EE5307
EXAM I
October 11, 2007
Name (Print): _ (Last) (First) I.D.: _ Solve ALL THREE problems. Time: 1 hr. 30 min. Maximum Score: 36 points. Problem 1 (a) Set up the state-variable description for the following circuit with input u, output y and state va
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University of Texas at Arlington
EE5307 Linear Systems Engineering
Fall 2016
Solution homework 2
Murilo Augusto Pinheiro
Solution Homework #2 EE5307
Murilo Augusto Pinheiro
September 16th 2016
! Considering the following differential equation
! + 6! +
EE 5307 homework 1 solution
The mathematical model of a nonlinear system was given as
x 1 = (x1 1) sin (x2 ) + u
x 2 = x1 ex2
y = 2x1 (x2 + 1)
a. To find the equilibrium points of the system we set x 1 = x 2 = 0 and u = 0:
(x1 1) sin (x2 ) = 0
x1 ex2 = 0
EE 5307 Homework 3 Solution.
1. BIBO stability and natural modes.
Express H1 (s) as
s+2
s(s2 + 1)
H1 (s) =
and note that the poles are s = 0, s = i. The natural modes of the system are 1, sin(t), cos(t).
As the poles are over the imaginary axis of the com
Homework 3 EE5307
Fall 2016
1- Consider the systems described by the following transfer functions.
H 1(s )
s 2
s3 s
H 2 (s )
s 5
s 2 2s 1
H 3 (s )
4
s2 4
a) Find natural modes for each system.
b) Investigate their BIBO stability.
2- Consider the linear
EE 5307 Homework 4 solution.
1a. For the system
x(k + 2) 6x(k + 1) 55x(k) = (3)k
we find the Z-transform as
z 2 X(z) z 2 x(0) zx(1) 6 (zX(z) zx(0) 55X(z) =
z
z+3
z
z
=
X(z) =
(z + 3)(z 2 6z 55)
(z + 3)(z 11)(z + 5)
z
z+3
z 2 X(z) 6zX(z) 55X(z) =
Then,
EE 5307 Homework 6 solution.
1a. For the system
x =
0
0
1
u
x+
2
2 3
y= 1 0 x
we obtain the state transition matrix as
(
1 )
s 1
eAt = L1 (sI A)1 = L1
= L1
2 s+3
s+3
s2 +3s+2
1
s2 +3s+2
2
s2 +3s+2
s
s2 +3s+2
Considering that the system poles are s =
EE 5307 homework 2 solution.
The first system will be solved:
x
+ 4x + 6x = 5u
1. To find the state space equations, we define
x1 = x,
x2 = x
Then, we have
x 1 = x2
x 2 = 6x1 4x2 + 5u
Considering y = x, the matrix representation of this system is
0
1
0
EE 5307 homework 1 solution.
The mathematical model of a nonlinear system was given as
1 (t) + sin (1 (t) + (51 (t) 22 (t) = 0
2 (t) (61 (t) 82 (t) = f (t)
1. To find the state space representation of the system, define the state variables as
x1
x2
x3
x4