EECE 340 Homework 6, Spring 2016
American University of Beirut · Signals and Systems · 3 pages · Prof. Karim Kabalan
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AMERICAN UNIVERSITY OF BEIRUT
ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT
EECE 340
Homework VI – State Variables Analysis
Problem 1
Let h(t) be the impulse response of a LTI system and its Laplace transform is given
by:
)
)(s
s
(
)
( s
H( s )
1
10
1
10
a. Find the differential equation describing the system
b. Represent this system in a state-space form by writing the state equations
and the output equation .
Problem 2
a. Draw the block diagram for a circuit that implements the following transfer
function. You may use amplifiers, adders, and integrators.
5
6s
2s
s
12
12
6s
s
4
)s(
X
)s(
Y
2
3
2
b. Determine the state equation for this system.
c. Determine the output equation for this system.
d. Is the state representation for an LTI system unique? Explain.
Problem 3
The dynamic equations of a linear time-invariant system whose output is y(t) and
input r(t) are given by:
Determine the Differential equation representing this system.
1
1
2
2
1
2
( )
( )
0
4
1
( )
( )
( )
4
2
1
( )
( )
1
0
( )
X t
X t
r t
X
t
X
t
X t
y t
X
t
Problem 4
A LTI system with input r(t) and output y(t) is represented by the DE equation
𝑑3𝑦(𝑡)
𝑑𝑡3
− 2 𝑑2𝑦(𝑡)
𝑑𝑡2
+ 6 𝑑𝑦(𝑡)
𝑑𝑡
− 10𝑦(𝑡) = −2 𝑑2𝑟(𝑡)
𝑑𝑡2
+ 5𝑟(𝑡)
Write the state and the output equations of the system
Problem 5
The open-loop transfer function of a unity feedback control system is given by:
s
2s
s
10
3s
)s(
G
2
3
1. Determine the Differential equation representing this system.
2. Represent this system in a state variable form by determining the state
equations and the output equation in matrix format.
Problem 6
A LTI system with input y(t) and output y(t) is represented by the DE
12 ( )
( )
6 ( )
( )
3
)
(
2
2
2
2
x t
dt
d x t
y t
dt
t
dy
dt
d y t
a. Determine the system transfer function
b. Represent the system is a state space format.
Problem 7
Consider the following state-space representation:
Obtain the system transfer function by using two different methods.
Problem 8
A linear time-invariant system, whose output is y(t) and input r(t), is represented
by the following transfer function
56
8s
s
2
6
s
10
)s(
R
)s(
Y
2
Write the state and output equations of this system
Problem 9
Consider an LTI system described by the following input-output given below with
r(t) is the input signal and y(t) is the output signal
)t(r
2
dt
)t(r
d
6y )t(
dt
dy )t(
3
dt
y )t(
d
2
2
2
2
Find the state space representation in this system