Download - Multivariable Control Systems
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Multivariable Control Multivariable Control SystemsSystems
Ali Karimpour
Assistant Professor
Ferdowsi University of Mashhad
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Ali Karimpour Sep 2009
Chapter 1Chapter 1
Vector Spaces
Norms
Unitary, Primitive and Hermitian Matrices
Positive (Negative) Definite Matrices
Inner Product
Singular Value Decomposition (SVD)
Relative Gain Array (RGA)
Matrix Perturbation
Linear Algebra
Topics to be covered include:
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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.
(R) numbers real of field over thenR
Some important vector spaces are:
(C) numberscomplex of field over thenC
(R) numbers real of field over the[0, interval on the functions Continuous
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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.
RF:.
Norm is a function that maps x to a nonnegative real number
A Norm must satisfy following properties:
0 x,0x Positivity 1
C and F x,xy Homogeneit 2 x
Fy x,,yx inequality Triangle 3 yx
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Ali Karimpour Sep 2009
Chapter 1
Norm of vectors
iiax
1 For p=1 we have 1-norm or sum norm
2/12
2
iiaxFor p=2 we have 2-norm or euclidian norm
ii
ax max For p=∞ we have ∞-norm or max norm
1
1
pax
pp
iip
p-norm is:
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Ali Karimpour Sep 2009
Chapter 1
Norm of vectors
2
1-
1
Let x
2)2,1,1max(x
6211xThen
4)211(x
222
2
1
1x2
1x 1
1x
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Ali Karimpour Sep 2009
Chapter 1
Norm of real functions
(R) numbers real of field over the
[0, interval on the functions continuousConsider
)(sup)( ]1,0[
tftft
1-norm is defined as
2
121
02)()(
dttftf2-norm is defined as
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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: ji
ijsumaA
,
Frobenius norm (extension of 2-norm of vectors) is:2
,
jiijF
aA
Max element norm (extension of max norm of vectors) is: ijji
aA,max
max
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Ali Karimpour Sep 2009
Chapter 1
Matrix norm
A norm of a matrix is called matrix norm if it satisfy
BAAB .
Define the induced-norm of a matrix A as follows:
pxipAxA
p1
max
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
iij
jxiaAxA maxmax
1111
Maximum column sum
If we put p=inf so we have
jij
ixiaAxA maxmax
1Maximum row sum
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Chapter 1
Unitary and Hermitian Matrices
A matrix nnCU is unitary if
A matrix nnCQ is Hermitian if
IUU H
QQ H
For real matrices Hermitian matrix means symmetric matrix.
1- Show that for any matrix V, Hand VVVV H are Hermitian matrices
2- Show that for any matrix V, the eigenvalues of Hand VVVV H
are real nonnegative.
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Chapter 1
Primitive Matrices
A matrix nnRA is nonnegative if whose entries are nonnegative numbers.
A matrix nnRA 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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Chapter 1
Primitive Matrices
primitive. is 1 and 2- seigenvalue with 11
20
A
primitive.not is 2 and 2- seigenvalue with 01
40
A
primitive.not is 1 and 1 seigenvalue with 11
01
A
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Chapter 1
Positive (Negative) Definite Matrices
A matrix nnCQ is positive definite if for any 0, xCx n
Qxx His real and positive
A matrix nnCQ is negative definite if for any 0, xCx n
Qxx His real and negative
A matrix nnCQ is positive semi definite if for any 0, xCx n
Qxx His real and nonnegative
Negative semi definite define similarly
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Chapter 1
Inner Product
yx,
An inner product is a function of two vectors, usually denoted by
CFF :.,.
Inner product is a function that maps x, y to a complex number
An Inner product must satisfy following properties:
xy,yx, :Symmetry 1
zybzxazbyax ,,, :Linearity 2
0 xF, x, positive is xx, :Positivity 3
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Chapter 1
Singular Value Decomposition (SVD)
1002.100
100100A
501.5
551A
1001.100
100100AA
)(1001.10
1010)( 111
AAAA
01.0
00A
55
55)( 1A
?
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Chapter 1
Singular Value Decomposition (SVD)
Theorem 1-1 mlCM : Let . Then there exist mlR and unitary matrices
llCY and mmCU such that
HUYM
00
0S0........21 r
r
S
...00
......
0...0
0...0
2
1
],......,,[],,......,,[ 2121 ml uuuUyyyY
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Ali Karimpour Sep 2009
Chapter 1
Singular Value Decomposition (SVD)
Example
824
143
121
M
H
M
27.055.079.0
53.077.035.0
80.033.050.0
.
000
053.40
0077.9
.
17.034.092.0
51.077.038.0
85.053.004.0
79.0
35.0
50.0
1u 11 77.9
92.0
38.0
04.0
77.9 yMu
55.0
77.0
33.0
2u 22 53.4
34.0
77.0
53.0
53.4 yMu
27.0
53.0
80.0
3u Has no affect on the output or 03 Mu
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Ali Karimpour Sep 2009
Chapter 1
Singular Value Decomposition (SVD)
Theorem 1-1 mlCM : Let . Then there exist mlR and unitary matrices
llCY and mmCU such that
HUYM HMMY of rseigenvecto from derived becan
MMU H of rseigenvecto from derived becan
HH MMMM or of seigenvalue nonzero of roots are ,...,, r21
3- Derive the SVD of
10
12A
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Ali Karimpour Sep 2009
Chapter 1
Matrix norm for matrices
If we put p=1 so we have
iij
jxiaAxA maxmax
1111
Maximum column sum
If we put p=inf so we have
jij
ixiaAxA maxmax
1Maximum row sum
pxipAxA
p1
max
If we put p=2 so we have
)()()(maxmax max1
2
2
121222
AAAx
AxAxA
xxi
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Chapter 1
Relative Gain Array (RGA)
The relative gain array (RGA), was introduced by Bristol (1966).
For a square matrix A
TAAAARGA )()()( 1
For a non square matrix A
T)A ()()( AAARGA†
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Chapter 1
Matrix Perturbation
1- Additive Perturbation
2- Multiplicative Perturbation
3- Element by Element Perturbation
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Chapter 1
Additive Perturbation
nmCA has full column rank (n). Then Suppose
Theorem 1-3
)()()(|min2
AAnArank nC nm
)(A
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Chapter 1
Multiplicative Perturbation
nnCA . Then Suppose
Theorem 1-4
)(
1)(|min
2 AnAIrank
nnC
)(A
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Chapter 1
Element by element Perturbation
)1
1(ij
ijijp aa
nnCA ij : Suppose is non-singular and suppose
is the ijth element of the RGA of A.
The matrix A will be singular if ijth element of A perturbed by
Theorem 1-5
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Chapter 1
Element by element Perturbation
Example 1-3
1002.100
100100A
500501
501500)(A
11a 002.1)1
1(11
Now according to theorem 1-5 if multiplied by
then the perturbed A is singular or
1002.100
1002.100
1002.100
100002.1*100PA