Question:
A process exhibits an inverse response to a step input. Which of the following statements is/are necessarily TRUE about the process?
A process exhibits an inverse response to a step input. Which of the following statements is/are necessarily TRUE about the process?
Show Hint
Inverse response is caused by a right-half-plane (positive) zero; it only affects the transient shape, not stability, which depends on pole location.
Updated On: Jul 17, 2026
- Its transfer function has a positive zero.
- Its transfer function has a negative zero.
- The initial direction of the step response is opposite to the final steady state value.
- The system is unstable due to a zero in the right half of the s-plane.
Show Solution
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The Correct Option is A, C
Solution and Explanation
Step 1: Recall what an inverse response means.
An inverse response is a phenomenon in which the process output initially moves in a direction opposite to where it eventually settles, before reversing and moving toward its true final steady state value. This is a purely dynamic (transient) effect, not a stability issue.
Step 2: Analyze option A - positive zero.
Consider G(s) = K(1 - tau_a s) / [(tau_1 s + 1)(tau_2 s + 1)], with tau_a, tau_1, tau_2 > 0. This zero at s = 1/tau_a is a positive (RHP) zero. Final value theorem gives positive final value K. Initial slope is negative, meaning the response first dips before rising. This is the standard textbook cause of inverse response. Statement A is TRUE.
Step 3: Analyze option B - negative zero.
A negative (LHP) zero only modifies speed and overshoot, never causes a sign reversal. Statement B is FALSE.
Step 4: Analyze option C.
This is exactly the defining symptom of inverse response: y(t) moves one way initially, then reverses toward the opposite final steady state. Statement C is TRUE.
Step 5: Analyze option D.
Stability depends only on pole locations, not zeros. A RHP zero changes transient shape but never makes the process unstable. Statement D is FALSE.
Step 6: Conclusion.
A and C are necessarily true. \[ \boxed{A, C} \]
An inverse response is a phenomenon in which the process output initially moves in a direction opposite to where it eventually settles, before reversing and moving toward its true final steady state value. This is a purely dynamic (transient) effect, not a stability issue.
Step 2: Analyze option A - positive zero.
Consider G(s) = K(1 - tau_a s) / [(tau_1 s + 1)(tau_2 s + 1)], with tau_a, tau_1, tau_2 > 0. This zero at s = 1/tau_a is a positive (RHP) zero. Final value theorem gives positive final value K. Initial slope is negative, meaning the response first dips before rising. This is the standard textbook cause of inverse response. Statement A is TRUE.
Step 3: Analyze option B - negative zero.
A negative (LHP) zero only modifies speed and overshoot, never causes a sign reversal. Statement B is FALSE.
Step 4: Analyze option C.
This is exactly the defining symptom of inverse response: y(t) moves one way initially, then reverses toward the opposite final steady state. Statement C is TRUE.
Step 5: Analyze option D.
Stability depends only on pole locations, not zeros. A RHP zero changes transient shape but never makes the process unstable. Statement D is FALSE.
Step 6: Conclusion.
A and C are necessarily true. \[ \boxed{A, C} \]
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