September 19, 2014
AAE 564, Fall 2014
Homework 3 - Solution
Exercise 1 To obtain a state space realization of the given transferfunction, it will rst be
transformed into a proper, rational form
s2 + 3s + 2
(s2 + 5s + 6) 2s 4
2s 4
G(s) = 2
=
= 2
+1
2 + 5s

October 10, 2012
AAE 564, Fall 2012
Homework 5 - Solution
Exercise 1 Given the matrix
0 1 0
A = 0 0 1
1 1 1
we need to identify the eigenvalue and eigenvectors rst before we decide if the matrix is
defective.
1
0
1
1
det(A I) = det 0
= (1 )(2 0) 1 det

November 26, 2014
AAE 564, Fall 2014
Homework 11 - Solution
Exercise 1 The system can be written as:
x1
x1
5
1 1
1 0
x2 = 1 3 1 x2 + 1 1 u1
u2
x3
x3
2 2 4
0 1
A
We determine controllability by checking
we rst identify:
5
1
AB =
2
5
2
1
A B = A (A B) =

October 23, 2014
AAE 564, Fall 2014
Homework 7 - Solution
Exercise 1 Considering the LTI system:
x1 =
x2
x2 = 2x1 3x2 +u
y = 3x1
x2
which can be written as
x1
x2
=
0
x1
+
u
x2
1
0
1
2 3
B
A
x1
x2
y = 3 1
+ 0 u
C
D
we would like to identify a persistent (d

1
Simple oscillator
(oscillator.mdl) Here we obtain a Simulink model of the simple spring-mass-damper system described by
m + cy + ky = 0
y
To obtain a Simulink model, we rst rearrange the equation as follows:
y=
c
k
y y
m
m
Simple oscillator (oscillator)

1
Simple oscillator
(oscillator.mdl) Here we obtain a Simulink model of the simple spring-mass-damper system described by
m + cy + ky = 0
y
To obtain a Simulink model, we rst rearrange the equation as follows:
y=
c
k
y y
m
m
Simple oscillator (oscillator)

December 15, 2008
AAE 564 Fall 2008
Test Two
Monday, December 15, 8-10am, BRNG B222
Problem 1 Consider the system with input w, output z, and state variables
x1 , x2 described by
x1 = x2 + w
x2 = 2x1 3x2 w
z = 2x1 + 2x2
Compute
2
0 |z(t)| dt
for the follo

November 5, 2008
AAE 564 Fall 2008
Test One
Problem 1 Obtain a state-space description of the following system.
q1 + q2 + sin q1 = u
q2 + q1 + q2 = 0
y = q1 + q2
Problem 2 Obtain a state space realization of the transfer function
G(s) =
1
s+1
s
s1
Problem

December 5, 2012
AAE 564 Fall 2011
Test Two
Thursday, December 13, 7-9pm, LILY G126
Problem 1 Without doing any integrations, compute
x1 (t)2 dt
0
for the system
x1 = x1 + x2
x2 = 2x1 + u
for the following two cases:
(a) x(0) =
1
0
,
u(t) 0
(b) x(0) = 0 ,

December 7, 2012
AAE 564, Fall 2012
Test 2 - Solution
Exercise 1 Having the system:
x1
x2
1 1
2 0
=
x1
0
+
u
x2
1
A
B
we can solve the integral using an Lyapunov equation:
x Q x dt with Q =
x1 (t)2 dt =
0
0
1 0
0 0
The state x will evolve as: x(t) = eAt x

N ame: ,
(Who are you !)
AAE 554, Fall 2008
Monday, December 15, 8—10am, BRNG B222
Attempt all questions
Read each question carefully and think before answering
Begin each solution on a new sheet (of paper) (bl: xl'l'w
a'cl =.- 4:12pm; -w
Z 7" aml'ﬂ’xl
I!

December 8, 2010
AAE 564 Fall 2010
Test Two
Problem 1 Consider the system with input u, output y, and state variables
x1, x2 described by
x1 = x2 + u
x2 = x1 + u
y = x1 + x2
(a) Is this system observable?
(b) If the system is unobservable, determine its u

December 15, 2008
AAE 564 Fall 2008
Test Two
Monday, December 15, 8-10am, BRNG B222
Problem 1 Consider the system with input w, output z, and state variables
x1 , x2 described by
x1 = x2 + w
x2 = 2x1 3x2 w
z = 2x1 + 2x2
Compute
2
0 |z(t)| dt
for the follo

