Integral controller behavior: In process control, what is the correct relationship between the input signal and the output of an ideal integral controller?
-
AThe output is proportional to the input
-
BThe rate of change of output is proportional to the input
-
CThe output is proportional to the rate of change of input
-
DNone of the above
-
EThe output is inversely proportional to the input
Answer
Correct Answer: The rate of change of output is proportional to the input
Explanation
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:
Start with transfer function: U(s)/E(s) = K_i / s.In time domain: u(t) = K_i ∫ e(t) dt.Differentiate: du/dt = K_i * e(t).Therefore, the rate of change of output is proportional to input.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.