Question

A proportional-integral-derivative (PID) controller is employed to stably control a plant with transfer function \[ P(s) = \frac{1}{(s + 1)(s + 2)}. \] Now, the proportional gain is increased by a factor of 2, the integral gain is increased by a factor of 3, and the derivative gain is left unchanged. Given that the closed-loop system continues to remain stable with the new gains, the steady-state error in tracking a ramp reference signal_________

• A. Remains unchanged
• B. Decreases by a factor of 2
• C. Decreases by a factor of 3
• D. Decreases by a factor of 5

Hint:

Increasing the integral gain in a PID controller reduces steady-state error for ramp inputs by improving the system’s ability to track such inputs.

Answer 1

The Correct Option is C

Step 1: Understanding the effect of PID gains on steady-state error.
In a PID controller, the steady-state error for different input types (e.g., step, ramp) is related to the type of system (based on the number of integrators in the open-loop transfer function). For a ramp input, the steady-state error is inversely proportional to the integral gain.
Step 2: Analyzing the system.
The transfer function of the system has two poles, so it’s a second-order system. Increasing the integral gain by a factor of 3 improves the system’s ability to track a ramp reference signal. The steady-state error decreases as the integral gain increases.
Step 3: Conclusion.
The steady-state error decreases by a factor of 3, as the integral gain has increased by a factor of 3. Thus, the correct answer is (C).