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
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
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.
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 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
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 #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 #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
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 #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 17: State Feedback Control
April 8, 2014
Controller Design
A continuous-time (or discrete-time) LTI system
x = Ax + Bu
or
x[k + 1] = Ax[k] + Bu[k]
Problem: Design input u so that behavior of system is better
Stabilize the system (if A is unstable
Lecture 19: Linear Quadratic Regulation: I
April 16, 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 = c
Lecture 14: Observability I
March 27, 2014
Motivation: State Estimation of D-T LTI Systems
A discrete-time n-state m-input p-output LTI system
x[k + 1] = Ax[k] + Bu[k],
x[0] = x0
y [k] = Cx[k] + Du[k]
Matrices A Rnn , B Rnm , C Rpn , D Rpm are known
Inp
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
ECE 602 Midterm 1
1. Enter your name in the space provided on this page below.
2. You have 75 minutes. Use the back of each page for rough work, if necessary.
3. You may only use a calculator and a single-side letter-size formula sheet in the exam, and
no
ECE 602 Homework #5
Due: 3/27/14
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
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
ECE 602 Homework #4
Due: 3/4/14
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 i
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
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 13: Controllability II
March 11, 2014
Controllability of C-T LTI Systems
A continuous-time n-state m-input LTI system
x = Ax + Bu,
x(0) = x0
(1)
Denition
The LTI system is called controllable at time tf > 0 if for any initial
state x0 Rn and any