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Multivariable Control Multivariable Control SystemsSystems
Ali KarimpourAssociate Professor
Ferdowsi University of Mashhad
Lecture 4
References are appeared in the last slide.
Dr. Ali Karimpour Feb 2014
Lecture 4
2
Poles and Zeros in Multivariable Systems
Topics to be covered include:
Multivariable Poles and Zeros
Direction of Poles and Zeros
Smith-McMillan Forms
Matrix Fraction Description (MFD) and Smith-McMillan Forms
Transmission Zero Assignment
Multivariable Poles and Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
3
Multivariable Poles (Through State Space Description )
Definition 4-1: The poles of a system with state-space description are eigenvalues of the matrix A.
The pole polynomial or characteristic polynomial is defined as niAi ,...,2,1,)( ),,( DandCBA
ip
AsIs )(
Thus the system’s poles are the roots of the characteristic polynomial
0)( AsIs
Note that if A does not correspond to a minimal realization then the poles by this definition will include the poles (eigenvalues) corresponding to uncontrollable and/or unobservable states.
Dr. Ali Karimpour Feb 2014
Lecture 4
4
Multivariable Poles (Though Rosenbrock’s System Matrix)
Let
)()()()(
)(PsWsRsQsP
s
Thus the system’s poles are the roots of the following system
0)()( sPs
DuCxyBuAxx
DBCAsI
s)(P
0)( AsIs 0)()( AsIsPs
So they are compatible.
Dr. Ali Karimpour Feb 2014
Lecture 4
5
Multivariable Poles (Thrugh Transfer Function Description )
Theorem 4-1: Finding pole polynomials through transfer function
The pole polynomial (s) corresponding to a minimal realization of a system with transfer function G(s) is the least common denominatorof all non-identically-zero minors of all orders of G(s).A minor of a matrix is the determinant of the square matrix obtained bydeleting certain rows and/or columns of the matrix.
Dr. Ali Karimpour Feb 2014
Lecture 4
6
Example 4-1
)1)(1()1)(1()2)(1()1(0)2)(1(
)1)(2)(1(1)(
2
sssssssss
ssssG
The non-identically-zero minors of order 1 are
21,
21,
11,
)2)(1(1,
11
ssssss
s
The non-identically-zero minor of order 2 are
2)2)(1()1(,
)2)(1(1,
)2)(1(2
sss
ssss
By considering all minors we find their least common denominator
)1()2)(1()( 2 ssss
Multivariable Poles (Thrugh Transfer Function Description )
Dr. Ali Karimpour Feb 2014
Lecture 4
7
Multivariable Poles
Note that if A corresponds to a minimal realization then the poles by both definition will be the same.
DuCxyBuAxx
0)( AsIs
Theorem 4-1: Finding pole polynomials through transfer functionThe pole polynomial (s) corresponding to a minimal realization of a system with transfer function G(s) is the least common denominatorof all non-identically-zero minors of all orders of G(s).A minor of a matrix is the determinant of the square matrix obtained bydeleting certain rows and/or columns of the matrix.
Otherwise if A does not correspond to a minimal realization then the poles by |sI-A|=0 will include the poles (eigenvalues) corresponding to uncontrollable and/or unobservable states.
Dr. Ali Karimpour Feb 2014
Lecture 4
8
Multivariable Poles
Exercise 4-1: Consider the state space realizationDuCxyBuAxx
000000
00111021
100111010321
0.108.06.700.36.18.0002.24.0004.08.2
DCBA
a- Find the poles of the system directly through state space form.
b- Find the transfer function of system ( note that the numerator and the denominator of each element must be prime ).
c- Find the poles of the system through its transfer function.
d- Compare poles from part “a” and “c” and explain the results.
Dr. Ali Karimpour Feb 2014
Lecture 4
9
Multivariable Zeros
• Dynamic response.
• Stability of inverse system.
• Closed loop poles are on the open loop zeros at high gain.
• Blocking the inputs.
• Not affected by feedback.
Importance of multivariable zeros
• Stability analysis by inverse Nyquist
Dr. Ali Karimpour Feb 2014
Lecture 4
10
Multivariable Zeros
Different definition of multivariable zeros
• Element zeros.
• Decoupling zeros (input and output).
• Transmission zeros or blocking zeros.
• System zeros.
• Invariant zeros.
Dr. Ali Karimpour Feb 2014
Lecture 4
11
Multivariable Zeros
Element zeros
Decoupling zeros (input and output)
Transmission zeros or blocking zeros
System zeros
They are not very important in MIMO design.
They are clearly subset of poles.
They are the zeros that block the output in special condition.
