(Solution) - As a newly hired engineer of the Ideal Gas Company -(2025 Original AI-Free Solution)

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As a newly hired engineer of the Ideal Gas Company, you are trying to make a reputation in the Process Control Group. However, this objective turns out to be a real challenge with I. M. Appelpolscher as your supervisor. At lunch one day, LM.A declares that a simple second-order process with a PI controller will always have a stability upper limit on Kc; that is, Kc is limited for all values of ?1 > 0. His best argument is that the open-loop process with the controller is third order. Furthermore, he claims that any critically damped second-order process will show he is right. Muttering ?au contraire,? you leave the table and quickly investigate the properties of

(a) What are the necessary and sufficient conditions for closed-loop stability for a PI controller?
(b) From these conditions, can you find a relationship for ?1 in terms of K that will guarantee stability? Show the stability region in a plot of ?1, versus Kc.
(c) Do some values of ?1 guarantee stability for all values of Kc? If so, what is the smallest value?