Lectures 16,17,18
6
Inner Product Spaces
6.1
Basic Definition
Parallelogram law, the ability to measure angle between two vectors and in particular, the
concept of perpendicularity make the euclidean space quite a special type of a vector space.
Essential
7
7.1
Eigenvalues and Eigenvectors
Introduction
The simplest of matrices are the diagonal ones. Thus a linear map will be also easy to
handle if its associated matrix is a diagonal matrix. Then again we have seen that the
matrix associated depends upon th
Lipschitz continuity
Invariant sets
Phase portratis
Autonomous systems
Control of Nonlinear Dynamical Systems
Ravi N Banavar
ravi.banavar@gmail.com
1
1
Systems and Control Engineering,
IIT Bombay, India
Course offered in the Spring semester, 2016
IIT Bomb
Lecture 13
5
5.1
Linear Transformations
Basic Definitions and Examples
We have already come across with the notion of linear transformations on euclidean spaces.
We shall now see that this notion readily extends to the abstract set up of vector spaces
alo
Lecture 10
4
4.1
Vector Spaces
Basic Definition and Examples
Throughout mathematics we come across many types objects which can be added and multiplied by scalars to arrive at similar types of objects. We have been doing this with R,
C, Mm,n (R), Mm,n (C)
Lecture 1
1
Introduction:
This is the second course in mathematics for you in IIT. In the first course itself you have
studied some basic properties such as continuity of functions of several variables. The central
theme of this course is to study thoroug
Linearization
Submanifolds
Center manifold theory
Control of Nonlinear Dynamical Systems
Ravi N Banavar
ravi.banavar@gmail.com
1
1
Systems and Control Engineering,
IIT Bombay, India
Course offered in the Spring semester, 2016
IIT Bombay
March 9, 2016
Slid
Region of Attraction and Linearization
R. N. Banavar,
February 4, 2016
These notes are based on my readings of the book
Nonlinear Systems, H. Khalil, Prentice Hall, 1996.
From a control engineers perspective, it is essential that an idea of the domain ar
Flows, Stability and Invariance
Ravi N Banavar
October 29, 2013
Consider a dynamical system of the form
x = f (x) + g1 (x)u1 + . . . gm (x)um
x(t) U Rn
where f (), g() are smooth functions and u1 , . . . um take values in a subset of R. The vector
f (x) i
Linearization, the Jacobian and the Center Manifold
R. N. Banavar,
February 4, 2016
These notes are based on my readings of the two books
Ordinary Differential Equations, V. I . Arnold, Springer, 1992
Nonlinear Systems, H. Khalil, Prentice Hall, 1996
1
SC 602 - Assignment 2
Question 1
Let A be an n n matrix having n distinct real eigenvalues 1 , . . . , n . Prove that there
exists an invertible n n matrix Q such that
QAQ1 = diagcfw_1 , . . . , n .
Question 2
Let the n n matrix A have n distinct real eig
SC 602 Assignment 4
Name:
Roll no.:
Question 1
Consider the system
x 1 = x1 ,
x 2 = (x1 x2 1)x32 + (x1 x2 1 + x21 )x2
a Show that x = 0 is the unique equilibrium point.
b Show, by using linearization, that x = 0 is asymptotically stable.
c Show that = cfw
SC 602 Assignment 1
Name:
Roll no.:
Question 1:
A simple population growth model is
x = (a
bx)x
c
where a, b, c are constants. The term (a bx)x depicts a population growth rate with a
resource constraint, and the term c reflects the harvesting rate. For t
SC 602 Assignment 5
Name:
Roll no.:
Question 1
For each of the following systems, investigate stability of the origin using the center manifold
theorem:
1.
x 1 = x1 x2 3
x 2 = x2 x21 + 2x81
2.
x 1 = x1 + x2 3 (x1 + x2 1)
x 2 = x32 (x1 + x2 1)
3.
x 1 = x2