ECES 521 Practice Final
1. A particular large course at a university has 2 lecture times. Each student will be enrolled in exactly one of these
lecture times. N students enroll in the course.
(a) Suppose the students are allowed to enroll in the lecture t

ECES 521, Homework 7
due November 13, 2013
Name:
Student ID:
1
1. Let X Expo(1 ) and Y Expo(2 ) be two independent continuous random variables, and let the random variable
Z =X +Y.
(a) Find the joint CDF FX,Y (x, y ) and joint PDF fX,Y (x, y ).
(b) Find t

ECES 521, Homework 9
due December 4, 2013
Name:
Student ID:
1
1. Derive the moment generating function for X N (, 2 ). Show the full derivation in your answer.
(a) Use your expression to calculate the Kurtosis
E[(X )4 ]
(E[(X )2 ])2
3 of X .
2. Derive th

ECES 521, Homework 1
September 26, 2013
Name:
Student ID:
1
1. (10 points) Construct a sample space and probability P to model an unfair six-sided die in which faces 1 to 5 are
equally likely, but face 6 has probability 1/3. Using this model, compute the

clearly x is a right eigl
t he operator d o n th
obtain
d (xy):
Hence Ai + Ilj is an
See Definition 2-12.
N ext we prove that a
t hat 11k is a n eigenvalue (
F
On the Matrix Equation
A M+MB=N
or
We now show that th
common. We prove thi
~(s
be the characte

354
STATE FEEDBACK A ND STATE ESTIMATORS
In Section 5-8, we discussed a different method of transforming an equation
into a controllable form without explicitly computing AkB. The method is first
to transform A into a Hessenberg form as in (5-89), a nd th

334
STATE FEEDBACK A ND STATE ESTIMATORS
where the X 's a gain denote possible nonzero elements a nd unfilled positions are
all zeros. T he m atrices A and B can be verified by inspection. T he m atrix C
in b oth cases are to be c omputed from CQ.
By c om

C ANONICAL-FORM DYNAMICAL EQUATIONS
.
)
developed from qualitative analyses
feedback control systems by using
ility study of feedback systems. In
properties of dynamical equations:
:hapter we shall study their practical
hniques from them.
.riant dynamical

~
182
CONTROLLABILITY AND OBSERVABILITY OF LINEAR DYNAMICAL EQUATIONS
CON'
Definition 5 -4
b (t)
T he d ynamical e quation E is s aid t o b e uniformly controllable if a nd o nly if there
exist a positive u , a nd positive ai t hat d epend o n u , s uch t

ECE-S512: Systems II
Winter 2011-2012
Assignment 5
Reading Assignment:
Chen, Linear System Theory and Design Chapter 7, pages 339-354
Written assignment:
Consider Method I and Method III for multivariate state feedback. For this assignment, we will apply

ECE-S512: Systems II
Winter 2011-2012
Assignment 2
Reading Assignment:
Chen Chapter 5 pages 187-199
Written assignment:
1. Controllability of an LTI System
In the last lecture we studied the concept of observability of a system, and found that if a
system

ECE-S 512: Systems II
Winter 2011-2012
Assignment 4
Reading Assignment:
Chen, Linear System Theory and Design Chapter 7, pages 325-331
Chen, Linear System Theory and Design Chapter 7, pages 334-339
Written Assignment
1. Controllable Canonical Form
Conside

ECE-S512: Systems II
Winter 2011-2012
Assignment 6
Reading Assignment:
Chen, Linear System Theory and Design Chapter 7, pages 354-365
Written assignment:
Consider the 2-input, 2-output system with the following dynamical equation FE:
1
2
=
2
1
=
0 1
1 0

ECE-S512: Systems II
Winter 2011-2012
Assignment 1
Reading Assignment:
Chen, Linear System Theory and Design Chapter 5, pages 168-187
Chen, Linear System Theory and Design Chapter 5, pages 192-194
Written assignment:
This weeks written assignment requires

Canonical Decomposition
Jordan-Canonical Form
ECES 512 Winter 2012
Ray Canzanese
30 January 2012
Data Fusion Laboratory
Dept. of Electrical and Computer Engineering
w w w. d a t a f u s i o n l a b . o r g
Summary
A state equation is said to be state con

Fundamentals of Systems II ECES 512
Homework 4 Solutions
2/17/2012
1. Controllable Canonical Form
The system is given by:
3
9
0
2
0
4
1
47 38 105 19 24
x 1
1
3
0
1 x 0 u
3
9
1
2
0
4
1
46 37 101 18 22
y 2 0 1 1 1x
We check controllability

Fundamentals of Systems 11 ECES 512
Homework 2 Solutions
2/6f2913
1. a. To solve for x0, we substitute for the given values of y(t) and t in equations 5-37 and 5-38, to
obtain the following system of equations:
013533 (1002479 (1002479 x10) -0115505
(

Systems II Winter 2011-2012
Homework Solutions
Fundamentals of Systems II Homework 5 Solutions
1. We note that the system matrix is not cyclic therefore we need to find a matrix K1 such that
A BK1 is cyclic. K1 should be a 3x6 matrix.
We note that the co

Systems II Winter 2011-2012
Homework Solutions
Fundamentals of Systems II Homework 6 Solutions
1. The Open Loop Estimator
The eigenvalues of the system matrix are: i 1;1;
1
1
1.94i; 1.94i
2
2
Clearly the two complex eigenvalues are in the RHP plane and

Systems II Winter 2011-2012
Homework Solutions
Fundamentals of Systems II Homework 7 Solutions
1. State Estimator Controller Combination
Since we only need to estimate 3 states, the system observer has to be a 3x3 system.
For estimator poles at -5, I chos

ECE-S 512: Systems II
Winter 2011-2012
Assignment 7
Reading Assignment:
Chen, Linear System Theory and Design Chapter 7, pages 365-377
Written Assignment
1. State feedback and State Estimator Combination
Consider the inverted pendulum example studied in t

20. Canonical Decomposition Theorem
Topics
Be able to
.
decompose a system into its controllable and uncontrollable states.
.
.
do the same for the observable and unobservable subsystems.
find the minimal realization of a system.
Ref. Section 6.8.
Fall 20

Global Linearization
We now design a feedback linearizing control such that the original nonlinear system is transformed
to an equivalent globally linear system. Roughly speaking, this is done by canceling out the
nonlinearities in the system with an appr

ECE-S513: Systems III
Spring 2012-2013
Homework 5 - Solutions
Written assignment:
1. Stability of Linear Systems Autonomous Systems
We developed a method to check the stability of linear autonomous systems using the Lyapunov
matrix equation. Verify the re