AERO 371 – Modelling and Control Systems - 1/1 - Quiz #2, 11:45 – 12:35, March 30, 2026
CONCORDIA UNIVERSITY
GINA CODY SCHOOL OF ENGINEERING AND COMPUTER SCIENCE
DEPARTMENT OF MECHANICAL, INDUSTRIAL AND AEROSPACE ENGINEERING
AERO 371 – Modelling and Control Systems
Quiz #2 Instructor: Youmin Zhang Date: March 30, 2026
______________________________________________________________________________
Answer all two (2) questions. Time: 11:45 - 12:35 Location: H 421
Please put your name and ID number in all pages.
For all questions:
Justify clearly all your steps.
Clearly show all equations and mathematical steps.
Answers without mathematical steps and/or justifications will be given a mark of zero.
Good Luck !
Problem 1 [10 marks]
Consider the following closed-loop system with non-unity feedback
R(s) + Y(s)
-
1) Obtain the closed-loop system transfer function Φ(s);
2) Find conditions for K to satisfy the stability condition of the closed-loop system using
Routh-Hurwitz criterion;
3) Please draw (scratch) Root Locus of the above system (note that necessary calculations and
steps need to be showed for constructing the Root Locus).
Problem 2 [10 marks]
For the same closed-loop system given in the above block diagram in Problem 1
1) When K = 1, please draw (scratch) Nyquist Plot/Diagram of the system (note that
necessary calculations and steps need to be showed for sketching the Nyquist Plot);
2) When K = 1, using Nyquist Stability Analysis method for determining closed-loop system
a) Absolute Stability, and
b) Relative Stability with the Gain Margin (G.M.);
c) How about system’s Absolute Stability when K = 10? Explain your conclusion with
necessary analysis.
, CONCORDIA UNIVERSITY
Gina Cody School of Engineering and Computer Science
Department of Mechanical, Industrial and Aerospace Engineering
AERO 371: Modelling and Control Systems
Quiz #2 Solution Manual – March 30, 2026
Problem 1 [10 marks]
1) Closed-loop System Transfer Function Φ(s) [2 marks]
K 1
Given the block diagram with forward path G(s) = s(s+2)
and feedback H(s) = s+1
. The closed-
loop transfer function is defined as:
Y (s) G(s)
Φ(s) = =
R(s) 1 + G(s)H(s)
Substituting the given functions:
K K
s(s+2) s(s+2)
Φ(s) = = s(s+2)(s+1)+K
K 1
1+ s(s+2) s+1 s(s+2)(s+1)
Simplifying the expression:
K(s + 1) K(s + 1)
Φ(s) = = 3
s(s + 2)(s + 1) + K s + 3s2 + 2s + K
2) Stability Analysis using Routh-Hurwitz Criterion [2 marks]
The characteristic equation is s3 + 3s2 + 2s + K = 0. We construct the Routh array:
s3 1 2
s2 3 K
(3)(2)−(1)(K) 6−K
s1 3
= 3
0
s0 K
For stability, all elements in the first column must be strictly positive:
1. K > 0
6−K
2. 3
> 0 =⇒ 6 − K > 0 =⇒ K < 6
Condition for stability: 0 < K < 6.
1
CONCORDIA UNIVERSITY
GINA CODY SCHOOL OF ENGINEERING AND COMPUTER SCIENCE
DEPARTMENT OF MECHANICAL, INDUSTRIAL AND AEROSPACE ENGINEERING
AERO 371 – Modelling and Control Systems
Quiz #2 Instructor: Youmin Zhang Date: March 30, 2026
______________________________________________________________________________
Answer all two (2) questions. Time: 11:45 - 12:35 Location: H 421
Please put your name and ID number in all pages.
For all questions:
Justify clearly all your steps.
Clearly show all equations and mathematical steps.
Answers without mathematical steps and/or justifications will be given a mark of zero.
Good Luck !
Problem 1 [10 marks]
Consider the following closed-loop system with non-unity feedback
R(s) + Y(s)
-
1) Obtain the closed-loop system transfer function Φ(s);
2) Find conditions for K to satisfy the stability condition of the closed-loop system using
Routh-Hurwitz criterion;
3) Please draw (scratch) Root Locus of the above system (note that necessary calculations and
steps need to be showed for constructing the Root Locus).
Problem 2 [10 marks]
For the same closed-loop system given in the above block diagram in Problem 1
1) When K = 1, please draw (scratch) Nyquist Plot/Diagram of the system (note that
necessary calculations and steps need to be showed for sketching the Nyquist Plot);
2) When K = 1, using Nyquist Stability Analysis method for determining closed-loop system
a) Absolute Stability, and
b) Relative Stability with the Gain Margin (G.M.);
c) How about system’s Absolute Stability when K = 10? Explain your conclusion with
necessary analysis.
, CONCORDIA UNIVERSITY
Gina Cody School of Engineering and Computer Science
Department of Mechanical, Industrial and Aerospace Engineering
AERO 371: Modelling and Control Systems
Quiz #2 Solution Manual – March 30, 2026
Problem 1 [10 marks]
1) Closed-loop System Transfer Function Φ(s) [2 marks]
K 1
Given the block diagram with forward path G(s) = s(s+2)
and feedback H(s) = s+1
. The closed-
loop transfer function is defined as:
Y (s) G(s)
Φ(s) = =
R(s) 1 + G(s)H(s)
Substituting the given functions:
K K
s(s+2) s(s+2)
Φ(s) = = s(s+2)(s+1)+K
K 1
1+ s(s+2) s+1 s(s+2)(s+1)
Simplifying the expression:
K(s + 1) K(s + 1)
Φ(s) = = 3
s(s + 2)(s + 1) + K s + 3s2 + 2s + K
2) Stability Analysis using Routh-Hurwitz Criterion [2 marks]
The characteristic equation is s3 + 3s2 + 2s + K = 0. We construct the Routh array:
s3 1 2
s2 3 K
(3)(2)−(1)(K) 6−K
s1 3
= 3
0
s0 K
For stability, all elements in the first column must be strictly positive:
1. K > 0
6−K
2. 3
> 0 =⇒ 6 − K > 0 =⇒ K < 6
Condition for stability: 0 < K < 6.
1
