Transcript No Slide Title

Multivariable Control
Systems
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
Chapter 1
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
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Ali Karimpour Sep 2009
Chapter 1
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:
Rn over thefield of real numbers(R)
C n over thefield of complex numbers(C)
Continuous functions on the interval[0, over thefield of real numbers(R)
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Ali Karimpour Sep 2009
Chapter 1
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
. : F  R
A Norm must satisfy following properties:
1 Positivity x  0 ,  x  0
2  Homogeneity x   x ,  x  F and   C
3  Triangleinequality x  y  x  y ,  x, y  F
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Ali Karimpour Sep 2009
Chapter 1
Norm of vectors
p-norm is:
x
p

p
   ai 
 i

1
p
For p=1 we have 1-norm or sum norm
p 1


x 1    ai 
 i

1/ 2
For p=2 we have 2-norm or euclidian norm
For p=∞ we have ∞-norm or max norm

2
x 2    ai 
 i

x

 max  ai
i

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Ali Karimpour Sep 2009
Chapter 1
Norm of vectors
x 1  (1  1  2)  4
1
Let x - 1
 2 
T hen
x 2  12  12  2 2  6
x
x 2 1

x
 max(1,1,2)  2

1
x 1 1
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Ali Karimpour Sep 2009
Chapter 1
Norm of real functions
Consider continuous functions on the interval[0,
over thefield of real numbers(R)
1-norm is defined as
f (t )   sup f (t )
t[ 0,1]
2-norm is defined as
1
2

f (t ) 2    f (t ) dt 
 0

1
2
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Ali Karimpour Sep 2009
Chapter 1
Norm of matrices
We can extend norm of vectors to matrices
Sum matrix norm (extension of 1-norm of vectors) is:
A sum   aij
i, j
Frobenius norm (extension of 2-norm of vectors) is:
A
F

a
2
ij
i, j
aij
Max element norm (extension of max norm of vectors) is: A max  max
i, j
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Ali Karimpour Sep 2009
Chapter 1
Matrix norm
A norm of a matrix is called matrix norm if it satisfy
AB  A . B
Define the induced-norm of a matrix A as follows:
A ip  max Ax
x p 1
p
Any induced-norm of a matrix A is a matrix norm
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Ali Karimpour Sep 2009
Chapter 1
Matrix norm for matrices
If we put p=1 so we have
A i1  max Ax 1  max aij
x 1 1
j
Maximum column sum
i
If we put p=inf so we have
A i  max Ax   max aij
x

1
i
Maximum row sum
j
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Ali Karimpour Sep 2009
Chapter 1
Unitary and Hermitian Matrices
A matrix U  C nn is unitary if
U HU  I
A matrix Q  C nn is Hermitian if
QH  Q
For real matrices Hermitian matrix means symmetric matrix.
1- Show that for any matrix V, V HV and VV H are Hermitian matrices
2- Show that for any matrix V, the eigenvalues of V HV and VV H
are real nonnegative.
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Ali Karimpour Sep 2009
Chapter 1
Primitive Matrices
A matrix A  R nn is nonnegative if whose entries are nonnegative numbers.
A matrix A  R nn is positive if all of whose entries are strictly positive numbers.
Definition 2.1
A primitive matrix is a square nonnegative matrix some power (positive integer) of
which is positive.
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Ali Karimpour Sep 2009
Chapter 1
Primitive Matrices
0 2
A

1
1


0 4
A

1
0


1 0
A

1
1


witheigenvalues - 2 and1 is primitive.
witheigenvalues - 2 and 2 is not primitive.
witheigenvalues 1 and1 is not primitive.
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Ali Karimpour Sep 2009
Chapter 1
Positive (Negative) Definite Matrices
A matrix Q  C nn is positive definite if for any x  C n , x  0
x H Qx
is real and positive
A matrix Q  C nn is negative definite if for any x  C n , x  0
x H Qx
is real and negative
A matrix Q  C nn is positive semi definite if for any x  C n , x  0
H
x Qx
is real and nonnegative
Negative semi definite define similarly
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Ali Karimpour Sep 2009
Chapter 1
Inner Product
An inner product is a function of two vectors, usually denoted by
 x, y 
Inner product is a function that maps x, y to a complex number
 .,.  : F  F  C
An Inner product must satisfy following properties:
1  Symmetry:  x, y  y,x 
2  Linearity:  ax  by, z  a  x, z  b  y, z 
3  Positivity:  x, x  is positive,  x  F, x  0
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Ali Karimpour Sep 2009
Chapter 1
Singular Value Decomposition (SVD)
100
A
100.2
 5
A 
5.01
100
100
100
A  A  
100.1
1
100
100
5
 5
 0 0
A  

