Concordia University
Department of Electrical and Computer Engineering
Fundamentals of Control Systems (ELEC372)
Fall 2024
S. Hashtrudi Zad
Assignment 7
Due: Wednesday, Nov. 6, 2024 (11:59pm)
1. In the feedback system of Fig. 1,
K(s) =
k(s + 3)
s+4
G(s) =
1
(s + 1)(s + 2)
Follow the steps given below to plot the root locus of the closed–loop system for k > 0.
(a) Write the C.E. in the form
1+k
(b) List the zeros and poles of
N (s)
= 0.
D(s)
N (s)
.
D(s)
(c) How many branches does the RL plot have?
(d) How many asymptotes does the RL have? Determine the centre (σA ) and angles
of the asymptotes (ϕA ).
(e) Determine the segments of the real axis that belong to the RL plot for k > 0.
(f) Plot the root locus for k > 0. The branches of the RL plot must be clearly drawn
with thick lines with arrows.
The calculations of breakaway points and angles of departure from poles and arrival
at zeros are not required.
R(s)
+
K(s)
G(s)
−
Figure 1: Feedback system.
1
Y (s)
2. In the feedback system of Fig. 1
K(s) =
k(s + 3)2
s
G(s) =
1
s+1
Plot the root locus of the closed–loop system for k > 0, showing all relevant steps
clearly. Calculate the breakaway points. The calculations of angles of departure from
poles and arrival at zeros are not required. The branches of the RL plot must be clearly
drawn with thick lines with arrows.
3. In the feedback system of Fig. 1
K(s) =
k
s
G(s) =
1
s2 + 2s + 5
(a) Plot the root locus of the closed–loop system for k > 0, showing all relevant steps
clearly. Calculate the angles of departure from the complex poles. The branches
of the RL plot must be clearly drawn with thick lines with arrows.
(b) Determine the gain k for which the system becomes marginally stable. If the reference input is zero and initial conditions are non zero, what will be the frequency
of output oscillaltions.
4. In the feedback system of Fig. 1,
K(s) =
s2 + 1
s+a
G(s) =
4
s
Plot the root locus of the closed–loop system for a > 0, showing all relevant steps
clearly. Calculate the breakaway point(s). The angles of departure from poles and
arrival at zeros are not required.
Hint. This problem is not in the standard form. Rewrite the CE in the form of:
1+k
2
N (s)
= 0.
D(s)