The transfer function of a Proportional-Integral (PI) controller GC(s) is given by
GC(s) = KC (1 + 1
πIs)
where KC is the controller gain, πI is the controller integral time constant and s is
the Laplace variable. The role of the integral component of the controller is to
____________.
Options:
(A) integrate the difference between the process and manipulated variable
(B) integrate the difference between the set point and disturbance variable
(C) integrate the difference between the set point and the measured variable
(D) integrate the difference between the input and output variable
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2026 Q.15)**
**1. Governing Concept**:
- **Topic**: Bioprocess Control & Transfer Functions of Controllers
- A **Proportional-Integral (PI)** controller generates an output proportional to the error $e(t)$ plus the integral of the error:
$p(t) = \bar{p} + K_c e(t) + \frac{K_c}{\tau_I} \int_0^t e(t')\,dt'$
- Taking the Laplace transform:
$P(s) = K_c E(s) + \frac{K_c}{\tau_I s} E(s) = K_c \left(1 + \frac{1}{\tau_I s}\right) E(s)$
- The controller transfer function is:
$G_c(s) = \frac{P(s)}{E(s)} = K_c \left(1 + \frac{1}{\tau_I s}\right) = K_c \left(\frac{\tau_I s + 1}{\tau_I s}\right)$
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Represents a pure proportional-derivative (PD) controller ($K_c(1 + \tau_D s)$).
- **Option (B) is INCORRECT**: Represents a pure proportional controller ($K_c$).
- **Option (C) is CORRECT**: Canonical Laplace domain transfer function form for a PI controller: $K_c \left(1 + \frac{1}{\tau_I s}\right)$.
- **Option (D) is INCORRECT**: Inverts the integral time constant placement.
**3. Final Answer**:
- **Correct Choice**: **C**
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