A step response of the flight path angle control system with the LQRI controller settings presented and a setpoint increment of \(5^{\circ}\), resulted in a settling time of 11.96 s. For smaller inputs, this time interval may be slightly shorter.