# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 4 Results and Discussion
### Kinematic and Dynamic Characteristics
The larger the error, the stronger the control action. A large \(K_{p}\) accelerates the response, but if too large, it can cause overshoot. * Integral term \(K_{i}\cdot\int e(t)dt\)--accumulates the error over time and eliminates systematic deviation (for example, due to a constant wind moment). However, an excessively large \(K_{i}\) makes the system slow and prone to oscillation. * Differential term \(K_{d}\cdot\frac{de(t)}{dt}\)--works as a predictor, compensating for sudden changes in error. It effectively dampens oscillations and improves the transient process. Thus, correctly selecting the parameters \(K_{p}\), \(K_{i}\), \(K_{d}\) ensures stable, fast, and accurate control of the flap position under a wide range of wind conditions. In the following sections, numerical modeling and tuning of these coefficients are carried out to achieve optimal transient characteristics. For the initial tuning of the PID controller parameters, the Ziegler-Nichols empirical method was used, which quickly provides approximate values of the coefficients that ensure acceptable system dynamics in the linear operating range. This method is particularly effective for systems with well-defined inertia and feedback, such as the mechanical part of the aerodynamic flap being considered. The Ziegler-Nichols tuning method involves initially turning off the integral and differential components (\(K_{i}=0\), \(K_{d}=0\)) and gradually increasing the proportional gain \(K_{p}\) until the system begins to exhibit sustained oscillations. At this point, the following conditions apply:  
\(K_{u}\)--the critical value of the gain coefficient at which the system starts oscillating without increasing or decreasing amplitude;  
\(T_{u}\)--the period of stable oscillations measured in this critical mode. During the numerical experiment and analysis of the closed-loop system's transient response (based on the complete transfer function \(H(s)\)), the following was established:  
\[K_{p}=0.6\cdot Ku,\ K_{i}=\frac{2\cdot K_{p}}{T_{u}},\ K_{d}=\frac{K_{p}\cdot T _{u}}{8} \tag{46}\]  
where \(K_{u}=3.5\) is the critical gain value, at which point the system begins to oscillate; \(T_{u}=0.8\) s is the period of stable oscillations. Substituting the values, \(K_{p}=2.1,\ K_{i}=5.25,\ K_{d}=0.21\). The obtained coefficients provide a stable system response with moderate settling time and an acceptable level of overshoot. These coefficients are applicable under moderate wind loads (up to 15 m/s) and serve as a basis for further adaptive and predictive tuning of the controller. This approach simplifies the initial calibration of the control system and provides a foundation for further adaptive adjustment of parameters as external conditions vary. The transfer function of the controller, according to Equation (43), takes the following form:  
\[G(s)=\frac{6.28}{s(0.12s+1)} \tag{38}\]0  
Thus, by substituting the coefficients calculated using the Ziegler-Nichols method into Equation (44), we obtain the complete transfer function of the closed-loop system for controlling the position of the aerodynamic flap:  
\[G(s)=\frac{6.28}{s(0.12s+1)} \tag{38}\]1  
This transfer function describes the behavior of the closed-loop system under the influence of the input control signal. The numerator reflects the combined action of the proportional, integral, and differential components of the controller, as well as the dynamics of the mechanical part of the system, including the inertial properties of the actuator. The denominator reflects the presence of negative feedback, ensuring stability and error correction. A key feature of this structure is the integral component, responsible for eliminating static error, as well as the differential component, which improves transient characteristics and suppresses high-frequency oscillations. This type of controller allows for precise flap positioning even under varying wind loads.