Invariant zeros
Invariant zeros=System zeros (In the case of square plant)
zeros i.o.d.zeros o.d.zeros i.d.zeroson Transmissizeros System
Dr. Ali Karimpour Feb 2014
Lecture 4
12
Multivariable Zeros (Through State Space Description )
DC
BAsIsP
yux
sP )(,0
)(DuCxyBuAxx
Laplace transform
The system zeros are then the values of s=z for which the Rosenbrock’s system matrix P(s) (if P(s) is square) loses rank resulting in zero outputfor some nonzero input.
000
,I
IDCBA
M g
Let
0)(
z
zg u
xMzI
Then the zeros are found as non trivial solutions of
This is solved as a generalized eigenvalue problem.
Rosenbrock system matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
13
DCBAsI
sP )(
Rosenbrock system matrix
Input decoupling zeros
System zero
rank. losses )( that of valueThe sPs
rank. losses that of valueThe A BsIs
Output decoupling zeros
rank. losses that of valueThe
C
AsIs
Invariant zeros
)(),(min)(result that s of values thoseare zerosInvariant
CnBnsP
)( of minorsgreatest all of gcdOr sP
Multivariable Zeros (Through State Space Description )
Dr. Ali Karimpour Feb 2014
Lecture 4
14
Multivariable Zeros
Element zeros
Decoupling zeros (input and output)
uxy
uxx
501010011
4000030000200001
1655
)4)(3)(2)(1()4)(3)(2()(
ss
ssssssssg
s= -2 and -4 are output decoupling zero (o.d.z)s= -3 and -4 are input decoupling zero (i.d.z)s= -4 is input-output decoupling zero (i.o.d.z)
Transmission zeros or blocking zeros
s= -6/5 is transmission or blocking zeroSystem zeros and invariant zeros
s= -6/5, s=-2, s=-3 and s=-4 are system zero(invariant zeros)
Dr. Ali Karimpour Feb 2014
Lecture 4
15
Let
Decoupling zeros (input and output)
uxy
uxx
1210112
1000
300020005
s= -2 is output decoupling zero (o.d.z)s= -5 is input decoupling zero (i.d.z)
Transmission zeros or blocking zeros
s= -4 is transmission or blocking zeroInvariant zeros
s= -4, s=-5 are invariant zeros.
Element zeros
34
382)(
ss
sssG
s= -4, -4
System zeross= -4, s=-5, s=-2 are system zeros.
1210112300
1002000005
)(s
ss
sP
Multivariable Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
16
Generally we have:
zeros i.o.d.zeros o.d.zeros i.d.zeroson Transmissizeros System
zeros SystemzerosInvariant zeroson Transmissi
System zerosInvariant zeros
Transmission zeros
Decoupling zeros
Multivariable Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
17zerosmnExactlymCBrankandDzerosCBrankmnmostAtD
zerosDrankmnmostAtD
:)(0)(2:0
)(:0
For square systems with m=p inputs and outputs and n states, limits on the number of transmission zeros are:
Multivariable Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
18
zerosmnExactlymCBrankandDzerosCBrankmnmostAtD
zerosDrankmnmostAtD
:)(0)(2:0
)(:0
Example 4-2: Consider the state space realizationDuCxyBuAxx
0014100
560100010
DCBA
0000010000100001
000
00141560
01000010
II
DCBA
M g
),( gIMeig
4z
10123 zerosofNumber
Multivariable Zeros (Through State Space Description )
Dr. Ali Karimpour Feb 2014
Lecture 4
19
Definition 4-2
zi is a zero (transmission zero) of G(s) if the rank of G(zi) is less than
the normal rank of G(s). The zero polynomial is defined as
)()( 1 ini zssz z
Where nz is the number of finite zeros (transmission zero) of G(s).
Multivariable Zeros (Through Transfer Function Description )
Dr. Ali Karimpour Feb 2014
Lecture 4
20
The zero polynomial z(s) corresponding to a minimal realization of the
system is the greatest common divisor of all the numerators of all
order-r minors of G(s) where r is the normal rank of provided that
these minors have been adjusted in such a way as to have the pole
polynomial as their denominators.)(s
Theorem 4-2
Multivariable Zeros (Through Transfer Function Description )
Dr. Ali Karimpour Feb 2014
Lecture 4
21
Example 4-3
)1)(1()1)(1()2)(1()1(0)2)(1(
)1)(2)(1(1)(
2
sssssssss
ssssG
according to example 4-1 the pole polynomial is:
)1()2)(1()( 2 ssss
The minors of order 2 with as their denominators are)(s
)1()2)(1()1(,
)1()2)(1()2)(1(,
)1()2)(1()2)(1(2
2
2
22
ssss
sssss
sssss
The greatest common divisor of all the numerators of all order-2 minors is
1)( ssz
Multivariable Zeros (Through Transfer Function Description )
Dr. Ali Karimpour Feb 2014
Lecture 4
22
Multivariable Zeros
Exercise 4-3: Consider the state space realizationDuCxyBuAxx
000000
00111021
100111010321
0.108.06.700.36.18.0002.24.0004.08.2
DCBA
a- Find the transmission zeros of the system directly through state space form.
b- Find the transfer function of system ( note that the numerator and the denominator of each element must be prime ).
c- Find the transmission zeros of the system through its transfer function.
