Question

A proportional-integral (PI) controller eliminates:

• A. Offset at steady state
• B. Oscillation in the closed-loop response
• C. Derivative kick
• D. Dead time effects

Hint:

Whenever "Integral action" or "I controller" is mentioned, its main purpose is always the elimination of steady-state error (offset).
However, the trade-off is that integral action tends to make the system more oscillatory and less stable.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks about the primary functional advantage of using a Proportional-Integral (PI) controller in a feedback loop.
Controllers are used to modify the behavior of a process to match a desired setpoint, and different controller modes (P, I, D) address different response characteristics.

Step 2: Key Formula or Approach:
A proportional controller output is proportional to the error: \[ p(t) = K_c e(t) + \bar{p} \] A proportional-integral (PI) controller output includes both proportional and integral terms: \[ p(t) = K_c \left[ e(t) + \frac{1}{\tau_I} \int_0^t e(t) \, dt \right] + \bar{p} \] In the Laplace domain, the transfer function of a PI controller is: \[ G_c(s) = K_c \left( 1 + \frac{1}{\tau_I s} \right) = K_c \left( \frac{\tau_I s + 1}{\tau_I s} \right) \]

Step 3: Detailed Explanation:
A Proportional (P) controller alone always results in a steady-state error (called offset) for a step change in setpoint or disturbance because a non-zero error is required to sustain a controller output different from the bias value $\bar{p}$.
When Integral (I) action is added, the controller integrates the error over time.
Even if the error $e(t)$ is extremely small, the integral term $\int e(t) \, dt$ will accumulate over time, changing the controller output to drive the process toward the setpoint.
The steady-state error is eliminated completely when the integral term stops changing, which only occurs when $e(t) = 0$ at steady state.
Mathematically, the introduction of the integral term adds a pole at the origin ($s = 0$) to the open-loop transfer function, which increases the system type by 1, thereby eliminating steady-state error (offset) for step inputs.

Step 4: Final Answer
Therefore, a PI controller eliminates offset at steady state, corresponding to option (A).