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A laboratory experiment can be modelled as a system with an open-loop transfer function Q3 1 P(s) (s + s)(Z + s) Consider a PI controller C(s) with the transfer function K(s + z1) C(s) (a) For the controller C(s) defined above with z, = 4: Continued overleaf Page 3 of 10 Determine the open loop transfer function G(s) = C(s)P(s), and identify the open-loop poles and open-loop zeros of the system. (i) (ii) For the root locus of G(s), determine the number of asymptotes and where they meet, and calculate the location of any double point(s). 1 (iii) Sketch the root-locus for this system, using the information derived in (i)-(ii) and comment on the characteristics of this system for different values of K, e.g. for small K and for larger values of K. Hint: You may find it useful to know that the equation s3 + 9.5s2 + 28s + 20 = 0 has the solutions x, = -4.23 + 1.14i, x2 = -4.23 - 1.14i, and x3 = -1.04.

A laboratory experiment can be modelled as a system with an open-loop transfer function Q3 1 P(s) (s + s)(Z + s) Consider a PI controller C(s) with the transfer function K(s + z1) C(s) (a) For the controller C(s) defined above with z, = 4: Continued overleaf Page 3 of 10 Determine the open loop transfer function G(s) = C(s)P(s), and identify the open-loop poles and open-loop zeros of the system. (i) (ii) For the root locus of G(s), determine the number of asymptotes and where they meet, and calculate the location of any double point(s). 1 (iii) Sketch the root-locus for this system, using the information derived in (i)-(ii) and comment on the characteristics of this system for different values of K, e.g. for small K and for larger values of K. Hint: You may find it useful to know that the equation s3 + 9.5s2 + 28s + 20 = 0 has the solutions x, = -4.23 + 1.14i, x2 = -4.23 - 1.14i, and x3 = -1.04.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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A laboratory experiment can be modelled as a system with an open-loop transfer function Q3 1 P(s) = (s + 2)(s + 5) Consider a PI controller C(s) with the transfer function K(s + z1) C(s) = (a) For the controller C(s) defined above with z, = 4: Continued overleaf Page 3 of 10 Determine the open loop transfer function G(s) = C(s)P(s), and identify the open-loop poles and open-loop zeros of the system. (i) For the root locus of G(s), determine the number of asymptotes and where they meet, and calculate the location of any double point(s). (ii) Sketch the root-locus for this system, using the information derived in (i)-(ii) and comment on the characteristics of this system for different values of K, e.g. for small K and for larger values of K. (iii) Hint: You may find it useful to know that the equation s3 + 9.5s² + 28s + 20 = 0 has the solutions x, = -4.23 + 1.14i, x2 = -4.23 – 1.14i, and x3 = -1.04. (b) The Bode diagram of G(s) is shown in Figure Q3 below. Determine approximate values for the gain and phase margins from this plot. Bode Diagram 40 20 -40 -135 -180 10 100 10' 102 Frequency (radis) Figure Q3 Using root locus analysis, for the plant P(s) and controller C(s) presented at the start of the question, determine the smallest value of z, for which the closed loop system may become unstable for large values of K. Sketch a corresponding root locus diagram and discuss the differences to the diagram which you obtained in (a). (c) [51 (Gap) aseyd
Transcribed Image Text:A laboratory experiment can be modelled as a system with an open-loop transfer function Q3 1 P(s) = (s + 2)(s + 5) Consider a PI controller C(s) with the transfer function K(s + z1) C(s) = (a) For the controller C(s) defined above with z, = 4: Continued overleaf Page 3 of 10 Determine the open loop transfer function G(s) = C(s)P(s), and identify the open-loop poles and open-loop zeros of the system. (i) For the root locus of G(s), determine the number of asymptotes and where they meet, and calculate the location of any double point(s). (ii) Sketch the root-locus for this system, using the information derived in (i)-(ii) and comment on the characteristics of this system for different values of K, e.g. for small K and for larger values of K. (iii) Hint: You may find it useful to know that the equation s3 + 9.5s² + 28s + 20 = 0 has the solutions x, = -4.23 + 1.14i, x2 = -4.23 – 1.14i, and x3 = -1.04. (b) The Bode diagram of G(s) is shown in Figure Q3 below. Determine approximate values for the gain and phase margins from this plot. Bode Diagram 40 20 -40 -135 -180 10 100 10' 102 Frequency (radis) Figure Q3 Using root locus analysis, for the plant P(s) and controller C(s) presented at the start of the question, determine the smallest value of z, for which the closed loop system may become unstable for large values of K. Sketch a corresponding root locus diagram and discuss the differences to the diagram which you obtained in (a). (c) [51 (Gap) aseyd
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