Dr. Ali Karimpour Feb 2014
Lecture 4
23
Poles and Zeros in Multivariable Systems
Topics to be covered include:
Multivariable Poles and Zeros
Direction of Poles and Zeros
Smith-McMillan Forms
Matrix Fraction Description (MFD) and Smith-McMillan Forms
Transmission zero assignment
Direction of Poles and Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
24
Directions of Poles and Zeros
Zero directions:
Let G(s) have a zero at s = z, Then G(s) losses rank at s = z and
there will exist nonzero vectors and such that zu zy
0)(0)( zGyuzG Hzz
uz is input zero direction and yz is output zero direction
We usually normalize the direction vectors to have unit length
12zu 1
2zy
Dr. Ali Karimpour Feb 2014
Lecture 4
25
Pole directions:
up is input pole direction and yp is output pole direction
Let G(s) have a pole at s = p. Then G(p) is infinite and we may
somewhat crudely write
)()( pGyupG Hpp
Hp
Hppp pqAqptAt
pH
ppp qBuCty
Directions of Poles and Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
26
Example4-4:
)1(25.4
412
1)(s
ss
sG
It has a zero at z = 4 and a pole at p = -2 . H
GsG
653.0757.0757.0653.0
271.000475.1
588.0809.0809.0588.0
45.442
51)3()( 0
H
GzG
6.08.08.06.0
00050.1
55.083.083.055.0
65.443
61)4()(
zuInput zero direction
zyOutput zero directionNow let the input as:
)()()( susGsy tetu 4
6.08.0
)(
41
6.08.0
)1(25.441
21
ss
ss
2.18.0
21
s
t
t
ee
ty 2
2
2.18.0
)( which does not contain any component of the input signal e4t
Directions of Poles and Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
27
Example4-5:
)1(25.4
412
1)(s
ss
sG
It has a zero at z = 4 and a pole at p = -2 .
)2()( GpG
)3(25.4431)2()(
GpG
pu
Input pole direction
py
Output pole direction
??!!
H
G
6.08.08.06.0
00.0009010
55.083.083.055.0
)001.02(
001.0 examplefor Let
Directions of Poles and Zeros
Dr. Ali Karimpour Feb 2014
Lecture 4
28
One Property of Zero Direction
DCBAsI
sP )(
Rosenbrock system matrix
System zero
rank. losses )( that of valueThe sPs
Thus we have a zero at s=z:
0
z
z
ux
DCBAzI
0
z
z
ux
DuCxyBuAxx
Now show that the output of following system is zero for all t
ztzz eutuxx )(,)0(
0,0)( tallforty
Dr. Ali Karimpour Feb 2014
Lecture 4
29
Minimality
A state space system (A, B, C and D) is minimal if it is a system with the least number of states giving its transfer function.
If there is another system (A1, B1, C1 and D1) with fewer states having the same transfer function, the given system is not minimal.
For SISO systems, a system is minimal if and only if the transfer function numerator polynomial and denominator polynomial have no common roots.
For MV systems one must use the definition of MV zeros.
Theorem4-3 . A system (A, B, C and D) is minimal if and only if it has no input decoupling zeros and no output decoupling zeros.
Dr. Ali Karimpour Feb 2014
Lecture 4
30
Poles and Zeros in Multivariable Systems
Topics to be covered include:
Multivariable Poles and Zeros
Direction of Poles and Zeros
Smith-McMillan Forms
Matrix Fraction Description (MFD) and Smith-McMillan Forms
Transmission zero assignment
Smith-McMillan Forms
Dr. Ali Karimpour Feb 2014
Lecture 4
31
Smith Form of a Polynomial Matrix
)(sSuppose that is a polynomial matrix.
Smith form of is denoted by , and it is a pseudo diagonalin the following form
)(s )(ss
000)(
)(s
s dss
)(,,........)(,)()( Where 21 sssdiags rds
)(si )(1 siis a factor of and1
)(
i
ii s
Dr. Ali Karimpour Feb 2014
Lecture 4
32
)(,,........)(,)()( Where 21 sssdiags rds
)(si )(1 siis a factor of and1
)(
i
ii s
10
1 gcd {all monic minors of degree 1}
2 gcd {all monic minors of degree 2}
r gcd {all monic minors of degree r}
………………………………………………………..
