Question

Which of the following controllers has more oscillatory behaviour in response?

• A. P-controller
• B. PI-controller
• C. PD-controller
• D. PID-controller

Hint:

A PI-controller can exhibit more oscillations due to the integral term, which accumulates error, while a PID-controller’s derivative term helps dampen oscillations.

Answer 1

The Correct Option is B

Step 1: Understand controller types and their responses. 
Controllers adjust the manipulated variable to maintain the controlled variable at the set point. Their response behavior (e.g., oscillatory, stable) depends on their structure:
P-controller (Proportional): Adjusts the output proportional to the error (\( u = K_p e \)). It is simple but often results in a steady-state offset and minimal oscillations.
PI-controller (Proportional-Integral): Includes an integral term to eliminate steady-state offset (\( u = K_p e + K_i \int e \, dt \)). The integral action can introduce oscillations, especially if the integral gain \( K_i \) is high, as it accumulates past errors.
PD-controller (Proportional-Derivative): Includes a derivative term to dampen oscillations (\( u = K_p e + K_d \frac{de}{dt} \)). The derivative action predicts error changes, reducing overshoot and oscillations.
PID-controller (Proportional-Integral-Derivative): Combines all three terms (\( u = K_p e + K_i \int e \, dt + K_d \frac{de}{dt} \)). The derivative term helps dampen oscillations introduced by the integral term, making the response more balanced. 
Step 2: Analyze oscillatory behavior. 
P-controller: Produces a quick response but typically does not oscillate much because it lacks integral or derivative action. However, it leaves a steady-state offset.
PI-controller: The integral term eliminates the offset but can cause oscillations. The accumulation of error in the integral term can lead to overshoot and sustained oscillations, especially in systems with time delays or high integral gain.
PD-controller: The derivative term reduces oscillations by anticipating error changes, leading to a more damped response compared to a P-controller.
PID-controller: The derivative term counteracts the oscillatory tendency of the integral term, resulting in a response that is less oscillatory than a PI-controller but more responsive than a P-controller. Among these, the PI-controller tends to have the most oscillatory behavior because the integral action can overshoot and oscillate while trying to correct the steady-state error, and there is no derivative term to dampen these oscillations. 
Step 3: Evaluate the options. 
(1) P-controller: Incorrect, as it has minimal oscillations but leaves a steady-state offset. Incorrect.
(2) PI-controller: Correct, as the integral term introduces more oscillatory behavior by accumulating error, without a derivative term to dampen it. Correct.
(3) PD-controller: Incorrect, as the derivative term reduces oscillations, making the response more damped. Incorrect.
(4) PID-controller: Incorrect, as the derivative term mitigates the oscillations caused by the integral term, resulting in a less oscillatory response than a PI-controller. Incorrect.
Step 4: Select the correct answer. 
The PI-controller has more oscillatory behavior in its response, matching option (2).