ECE 602 Midterm 1 Solution
Problem 1. (20 pts) Suppose a matrix A R33 has the characteristic equation
A () = 3 + 22 + .
(a) (10 pts) Write eA as a proper linear combination of I, A, A2 .
(b) (5 pts) Find the determinant of matrix eA .
(c) (5 pts) Based on
ECE 602 Final Exam Solution
Problem 1. (40 pts) Consider the continuous-time linear time-invariant system
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t),
where x(t) R3 , u(t), y(t) R, and matrices A, B, C are given by
1 0 0
0
A = 1 0 0 , B = 1 , C = 1 1 0 .
1 1 1
1
ECE 602 Final Solution
Problem 1. (10 pts) Suppose a square matrix A Rnn has two distinct eigenvalues
1 = 1,
2 = 0,
with the following additional information:
dim(A 1 I)k
2
3
3
3
k
1
2
3
4
dim(A 2 I)k
3
5
6
6
(a) (6 pts) Find the number and sizes of all J
ECE 602 Homework #2
Due: 2/11/14
Problem 1. Suppose that we know a matrix A R1212 has two distinct eigenvalues 1 and 2 ,
with the following additional information:
k
1
2
3
4
5
6
7
dim N (A 1 I)k
2
3
4
5
5
5
5
dim N (A 2 I)k
2
4
5
(a) What are the numbers
HOMEWORK 7
EE-602
DUE: OCTOBER 13, 2011
PROBLEMS: EXISTENCE AND UNIQUENESS
REQUIRED PROBELMS: 3, 7, 8, 9
!
1. Consider the differential equation x = 1.5 x1/ 3, x(0 ) = 0
(a) there does not exist a local solution;
(b) there exists a locally unique solution
Lecture 7: Autonomous LTV Systems
February 11, 2014
Continuous-Time Autonomous LTV Systems
Consider an autonomous linear time-varying (LTV) system
x(t) = A(t)x(t),
t0
with the initial condition x(0) Rn .
The system matrix A(t) Rnn is a function of t 0
T
EE 602 Homework #5
Solutions
Fall 08
Observability Problems
1. (a) Note that
e At
" et
$
=$ 0
$
$
#0
0
e! t
0
0%
'
0'
'
e2 t &
'
for t 0, From the solution form
t
x(t ) = e A(t !1) x(1) + " e A(t ! q ) Bu (q ) dq
1
# 2 et
%
= % e2 ! t
% 2t
%
$e
#
!1+ t
&
Discussion Quiz
EE-602
Fall 09
1. Using a forward Euler approximation with h = 0.1,
!
x (t k ) =
to the differential equation,
x (t k + h ) ! x (t k )
h
dx !1
=
,! x (0 ) = 0.1, compute an
dt 2 x
estimate for x(0.1).
2. Using separation of variables, solv
ECE 602 Final Exam
1. Enter your name in the space provided on this page below.
2. You have 120 minutes. Use the back of each page for rough work, if necessary.
3. You may use a calculator and a double-side formula sheet, and nothing else, in the exam.
4.
ECE 602 Homework #3
Due: 2/20/14
Problem 1. Consider the matrix (note that this is the same matrix in problem 3 of HW #2)
2 1 3
A = 0 1 0 .
0 1 1
(a) For the LTI system x = Ax, nd all the modes of the system.
(b) Suppose x(0) = 1 1 0
modes in (a).
T
. Wri
ECE 602 Homework #4 Solution
Problem 1. (Continued from Problem 3 of HW #3) Consider three cars moving on the
same lane, whose initial locations at time t = 0 are x1 (0) = x2 (0) = x3 (0) = 0.
(a) Suppose the cars have the same rendezvous dynamics as in P
Name: _
EE-602
Exam II
October 9, 2008
140 Point Exam
1 hour and 30 minutes
INSTRUCTIONS
This is a closed book, closed notes exam. You are not permitted a calculator. Work
patiently, efficiently, and in an organized manner clearly identifying the steps yo
ECE 602 Homework #5 Solution
Problem 1. Consider the following linear
0
0
x[k + 1] =
0
0
time-invariant discrete-time system
1 0 0
0
1
0 0 0
x[k] + u[k].
0
0 1 0
0 1 1
1
(a) Is the system controllable?
(b) Assume that x[0] = 0. What is the set of stat
ECE 602 Homework #2 Solution
Problem 1. Suppose that we know a matrix A R1212 has two distinct eigenvalues 1 and 2 ,
with the following additional information:
dim N (A 1 I)k
2
3
4
5
5
5
5
k
1
2
3
4
5
6
7
dim N (A 2 I)k
2
4
5
(a) What are the numbers of J
HOMEWORK 4
DUE: SEPTEMBER 22, 2011
REQUIRED PROBELMS. General: 1, 3, 8, 9, 12, 18, 22
Pole Placement: 1, 5
Before beginning, carefully read through your notes and be able to reproduce all derivations
without your notes.
GENERAL PROBLEMS
1. (Review. Comple
ECE 602 Homework #3 Solution
Problem 1. Consider the matrix (note that this is the same matrix in problem 3 of HW #2)
2 1 3
A = 0 1 0 .
