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
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
ECE 486
Assignment # 1
Issued: Aug 25
Due: Sep 1, 2016
Reading Assignment:
FPE, 6th ed., Sections 1.1, 1.2, 2.12.4, 7.2, 9.2.1.
Problems:
(the first two problems are designed to test your background)
1. Compute the characteristic polynomial P ( ) = det(A
ECE 528/ME 546/GE 520 ANALYSIS OF NONLINEAR SYSTEMS
SPRING 2016
Section 1
MW 11-12:20pm
3020 ECE Bldg.
Instructor:
Office:
Email:
Office Hours:
Prof. Prashant G. Mehta
359 CSL
mehtapg@illinois.edu
MW 5:00-6:00pm, or by appointment
Regular office hours in
ECE 528/ME 540/GE 520 Homework 3
February 21, 2016
Due Wednesday, March 2 2016
Recommended reading: Sections 3.4, 4.1 and 4.2 in Khalil.
1. Problem 3.15 on page 107, Khalil.
2. Problem 3.17 on page 107, Khalil.
3. Investigate whether or not the following
ECE 528/ME 540/GE 520 Homework 6
April 9, 2016
Due Wednesday, April 20
Recommended reading: Chapter 6 and Section 7.1 of Khalil.
1. Problem 6.1 on page 259 of Khalil.
2. Problem 6.2 on page 259 of Khalil.
3. Problem 6.10 on page 260 of Khalil.
4. Problem
ECE 528/ME 540/GE 520 Homework 5
April 1, 2016
Due Wednesday, April 13
Recommended reading: Chapter 5 of Khalil.
1. Consider the LTI system
x = Ax + Bu
y = Cx
Suppose that there is a positive semidefinite matrix P that solves the Riccati equation
P A + AT
ECE 528/ME 540/GE 520 Homework 2
February 6, 2016
Due Wednesday, February 15 2016
Recommended reading: Appendix B and Chapter 3 of Khalil.
1. Using the CMT, prove that the sequence defined recursively by
xk+1 =
2
xk
+
2
xk
converges to 2 for every x0 [
ECE 528/ME 540/GE 520 Homework 4
March 8, 2016
Due Wednesday, March 30 2016
Recommended reading: Chapter 4 of Khalil.
1. Problem 4.9 on page 182, Khalil.
2. Problem 4.15 on page 184, Khalil.
3. Problem 4.21 on page 185, Khalil.
4. Problem 4.39 on page 188
ECE 528, Spring 15
TAKE-HOME EXAM SOLUTIONS
Friday, May 8
Note: since the exam problems are not locally Lipschitz, their solutions are not necessarily unique. Im
not attempting to list all correct solutions here.
1. Consider the system x = f (x) where x R
ECE 486
Assignment # 2
Issued: September 1
Due: September 8, 2016
Reading Assignment:
FPE, Section 3.1.
Problems:
1.
(i) Compute by hand the Laplace transforms Fi = L(fi ) of
f1 (t) = sin(t) (recall Eulers formula)
f2 (t) = e
t
f3 (t) = sin(t)+e
t
(ii) Us
Plan of the Lecture
I
Review: prototype 2nd-order system
I
Todays topic: transient response specifications
Goal: develop formulas and intuition for various features of the
transient response: rise time, overshoot, settling time.
Reading: FPE, Sections 3.3
ECE 517, Fall 16
PROBLEM SET 1
Due Wednesday, Sep 7
1. Consider the 2-D system
6x1
+ 2x2
(1 + x21 )2
2(x1 + x2 )
x 2 =
(1 + x21 )2
x 1 =
and the candidate Lyapunov function
V (x1 , x2 ) =
x21
+ x22
1 + x21
Compute the derivative of this V along solution
ECE 517:
Nonlinear and Adaptive Control
Fall 2015 Lecture Notes
Daniel Liberzon
October 28, 2015
2
DANIEL LIBERZON
Disclaimers
I dont recommend printing the entire file in advance, since I continue to edit the notes
during the semester.
These lecture note
ECE 486
Assignment # 3
Issued: September 8
Due: September 15, 2016
Reading Assignment:
FPE, Sections 3.3-3.6.
Problems:
(unless otherwise noted, you can use a calculator/computer to arrive at numerical answers)
1. Consider the system given by the block di
Plan of the Lecture
I
Review: control, feedback, etc.
I
Todays topic: state-space models of systems; linearization
Goal: a general framework that encompasses all examples of
interest. Once we have mastered this framework, we can
proceed to analysis and th
Plan of the Lecture
I
Review: state-space models of systems; linearization
I
Todays topic: linear systems and their dynamic response
Goal: develop a methodology for characterizing the output of a
given system for a given input.
Reading: FPE, Section 3.1,
Plan of the Lecture
I
Todays topic: what is feedback control? past, present,
future
Goal: get comfortable with the idea of feedback control as a
means of getting unreliable or unstable components to behave
reliably.
Recommended reading:
I
I
FPE, Chap. 1 s
ECE 517, Fall 16
PROBLEM SET 2
Due Tuesday, Sep 271
1. 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 help o
ECE 528/ME 540/GE 520 Homework 1
February 2, 2016
Due Wednesday, February 3 2016
Recommended reading: Appendix A and Chapter 2 of Khalil.
1. Prove that the sequence defined recursively by
xk+1 =
2
xk
+
2
xk
converges to 2 for every x0 [ 2, 2 2].
2. Let