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A transfer function of control system does not have pole-zero cancellation. Which one of the following statements is true?

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A transfer function of control system does not have pole-zero cancellation. Which one of the following statements is true?

(a) System is neither controllable nor observable

(b) System is completely controllable and observable

(c) System is observable but uncontrollable

(d) System is controllable but unobservable

This question was addressed to me in a national level competition.

Question is from State Variable Analysis topic in portion Time Response Analysis, Design Specifications and Performance Indices of Control Systems

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The correct choice is (b) System is completely controllable and observable

The best I can explain: If the input-output transfer function of a linear time-invariant system has pole-zero cancellation, the system will be neither controllable nor observable.

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