0 1 1
(a) For the LTI system x = Ax, nd all the modes of the system.
(b) Suppose x(0) = 1 1 0
modes in (a).
T
. Write t
ECE 602 Homework #1
Due: 1/28
Problem 1. (DeCarlo) Suppose a physical system is represented by the following system of
equations, where u(t) is the input and y(t) is the output:
d
dt
z1 (t)
z2 (t)
=
z1 (t 1) + u(t)
z1 (t)
,
y(t) = z1 (t + 1) + z2 (t) u(t)
Cauley Jan 23, 2009
EE602 Homework #1 Solutions
Note: Most of the solutions provided below are terse. Your work is expected to be much more complete. 1. Determine if the following systems are linear, time-invariant, and causal: (a) y(t) = t, if |u(t
Lecture 15: Observability II
March 30, 2016
1 / 10
C-T LTI Systems
A continuous-time n-state m-input p-output LTI system
x = Ax + Bu
y = Cx + Du
Matrices A Rnn , B Rnm , C Rpn , D Rpm are known
Can we determine x(0) from u and y over the time interval [
ECE 602 Homework #3 Solution
Problem 1. For A =
1
1
, find eAt using at least two different methods.
0 2
Solution: We can use the Laplace transform method:
e
At
=L
1
1
(sI A)
1
=L
s 1 1
0
s+2
1
"
1
=L
1
s1
1
(s1)(s+2)
1
s+2
0
#
et
=
0
e2t )
.
e2t
1 t
3 (
ECE 602 Homework #3
Due: 2/18
Problem 1. For A =
1
1
, find eAt using at least two different methods.
0 2
Problem 2. Given A Rnn , recall that a subspace V Rn is called A-invariant if for any v V ,
we have Av V . Let M Rnr for some 1 r n, and denote by R(
Lecture 20: Linear Quadratic Regulation: II
April 21, 2014
LQR Problem Formulation
A discrete-time LTI system
x[k + 1] = Ax[k] + Bu[k],
x[0] = x0
y [k] = Cx[k] + Du[k]
Problem: Given a time horizon k cfw_0, 1, . . . , N, nd the optimal input
sequence U =
Lecture 18: Output Feedback Observer Design
April 15, 2014
State Observer Problem
A continuous-time (or discrete-time) LTI system
x = Ax
y = Cx
or
x[k + 1] = Ax[k]
y [k] = Cx[k]
Problem:
A and C are known
Input u and output y , but not state x, can be m
Lecture 16: Minimality, BIBO Stability, and
Canonical Forms
April 3, 2014
Kalman Decomposition
For any continuous-time n-state m-input p-output LTI system
x = Ax + Bu,
Its Kalman Canonical Form is
Aco
0
A21 Ac o
x =
0
0
y=
0
Cco
0
0
y = Cx + Du
( = Tx
Lecture 21: Linear Quadratic Regulation: III
April 23, 2014
C-T LQR Problem Formulation
A continuous-time LTI system
x = Ax + Bu,
x(0) = x0
Problem: Given a time horizon t [0, tf ], nd the optimal input u(t),
t [0, tf ], that minimizes the cost function
t
Lecture 22: Model Order Reduction
April 29, 2014
A Mechanical System Example
Figure : A four-mass mechanical system.1
M q + G q + Kq = Du, y = Pq + Q q
M = diag (m1 , m2 , m3 , m4 ),
k1 + k2
k2
0
0
k2
G = diag (b1 , 0, 0, b5 ),
k2 + k3
k3
0
K =
T
0
k3
ECE 602 Homework #1 Solution
Problem 1. (DeCarlo) Suppose a physical system is represented by the following system of
equations, where u(t) is the input and y(t) is the output:
d z1 (t)
z1 (t 1) + u(t)
=
, y(t) = z1 (t + 1) + z2 (t) u(t).
z1 (t)
dt z2 (
ECE 602 Midterm 1 Solution
Problem 1. (25 pts) A matrix A R33
2
is given by A = 2 1 1 2 .
0.3
(a) (5 pts) Find all the eigenvalues of A;
(Hint: A has rank one; and it has one eigenvector that is very easy to spot).
(b) (5 pts) Find Ak for k = 0, 1, 2 . .
ECE 602 Homework #2
Due: 2/4
Problem 1. The matrix Ac given by
a1 a2
1
0
0
1
Ac =
.
.
.
.
0
0
an1 an
0
0
0
0
.
.
.
.
1
0
is called a top companion matrix (recall the dynamics matrix of the controller canonical form).
Show that the characteristic
ECE 602 Homework #1
Due: 1/28
Problem 1. (DeCarlo) Suppose a physical system is represented by the following system of
equations, where u(t) is the input and y(t) is the output:
d z1 (t)
z1 (t 1) + u(t)
=
, y(t) = z1 (t + 1) + z2 (t) u(t).
z1 (t)
dt z2
Lecture 20: Linear Quadratic Regulation
April 20, 2016
1 / 31
LQR Problem Formulation
A discrete-time LTI system
x[k + 1] = Ax[k] + Bu[k],
x[0] = x0
Problem: Given a time horizon k cfw_0, 1, . . . , N, find the optimal input
sequence U = cfw_u[0], . . . ,