December 8, 2014
AAE 564 Fall 2014
Homework Thirteen
Due: NOT!
Exercise 1 Consider a system described by
x1 = x2 + u
x2 = x1 + u + w
z = x1
where u is a control input, w is a unknown constant disturbance input, x1 , x2 are state
variables, and z is a perf

November 26, 2014
AAE 564 Fall 2014
Homework Twelve
Due: Monday, December 8
Exercise 1 (By hand.) Consider the system described by
x1 = x1 + x2 + u
x2 = u
Obtain (by hand) a state feedback controller which results in a closed loop system which is
asymptot

October 10, 2012
AAE 564, Fall 2012
Homework 6 - Solution
Exercise 1 Suppose A is a 3 3 matrix and:
det(sI A) = s3 + 2s2 + s + 1
sh is
ar stu
ed d
vi y re
aC s
ou
ou rc
rs e
eH w
er as
o.
co
m
Using the Cayley-Hamilton theorem, which says, that the matrix

September 11, 2015
AAE 564, Fall 2015
Homework 1 - Solution
Exercise 1 To transform the linear system into a state space description, we need to relabel
the variables accoring to the following rule:
xi = q (i1)
,
i = 1.n
Hence xi refers to the (i-1)-th de

September 25, 2015
AAE 564 Fall 2015
Homework Five
Due: Monday, October 5
Exercise 1 Determine whether or not the following matrix is nondefective.
0 1 0
A=
0 0 1
1 1 1
Exercise 2 What is the companion matrix whose eigenvalues are 1, 2, and 3?
Exercise

August 28, 2015
AAE 564 Fall 2015
Homework One
Due: Friday, September 4
Exercise 1 Consider a system described by a single nth -order linear differential equation of
the form
q (n) + an1 q (n1) + . . . a1 q + a0 q = u
where q(t) IR and q (n) := ddtnq . By

November 13, 2015
AAE 564 Fall 2015
Homework Eleven
Due: Monday, November 23
Exercise 1 (By hand.) Determine whether or not the following system is controllable.
x 1 = 5x1 + x2 x3 + u1
x 2 = x1 + 3x2 x3 + u1 + u2
x 3 = 2x1 2x2 + 4x3 + u2
If the system is

October 23, 2015
AAE 564 Fall 2015
Homework Eight
Due: Monday, November 2
Exercise 1 Determine (by hand) whether each of the following systems is asymptotically
stable, stable, or unstable.
x 1 = x1 + 2001x2
x 2 = x1
x 1 = x1
x 2 = x2
x 1 = x1 + x2
x 2 =

November 6, 2015
AAE 564 Fall 2015
Homework Ten
Monday, November 16
Exercise 1 Determine (by hand) whether or not each of the following systems are observable.
x 1 = x1
x 1 = x1
x 1 = x1
x 1 = x2
x 2 = x2 + u
x 2 =
x2 + u
x 2 = x2 + u
x 2 = 4x1 + u
y = x1

October 30, 2015
AAE 564 Fall 2015
Homework Nine
Due: Monday, November 9
Exercise 1 Determine (by hand) the maximum singular value of the following matrices.
3
4
A=
!
A=
,
3 4
A=
,
0 1
1 0
!
Exercise 2 Determine (by hand) the singular value decomposition

September 6, 2015
AAE 564 Fall 2015
Homework Two
Due: Friday, September 11
Exercise 1 Obtain the A, B, C, D matrices for a state space representation of the following
systems:
(a)
u = a0 q + a1 q + . . . + an1 q (n1) + q (n)
y = 0 q + 1 q + . . . + n1 q (

September 10, 2015
AAE 564 Fall 2015
Homework Three
Due: Monday, September 21
Exercise 1 Obtain the transfer function of the following system
x(t)
= x(t) + 2x(t h) + u(t)
y(t) = x(t)
Exercise 2 The two pendulum cart system. Unless otherwise specified, fro

October 14, 2015
AAE 564, Fall 2015
Homework 5 - Solution
Exercise 1 Given the matrix
0 1 0
A = 0 0 1
1 1 1
we need to identify the eigenvalue and eigenvectors rst before we decide if the matrix is
defective.
1
0
1
1
= (1 )(2 0) 1 det
det(A I) = det 0

November 4, 2015
AAE 564, Fall 2015
Homework 9 - Solution
Exercise 1 The singular values i of a matrix A can be determined by solving for the
eigenvalues i of A A and using the relation: i = i as shown on p. 252 of the notes
(a)
AA= 3 4
3
= 25
4
max =
25