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ELECTENG 722 ELECTRICAL AND ELECTRONIC ENGINEERING

创建日期：2024-04-13 16:19:19

所属分类：电气工程electric engineering

标签： ELECTENG 722

ELECTRICAL AND ELECTRONIC ENGINEERING

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ELECTENG 722

ELECTRICAL AND ELECTRONIC ENGINEERING
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.
Page 1 of 7
ELECTENG 722
1. (a) Consider the system shown in Fig-1.
Figure 1
(i) Express this system in state variable form considering x1 and x2 as state vari-
ables. (4 marks)
(ii) Determine the transfer function of this system from the state variable
model (6 marks)
(b) Determine the output response for the system
x˙1 = x2
x˙2 = −8x1 − 6x2 + u
y = x1 + x2 + 2u
The input u is an impulse function. Assume x1(0) = 0 and x2(0) = 0. (10 marks)
Page 2 of 7
ELECTENG 722
2. (a) Discuss about four important features of nonlinear systems. (4 marks)
(b) Consider a nonlinear system described by the state equations
x˙1 = x2
x˙2 = −x1 + x2(1− x21 + 0.1x41)
Find all the equilibrium point/points of the system and determine the nature of each isolated
equilibrium point/points. (8 marks)
(c) Define describing function of a nonlinear element and determine the describing function of
ON-OFF nonlinearity. (4 marks)
(d) How would you predict the stability or instability of limit cycle oscillations of a nonlinear
system using describing function method. (4 marks)
Page 3 of 7
ELECTENG 722
3. An open loop system with an integrator is described by:
x˙(t) =
[
x˙1(t)
x˙2(t)
]
=
[
1 0
−1 0
] [
x1(t)
x2(t)
]
+
[
1
0
]
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.
(a) Assuming all the state variables are measurable, design a LQR controller that minimises
the performance index
J =
∫ ∞
0
[
x21(t) + x
2
2(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
(12 marks)
(b) Using the optimal state feedback gains computed in 3(a), find the closed loop pole posi-
tions for the control system. Compare these with the poles of the open loop system and
comment. (5 marks)
(c) Suppose now that x2(t) cannot be measured. Explain how the controller you have calculated
above could be extended to cope with this situation. (3 marks)
Page 4 of 7
ELECTENG 722
4. (a) Given a discrete plant
x(k + 1) = Ax(k) + Bu(k)
y(k) = Cx(k)
where A =
[
0.1 0
0.6 0.3
]
, B =
[
0.1
0
]
, C = [1 1] and x(k) =
[
x1(k)
x2(k)
]
.
(i) Determine if the discrete system is BIBO stable, controllable and observ-
able. (2 marks)
(ii) Design an observer-based control for the discrete system by placing the eigenvalues of
(A−BK) at −0.1 and −0.2, and the eigenvalues of (A−LC) at 0 and 0. (8 marks)
(iii) Assuming that the system and the controller (including observer) are given by the
following equation:
x(k + 1) = Ax(k) + Bu(k); y(k) = Cx(k)
xˆ(k + 1) = Axˆ(k) + Bu(k) + L(y(k)− yˆ(k)); yˆ(k) = Cxˆ(k)
u(k) = Nr −Kxˆ(k)
where matrices K and L are computed in Q4(iii), compute N such that y(∞) =
limk 7→∞ y(k) = r, where r is an arbitrary constant. (2 marks)
(b) A first order system is given by
x(k + 1) = 2x(k) + u(k)
y(k) = 2x(k)
where x(k) is the state variable, y(k) is the output and u(k) is the control input. Design a
tracking controller with an integral action for this first order system to yield a dead beat
response. (8 marks)
Page 5 of 7
ELECTENG 722
5. (a) (i) Find the solution of the difference equation
x(k + 2) + 6x(k + 1) + 8x(k) = 0, x(0) = 0, x(1) = 1
(3 marks)
(ii) Sketch the impulse response of a zero order hold device and compute its transfer
function. Show that s = 0 is not a pole of this transfer function (3 marks)
(iii) Determine C(z)
R(z)
for the feedback control system shown in Fig-2. (4 marks)
Figure 2
(b) Consider the following plant
x˙(t) =
[
x˙1(t)
x˙2(t)
]
=
[
0 1
0 0
] [
x1(t)
x2(t)
]
+ w(t)
y(t) =
[
1 0
] [ x1(t)
x2(t)
]
+ v(t)
where x1(t) and x2(t) are the state variables, y(t) is the output, w(t) and v(t) are zero-
mean stochastic Gaussian processes uncorrelated in time and with each other. Assume
the plant has the measurement noise covariance, V = 1, and the process noise covariance,
W =
[
1 0
0 1
]
. Design a Kalman filter to estimate the state x(t) by xˆ(t) such that the
estimate error covariance is minimized, 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 problem is
˙ˆx(t) = Axˆ(t) + PoC
TV −1[y(t)− Cxˆ(t)]
where Po > 0 is a solution of the algebraic Riccati equation:
APo + PoA
T − PoCTV −1CPo + W = 0
(10 marks)
Page 6 of 7
ELECTENG 722
Table 1: Table of Z-Transform
X(S) x(t) x(kT ) or x(k) X(z)
1 - - Kronecker Delta, δ0(k) 1
2 1s 1(t) 1(k)
1
1−z−1 =
z
z−1
3 1s+a e
−at e−akT 1
1−e−aT z−1 =
z
z−e−aT
4 1
s2
t kT Tz
−1
(1−z−1)2 =
Tz
(z−1)2
5 ω
s2+ω2
sinωt sinωkT z
−1sinωT
1−2z−1cosωT+z−2 =
zsinωT
z2−2zcosωT+1
6 s
s2+ω2
cosωt cosωkT 1−z
−1cosωT
1−2z−1cosωT+z−2 =
z2−zcosωT
z2−2zcosωT+1
7 - - ak 1
1−az−1 =
z
z−a
8 - - ak−1, k = 1, 2, 3 z
−1
1−az−1 =
1
z−a
9 Z [x(k + n)] = znX(z)− znx(0)− zn−1x(1)− zn−2x(2)− zx(n− 1)

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