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Q3. A process can be represented by the transfer functions: y*(s): 4 (2s + 1)(5s - 1) 42 (2s + 1)(5s − 1)u* (s) a) Sketch the response you'd expect to see when changes by a unit step. b) Proportional control is to be added to the process. What is the range of gains that will give a stable and non-oscillatory controlled response?

Q3. A process can be represented by the transfer functions: y*(s): 4 (2s + 1)(5s - 1) 42 (2s + 1)(5s − 1)u* (s) a) Sketch the response you'd expect to see when changes by a unit step. b) Proportional control is to be added to the process. What is the range of gains that will give a stable and non-oscillatory controlled response?

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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Q3. A process can be represented by the transfer functions: y*(s) = 4 (2s + 1)(5s − 1)` d* (s) 42 (2s + 1)(5s − 1) - -u* (s) a) Sketch the response you'd expect to see when changes by a unit step. b) Proportional control is to be added to the process. What is the range of gains that will give a stable and non-oscillatory controlled response?
Transcribed Image Text:Q3. A process can be represented by the transfer functions: y*(s) = 4 (2s + 1)(5s − 1)` d* (s) 42 (2s + 1)(5s − 1) - -u* (s) a) Sketch the response you'd expect to see when changes by a unit step. b) Proportional control is to be added to the process. What is the range of gains that will give a stable and non-oscillatory controlled response?
3. a) The open-loop transfer functions are second-order. The open-loop response is unstable since there is a positive pole. The roots are real so the response is not oscillatory. Therefore, the open- loop response to a step disturbance looks similar to an exponential rise. b) Proportional control is to be added to the process. What is the range of gains that will give a stable and non-oscillatory controlled response? Unstable process - add proportional control to stabilise. Note: direct acting. We proceed just as we did for first order processes, but now subbing in the 2nd order TFs, G, and GD. We find the following closed-loop response: y* (s) = = 4 (2s+ 1)(5s − 1) + 42Kc d* (s) + 42Kc (2s + 1) (5s − 1) +42Kc To check for stability and oscillation-free response we need to look at the roots of the characteristic equation. 1 For stability we need to avoid positive roots, thus, Kc> 42 1 42 1.225 To avoid oscillation, we need to avoid complex roots, thus, Kc ≤ 42 So, the acceptable range for the gain is: 1.225 42 <Kc ≤. -YSP (S) In fact, we would generally prefer to have an oscillatory response, since it will allow us to reach steady state more quickly (c.f. week 8 activities on controller tuning).
Transcribed Image Text:3. a) The open-loop transfer functions are second-order. The open-loop response is unstable since there is a positive pole. The roots are real so the response is not oscillatory. Therefore, the open- loop response to a step disturbance looks similar to an exponential rise. b) Proportional control is to be added to the process. What is the range of gains that will give a stable and non-oscillatory controlled response? Unstable process - add proportional control to stabilise. Note: direct acting. We proceed just as we did for first order processes, but now subbing in the 2nd order TFs, G, and GD. We find the following closed-loop response: y* (s) = = 4 (2s+ 1)(5s − 1) + 42Kc d* (s) + 42Kc (2s + 1) (5s − 1) +42Kc To check for stability and oscillation-free response we need to look at the roots of the characteristic equation. 1 For stability we need to avoid positive roots, thus, Kc> 42 1 42 1.225 To avoid oscillation, we need to avoid complex roots, thus, Kc ≤ 42 So, the acceptable range for the gain is: 1.225 42 <Kc ≤. -YSP (S) In fact, we would generally prefer to have an oscillatory response, since it will allow us to reach steady state more quickly (c.f. week 8 activities on controller tuning).
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