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). ** (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.

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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 + z,)
C(s) =
(a)
For the controller C(s) defined above with z, = 4:
Continued overleaf
Page 3 of 10
(i)
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.
[3
(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).
(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 x = -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
-135
-180
100
Frequency (rad/s)
10
10
102
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)
(Bep) eseyd
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 + z,) C(s) = (a) For the controller C(s) defined above with z, = 4: Continued overleaf Page 3 of 10 (i) 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. [3 (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). (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 x = -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 -135 -180 100 Frequency (rad/s) 10 10 102 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) (Bep) eseyd
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