1. Derivative Control Action ($D$): Derivative control produces an output based on the
rate of change of the error signal. Mathematically, it is represented as $D = K_d \frac{de}{dt}$. In the Laplace domain, the transfer function is $G_d(s) = K_d s$.
2. High-Pass Characteristics:
• Slow changes (Low Frequency): If the error changes very slowly, the derivative is small, and the output is low. It "blocks" or ignores low-frequency signals.
• Fast changes (High Frequency): If the error changes rapidly, the derivative is large, and the output is very high. It "passes" or amplifies high-frequency signals.
This behavior of responding more strongly to high frequencies and ignoring low ones is exactly how a
high-pass filter behaves.
3. Contrast with Integral Control: Integral control ($\int e \, dt$ or $1/s$) acts as a
low-pass filter. It averages out fast fluctuations (noise) and responds strongly to slow, sustained changes.
Because derivative action is sensitive to rapid changes, it is also highly sensitive to measurement noise, which is why "derivative filtering" is almost always used in practical PID applications.