Which of the following statements are CORRECT for a controller?
P. In a proportional controller, a control action is proportional to the error
Q. In an integral controller, a control action is proportional to the derivative
of the error
R. There is no “offset” in the response of the closed-loop first-order process
with a proportional controller
S. There is no “offset” in the response of the closed-loop first-order process
with a proportional-integral controller
Options:
(A) P and Q only
(B) P and R only
(C) P and S only
(D) Q and S only
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2022 Q.41)**
**1. Governing Concept**:
- **Topic**: Bioprocess Control & Feedback Controller Characteristics
- Evaluating controller principles:
- **P. Proportional controller**: Control action is directly proportional to current error ($p(t) = \bar{p} + K_c e(t)$) $\rightarrow$ **CORRECT**.
- **Q. Integral controller**: Control action is proportional to the **integral of the error** ($\int e(t) dt$), not derivative $\rightarrow$ **INCORRECT**.
- **R. Offset in P-controller**: A pure proportional controller operating on a first-order process always exhibits a permanent non-zero steady-state error (**offset**) in response to a step change in load or setpoint $\rightarrow$ **INCORRECT**.
- **S. Offset elimination in PI-controller**: The addition of integral action introduces reset action that continues adjusting the manipulated variable until error is driven strictly to zero, completely **eliminating offset** $\rightarrow$ **CORRECT**.
- Therefore, statements **P and S only** are correct.
**2. Option Analysis**:
- **Option (A) [P and Q only] is INCORRECT**: Q confuses integral with derivative control.
- **Option (B) [P and R only] is INCORRECT**: R falsely claims no offset in P-controllers.
- **Option (C) [P and S only] is CORRECT**: P accurately defines proportional mode, and S correctly confirms offset elimination by integral mode.
- **Option (D) [Q and S only] is INCORRECT**: Q is factually incorrect.
**3. Final Answer**:
- **Correct Choice**: **C**
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