Name: _
Problem #1: _ 2: _ 3: _ 4: _
ECE 604 LINEAR SYSTEMS
Midterm Exam Thursday, October 30, 2008 7:00 P.M. - 9:00 P.M.
5: _ Total: _
The exam is open book and open notes. You must show your work to receive partial credit. The problem values are: Proble
Problem #1: _ 2: _ 3: _ 4: _ 5: _
ECE 604 LINEAR SYSTEMS
Fall 2008 Final Exam Issued: December 10, 2009
6: _ Total: _
Due: 5:00 P. M., December 16, 2009
Consistent with the ECE Honor Code, you are asked to read the following voluntary statement carefully
ECE 604 STATE VARIABLE ANALYSIS
Fall, 2009 Solution to Problem Set #4 Problem 1: (Rugh 9.1) Controllable: The controllability matrix is
! 1 2+% C = ! B AB A 2B # = & 1 1 " $ & & 1 0 " 2 + 2% 1 0 # ' ' ' $
Take the determinant:
" 2+! det C = 1idet $ # 1
2
ECE 604 STATE VARIABLE ANALYSIS
Fall, 2014
Solution to Problem Set #5
Problem 1:
(Homework)
a) Define = eA . Show that we can do this if and only if:
C
C
rank C 2 = n
n1
C
(
)
i.e., the sampled-data system is observable ( eA ,C is observable).
Sup
Lecture 14
ECE 604
State Variable Analysis
Doug Looze
Oct. 3, 2014
Linear Systems Douglas
Looze
1
Announcements
Problem set 3 is available
Due Friday, October 10
Oct. 3, 2014
Linear Systems Douglas
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2
Last Time
Modal decomposition
Assume full set of
Lecture 16
ECE 604
State Variable Analysis
Doug Looze
Oct. 8, 2014
Linear Systems Douglas
Looze
1
Announcements
Problem set 3 is available
Due Friday, October 10
Oct. 8, 2014
Linear Systems Douglas
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2
Conditions
Equivalent
(A(t), B(t) controllable
In
Lecture 13
ECE 604
State Variable Analysis
Doug Looze
Oct. 1, 2014
Linear Systems Douglas
Looze
1
Last Time
Eigenstructure
Oct. 1, 2014
Linear Systems Douglas
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2
Today
LTI systems
Modal decomposition
Reading
Ch. 5: 7481
Oct. 1, 2014
Linear Systems Do
Lecture 15
ECE 604
State Variable Analysis
Doug Looze
Oct. 6, 2014
Linear Systems Douglas
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1
Announcements
Problem set 3 is available
Due Friday, October 10
Oct. 6, 2014
Linear Systems Douglas
Looze
2
Last Time
Characterizations of
Controllability
Ob
Lecture 17
ECE 604
State Variable Analysis
Doug Looze
Oct. 10, 2014
Linear Systems Douglas
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1
Announcements
Problem set 3 is due
Problem set 4 is available
Due Friday, October 24
Next class: Tuesday, October 14
Midterm
Tuesday, October 28
79 PM
Marcu
Lecture 12
ECE 604
State Variable Analysis
Doug Looze
Sept. 29, 2014
Linear Systems Douglas
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1
Last Time
Inhomogeneous linear system
d
x ( t ) =A ( t ) x ( t ) +B ( t ) u ( t )
dt
y ( t ) =C ( t ) x ( t ) +D ( t ) u ( t )
x ( t0 ) =x 0
Variation of c
Lecture 11
ECE 604
State Variable Analysis
Doug Looze
Sept. 26, 2014
Linear Systems Douglas
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1
Announce
Problem set 2 due today
Sept. 26, 2014
Linear Systems Douglas
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2
Last Time
Self-commuting matrices
Sept. 26, 2014
Linear Systems Douglas
Looz
Lecture 8
ECE 604
State Variable Analysis
Doug Looze
Sept. 19, 2014
Linear Systems Douglas
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1
Announce
Problem set 1 due
Problem set 2 available on Moodle
Due Friday, September 26
Sept. 19, 2014
Linear Systems Douglas
Looze
2
Last Time
Self commuting
Lecture 10
ECE 604
State Variable Analysis
Doug Looze
Sept. 24, 2014
Linear Systems Douglas
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1
Announce
Problem set 2 available on Moodle
Due Friday, September 26
Sept. 24, 2014
Linear Systems Douglas
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2
Last Time
Discrete-time systems
Lineariza
Lecture 7
ECE 604
State Variable Analysis
Doug Looze
Sept. 17, 2014
Linear Systems Douglas
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1
Announce
Problem set 1 available on Moodle
Due Friday, September 19
Sept. 17, 2014
Linear Systems Douglas
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2
Last Time
Homogeneous systems
State equati
Lecture 9
ECE 604
State Variable Analysis
Doug Looze
Sept. 22, 2014
Linear Systems Douglas
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1
Announce
Problem set 1 graded
Problem set 2 available on Moodle
Due Friday, September 26
Office hours this week
Tuesday, 35 PM
Sept. 22, 2014
Linear Systems
Lecture 6
ECE 604
State Variable Analysis
Doug Looze
Sept. 15, 2014
Linear Systems Douglas
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1
Announce
Problem set 1 available on Moodle
Due Friday, September 19
Sept. 15, 2014
Linear Systems Douglas
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2
Last Time
Homogeneous systems
State equati
Lecture 18
ECE 604
State Variable Analysis
Doug Looze
Oct. 14, 2014
Linear Systems Douglas
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1
Announcements
Problem set 4 is available
Due Friday, October 24
No office hours this week
Midterm
Tuesday, October 28
79 PM
Marcus 15
Oct. 14, 2014
Linear S
ECE 604 LINEAR SYSTEMS
Problem Set #1
Issued: Friday, September 12, 2014
Due: Friday, September 19, 2014
Problem 1: Determine whether the following sets of vectors are linearly independent:
4 2 2
a) 9 , 13 , 4 . with scalars R .
