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Multivariable Control Multivariable Control SystemsSystems
Ali KarimpourAssociate Professor
Ferdowsi University of Mashhad
Lecture 2
References are appeared in the last slide.
Dr. Ali Karimpour Feb 2014
Lecture 2
2
Vector Spaces, Norms
Unitary, Primitive, Hermitian and positive(negative) definite Matrices
Inner Product
Singular Value Decomposition (SVD)
Relative Gain Array (RGA)
Matrix Perturbation
Linear Algebra
Topics to be covered include:
Vector Spaces, Norms
Dr. Ali Karimpour Feb 2014
Lecture 2
3
Vector Spaces
A set of vectors and a field of scalars with some propertiesis 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 theon functions Continuous
Dr. Ali Karimpour Feb 2014
Lecture 2
4
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:
0for x0 and 0 x,0x Positivity 1 x
C and F x,xy Homogeneit 2 x
Fy x,,yx inequality Triangle 3 yx
Dr. Ali Karimpour Feb 2014
Lecture 2
5
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
iiax max
For p=∞ we have ∞-norm or max norm
1
1
pax
pp
iip
p-norm is:
Dr. Ali Karimpour Feb 2014
Lecture 2
6
Norm of vectors
21-1
Let x
2)2,1,1max(x
6211xThen
4)211(x 222
2
1
1x2
1x 1
1x
Dr. Ali Karimpour Feb 2014
Lecture 2
7
Norm of real functions
(R) numbers real of field over the[0, interval on the functions continuousConsider
)(sup)( ]1,0[
tftft
∞-norm is defined as
21
21
02)()(
dttftf2-norm is defined as
Any idea for 1-norm ?
Dr. Ali Karimpour Feb 2014
Lecture 2
8
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
,ji
ijFaA
Max element norm (extension of max norm of vectors) is: ijjiaA
,maxmax
Dr. Ali Karimpour Feb 2014
Lecture 2
9
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
p 1max
Any induced-norm of a matrix, is a matrix norm
Dr. Ali Karimpour Feb 2014
Lecture 2
10
Matrix norm for matrices
If we put p=1 so we have
i
ijjxiaAxA maxmax
1111
Maximum column sum
If we put p=inf so we have
jijixiaAxA maxmax
1Maximum row sum
Dr. Ali Karimpour Feb 2014
Lecture 2
11
Vector Spaces, Norms
Unitary, Primitive, Hermitian and positive(negative) definite Matrices
Inner Product
Singular Value Decomposition (SVD)
Relative Gain Array (RGA)
Matrix Perturbation
Linear Algebra
Dr. Ali Karimpour Feb 2014
Lecture 2
12
Unitary and Hermitian Matrices
A matrix nnCU is unitary if
A matrix nnCQ is Hermitian if
IUU H
QQH
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.
Dr. Ali Karimpour Feb 2014
Lecture 2
13
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 2.1A primitive matrix is a square nonnegative matrix where some power(positive integer) of which is positive.
Dr. Ali Karimpour Feb 2014
Lecture 2
14
Primitive Matrices
primitive. is 1- and 2 seigenvalue with 1120
1
A
primitive.not is 2 and 2- seigenvalue with 0140
3
A
primitive.not is 1 and 1 seigenvalue with 1101
4
A
primitive.not is 0 and 4 seigenvalue with 4000
2
A
Dr. Ali Karimpour Feb 2014
Lecture 2
15
Positive (Negative) Definite Matrices
A matrix nnCQ is positive definite if for any 0, xCx n
QxxH is real and positive
A matrix nnCQ is negative definite if for any 0, xCx n
QxxH is real and negative
A matrix nnCQ is positive semi definite if for any 0, xCx n
QxxH is real and nonnegative
Negative semi definite define similarly
Dr. Ali Karimpour Feb 2014
Lecture 2
16
Vector Spaces, Norms
Unitary, Primitive, Hermitian and positive(negative) definite Matrices
Inner Product
Singular Value Decomposition (SVD)
Relative Gain Array (RGA)
Matrix Perturbation
Linear Algebra
Dr. Ali Karimpour Feb 2014
Lecture 2
17
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
Dr. Ali Karimpour Feb 2014
Lecture 2
18
Vector Spaces, Norms
Unitary, Primitive, Hermitian and positive(negative) definite Matrices
Inner Product
Singular Value Decomposition (SVD)
Relative Gain Array (RGA)
Matrix Perturbation
Linear Algebra
Dr. Ali Karimpour Feb 2014
Lecture 2
19
Singular Value Decomposition (SVD)
1002.100100100
A
501.5551A
1001.100100100
AA
)(1001.101010
)( 111
AAAA
01.000
A
5555
)( 1A
?
