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EE 6300 Exam#1 Oct. 23, 2014
75 minutes
Copyright c© 2014 by L. Acar. All rights reserved. No parts of this document may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means without the written permission of the copyright holder(s).
1. The block diagram of a control system is given below.
s + 2
s + 1
1
s
1
s
u y
Obtain a state-space representation of the system without any block-diagram reduction. (25pts)
2. A control system is described by
x(t) =
[
−2 −8−2 −2
]
x(t) +
[
21
]
u(t),
andy(t) =
[
1 0]
x(t),
where u, x, and y are the input, the state, and the output variables, respectively.
(a) Determine x(t) for t ≥ 0, when x(0) =[
1 −1]T
, and u(t) = 0 for t ≥ 0. Show all your workclearly. (10pts)
(b) Determine x(t) for t ≥ 0, when x(0) = 0, and u(t) = 1 for t ≥ 0. Show your work clearly.(10pts)
(c) Determine x(t) for t ≥ 0, when x(0) =[
1 −1]T
, and u(t) = 1 for t ≥ 0. Show your workclearly. (10pts)
3. A time-varying linear control system is described by
x(t) =
[
−t 10 −t
]
x(t) +
[
01
]
u(t),
where u and x are the input and the state variables, respectively. Determine the state transitionmatrix Φ(t, t0). Show all your work clearly. (25pts)
4. A control system is described in state-space representation, such that
x(t) = Ax(t) + Bu(t),
y(t) = Cx(t) + Du(t),
where u, x, and y are the input, the state, and the output variables, respectively. For the followingA, B, C, and D matrices, determine whether the system is asymptotically stable, marginally stable,or unstable in the state sense; and whether it is bounded-input-bounded-output stable or not. Justifyyour answer.
1
(a)
A =
0 −1 1 0 01 0 0 1 00 0 0 −1 00 0 1 0 00 0 0 0 0
, B =
11000
, C =[
0 0 1 0 1]
, and D = 0.
(05pts)
(b)
A =
0 −1 1 0 01 0 0 1 00 0 0 −1 00 0 1 0 00 0 0 0 −1
, B =
11001
, C =[
0 0 1 0 1]
, and D = 0.
(05pts)
(c)
A =
0 −1 0 0 01 0 0 0 00 0 0 −1 00 0 1 0 00 0 0 0 0
, B =
11001
, C =[
0 0 1 0 1]
, and D = 0.
(05pts)
(d)
A =
0 −1 0 0 01 0 0 0 00 0 0 −1 00 0 1 0 00 0 0 0 1
, B =
11001
, C =[
0 0 1 0 1]
, and D = 0.
(05pts)
2