Question:
If the transfer function of a controller is given by
\[
G_c(s)=\frac{K_1s^2+K_2s+K_3}{s},
\]
then the proportional, integral, and derivative constants are:
If the transfer function of a controller is given by
\[
G_c(s)=\frac{K_1s^2+K_2s+K_3}{s},
\]
then the proportional, integral, and derivative constants are:
Show Hint
To easily identify the parameters: The constant term independent of \(s\) is always the proportional gain (\(K_p\)), the term divided by \(s\) is the integral gain (\(K_i\)), and the coefficient of \(s\) is the derivative gain (\(K_d\)).
Updated On: Jun 25, 2026
- \(K_1, K_2\) and \(K_3\) respectively
- \(K_3, K_2\) and \(K_1\) respectively
- \(K_2, K_1\) and \(K_3\) respectively
- \(K_2, K_3\) and \(K_1\) respectively
Show Solution
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The Correct Option is D
Solution and Explanation
Concept:
A Proportional-Integral-Derivative (PID) controller creates an output command that combines proportional, integral, and derivative terms. The standard parallel-form transfer function of a PID controller is defined as:
\[
G_{PID}(s) = K_p + \frac{K_i}{s} + K_d s
\]
where:
• \(K_p\) is the Proportional Gain Constant.
• \(K_i\) is the Integral Gain Constant.
• \(K_d\) is the Derivative Gain Constant.
Step 1: Deconstruct the given controller transfer function into independent fractional elements. The provided expression is: \[ G_c(s) = \frac{K_1s^2 + K_2s + K_3}{s} \] Divide each term in the numerator individually by the common denominator \(s\): \[ G_c(s) = \frac{K_1s^2}{s} + \frac{K_2s}{s} + \frac{K_3}{s} \] Simplifying the fractions: \[ G_c(s) = K_1s + K_2 + \frac{K_3}{s} \]
Step 2: Match terms with standard PID gain parameters. Rearranging the terms to align with the standard form: \[ G_c(s) = K_2 + \frac{K_3}{s} + K_1s \] Comparing this directly with the standard equation \(G_{PID}(s) = K_p + \frac{K_i}{s} + K_d s\):
• Proportional Constant (\(K_p\)) = \(K_2\)
• Integral Constant (\(K_i\)) = \(K_3\)
• Derivative Constant (\(K_d\)) = \(K_1\) Thus, the respective sequence of constants is \(K_2, K_3\) and \(K_1\), which corresponds to option (D).
• \(K_p\) is the Proportional Gain Constant.
• \(K_i\) is the Integral Gain Constant.
• \(K_d\) is the Derivative Gain Constant.
Step 1: Deconstruct the given controller transfer function into independent fractional elements. The provided expression is: \[ G_c(s) = \frac{K_1s^2 + K_2s + K_3}{s} \] Divide each term in the numerator individually by the common denominator \(s\): \[ G_c(s) = \frac{K_1s^2}{s} + \frac{K_2s}{s} + \frac{K_3}{s} \] Simplifying the fractions: \[ G_c(s) = K_1s + K_2 + \frac{K_3}{s} \]
Step 2: Match terms with standard PID gain parameters. Rearranging the terms to align with the standard form: \[ G_c(s) = K_2 + \frac{K_3}{s} + K_1s \] Comparing this directly with the standard equation \(G_{PID}(s) = K_p + \frac{K_i}{s} + K_d s\):
• Proportional Constant (\(K_p\)) = \(K_2\)
• Integral Constant (\(K_i\)) = \(K_3\)
• Derivative Constant (\(K_d\)) = \(K_1\) Thus, the respective sequence of constants is \(K_2, K_3\) and \(K_1\), which corresponds to option (D).
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