GATE EE 2015 Set 1 — Question 65
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Control Systems → Frequency Response & Root Locus → Root Locus Construction
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Question
The open loop poles of a third order unity feedback system are at . Let the frequency corresponding to the point where the root locus of the system transits to unstable region be . Now suppose we introduce a zero in the open loop transfer function at , while keeping all the earlier open loop poles intact. Which one of the following is TRUE about the point where the root locus of the modified system transits to unstable region?
Correct answer
(D) Root locus of modified system never transits to unstable region
Solution
For the original system with poles at , the characteristic equation is , which is . Using Routh-Hurwitz, the system becomes unstable for . When a zero is added at , the open-loop transfer function becomes . The characteristic equation is . The Routh table is:
Since all elements in the first column () are positive for all , the system is stable for all positive values of . Therefore, the root locus never enters the right-half plane (unstable region). This matches option (D).
| Col 1 | Col 2 | |
|---|---|---|
| 1 | ||
| 3 | ||
| 0 | ||
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