ECE 515
Assignment # 1
Issued: January 18
Due: January 28, 2010
Reading Assignment:
Lecture Notes Ch. 1 and 2. See also Brogan, 3.13.4, 5.15.8, or Chen 2.12.8, 3.33.5.
Problems:
1. Based on Sections 1.1 and 1.2 of the lecture notes : You are given a nonli
ECE 515
Correspondence # 26
FALL 2013
November 6, 2013
ADDITIONAL NOTES ON FEEDBACK
Hitherto, we have seen the role of feedback in the regulation of LTI systems; that is, how to choose
u = Kx or Ky or K x (where x is generated by a dynamic compensator) so
ECE 515
Correspondence # 33
FALL 2013
November 27, 2013
BASICS OF FINITE-DIMENSIONAL OPTIMIZATION
In the last lecture (November 21), I introduced the Hamilton-Jacobi-Bellman (HJB) equation
in connection with obtaining the optimal control function, u[0,tf
ECE 515
Correspondence # 30R
FALL 2013
November 20, 2013
LINEAR QUADRATIC OPTIMAL REGULATOR
In Correspondence # 26, we have seen a Lyapunov function-based approach for selecting stabilizing state-feedback controllers for LTI systems, which has involved th
ECE 515 Assignment 3
22 Sept 2016
1
Question 1
It is sufficient to prove that the eigenvectors are linearly independent. Denote by z1 , z2 , .zN the
eigenvalues and corresponding eigenvectors
v1 , .,
v
N . For the linear combination 1 v1 +2 v2 +3 v3 +
.
Problem Set # 1
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, belabbas@illinois.edu, CSL 166
Teaching Assistant:Xiaobin Gao, xgao16@illinois.edu
Due date: Tue Sep 15 2015
Readings: Class notes Chapters 2 and 3. Linear algebra review
Problem Set # 6
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, belabbas@illinois.edu, CSL 166
Teaching Assistant:Xiaobin Gao, xgao16@illinois.edu
Due date: Thu Nov 19 2015
Readings: Class notes Chapters 10-11 Do the following problem
Problem Set # 4
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, belabbas@illinois.edu, CSL 166
Teaching Assistant:Xiaobin Gao, xgao16@illinois.edu
Due date: Thu Oct 15 2015
Readings: Class notes Chapters 5 and 6. Also, review some con
Problem Set # 3
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, belabbas@illinois.edu, CSL 166
Teaching Assistant:Xiaobin Gao, xgao16@illinois.edu
Due date: Tue Oct 6 2015
Readings: Class notes Chapters 4 and 5. Also, review some conc
ECE 515 homework 5 solution
Author: Xiaobin Gao
Problem 1; Solution:
1 = 2
cfw_ 2 = 31 22 +
= 1 + 2
Taking Laplace Transform on both sides and assume zero initial conditions (since we only want to find
the relationship between and ), we have
1 () = 2
ECE 515: Control System Theory and Design
Fall 2015
Problem Set 6
Author: Xiaobin Gao
Due Date: 11/19
1. Problem 1
Since the cost function is given by
Z
L(x, x)
=
1
(x2 + x 2 )dt
0
The Lagrangian L(x, x)
= x2 + x 2 . By Euler-Lagrange equation
Lx =
d
dt
Problem Set # 5
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, belabbas@illinois.edu, CSL 166
Teaching Assistant:Xiaobin Gao, xgao16@illinois.edu
Due date: Tue Nov 10 2015
Readings: Class notes Chapters 6-8 Do the following problems.
Problem Set # 5
ECE 515: Control System Theory and Design
Instructor:M.-A. Belabbas, belabbas@illinois.edu, CSL 166
Teaching Assistant:Xiaobin Gao, xgao16@illinois.edu
Due date: Tue Nov 10 2015
Readings: Class notes Chapters 6-8 Do the following problems.
ECE 515, Spring 2012
PROBLEM SET 10
Due Thursday, Apr 19
Reading: Class Notes, Sections 10.1, 10.2; my optimal control lecture notes (posted on the class website),
Section 5.1.
Problems:
1. Consider the optimal control problem
x
= u,
V =
!
t1
t0
"
#
x4 (
ECE 515, Spring 2012
PROBLEM SET 10
Due Thursday, Apr 19
Solutions:
1.
a) System state space representation is
0 1
0
x =
x+
u.
0 0
1
Using HJB,
h
@V
@V x2 i
4
2
= min x1 + u +
u
u
@t
@x
h
@V
@V i
= min x41 + u2 +
x2 +
u
u
@x1
@x2
with
V (x(t1 ), t1 ) = m(
Controllable subspace decomposition
M.-A. Belabbas
1. We deal here with linear time-invariant systems
x = Ax + Bu
where A Rnn and B Rn . The restriction that B be a column vector is not an essential
one, it does make the notation a tad simpler though.
2.
ECE 515
Correspondence # 25
FALL 2013
November 6, 2013
REDUCED ORDER OBSERVERS
The following is a write-up on reduced order observers which I will briey discuss in class on
Thursday (tomorrow). It also includes material on controller design using reduced
ECE 515
Correspondence # 21
FALL 2013
October 24, 2013
SUPPLEMENTARY NOTES ON FEEDBACK
The following example, which is also discussed (partially) in the Lecture Notes, Chapter 7, serves to
illustrate most of the concepts we will be introducing during the
ECE 515, Spring 2012
PROBLEM SET 5
Due Thursday, Feb 23
Reading: Class Notes, Sections 4.24.4, 4.6, 4.7, 5.1.
Problems:
1. Consider the system
x1 = x 2 ,
x2 = x2 sin x1
and the three candidate Lyapunov functions
V1 (x) = 1 x2 cos x1 ,
22
V2 (x) = 1 + 1 x2
Lecture notes: Applications of Diagonalization
Math 54, GSI: Alan Tarr, July 27th
The biggest direct application of diagonalization is that it gives us an easy way
to compute large powers of a matrix A, which would be impossible otherwise.
If A is diagona
Linear Systems Notes
1
1
Lecture 1
Introduction Consider the following linear system
x = Ax + Bx
(1)
where x Rn , which describes the dynamics of x, given by two components: Ax the position of
x (A L(Rn , Rn ), a linear mapping), and Bu, where u Rm is cal