Integral controller behavior: In process control, what is the correct relationship between the input signal and the output of an ideal integral controller?

Introduction / Context:
Integral controllers are a key component in PID control systems. Their main role is to eliminate steady-state error by integrating the error signal over time.

Given Data / Assumptions:

• Ideal continuous-time integral controller.
• Transfer function is K_i / s.

Concept / Approach:

Mathematically: if e(t) is the input error, the controller output u(t) = K_i ∫ e(t) dt. Differentiating both sides shows du/dt = K_i * e(t), meaning the rate of change of output is proportional to the input error.

Step-by-Step Solution:

Verification / Alternative check:

Block-diagram modeling of PID confirms the integral action has infinite DC gain, accumulating input over time.

Why Other Options Are Wrong:

• (a) Proportional controller, not integral.
• (c) Describes a derivative controller.
• (d) Incorrect since the integral has a defined relationship.

Common Pitfalls:

• Confusing integral with proportional action.
• Ignoring the fact that the integral term builds gradually over time.

Final Answer:

The rate of change of output is proportional to the input.