ELEN90064 Advanced Control Systems
Student Number:......................................................
The University of Melbourne
Department of Electrical and Electronic Engineering
Semester 2 Assessment, 2016
ELEN90064 Advanced Control Systems
Reading Time: 15 minutes
Writing Time: 3 hours
This paper has 8 pages, including this page
Authorized Materials:
• Melbourne School of Engineering approved electronic calculators may be
used (with no data stored in memory).
• This is an open book exam: books, lecture notes and worked examples
are all allowed.
Instructions to Invigilators:
• This examination paper is to be collected together with answer script books.
• This exam paper should not be lodged with the Baillieu library.
Instructions to Students:
• Students should attempt all 6 questions.
• Students should read the instructions for each question carefully.
• Students should answer all questions in script books provided.
• Marks for each question are shown in parentheses.
• Students should show all calculations and mathematical manipulations fully.
• Students should submit this paper together with answer script books.
• Answers to each question should be started on a new page.
• Maximum possible mark is 60.
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Question 1 (10 Marks) Consider a discrete-time system with the fol-
lowing state space model:
x1(k + 1) = 3x1(k) + u(k)
x2(k + 1) = 2x1(k)− 0.5x2(k) .
Suppose that your manager told you that you have to stabilize this
system and keep the budget as low as possible as the company is going
through financial difficulties. The above model includes the plant and
actuator and u(k) is the input to the actuator. You did some enquiries
and you found that you have a choice of two sensors. The first sensor
costs $300 and it measures
y(k) = x1(k) + x2(k) + ν(k),
where > 0 is a very small number and ν(k) is the measurement
noise which can not be neglected. The second sensor costs $500 and
it measures
y(k) = x1(k) + ν(k) ,
where again ν(k) is the measurement noise that can not be neglected.
It is assumed that in both sensors, |ν(k)| ≤ N,∀k. Answer the fol-
lowing questions:
1. (5 Marks) Which sensor would you choose and why? (please
present a detailed analysis that is backing up your choice)
2. (5 Marks) With the chosen sensor, design a controller that sta-
bilises the system. In particular, consider the effects of the noise
on the closed-loop behaviour.
Please turn over the page.
2
Figure 1: RLC circuit
Question 2 (10 Marks) Consider the electrical circuit in Figure 1.
The input to the system is the current u of the current source and the
output of the system is the voltage y shown in Figure 1. If we choose
state variables x1 := iL and x2 := vC, we can write the state space
model of the system in the following form:
x˙1 = −2R
L
x1 +
1
L
x2 +
R
L
u
x˙2 = − 1
C
x1 +
1
C
u
y = −Rx1 + x2 + Ru .
1. (2 Marks) Check how controllability of the system depends on
R,L,C. Explain your findings.
2. (2 Marks) Check how observability of the system depends on R,L,C.
Explain your findings.
3. (4 Marks) For R2 = LC , find the transfer function from u to y
and draw a block diagram of the system that corresponds to the
Kalman decomposition.
4. (2 Marks) Supposing that 1 = R2 = LC and that u(t) = sin(t), find
y(t) for arbitrary initial conditions.
Please turn over the page.
3
Question 3 (10 Marks) Consider a heat exchanger given in Figure 2
that consists of a radiator and a fan. Heated water is passed through
the radiator and the fan blows air across it. The water circulation
system consists of a pump and a heater tank. The control objective
is to control the temperature drop across the radiator together with
the air flow rate across it by adjusting the inputs to the heater and
the fan. This is a two-input two-output system, see Figure 2, and
experiments have shown that the subsystems G12 and G22 are linear
whereas the subsystem G11 is nonlinear and its model was identified
from experiments to be:
y(k + 1) = 2 + y(k) + 0.5u(k) + 0.05u(k − 1)− 0.01y2(k)− 0.01u2(k)
−0.002y2(k)u(k)− 0.002u3(k) ,
where u is the input to the fan and y is the air flow rate. With
reference to this model for subsystem G11, address the following:
1. (2 Marks) Taking x1(k) = y(k) and x2(k) = u(k − 1), derive a
corresponding state-space model;
2. (2 Marks) Find all equilibria of this state-space model;
3. (2 Marks) Linearize the state-space model around an equilibrium
with constant input u∗ = 1;
4. (4 Marks) Design a full state feedback controller that achieves
dead beat behaviour for the linearized system from the previous
question (assuming you can measure x1 and x2).
Please turn over the page.
4
Figure 2: Heat exchanger system and its block diagram
Please turn over the page.
5
Question 4 (10 Marks) A DC motor needs to operate at reference
angular velocity of ωr = 10rad/sec. Its transfer function from the
actuating input to the angle of the shaft is given by:
Θ(s)
U(s)
= G(s) =
2
s(s + 1)
By precise measurements, you have found that the radius of the shaft
is not perfect and it changes as a function of angle θ as shown in Fig-
ure 3 (i.e. r = r(θ)). This imperfection is known to cause an input
Figure 3: Radius versus angle of the shaft
disturbance of the magnitude proportional to r(θ)− 0.1. Approximate
this disturbance as a sinusoid of a certain frequency and design a con-
troller that robustly attenuates the effect of this disturbance. Assume
that you have two sensors that measure the angle θ and the angular
velocity ω of the shaft.
Note: if your calculations involve matrices n × n, n > 2, then you
do not need to solve such equations analytically but just explain the
steps that you need to follow to answer the question.
Please turn over the page.
6
Question 5 (10 Marks) Consider a nonlinear system
x˙ = f (x)
suppose that you have found a positive definite Lyapunov function
that satisfies:
∂V
∂x
f (x) ≤ 0
and there exists a non-zero row matrix H such that:
f (x) = 0,∀x ∈ {x : Hx = 0} .
Show that the origin of this system is stable but not attractive.
Please turn over the page.
7
Question 6 (10 Marks) Consider a model of a water tank whose
actuator (adjustable valve) is far away from the tank and this causes
a transport delay. The transfer function of the linearized model is
Y (s)
U(s)
= G(s) =
e−s
s + 1
,
where u is the control input and y is the water level in the tank.
Assume that the system is to be digitally controlled with a sampling
period ∆ = 0.5sec.
1. (2 Marks) Find the input-output model of the continuous-time
plant (i.e. delay-differential equation).
2. (2 Marks) Assuming that D/A and A/D converters are mod-
elled as the zero order hold and ideal sampler respectively, find
a discrete-time model of the plant in a state-space form.
3. (6 Marks) Design a discrete-time LQG controller assuming you
measure only y. Assume that the matrix corresponding to the
output covariance is V = 0.2 and the matrix corresponding to the
input covariance is W = 0.1I, where I is the identity matrix.
Note: if you are unable to solve some Ricatti equations, just write
them down in as much detail as you can and explain the steps you
need to follow to solve the problem.
End of the examination paper.
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