GATE EC 2025 Set 1 — Question 36
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Control Systems → Compensators & Controllers → PID Controllers
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Question
Let be the transfer function of a second-order system. A controller is connected to the system in the configuration shown below.Consider the following statements.
(i) There exists no controller of the form , where is a positive real number, such that the closed loop system is stable.
(ii) There exists at least one controller of the form , where and are positive real numbers, such that the closed loop system is stable.Which one of the following options is correct?
(i) There exists no controller of the form , where is a positive real number, such that the closed loop system is stable.
(ii) There exists at least one controller of the form , where and are positive real numbers, such that the closed loop system is stable.Which one of the following options is correct?
Correct answer
(D) Both (i) and (ii) are TRUE
Solution
The characteristic equation of the closed-loop system is . (i) For , . This equation has roots in the right-half plane, so the system is unstable. Statement (i) is TRUE. (ii) For , . For , all coefficients are positive, and for a second-order system, this is sufficient for stability. Statement (ii) is TRUE. Thus, both (i) and (ii) are TRUE.
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