multivariable control systems - ali...

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:



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


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


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:


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


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 ?


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


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


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


Lecture 2

11



Inner Product



Matrix Perturbation

Linear Algebra


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.


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.


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


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


Lecture 2

16



Inner Product



Matrix Perturbation

Linear Algebra


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


Lecture 2

18



Inner Product



Matrix Perturbation

Linear Algebra


Lecture 2

19


1002.100100100

A

501.5551A

1001.100100100

AA

)(1001.101010

)( 111

AAAA

01.000

A

5555

)( 1A

?


Lecture 2

20


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


Lecture 2

21


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


Lecture 2

22


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


Lecture 2

23

Matrix norm for matrices


i

ijjxiaAxA maxmax

1111

Maximum column sum

If we put p=inf so we have

jijixiaAxA maxmax

1Maximum row sum

pxipAxA

p 1max


)()()(maxmax max12

2

121222

AAAxAx

AxAxxi


Lecture 2

24



Inner Product



Matrix Perturbation

Linear Algebra


Lecture 2

25


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†


Lecture 2

26



Inner Product



Matrix Perturbation

Linear Algebra


Lecture 2

27

Matrix Perturbation

1- Additive Perturbation

2- Multiplicative Perturbation

3- Element by Element Perturbation


Lecture 2

28

Additive Perturbation

nmCA has full column rank (n). Then Suppose

Theorem 2-2

)()()(|min2

AAnArank nC nm

)(A


Lecture 2

29

Multiplicative Perturbation

nnCA . Then Suppose

Theorem 2-3

)(

1)(|min2 A

nAIranknnC

)(A


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


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


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

.


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


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.


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

multivariable control systems - ali...

Documents