ELECTENG 704
THE UNIVERSITY OF AUCKLAND
SECOND SEMESTER, 2020
Campus: City
ELECTRICAL AND ELECTRONIC ENGINEERING
Advanced Control Systems
(Time allowed: THREE hours)
NOTE: Answer ALL FIVE questions.
All questions are of equal mark value (20 marks).
Show ALL workings unless instructed otherwise.
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ELECTENG 704
1. (a) An open loop system is described by:
x˙(t) =
[
x˙1(t)
x˙2(t)
]
=
[
0 1
0 0
] [
x1(t)
x2(t)
]
+
[
0
1
]
u(t))
y(t) =
[
1 0
] [ x1(t)
x2(t)
]
where x1(t) and x2(t) are the state variables, y(t) is the output, and u(t) is the input.
Assuming that all the state variables are measurable, design a LQR controller that
minimises the performance index
J =
∫ ∞
0
[
x21(t) + u
2(t)
]
dt
Note that the solution of the standard LQR problem is
u(t) = −R−1BTPx(t)
where P > 0 is a solution of the algebraic Riccati equation:
ATP + PA− PBR−1BTP +Q = 0
(10 marks)
(b) Assuming that a system has the process noise covariance matrix Qe =
[
1 0
0 1
]
, and the
measurement noise covariance, Re = 1, design a Kalman filter to estimate the state x(t)
by xˆ(t) such that the estimate error covariance is minimised, that is, the following index
is minimized:
Ja = E
[
{x(t)− xˆ(t)}T {x(t)− xˆ(t)}
]
Note that the solution of the Kalman filter problem is
˙ˆx(t) = Axˆ(t) + PeC
TR−1e (y(t)− Cxˆ(t))
where Pe > 0 is a solution of the algebraic Riccati equation
APe + PeA
T − PeCTR−1e CPe +Qe = 0
(10 marks)
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ELECTENG 704
2. (a) Consider the following linear system
x˙(t) = Ax(t) +B1w(t) +B2u(t)
z(t) = C1x(t) +D12u(t)
y(t) = C2x(t) +D21w(t)
where x(t), u(t), w(t), y(t), and z(t) denote respectively, the state, the input, the dis-
turbance, the output and the regulated output. Further, if
A =
[
0 1
0 0
]
, B1 =
[
0 0
0 0.5
]
, B2 =
[
0
3
]
, C1 =
[
0.5 0
0 0
]
,
D12 =
[
0
1
]
, C2 =
[
3 0
]
and D21 =
[
1 0
]
design an H∞ output feedback controller such that the ∞ norm of the transfer function
from w(t) to z(t) is less than 0.5.
Hint: An H∞ output feedback controller can be found by solving the following Riccati
equations
ATP + PA+ P
(
B1B
T
1
γ2
−B2BT2
)
P + CT1 C1 = 0
AQ+QAT +Q
(
CT1 C1
γ2
− CT2 C2
)
Q+B1B
T
1 = 0
and
Q−1 >
P
γ2
The H∞ output feedback controller is given by
˙ˆx(t) =
(
A+
B1B
T
1
γ2
P
)
xˆ(t) + ZQCT2 (y(t)− C2xˆ(t)) +B2u(t)
u(t) = −BT2 Pxˆ(t)
where Z =
(
I − QP
γ2
)−1
. (10 marks)
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ELECTENG 704
(b) Consider the following linear system x˙1(t)x˙2(t)
x˙3(t)
 =
 0 1 01 0 0
0 1 1
 x1(t)x2(t)
x3(t)
+
 00
1
u(t)
y(t) =
[
1 0 0
0 1 1
] x1(t)x2(t)
x3(t)
 .
Design a reduced order observer with its pole at −5. (10 marks)
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ELECTENG 704
3. (a) Consider the following linear time-invariant system:[
x˙1(t)
x˙2(t)
]
=
[
0 1
1 1
] [
x1(t)
x2(t)
]
+
[
0 0
1 1
]
u(t)
y(t) =
[
1 1
1 0
] [
x1(t)
x2(t)
]
or in the vector form
x˙(t) = Ax(t) +Bu(t); y(t) = Cx(t)
(i) Determine if the system is controllable.
