ELEC ENG 3101 Control
ELEC ENG 7082 Principles of Control Systems
In-Class Test 2 Sample
Total Duration:
45 minutes
Permitted Materials: Calculator; One A4 single-sided sheet with own notes.
Total Mark:
12 marks
ο§ There are 12 questions (child questions) distributed within 3 main questions,
each question worth 1 mark.
Question 1 [2 marks] (Dominant poles)
Consider the following loop transfer function and unity negative feedback:
πΏπΏ(π π ) =
200
π π (π π 2 + 44π π + 165)
200
1.1 – Show that the transfer function of the closed-loop system is ππ(π π ) =
(π π + 40)(π π 2 + 4π π + 5)
and determine the poles of ππ(π π ).
1.2 – For the given third-order system ππ(π π ), is the step response underdamped, overdamped, or
critically damped, and explain why based on the pole locations and their dominance? Refer to
Topic 1 - Dominant Poles (in Test 2 Review)
Question 2 [2 marks] (Transfer function and state space model)
Consider the differential equation of
9 ππ 3 π¦π¦(π‘π‘)
ππ 2 π¦π¦(π‘π‘) 90 ππππ(π‘π‘)
4.5 ππππ(π‘π‘)
ππ 2 π’π’(π‘π‘)
+ 18
+
+ 45π¦π¦(π‘π‘) = 54π’π’(π‘π‘) +
+9
7 πππ‘π‘ 3
πππ‘π‘ 2
14 ππππ
7 ππππ
πππ‘π‘ 2
representing the dynamic of a linear system.
ππ(π π )
2.1 – Using the Laplace transform, find the transfer function πΊπΊ(π π ) = ππ(π π ) of this system,
given β {π¦π¦(π‘π‘)} = ππ(π π ), β {π’π’(π‘π‘)} = ππ(π π ), β οΏ½π¦π¦Μ (π‘π‘) =
ππππ(π‘π‘)
ππππ
οΏ½ = π π π π (π π ).
2.2 – Obtain a state space model of the system above using phase variables, where y
is the output, π’π’ is the input. Refer to Topic 4 – Canonical forms (Test 2 Review).
Question 3 [3 marks] (Rough-Hurwitz stability criterion)
3.1 – Consider the following feedback control system:
R(s)
+_
1
π π (π π − 3)
πΎπΎππ + πΎπΎππ π π
Y(s)
(a) Using the Routh-Hurwitz stability criterion to find the range of πΎπΎππ and πΎπΎππ for the stability
of the closed-loop system. πΎπΎππ > 3. π₯π₯π₯π₯ and πΎπΎππ > 0
(b) Choose a value of πΎπΎππ , determine πΎπΎππ to obtain the following performance:
The output response to the unit step input has 4.32% maximum peak, i.e. the percent
2
overshoot (P.O.) is 4.32, given P.O. = 100 ππ −ππππ/οΏ½1−ππ . Similar to Test 1.
3.2 – Consider the following feedback control system:
1
(0.45π π + 1)(π π + 1.5)
πΎπΎ
π π + 1.5
1
0.007π π + 1
Find the range of πΎπΎ for the stability of the closed-loop system. Answer: -2.xx < K < 18.xx
Question 4 [3 marks] (Optimal control with LQR design)
The following state space model is used in a position control application:
ππΜ (π‘π‘) = οΏ½
−0.8
0.8
0
2.5
οΏ½ ππ(π‘π‘) + οΏ½ οΏ½ π’π’(π‘π‘), π¦π¦(π‘π‘) = [0 1]ππ(π‘π‘).
0
0
The controllability matrix [π©π© π¨π¨π¨π¨] = οΏ½2.5 −2οΏ½ has full rank of 2 with the determinant being non-zero.
0
2
Consider the full state-feedback optimal control law π’π’(π‘π‘) = −π²π²π²π²(π‘π‘) that minimise the cost function
∞
π½π½ = οΏ½ (ππππ πΈπΈπΈπΈ + π’π’ππ πΉπΉπ’π’)ππππ
0
with πΈπΈ = οΏ½1
0
οΏ½ and two cases of πΉπΉ = [1.3] and πΉπΉ = [3], π²π² = πΉπΉ−ππ π©π©π»π» π·π·, where π·π· is the solution to
0 1.5
the algebraic Riccatti equation as follows:
(i) π·π· = οΏ½0.4357 0.4523οΏ½ when πΉπΉ = [1.3], and (ii) π·π· = οΏ½0.3746 0.3394οΏ½ when πΉπΉ = [3].
0.4523
1.1307
0.3394
0.8485
4.1 – Determine the state feedback gain π²π² for the two cases of πΉπΉ above (π²π² = πΉπΉ−ππ π©π©π»π» π·π·). Use ChatGPT.
4.2 – Using the characteristic equation det(π π π°π° − π¨π¨ + π©π©π©π©) = 0 of the closed-loop system to find the
ππ ππ
two poles for the two cases of π²π² in (a), noting that det οΏ½
οΏ½ = ππππ − ππππ. Use ChatGPT.
ππ ππ
4.3 – Are the transient responses underdamped or overdamped? Underdamped if complex conjugate
poles, overdamped if real poles.
4.4 – If the state vector ππ(π‘π‘) is not available for the control law π’π’(π‘π‘) = −π²π²π²π²(π‘π‘), the following full state
observer has been derived by Luenberger for the state-space system above:
π’π’(π‘π‘)
οΏ½.
π¦π¦(π‘π‘)
(a) Rewrite the state space model of the observer in the block diagram with the input vector of οΏ½
(b)
π’π’(π‘π‘)
οΏ½Μ (π‘π‘) = (π¨π¨ − π³π³π³π³)ππ
οΏ½ (π‘π‘) + [π©π© πͺπͺ] οΏ½
ππ
οΏ½
π¦π¦(π‘π‘)
What should the settling time of the observer be relatively to the settling time of the statefeedback closed-loop system? – At least half, one fifth is even better.
0
