# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 4 Results and Discussion
### Kinematic and Dynamic Characteristics
The obtained expression is used to build a model in a numerical simulation environment and for further analysis of system stability, settling time, overshoot, and sensitivity to disturbances. It forms the foundation for transitioning to adaptive tuning of control parameters, tailored to real-world wind scenarios. As shown by the dynamic analysis, the wind load torque \(M_{w}(t)\) increases proportionally to the square of the wind speed (\(\propto v^{2}\)). This means that even a slight increase in wind speed leads to a sharp increase in the external disturbing torque. Such enhancement of external influence significantly impacts the behavior of the control system:  
* The control error increases due to insufficient torque compensation;* The inertial effect manifests--the flap reacts more slowly to control inputs;
* The risk of oscillations and overshoot increases, especially when using fixed controller parameters. Under strong and unstable wind conditions (e.g., when \(v>15\) m/s), fixed PID controller coefficient values become ineffective. In such situations, it is recommended to use an adaptive control strategy, where the controller parameters dynamically change depending on the current wind torque. Adaptive parameter correction is implemented as follows:
* Decreasing \(K_{p}\) reduces the system's sensitivity and lowers the likelihood of sharp responses, which is especially important when inertial and mechanical constraints are present;
* Increasing \(K_{d}\) enhances the damping effect, smooths transient processes, and suppresses oscillations caused by impulse wind gusts. To implement this concept, an adaptive PID controller is used, and its transfer function takes the following form: \[G(s)=\frac{6.28}{s(0.12s+1)} \tag{38}\]2 (49) where \(K_{wind}\) is the adaptation coefficient and \(M_{max}\) is the maximum wind torque. To improve the stability of the control system under sudden changes in wind load, an adaptive tuning strategy for the PID controller coefficients is used. The main idea is to modify the values of the coefficients \(K_{p}\), \(K_{i}\), and \(K_{d}\) based on the current value of the wind pressure torque \(M_{w}(t)\). As the load increases, the proportional and integral coefficients are decreased, reducing the system's sensitivity, while the differential coefficient is increased to enhance damping. This adaptive correction is implemented using the following dependencies: \[G(s)=\frac{6.28}{s(0.12s+1)} \tag{38}\]3 \[G(s)=\frac{6.28}{s(0.12s+1)} \tag{38}\]4 (50) \[G(s)=\frac{6.28}{s(0.12s+1)} \tag{38}\]5 where \(K_{p}(t)\) is the current values of the PID controller coefficients; \(K_{p,base}\), \(K_{i,base}\), \(K_{d,base}\) are the base (calculated) values of the coefficients, determined using the Ziegler-Nichols method; \(\alpha_{p}\), \(\alpha_{i}\), \(\alpha_{d}\) are the adaptive scaling coefficients; \(K_{wind}\) is the adaptation coefficient, reflecting sensitivity to the wind torque; \(M_{w}(t)\) is the current wind pressure torque; \(M_{max}\) is the upper boundary of the wind torque, corresponding to critical conditions. Under strong gusts, the Kp gain is reduced to decrease the system's response. In this study, the dynamics of the wind turbine flap control system were modeled using a PID controller with adaptive coefficients. The objective of the simulation was to identify the optimal values of the adaptation coefficients for the proportional (\(\alpha_{p}\)), integral (\(\alpha_{i}\)), and differential (\(\alpha_{d}\)) components, at which the control system demonstrates the best transient characteristics: minimum settling time, absence of sustained oscillations, and minimal overshoot. In this section, the effect of varying the adaptation coefficients \(\alpha_{p}\), \(\alpha_{i}\), and \(\alpha_{d}\) in Equation (50) is systematically explored to assess their influence on control performance. By analyzing the system response under different parameter combinations, we identify trends and trade-offs that inform the selection of gain scaling in the adaptive PID structure.