# Inverse Optimal Design for Position Control of a Quadrotor
## 4 Simulation Results and Analysis
### Simulation Results
In order to verify the performance of the proposed controller in this paper, we have conducted simulation to compared the performance with other control techniques. In this simulation, the ascending circular trajectory is set as the reference trajectory, we tested whether the quadrotor track the trajectory well in nonlinear motion. We compared the proposed inverse optimal control, LQR control that is a typical optimal control, and PD control. Since, this paper focuses on position control, so all of the quadrotor attitude control uses the same PID control and same PID gains of attitude. The PID gains of attitude is equal to the gains of the previous simulation. For the PD position control, the control performance depends on the PD gain. Therefore, proper PD gain setting is important for proper performance comparison. To select the proper PD gain, the range of the reference attitude angle and the magnitude of the altitude input were selected in comparison with the other two control methods. Since, the range of the reference attitude angle for position control changes according to the position control gain. Figures 11 and 12 show trajectory tracking control simulation results for three control methods, when the initial position is \((0,1,0)\) with the ascending circular reference trajectory. More details, the time histories of the \(x,\,y,\,z\) for each motion of a quadrotor are shown in Figure 12. From the simulation results, we verify that the proposed inverse optimal control has better trajectory tracking performance than other control methods. Figure 13 shows the reference attitude angle for trajectory tracking. In Figure 13, we can see that the attitude of the quadrotor changes continuously around 40 degrees, and if the attitude angle changes greatly, the linear control method like LQR control, PD control does not perform well. The proposed inverse optimal control is an optimal control method for nonlinear system, unlike LQR control method, it can be expected good performance even in the circular trajectory like Figure 4:0 Looking at the range of attitude angles of Figure 13 and the magnitudes of the inputs of Figures 14 and 15, we can see that the three control methods are simulated in very similar conditions. Figure 4:1 The attitude response for the trajectory tracking control. The dashed line indicates the reference attitude angles that enable the quadrotor to converge in the desired trajectory, and the solid line is the result of the simulation about attitude responses of a quadrotor: (**a**) \(\theta\); (**b**) \(\psi\); (**c**) \(\phi\). Figure 4:2 Circular trajectory tracking control results: (**a**) Inverse optimal control; (**b**) LQR control; (**c**) PD control. Figure 4:3 _Cont._  
Figure 4:4 Circular trajectory tracking control results for each of the three axes: (**a**) Inverse optimal control; (**b**) LQR control; (**c**) PD control. Figure 4:5 The attitude response for the trajectory tracking control: (**a**) Inverse optimal control; (**b**) LQR control; (**c**) PD control. Figure 4:6 The control inputs for the attitude of a quadrotor: (**a**) Inverse optimal control; (**b**) LQR control; (**c**) PD control. Figure 4:7 The control input for the altitude of a quadrotor: (**a**) Inverse optimal control; (**b**) LQR control; (**c**) PD control. Along with the simulation results, we can say the advantage of inverse optimal is in terms of nonlinearity and optimality. The proposed inverse optimal control is of great significance that we dealt with the optimal concept in a nonlinear system and we can find an optimal solution. Simulation results show that the inverse optimal control is better performance because it overcomes the limitation of LQR that are designed based on linear systems. It is an advantage of inverse optimal control from a nonlinear point of view.