ECE 490: Introduction to Optimization
Spring 2014
Midterm II
R. Srikant
April 24, 7:00-8:30 pm
No calculators are allowed. You are allowed two 8.5x11 sheets (four pages) of notes.
There are ve problems, each worth 20 points.
1. Consider the following opti
ECE 490: Introduction to Optimization
Spring 2014
Midterm I
R. Srikant
Mar. 6, 7:00-8:30 pm
No calculators are allowed. You are allowed two 8.5x11 sheets (four pages) of notes.
There are ve problems, each worth 20 points.
1. Consider the problem of minimi
1
ECE517, Fall 2015
PROBLEM SET 2 Solution
1. (10 Points) Consider the class of scalar plants
y = ay + bu,
a R, b > 0
(1)
In Section 3.1.1 of class notes, it is shown that the controller (19) is a universal regulator for this class of plants,
with the hel
1
ECE517, Fall 2015
PROBLEM SET 1 Solution
1. (20 Points) Consider the 2-D system
6x1
+ 2x2 ,
(1 + x2 )2
1
2(x1 + x2 )
x2 =
(1 + x2 )2
1
x1 =
and the candidate Lyapunov function
V (x1 , x2 ) =
x2
1
+ x2 .
2
1 + x2
1
Compute the derivative of this V alon
ECE/CS 541: COMPUTER SYSTEMS ANALYSIS
Homework #4
Due Tuesday, October 6, 2009
The goals of this exercise are for you to become familiar with the mechanics of modeling using
Mbius and analysis of systems using Fault Trees. Completing this exercise will as
ECE/CS 541: COMPUTER SYSTEM ANALYSIS
Homework #1 Due Tuesday, September 1, Beginning of Class
August 25, 2008
The intent of this homework is to review material you should have learned in your probability and statistics course (e.g., ECE 413) and explore n
ECE 541 Homework 2
Assigned Tuesday, September 1, 2009
Due Tuesday, September 8, 2009
Assume that all component failures are independent unless otherwise stated.
1. A monitoring system consists of a group of four sensors, a main data processor, a backup d
CS/ECE 541 Homework # 3, Due in class September 15
1. A random walk on a ring is a stochastic process where there are N states, numbered 0 to
N 1. Each step, a walker in state i steps to state (i + 1) mod N with probability p, and to
state (i 1) mod N wit
University of Illinois Spring 2011
ECE534 Moulin
Final Exam
Monday, May 9, 2011
Sow-ﬂow SET
Name
Score
Please do not turn this page until requested to do so.
You may use three cheat sheets for this exam. Make sure to show all of your work on the longer
pr
ECE 534: Random Processes
Spring 2011
Solutions to Probability Review Quiz
1. Suppose youre on a game show and given a choice of three doors. A car is behind one of
the doors with equal probability. After you pick a door, the host opens one of the other
t
ECE 534: Random Processes
Spring 2013
Probability Review Quiz
R. Srikant
Feb 7 2013, 7:00-8:15
Each problem is worth 25 points. No calculators allowed; you are allowed two sheets of handwritten
notes.
1. Consider the following CDF of a random variable X:
ECE 534: Random Processes
Fall 2010
Solutions to Midterm 2
1. You are given the magnitude of a normal random variable X N (0, 1). Find the
estimator X = a + b|X| that minimizes MSE = E[(X X)2 ], and give the resulting
MSE.
Solution: We have
Cov(X, |X|)
X
ECE 534: Random Processes
Fall 2010
Solutions to Midterm 1
1. The pair of random variables (Xn , Yn ) R2 converges in some sense (a.s., or m.s., or d.)
to (X, Y ) as n . Dene Zn = Xn + Yn and Z = X + Y . Are the following three
statements true?
d
(Xn , Y
ECE 534: Random Processes
Spring 2011
Solutions to Midterm
1. Let cfw_Xn be a sequence of d dimensional vectors and let X be a random vector, all dened
2
on the same sample space. The Euclidean norm of a vector v Rd is v = ( d vi )1/2 .
i=1
We say that c
ECE 534: Random Processes
Spring 2013
HW VII
R. Srikant
1. Consider a discrete-time queueing system with Ber() arrivals and Ber() service with < .
Assume departures occur before arrivals in each time slot.
(a) Write down the queueing dynamics of this syst
University of Illinois
Fall 2011
ECE 534: Exam II
Monday November 14, 2011
7:00 p.m. 8:15 p.m.
103 Talbot
1. [15 points] Suppose Xt = V exp(U t) where U and V are independent, and each is uniformly
distributed over the interval [0, 1].
(a) Find the mean f
ME540/ECE515 HW#1 Solutions
1. Depicted below is a Pendubot with massless links, and masses at their ends. As in the course
notes a torque is applied at the base of the rst link, and is available for control.
The equations of motion for the system are giv
ECE515-ME540: Assignment #4
Due: Apr 2
Instructor: Geir E. Dullerud
Problems:
1. Given > 0 dene the subset of the complex numbers C = cfw_z C : Re(z) < . Prove that if
eig(A) C and Q is a positive denite matrix, then there is a unique positive denite solu
ECE515-ME540: Assignment #3
Due: Mar 12
Instructor: Geir E. Dullerud
Problems:
1. Given arbitrary constants a, b, and the matrix
(a 1)
1
0
(1 + a)
0 .
A = 1
1
(1 + a b) b
(a) Determine the eigenvalues of this matrix.
(b) Find 3 linearly independent genera
ECE515-ME540: Assignment #2
Due: Feb 19
Instructor: Geir E. Dullerud
Problems:
(1) Complete exercise 2.11.10 in the course notes.
(2) Complete exercise 2.11.19 in course notes.
(3) Jordan decomposition. In some sense, matrices that have non-scalar Jordan
ECE515-ME540: Assignment #1
Due: Feb 5
Instructor: Geir E. Dullerud
Problems:
(1) Depicted below is a Pendubot with massless links, and masses at their ends. As in the course notes a
torque is applied at the base of the rst link, and is available for cont
ECE 550 Final Exam
Instructions: This is an closed-book, closed-notes exam. Only 2 8.5x11 crib sheets are
allowed during the exam. Moreover, calculators, computers and other electronic aids are not
allowed in this exam. Please make sure to state your assu
ME 540/ECE515 Midterm - Spring 2011
Name:
A table of Laplace transforms and other useful formulas appears on the last page of this
booklet. If you separate the other pages of this book, write your name on every page. Calculators are not allowed. The point