Dr. Ali Karimpour Feb 2014
Lecture 2
20
Singular Value Decomposition (SVD)
Theorem 2-1 mlCM : Let . Then there exist mlR and unitary matrices
llCY and mmCU such that
HUYM
000S
0........21 r
r
S
...00......0...00...0
2
1
],......,,[],,......,,[ 2121 ml uuuUyyyY
Dr. Ali Karimpour Feb 2014
Lecture 2
21
Singular Value Decomposition (SVD)
Example 2-1
824143121
M
H
M
27.055.079.053.077.035.080.033.050.0
.000053.400077.9
.17.034.092.0
51.077.038.085.053.004.0
79.035.050.0
1u 11 77.992.038.004.0
77.9 yMu
55.077.033.0
2u 22 53.434.077.053.0
53.4 yMu
27.053.080.0
3u Has no affect on the output or 03 Mu
Dr. Ali Karimpour Feb 2014
Lecture 2
22
Singular Value Decomposition (SVD)
Theorem 2-1 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
Exercise 2-1 Derive the SVD of
10
12A
Dr. Ali Karimpour Feb 2014
Lecture 2
23
Matrix norm for matrices
If we put p=1 so we have
i
ijjxiaAxA maxmax
1111
Maximum column sum
If we put p=inf so we have
jijixiaAxA maxmax
1Maximum row sum
pxipAxA
p 1max
If we put p=2 so we have
)()()(maxmax max12
2
121222
AAAxAx
AxAxxi
Dr. Ali Karimpour Feb 2014
Lecture 2
24
Vector Spaces, Norms
Unitary, Primitive, Hermitian and positive(negative) definite Matrices
Inner Product
Singular Value Decomposition (SVD)
Relative Gain Array (RGA)
Matrix Perturbation
Linear Algebra
Dr. Ali Karimpour Feb 2014
Lecture 2
25
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†
Dr. Ali Karimpour Feb 2014
Lecture 2
26
Vector Spaces, Norms
Unitary, Primitive, Hermitian and positive(negative) definite Matrices
Inner Product
Singular Value Decomposition (SVD)
Relative Gain Array (RGA)
Matrix Perturbation
Linear Algebra
Dr. Ali Karimpour Feb 2014
Lecture 2
27
Matrix Perturbation
1- Additive Perturbation
2- Multiplicative Perturbation
3- Element by Element Perturbation
Dr. Ali Karimpour Feb 2014
Lecture 2
28
Additive Perturbation
nmCA has full column rank (n). Then Suppose
Theorem 2-2
)()()(|min2
AAnArank nC nm
)(A
Dr. Ali Karimpour Feb 2014
Lecture 2
29
Multiplicative Perturbation
nnCA . Then Suppose
Theorem 2-3
)(
1)(|min2 A
nAIranknnC
)(A
Dr. Ali Karimpour Feb 2014
Lecture 2
30
Element by element Perturbation
)11(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 2-4
Dr. Ali Karimpour Feb 2014
Lecture 2
31
Element by element Perturbation
Example 2-2
1002.100100100
A
500501501500
)(A
11a 002.1)11(11
Now according to theorem 1-5 if multiplied by
then the perturbed A is singular or
1002.1001002.100
1002.100100002.1*100
PA
Dr. Ali Karimpour Feb 2014
Lecture 2
32
2-2- Show that any induced norm is a matrix norm.
Exercises
2-3 The spectral radius of a matrix is: iiA max)(
iwhere is the eigenvalue of A. Show that the spectral radius is not a norm.
2-4 Suppose A is Hermitian. Find the exact relation between the eigenvalues and singular values of A. Does this hold if A is not Hermitian?
2-5 Verify that if Q is Hermitian then its eigenvalues are real.
2-6 Show that Frobnius norm can be derived by )( AAtr H
.
Dr. Ali Karimpour Feb 2014
Lecture 2
33
Exercises (Continue).
2-7 Show that if rank(A)=1, then, 2AA
F
.
2-8 Suppose
by use of SVD:a) Find the null space of A.b) Find the range space of A.
572321743
A
75.22x 2
Axc) If what is the maximum and minimum of
.
2-9 Show that a matrix M is positive definite, if and only if the Hermitian matrix
2)( HMM
is positive definite.(Hint: use ) 2
)(2
)( HH MMMMM
Dr. Ali Karimpour Feb 2014
Lecture 2
34
Exercises (Continue).
..
1120
A2-10 Show that is primitive.
2-11 Find a non primitive matrix such that its spectral radius is a simple root of the characteristic polynomial and its spectral radius is strictly greater than the modulus of any other eigenvalues.
Dr. Ali Karimpour Feb 2014
Lecture 2
35
Evaluation and References
• Control Configuration Selection in Multivariable Plants, A. Khaki-Sedigh, B. Moaveni, Springer Verlag, 2009.
References
• Multivariable Feedback Control, S.Skogestad, I. Postlethwaite, Wiley,2005.
• Multivariable Feedback Design, J M Maciejowski, Wesley,1989.
• http://saba.kntu.ac.ir/eecd/khakisedigh/Courses/mv/
Web References
• http://www.um.ac.ir/~karimpor
• تحلیل و طراحی سیستم هاي چند متغیره، دکتر علی خاکی صدیق
Evaluation
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