EE6620 Assignment 4 (Sem B 2020/21)
Submission Instructions
1. Answer all questions.
2. Answers must be hand-written and calculation steps are required.
3. Generate a pdf file for your answers and submit via Assignment in CANVAS. Deadline
for submission: 11:59pm 30-Nov-2021 (Tuesday)
4. You should make sure that the pdf file is clear enough for marking.
5. No late submission penalty
TOTAL 100 marks
Q1.
This question refers to the following block diagram:
A system is given as
�̇�(𝒕) = 𝑨𝒙(𝒕) + 𝑩𝒖(𝒕)
𝒚(𝒕) = 𝑪𝒙(𝒕)
where 𝐴 = [
2.8 −0.5
0.5 0.25
], 𝐵 = [
−0.9
0.9
] and 𝐶 = [−0.9 −0.9].
(a) Let 𝑢 = 𝑝𝑟 − 𝐾𝑥, determine p and 𝐾 such that the desired eigenvalues are −1 ± 3𝑗 and 𝑦(𝑡)
can track a unit step input 𝑟 asymptotically. Express your answers in 3 decimal points.
[20 marks]
(b) Due to some errors, the gain of the detector for 𝑥1 has wrongly set as 1.1, not 1.0 (But no error
in the detector for 𝑥2). Compare the steady state error and transient performance (including peak
value, peak time, settling time and rise time) in this case and expected case (i.e. gain = 1.0), based
on the controller in (a) and a unit step input. [15 marks]
(c) To avoid the problem in (b), build a reduce-order observer to estimate 𝑥1 for the controller in (a).
[20 marks]
(d) Draw a block diagram for a realization including the system, the controller in (a) using observed
states from the reduce-order observer in (c). Only integrators, adders (can be + or -) and gains
are to be used in your diagram. [You are allowed to use Simulink to draw the diagram.]
[20 marks]
Q2.
Consider a system with state space representation
�̇�(𝑡) = [
0 −1
2 2
] 𝑥(𝑡) + [
0
1
] 𝑢(𝑡)
𝑦(𝑡) = [2 1]𝑥(𝑡) + 2𝑢(𝑡)
where 𝑥(𝑡) ∈ 𝑅2×1, 𝑢(𝑡) ∈ 𝑅, 𝑦(𝑡) ∈ 𝑅.
Design 𝑢(𝑡) to minimize the following performance index:
𝐽 = ∫ 𝑦2(𝑡) 𝑑𝑡
∞
0
[25 marks]
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