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, I.M.A. declares that a simple second-order process with a PI controller will always have a stability upper limit on K꜀;that is,K꜀ is limited for all values of ?I>0. His best argument is thatthe open-loop process with the controller is third order. Fur-thermore, he claims that any critically damped second-orderprocess will show he is right.Muttering “au contraire,” you leave the table and quickly investigate the properties of GᵥGₚGₘ=5/{(10s+1)}²(a)What are the conditions for closed-loop stability for a PI controller?(b)From these conditions, can you find a relationship for ?I in terms of K꜀ that will guarantee stability? Show the stability region in a plot of ?I versus K꜀ .(c)Do some values ofτIguarantee stability for all values of K꜀? If so, what is the smallest value?
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, I.M.A. declares that a simple second-order process with a PI controller will always have a stability upper limit on K꜀;that is,K꜀ is limited for all values of ?I>0. His best argument is thatthe open-loop process with the controller is third order. Fur-thermore, he claims that any critically damped second-orderprocess will show he is right.Muttering “au contraire,” you leave the table and quickly investigate the properties of GᵥGₚGₘ=5/{(10s+1)}²(a)What are the conditions for closed-loop stability for a PI controller?(b)From these conditions, can you find a relationship for ?I in terms of K꜀ that will guarantee stability? Show the stability region in a plot of ?I versus K꜀ .(c)Do some values ofτIguarantee stability for all values of K꜀? If so, what is the smallest value?
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