## 1 Introduction  
PID control is widely used in industry to regulate variables such as temperature, speed, and position. Its name derives from its three terms, proportional, integral, and derivative, each with a specific function and a specific gain, \(k_{P}\), \(k_{I}\), and \(k_{D}\). The P term produces an output proportional to the current error. The I term eliminates steady-state error, and the \(D\) term anticipates future errors using the \(D\) of error. \(D\) is considered valuable as it enhances response time and transient behavior and is particularly effective in shortening response time and accelerating recovery from system disturbances [1, 2, 3]. This improves response speed and reduces overshoot, quickly stabilizing the system. However, \(D\) also has drawbacks, such as noise amplification and difficulty in proper tuning. Its sensitivity to noise in measurement signals can lead to unstable responses, and it requires deep knowledge of the system to properly adjust \(k_{D}\). Additionally, there are phenomena such as the (_Ltd-D_) bandwidth-limited derivative control and the \"(_D-kick_) derivative kick\". The _Ltd-D_ occurs when a process-variable measurement contains excessive noise due to high-frequency components in the system. \(D\) amplifies this noise because it has a high-frequency gain, making it unable to respond effectively to rapid changes and limiting its performance. In practical scenarios, a highly noisy control signal can cause actuator damage, leading to poor actuator performance [4]. _D-kick_ occurs with abrupt changes in SP, resulting in a sharp peak in the control signal. This undesired behavior makes \(D\) excessively large, potentially causing overshoot or actuator saturation, complicating control and stability. In summary, \(D\) enhances a control system's performance but must be carefully implemented to avoid undesired effects. Various strategies have been proposed to mitigate the adverse effects of \(D\)[3; 4]. Among these, the pure differentiator enhances high-frequency gain and prevents impulses in control signals, often combined with filtering techniques to minimize undesired effects, as demonstrated by [PERSON] et al. [5]. Another widely adopted method is the linear low-pass filter, proposed by [PERSON] et al. [6], in which a first-order low-pass filter is cascaded with the differentiator. This approach is particularly effective in data acquisition processes, although it tends to slow down transient responses. The speed feedback technique [7], which employs PV instead of the e(t) signal, offers another alternative for refining \(D\). Similarly, setpoint filters, as implemented by [PERSON] et al. [8], reduce sensitivity to SP changes and mitigate overshoot. However, tuning these filters for stability and robustness remains a complex task. Pre-filters, analyzed by [PERSON] et al. [9], are often used to achieve smooth control during step changes in SP but fail to address disturbances effectively as they are not integrated into a feedback loop. Finally, nonlinear mean filters, evaluated by [PERSON] et al. [10], provide another solution by averaging data points to eliminate noise and disturbances. While these filters effectively smooth control signals and suppress noise, they often increase response times and excessively dampen underdamped processes, limiting their utility in dynamic systems. These remedies only provide partial solutions. They primarily involve calculating filters to attenuate the controller signal, which cancels \(D\) under certain operating conditions. Besides these approaches, advanced control strategies, such as FL and NN, have been proposed as alternatives to improve the controller's performance. [PERSON] et al. [11], for instance, demonstrated improved system behavior using FL in a specific plant. While these methods enhance performance, they rely on expert knowledge, demand significant computational resources, and require intricate rule definitions.