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
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
Lipschitz continuity
Invariant sets
Phase portratis
Autonomous systems
Control of Nonlinear Dynamical Systems
Ravi N Banavar
[email protected]
1
1
Systems and Control Engineering,
IIT Bombay, Ind
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 read
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
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 variable
Linearization
Submanifolds
Center manifold theory
Control of Nonlinear Dynamical Systems
Ravi N Banavar
[email protected]
1
1
Systems and Control Engineering,
IIT Bombay, India
Course offered in
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 per
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 , .
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, 199
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 .
Q
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,
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 constra
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 = x