(ii) Determine if the system is observable.
(iii) Design an observer-based controller for the system by placing the eigenvalues of
(A−BK) at −1 and −2, and the eigenvalues of (A−LC) at −10 and −20 , where
K and L are the gain matrices of the observer-based controller. (10 marks)
(b) The pressure deviation p(t) in a boiler from the normal operating pressure p is described
by the equation
p(t) =
0.4− 0.2z−1
(1− 0.9z−1)(1− 0.6z−1)u(t− 2) +
1− 0.7z−1
(1− 0.9z−1)(1− 0.6z−1)e(t)
where e(t) is a zero mean Gaussian white noise disturbance with variance 0.5.
(i) Determine the minimum variance control law for p(t). (4 marks)
(ii) Compute the variance of p(t) under minimum variance control. (1 mark)
(c) Consider a scalar system
x˙ = cosx− x3 + u
(i) Find the feedback linearizing control law which globally stabilizes the sys-
tem. (1 mark)
(ii) Design a controller using control lyapunov function (clf) approach considering
V (x) =
x2
2
, and Q(x) = x2 + x4
(3 marks)
(iii) Which of the above two controllers would you prefer and why ? (1 mark)
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ELECTENG 704
4. (a) Consider the system
x˙1 = −2x1 + x2
x˙2 = −5x1
(i) Compute the eigenvalues of the system and investigate its stability. (4 marks)
(ii) Investigate the stability of this system using Lyapunov’s direct method using the
Lyapunov function
V (x) =
1
2
(x21 + x
2
2)
(4 marks)
(iii) Does the stability information from Lyapunov’s direct method gives the correct
answer ? Make a critical judgment on your conclusions from the results of (i) and
(ii). (2 marks)
(b) Consider the nonlinear system
x˙1 = x1 + x2
x˙2 = 3x
2
1x2 + x1 + u
y = −x31 + x2
(i) Determine the relative degree of this system and find the input-output linearising
control law. (2 marks)
(ii) Determine the zero-dynamics of this system. (6 marks)
(iii) Is this system minimum phase ? (2 marks)
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ELECTENG 704
5. (a) Draw the schematics of Hammerstein and Wiener models used to represent a class of
nonlinear systems. (2 marks)
(b) Write the mathematical expression of Volterra series. (2 marks)
(c) Prove that the least squares estimates are biased if the data is corrupted by coloured noise
sequence. Show that the instrumental variable estimates can be unbiased. (4 marks)
(d) Let T1,T2,. . .,Tn be n-number of unbiased estimators of γ(θ) with variances σ
2
1,σ
2
2,. . .,σ
2
n.
Let T be another estimator of γ(θ), which is a linear combination of T1,T2,. . .,Tn, and is
given by
T = l1T1 + l2T2 + . . .+ lnTn
Find the condition, in terms of l1,l2,. . .,ln, such that T is an unbiased estimator of γ(θ).
Assuming that the estimators T1,T2,. . .,Tn are uncorrelated, calculate the variance of the
estimator T . (3 marks)
(e) Consider a system whose output y is related to the input u as:
y(t) = b0u(t) + b1u(t− 1) + e(t)
where e is the modelling error. Show that the parameters of this system can not be
uniquely identified ( i.e. parameter identifiability is lost) by exciting the system with a
step input. (3 marks)
(f) From the perspective of achieving the generalisation capability of a model, answer the
following:
(i) What is bias-variance dilemma ? (2 marks)
(ii) What is principle of cross validation ? (2 marks)
(iii) Explain the difference between the one step ahead prediction (OSA) and model
predicted output (MPO), considering the model
y(k) = θ1y(k − 1) + θ2y(k − 2) + θ3y(k − 3) + θ4u(k − 1) + e(k)
(2 marks)
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