1 10 1
1+ j
b
ECE 604 STATE VARIABLE ANALYSIS
Fall, 2014
Solution to Problem Set #1
Problem 1:
(From homework)
a) Linear combination:
4
2
2
4 2 2 1
9 1 + 13 2 + 4 3 = 9 13 4 2
1
10
1
1 10
1
3
A
The coefficients i can all be non-zero if and on
ECE 604 LINEAR SYSTEMS
Problem Set #6
Issued: Friday, November 7, 2014
Due: Friday, November 21, 2014
Problem 1:
Rugh, Problem 6.3, p. 125.
Problem 2:
Rugh, Problem 6.7, p. 111.
Problem 3:
a) Rugh, Problem 6.11, p. 112.
b) Rugh, Problem 7.3, p. 125.
Probl
Name: _
Problem #1: _
2: _
3: _
4: _
ECE 604 LINEAR SYSTEMS
5: _
Midterm Exam
Total: _
Wednesday, November 2, 2011
7:00 P.M. - 9:00 P.M.
The exam is open book and open notes. You must show your work to receive partial credit. The
problem values are:
Probl
Name: _
Problem #1: _
2: _
3: _
4: _
ECE 604 LINEAR SYSTEMS
Total: _
Midterm Exam
Tuesday, October 28, 2014
7:00 P.M. - 9:00 P.M.
The exam is open book and open notes. You must show your work to receive partial credit. The
problem values are:
Problem #1
P
2
Linear dierential equations
Systems of linear dierential equations form the focus of our rst line of investigation. In particular, we will develop a theory of existence and uniqueness of
solutions of homogeneous initial-value problems of the form x(t) =
ECE 604 STATE VARIABLE ANALYSIS
Fall, 2014
Solution to Problem Set #3
Problem 1:
a) The transformed system is:
( ) ( ( ) ( ) ( ) ( ) ( )
()
z t = P t A t + P t P 1 t z t
t
1
= e
1 et
0
0
=
t
1 e
z t = 0 0 z t
0 1
()
0
et
et
1
()
z0 = P 0 x 0
ECE 604 STATE VARIABLE ANALYSIS
Fall, 2014
Solution to Problem Set #4
Problem 1:
(Rugh 9.1)
Controllable:
The controllability matrix is
1 2+
C = B AB A 2B = 1
1
1
0
2 + 2
1
0
Take the determinant:
2+
det C = 1idet
1
2 + 2
= 2 + 2 2 =
1
The syste
ECE 604 LINEAR SYSTEMS
Problem Set #3
Issued: Friday, October 3, 2014
Due: Friday, October 10, 2014
Problem 1:
a) Consider the homogeneous system:
0
x (t ) =
et
et
x (t )
1
1
x (0) =
1
(1)
Define
et
P (t ) =
1
1
et
Let z ( t ) = P ( t ) x (
ECE 604 LINEAR SYSTEMS
Problem Set #6
Issued: Friday, October 31, 2014
Problem 1:
Due: Friday, November 7, 2014
Consider the p-output homogeneous system
x ( t ) = Ax ( t )
x ( 0 ) = x0
y ( t ) = Cx ( t )
( )
We would like to determine the initial state fr
ECE 604 LINEAR SYSTEMS
Problem Set #4
Issued: Friday, October 10, 2014
Problem 1: Rugh, Exercise 9.1, p. 152
Problem 2: Rugh, Exercise 9.2, p. 152.
Problem 3: Rugh, Exercise 9.6, p. 153.
Problem 4: Rugh, Exercise 9.7, p. 153.
Problem 5: Rugh, Exercise 9.1
Lecture 20
ECE 604
State Variable Analysis
Doug Looze
Oct. 17, 2014
Linear Systems Douglas
Looze
1
Announcements
Problem set 4 is available
Due Friday, October 24
Midterm
Tuesday, October 28
79 PM
Marcus 15
Oct. 17, 2014
Linear Systems Douglas
Looze
2
Las