Question

The transfer function of a Proportional-Integral (PI) controller \(G_C(s)\) is given by
\[ G_C(s) = K_C\left(1 + \frac{1}{\tau_I s}\right) \]
where \(K_C\) is the controller gain, \(\tau_I\) is the controller integral time constant and \(s\) is the Laplace variable. The role of the integral component of the controller is to .

• 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

Hint:

The integral term of a PI controller always acts on the feedback error, the gap between the set point and the measured process variable, not on the manipulated variable, disturbance, or a vague input/output pair.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the transfer function of a PI (Proportional-Integral) controller, \(G_C(s) = K_C\left(1 + \frac{1}{\tau_I s}\right)\), and asked what the integral term physically does inside a feedback control loop.

Step 2: Key Formula or Approach:
In any feedback control loop, the controller acts on the error signal \(e(t)\), defined as the difference between the set point (the desired value of the process variable) and the measured variable (the actual value read by the sensor).
\[ e(t) = y_{sp}(t) - y_{measured}(t) \]
The PI controller output is
\[ u(t) = K_C\, e(t) + \frac{K_C}{\tau_I}\int_0^t e(t')\, dt' \]

Step 3: Detailed Explanation:
The first term, \(K_C e(t)\), is the proportional action: it reacts to the error at the present instant.
The second term, \(\frac{K_C}{\tau_I}\int e(t')dt'\), is the integral action: it keeps adding up (accumulating) the error over time.
Because this term never forgets past error, it keeps growing until the error becomes exactly zero, which is what removes any steady state offset that a pure proportional controller would leave behind.
The quantity being integrated is always the error signal, the gap between what we want (set point) and what we actually measure (measured or process variable), never a gap between two inputs or two outputs.
Option (A) mixes up the process variable with the manipulated variable (the controller output itself, like a valve position); these are not the two quantities compared to form the error.
Option (B) wrongly pairs the set point with the disturbance variable; a disturbance is an external upset, not something subtracted from the set point to make the error signal.
Option (D) speaks of an input and output variable in a vague, non standard way that does not match the precise control engineering definition of error.
Only option (C), the set point minus the measured variable, is the correct error definition being integrated.

Step 4: Final Answer:
The integral component of a PI controller integrates the difference between the set point and the measured variable, so as to drive this error to zero over time. \[ \boxed{\text{(C) integrate the difference between the set point and the measured variable}} \]