## 3 Model and Research
### Sliding PID for Navigation System  
The UAV control systems are designed to execute navigation tasks. The navigation controllers are typically used as the outer loops for the attitude control system presented above. The structure of the high-level controllers may vary depending on airframe and actuator construction. The specifics of the modelling and control systems used are presented in [16]. Taking into account what is presented in the mentioned book, and operating on the basis of the study of approaches to design, the following systems are compiled. The typical (used in the most used civil autopilots) solution is the usage of the PID controller for a  
Figure 3: Pitch angle stabilization system. Figure 2: Motor block Simulink model. velocity loop and P or PI controller for an outer loop [17]. The other way of handling linear controllers of the aircraft is to provide the LQ controller for the trajectory that is presented in [18]. Although the research provided good results, the linear-quadratic system requires feedback linearization, for example [19]. This significantly complicates the problem and does not create a universal solution due to the fact that, in the case of VTOL drones such as FIXAR, the task is complicated by the wide range of disturbing influences. As a result, to simplify the system implementation, drones of this type use PID controllers to operate in \"aircraft\" and take-off and landing \"copter\" mode. The switching takes place either in sliding mode or under the control of an autopilot state machine. This section will discuss the application of such a system for the object under study. During the \"copter\" mode, the aircraft holds its position in 3-dimensional space, with disturbance from aerodynamic forces. At the same time, the system is required to work with the maximum wind speed 2-3 times higher than the aircraft ground speed. A typical solution for such a system is a multicopter position control system [20] in which the aircraft throttle channel controls the altitude, and the inclination of the entire aircraft achieves horizontal movement. While the PID controller controls the internal velocity loops, the P controller is used for position control, the coefficient of which determines the final dynamics of the system. For tuning, we simplify the system (3) with cascaded loops to the form presented in (6) using the approximation detailed in [21]. \[ \begin{split}\mathbf{W_{vcg}}=\frac{\mathbf{V_{vg}}}{\mathbf{C_{ P}}}&=\frac{\mathbf{\lambda_{vg}}}{s(\mathbf{s}\mathbf{T_{m}} \mathbf{s}^{3}+4\mathbf{T_{m}}\mathbf{s}^{2}+\mathbf{s}+\mathbf{K_{o}})}\\ \mathbf{W_{vg}}=\frac{\mathbf{V_{vg}}}{\mathbf{C_{R}}}& =\frac{\mathbf{\lambda_{vg}}}{s(\mathbf{s}\mathbf{T_{m}}\mathbf{s}^{3}+4 \mathbf{T_{m}}\mathbf{s}^{2}+\mathbf{s}+\mathbf{K_{o}})}\\ \mathbf{W_{zgv}}(\mathbf{s})=\frac{\mathbf{C_{T}}}{\mathbf{V_{ zg}}}&=\frac{\mathbf{4k}}{\mathbf{m}}\\ \end{split}  \tag{6}\]  
Parameter calculation resulted in the third-order oscillating link can being decomposed into an aperiodic link and a second-order oscillating link. Given this assumption, a PI controller tuned to a symmetric optimum is synthesized for all axes. Flight in the \"airplane\" mode is performed at a speed 1.5-2 times higher than maximum possible wind speed. In this state, aerodynamic forces create a significant part of the lifting force of the aircraft and can be used to control the object. Required in order to move efficiently, stabilization is provided by the airspeed of the aircraft. A typical aircraft navigation system, as applied to the object under study, is presented in Figure 4. In this system, the motors are tilted to the horizon and the longitudinal motion is distributed through the throttle/yaw channels and controlled using aerodynamic forces. Then, given the wide requirements on the range of perturbing influences (the ratio of the maximum wind to the aircraft cruise speed is 0.6), we apply the Ziegler-Nichols [22,23] frequency method. The results of the tuning and control systems schemes are given in Table 2 below.