0
.
1
0


?
 10
( A  A)  
10.01
1
10 
1
1

A


(
A
)

 10
 5 5 
( A )  

5

5


1
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Ali Karimpour Sep 2009
Chapter 1
Singular Value Decomposition (SVD)
Theorem 1-1 : Let
Y  C ll
and
M  C l m
U  C mm
. Then there exist
  R l m
and unitary matrices
such that
M  YU H
S 0 


0
0


 1 0
0 
2
S
.
.

0 0
0
... 0 
... . 

...  r 
...
1   2  ........  r  0
Y  [ y1 , y2 ,......,yl ], U  [u1 , u 2 ,......,u m ]
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Ali Karimpour Sep 2009
Chapter 1
Singular Value Decomposition (SVD)
Example
1 2  1
M  3 4 1 
4 2 8 
0  0.50  0.33  0.80
0.04  0.53  0.85 9.77 0
M  0.38  0.77 0.51  .  0 4.53 0  . 0.35  0.77 0.53
0.92 0.34  0.17  0
0
0 0.79 0.55
0.27 
0.50
u1  0.35
0.79
0.04
Mu1  9.770.38  9.77 y1
0.92
 0.80
u3   0.53 
 0.27 
 0.33
u 2   0.77
 0.55 
H
 0.53
Mu2  4.53 0.77  4.53y2
 0.34 
Has no affect on the output or
Mu3  0
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Ali Karimpour Sep 2009
Chapter 1
Singular Value Decomposition (SVD)
Theorem 1-1 : Let
Y  C ll
and
M  C l m .
U  C mm
Then there exist
  R l m
and unitary matrices
such that
M  YU H
Y can be derived from eigenvectors of M M H
U can be derived from eigenvectors of M H M
1 , 2 , ..., r are rootsof nonzero eigenvalues of M H M or M M H
3- Derive the SVD of
2 1 
A

0  1
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Ali Karimpour Sep 2009
Chapter 1
Matrix norm for matrices
A ip  max Ax
x p 1
p
If we put p=1 so we have
A i1  max Ax 1  max aij
x 1 1
j
Maximum column sum
i
If we put p=inf so we have
A i  max Ax   max aij
x
1

i
Maximum row sum
j
If we put p=2 so we have
A i 2  max Ax 2  max
x 2 1
x 2 1
Ax
x
2
  1 ( A)   max ( A)   ( A)
2
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Ali Karimpour Sep 2009
Chapter 1
Relative Gain Array (RGA)
The relative gain array (RGA), was introduced by Bristol (1966).
For a square matrix A
RGA( A)  ( A)  A  ( A1 )T
For a non square matrix A
†
RGA( A)  ( A)  A  ( A )T
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Ali Karimpour Sep 2009
Chapter 1
Matrix Perturbation
1- Additive Perturbation
2- Multiplicative Perturbation
3- Element by Element Perturbation
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Ali Karimpour Sep 2009
Chapter 1
Additive Perturbation
Theorem 1-3
Suppose
A  C mn
has full column rank (n). Then


min  2 | rank ( A  )  n   n ( A)   ( A)
C m n

 ( A) 

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Ali Karimpour Sep 2009
Chapter 1
Multiplicative Perturbation
Theorem 1-4
Suppose
A  C nn . Then


1
min
 2 | rank( I  A)  n 
n n
C
 ( A)
 (A) 
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Ali Karimpour Sep 2009
Chapter 1
Element by element Perturbation
Theorem 1-5 : Suppose A  C nn is non-singular and suppose ij
is the ijth element of the RGA of A.
The matrix A will be singular if ijth element of A perturbed by
aijp  aij (1 
1
ij
)
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Ali Karimpour Sep 2009
Chapter 1
Element by element Perturbation
Example 1-3
100
A
100.2
100
100
Now according to theorem 1-5 if
 500 501 
(A)  

 501  500
a11 multiplied by
(1 
1
11
)  1.002
then the perturbed A is singular or
100*1.002 100 100.2 100
AP  



100
.
2
100
100
.
2
100

 

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Ali Karimpour Sep 2009