ELEC4631 Lecture 8:
Observers, Output Feedback, and the
First Method of Lyapunov
Lecturer: Dr. Hendra Nurdin
Uncontrollable and unobservable systems
The system
)# 2 12&
#1&
S = +%
( , %( ,
$0'
*$1 5'
[1
2]
is observable and controllable.
Eigenvalues of
ELEC4631:
Review Lecture
Lecturer: Dr Hendra Nurdin
Today
Review of materials from the last twelve weeks
Information about final exam
Out-of-session consultation
ELEC4631, S1
2
Mind map
Dynamical
systems
Elementary
nonlinear control
Stability
theory
Ly
LIST OF FORMULAS
Controllability matrix: C = B AB . . . An1 B
C
CA
Observability matrix: O =
.
.
n1
CA
Transformation to control canonical
0 0
0 0
0 0
= . .
. .
0 1
1 p1
form: T = C 1 , with
.
.
.
.
.
0
0
1
.
.
0
1
p1
.
.
1
p1
p2
.
.
. . . pn4 p
ELEC4631 Solutions Tutorial 4
2
1. (a) Since (i) 2 0, D1 + 0 and 1 cos 0 then (ii) V (, ) = 0
2
2 = 0, D1 + , 1 cos = 0 = 0, = 0. From (i) and (ii), V is lpd
around the origin (in fact it is more, it is a positive definite function).
(b)
V (x) = V (x)T
Title page
THE UNIVERSITY OF NEW SOUTH WALES
MONTH OF EXAMINATION: JUNE, 2008
ELEC4631 CONTINUOUS TIME CONTROL SYSTEM DESIGN
(1) TIME ALLOWED: 3 HOURS
(2) TOTAL NUMBER OF QUESTIONS: 13
(3) ANSWER ALL QUESTIONS
(4) ALL QUESTIONS ARE OF EQUAL VALUE (5 point
Solutions
(1a) For
2 3 0 4
1
1
0 0 0
0
A=
, B =
0
0
1 0 0
0 0 1 0
0
the controllability matrix is
T = [B
AB
A2 B
1
0
A3 B ] =
0
0
2
1
0
0
1 4
2 1
1 2
0 1
Its easy to check rank(T ) = 4, so the system is controllable;
(1b) As T is upper triangular, b
ELEC4631 Lab 0
Introduction to Matlab, Simulink, QUARC
for Multivariable Control System
Semester 1, 2016
1
General information
1. Students must not arrive late at the lab by more than 10 minutes. Demonstrators
can refuse to let in and/or mark late student
ELEC4631 Tutorial 3
1. Compute the exponential of the following matrices:
9 8
(a) A =
.
12 11
1 1
(b) A =
.
1 1
0 1
0
1 .
(c) A = 0 0
0 0.5 0.5
2. Show that for the following ODEs the origin is an asymptotically stable equilibrium
point using the suggeste
ELEC4631 Lab 1
Semester 1, 2016
1
General Information
Students must not arrive late at the lab by more than 10 minutes. Demonstrators can
refuse to let in late students.
2
Preparatory tasks
Before attending the lab, learn the function of, and how to use,
ELEC4631 Continuous-Time Control System Design
Weeks 1 and 2: Brief Review of Linear Algebra and
Vector Calculus
Lecturer: Dr Hendra Nurdin
1
Introduction
The first two lectures provide a brief review of concepts from linear algebra and vector
calculus th
ELEC4631 Solutions Tutorial 1
1. (a) cA () = det(I A) = (4)(2). Two positive eigenvalues 2 and 4, matrix
is positive definite.
(b) cA ()
A) = 2 8 1. One positive and one negative eigenvalue
= det(I
4 + 17 and 4 17, respectively. Matrix indefinite (not d
ELEC4631 Tutorial 2
Do all of the following exercises by hand without using Matlab or any other
computational aid.
1. Consider the quadratic form in R3
f (x) = 4x21 + 6x1 x2 + 4x22 + 8x2 x3 + 4x23 .
(a) Write the quadratic form in the form f (x) = xT Ax f
ELEC4631 Lecture 4: State-Feedback
Control Using Lyapunov Functions
Lecturer: Dr Hendra I. Nurdin
(Adapted and expanded from earlier
lecture slides by Prof. Tuan D. Hoang)
Lecture outline
Controlled ODEs.
Lyapunov functions revisited.
State-feedback de
ELEC4631 Tutorial 1
Do all of the following exercises by hand without using Matlab or any other
computational aid.
1. Determine whether the following matrices have some definiteness property (positive
definite or semidefinite, negative definite or semidef
ELEC4631 Lecture 5:
Linear Time-Invariant State-Space
Systems
Lecturer: Dr. Hendra Nurdin
Input-output models
In ELEC3114 you have mostly encountered inputoutput systems represented by differential equations:
y (n) (t) + an1 y (n1) (t) + a0 y(t) = bm u(m