Q3 (a) A temperature control system for a laboratory experiment can be modelled as a system with an open-loop trans fer function of 1 G.(s) = K - (s + 1)(s + 5)(s + 10) (i) Identify the open-loop poles and zeros of the system, and the relative degree of the open-loop system. (ii) Determine the number of asymptotes in the root-locus and where they meet. (iii) Calculate the location of any double point(s) of the root-locus. (iv) Sketch the root-locus for this system, using the information derived m (i)-(iii). Comment on the characteristics of this system for small K and for larger values of K. What is the maximal value of K for which the closed-loop system is overdamped?
Q3 (a) A temperature control system for a laboratory experiment can be modelled as a system with an open-loop trans fer function of 1 G.(s) = K - (s + 1)(s + 5)(s + 10) (i) Identify the open-loop poles and zeros of the system, and the relative degree of the open-loop system. (ii) Determine the number of asymptotes in the root-locus and where they meet. (iii) Calculate the location of any double point(s) of the root-locus. (iv) Sketch the root-locus for this system, using the information derived m (i)-(iii). Comment on the characteristics of this system for small K and for larger values of K. What is the maximal value of K for which the closed-loop system is overdamped?
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![A temperature control system for a laboratory experiment can be modelled as a
system with an open-loop transfer function of
Q3
(a)
1
G.(s) = K
(s + 1)(s + 5)(s + 10)
Identify the open-loop poles and zeros of the system, and the relative
degree of the open-loop system.
(i)
(ii)
Determine the number of asymptotes in the root-locus and where they
meet.
(iii)
Calculate the location of any double point(s) of the root-locus.
(iv)
Sketch the root-locus for this system, using the information derived n
(i)-(iii). Comment on the characteristics of this system for small K and
for larger values of K. What is the maximal value of K for which the
closed-loop system is overdamped?
Hint: The equation s? + 10.67s + 21.67 = 0 has the solutions s, = -7.94 and
S2 = -2.73.
In a modified system, an ideal proportional-derivative controller with the
transfer function
(b)
C(s) = K + Ks
is used, so that the overall open-loop system is now
1
G.(s)
= C(s)
(s + 1)(s + 5)(s + 10)
Determine the number of asymptotes in the root-locus of this system and where
they meet, and sketch the approximate root-locus for this modified system. How
has the change affected the stability of the closed-loop system? (Note that you
are not required to calculate the exact location of any double point(s) for (b)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9147e2a-f1a4-4ee2-826b-508a5fd2d6bb%2Fd79277fd-38c3-41cf-992b-b6465bfec670%2F1h495ln_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A temperature control system for a laboratory experiment can be modelled as a
system with an open-loop transfer function of
Q3
(a)
1
G.(s) = K
(s + 1)(s + 5)(s + 10)
Identify the open-loop poles and zeros of the system, and the relative
degree of the open-loop system.
(i)
(ii)
Determine the number of asymptotes in the root-locus and where they
meet.
(iii)
Calculate the location of any double point(s) of the root-locus.
(iv)
Sketch the root-locus for this system, using the information derived n
(i)-(iii). Comment on the characteristics of this system for small K and
for larger values of K. What is the maximal value of K for which the
closed-loop system is overdamped?
Hint: The equation s? + 10.67s + 21.67 = 0 has the solutions s, = -7.94 and
S2 = -2.73.
In a modified system, an ideal proportional-derivative controller with the
transfer function
(b)
C(s) = K + Ks
is used, so that the overall open-loop system is now
1
G.(s)
= C(s)
(s + 1)(s + 5)(s + 10)
Determine the number of asymptotes in the root-locus of this system and where
they meet, and sketch the approximate root-locus for this modified system. How
has the change affected the stability of the closed-loop system? (Note that you
are not required to calculate the exact location of any double point(s) for (b)).
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