Multivariable Control Systems - PowerPoint PPT Presentation

About This Presentation
Title:

Multivariable Control Systems

Description:

Multivariable Control Systems Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Chapter 1 Vector Spaces Norms Unitary, Primitive and Hermitian Matrices ... – PowerPoint PPT presentation

Number of Views:369
Avg rating:3.0/5.0
Slides: 27
Provided by: profsite3
Category:

less

Transcript and Presenter's Notes

Title: Multivariable Control Systems


1
Multivariable Control Systems
  • Ali Karimpour
  • Assistant Professor
  • Ferdowsi University of Mashhad

2
Chapter 1
Linear Algebra
Topics to be covered include
  • Vector Spaces
  • Norms
  • Unitary, Primitive and Hermitian Matrices
  • Positive (Negative) Definite Matrices
  • Inner Product
  • Singular Value Decomposition (SVD)
  • Relative Gain Array (RGA)
  • Matrix Perturbation

3
Vector Spaces
A set of vectors and a field of scalars with some
properties is called vector space.
To see the properties have a look on Linear
Algebra written by Hoffman.
Some important vector spaces are
4
Norms
To meter the lengths of vectors in a vector
space we need the idea of a norm.
Norm is a function that maps x to a nonnegative
real number
A Norm must satisfy following properties
5
Norm of vectors
6
Norm of vectors
7
Norm of real functions
8
Norm of matrices
We can extend norm of vectors to matrices
9
Matrix norm
A norm of a matrix is called matrix norm if it
satisfy
Define the induced-norm of a matrix A as follows
Any induced-norm of a matrix A is a matrix norm
10
Matrix norm for matrices
If we put p1 so we have
Maximum column sum
If we put pinf so we have
Maximum row sum
11
Unitary and Hermitian Matrices
For real matrices Hermitian matrix means
symmetric matrix.
12
Primitive Matrices
Definition 2.1 A primitive matrix is a square
nonnegative matrix some power (positive integer)
of which is positive.
13
Primitive Matrices
14
Positive (Negative) Definite Matrices
Negative semi definite define similarly
15
Inner Product
An Inner product must satisfy following
properties
16
Singular Value Decomposition (SVD)
17
Singular Value Decomposition (SVD)
Theorem 1-1
18
Singular Value Decomposition (SVD)
Example
Has no affect on the output or
19
Singular Value Decomposition (SVD)
20
Matrix norm for matrices
If we put p2 so we have
21
Relative Gain Array (RGA)
The relative gain array (RGA), was introduced by
Bristol (1966).
For a square matrix A
For a non square matrix A
22
Matrix Perturbation
1- Additive Perturbation
2- Multiplicative Perturbation
3- Element by Element Perturbation
23
Additive Perturbation
Theorem 1-3
24
Multiplicative Perturbation
Theorem 1-4
25
Element by element Perturbation
Theorem 1-5
26
Element by element Perturbation
Example 1-3
then the perturbed A is singular or
Write a Comment
User Comments (0)
About PowerShow.com