# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 4 Results and Discussion
### Kinematic and Dynamic Characteristics
Thus, the transfer function of the mechanical link can also be described in a similar way--as a scaling factor that transforms the motion of the gearbox output into the corresponding motion of the flap:  
\[G_{red}(s)=\frac{1}{i_{mech}}=\frac{1}{1.875} \tag{40}\]  
Taking into account the previously defined characteristics of the servo motor, gearbox, and mechanical lever link, it is possible to derive the overall transfer function of the entire mechanical part of the control system--from the control input to the servo motor to the angular position of the aerodynamic flap at the output. The complete transfer function is formed as the sequential product of the transfer functions of the servo motor \(G\)(s), the gearbox \(G_{real}\)(s), and the mechanical linkage \(G_{mech}\)(s):  
\[G_{total}(s)=G(s)\ \cdot G_{red}(s)\cdot G_{mech}(s) \tag{41}\]  
Substituting the previously obtained expressions,  
\[G_{total}(s)=\frac{6.28}{s(0.12s+1)}\cdot\frac{1}{14}\cdot\frac{1}{1.875}=\frac {0.24}{s(0.12s+1)} \tag{42}\]  
The resulting transfer function \(G_{total}\)(s) reflects the inertial-integral nature of the entire mechanical part of the system. It accounts for the scaling of the control input caused by the cascade transmission of torque and speed and can be used in control system synthesis, particularly in the calculation of correct PID controller parameters or in numerical modeling of the aerodynamic flap's response to the input signal. After determining the overall transfer function of the mechanical part of the system, which includes the servo motor, gearbox, and lever mechanism, the next step is to proceed with the synthesis of the closed-loop control system. The control objective is to ensure accurate positioning of the aerodynamic flap under various external disturbances, including wind gusts, taking into account inertia and mechanical limitations. The control system is built according to the classical scheme with negative feedback. In this case, the controller used is a PID controller with a transfer function \(C(s)\), which includes proportional, integral, and differential components [41]:  
\[C(s)=K_{p}+\frac{K_{i}}{s}+K_{d}s \tag{43}\]  
Thus, the transfer function of the closed-loop system is given by  
\[H(s)=\frac{C(s)G_{total}(s)}{1+C(s)G_{total}(s)} \tag{44}\]  
The task of tuning the controller involves selecting the optimal values of the coefficients \(K_{p}\), \(K_{i}\), \(K_{d}\), which ensure the desired transient characteristics of the system. In this study, the following control quality criteria were established:  
* Overshoot should not exceed 5%;
* Settling time should be no more than 1 s.  
Further modeling and analysis of the system's time characteristics allow the determination of optimal PID controller parameters that provide stable, fast, and precise system response to the given control input under varying external conditions. To stabilize the flap position and compensate for external disturbances (including wind gusts), a classical control scheme with negative feedback based on the flap's angular position is used. To achieve effective and stable control of the aerodynamic flap's angular position, a classical PID controller is employed, generating the control torque based on the current deviation of the system from the desired position. The control input is generated based on the following equation [34]:  
\[M_{ctrl}(t)=K_{p}\cdot e(t)+K_{i}\cdot\int_{0}^{t}e(\tau)\,d\tau+K_{d}\cdot \frac{de(t)}{dt} \tag{45}\]  
where \(M_{ctrl}(t)\) is the control torque applied by the servo motor; \(e(t)=\theta_{set}(t)-\theta_{s}(t)\) is the control error (the deviation between the desired and current flap positions); \(K_{p}\) is the proportional coefficient, which amplifies the current deviation; \(K_{i}\) is the integral coefficient, which eliminates static error; \(K_{d}\) is the differential coefficient, which damps oscillations. In the equation, the following components are used:  
* Proportional term \(K_{p}\cdot e(t)\)--responds immediately to the deviation.