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 K_c; that is, K_c 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 K_c.

(c) Do some values of ?₁ guarantee stability for all values of K_c? If so, what is the smallest value?

Transcribed Image Text:

G.G,Gm (10r + 1)