Question

A process exhibits an inverse response to a step input. Which of the following statements is/are necessarily TRUE about the process?

• A. Its transfer function has a positive zero.
• B. Its transfer function has a negative zero.
• C. The initial direction of the step response is opposite to the final steady state value.
• D. The system is unstable due to a zero in the right half of the s-plane.

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.

Answer 1

The Correct Option is A, C

Take $G(s) = \dfrac{2(1 - 0.5s)}{(s+1)(2s+1)}$. Final value $y(\infty)=2$. Initial slope $\dot y(0)=-2$ (negative), so the response dips before rising to 2, confirming the RHP zero at $s=2$ causes inverse response (A) and the initial-vs-final direction mismatch (C). If the zero were negative instead, $\dot y(0)$ would share the sign of $y(\infty)$, no reversal, ruling out B. The poles at $s=-1,-0.5$ never move regardless of zero sign, so stability is unaffected, ruling out D. \[ \boxed{A, C} \]