………………………………………………………..
Smith Form of a Polynomial Matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
33
The three elementary operations for a polynomial matrix are used to find Smith form.
• Multiplying a row or column by a non-zero constant;
• Interchanging two rows or two columns; and
• Adding a non-zero polynomial multiple of a row or column to another row or column.
)().....()()()()().......()( 121122 sRsRsRssLsLsLs nns
)()()()( sRssLss
Smith Form of a Polynomial Matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
34
Example 4-6
21)2(2
)2(4)(
s
ss
10 1}1,2,2,1gcd{1 ss
)3)(1(}34gcd{ 22 ssss
1)(0
11
s )3)(1()(1
22 sss
)3)(1(0
01)(
ssss
Exercise 4-4: Derive R(s) and L(s) that convert Π(s)to Πs(s)
Smith Form of a Polynomial Matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
35
Theorem 4-4 (Smith-McMillan form)
)()(
1)( ssd
sG
Let be an m × p matrix transfer function, where
are rational scalar transfer functions, G(s) can be represented by:
)]([)( sgsG ij )(sg ij
Where Π(s) is an m× p polynomial matrix of rank r and d(s) is the least
common multiple of the denominators of all elements of G(s) .
)(~ sGThen, is Smith McMillan form of G(s) and can be derived directly by
000)(
000)(
)(1)(
)(1)(~ sMs
sds
sdsG ds
s
Smith Form of a Polynomial Matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
36
Theorem 4-4 (Smith-McMillan form)
)()(
1)( ssd
sG
000)(
000)(
)(1)(
)(1)(~ sMs
sds
sdsG ds
s
)()(,,........
)()(,
)()()(
2
2
1
1
ss
ss
ssdiagsM
r
r
)()()()(~ sRsGsLsG )(~)(~)(~)( sRsGsLsG
11 )()(~,)()(~ sRsRsLsL
and unimodular are )( and )(~,)(,)(~ matrices The sRsRsLsL
Smith Form of a Polynomial Matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
37
Example 4-7
)2)(1(21
)1(2
)1(1
)2)(1(4
)(
sss
ssssG
)2)(1()(,5.0)2(2
)2(4)(,)(
)(1)(
sssds
sss
sdsG
)3)(1(0
01)(
ssss
230
0)2)(1(
1
)()(
1)(~
ss
ssssd
sG s
Smith Form of a Polynomial Matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
38
Example 4-7
824824
11
)2)(1(1)(
)(1)(
22
22
ssssss
sss
sdsG
824824
11)(
22
22
sssssss
)4(30)4(30
11)(
2
21
sss
00)4(30
11)( 2
2 ss
00)4(30
01)( 2
3 ss
00
)4(001
)()( 24 sss s )()()()()()( 4312 sRsRssLsLss
)(~)(~)(~)(~)()(~)(
1)()(
1)( sRsGsLsRssLsd
ssd
sG s
00
120
0)2)(1(
1
)(~ss
ss
sG
Smith Form of a Polynomial Matrix
Dr. Ali Karimpour Feb 2014
Lecture 4
39
Poles and Zeros in Multivariable Systems
Topics to be covered include:
Multivariable Poles and Zeros
Direction of Poles and Zeros
Smith-McMillan Forms
Matrix Fraction Description (MFD) and Smith-McMillan Forms
Transmission zero assignment
Matrix Fraction Description (MFD) and Smith-McMillan Forms
Dr. Ali Karimpour Feb 2014
Lecture 4
40
Matrix Fraction Description (MFD)
Matrix Fraction Description for Transfer Matrix
)()(
1)(G sNsd
s Suppose G is an pq matrix so
Left Matrix Fraction Description (LMFD) )()()()()(G 11 sNsDsNIsds LLp
)()()()()(G 11 sDsNIsdsNs RRq Right Matrix Fraction Description
(RMFD)
But this forms are not irreducible.
Irreducible RMFD and LMFD can be derived directly through SMM form.
Dr. Ali Karimpour Feb 2014
Lecture 4
41
Matrix Fraction Description&
Smith-McMillan form
Let is a matrix and its the Smith McMillan is )(sG mm )(~ sG
0,...,0,)(,...,)()( 1 ssdiagsN r
1,...,1,)(,...,)()( 1 ssdiagsD r
Let define:
or)()()(~ 1 sDsNsG )()()(~ 1 sNsDsG
We know that
)(~)(~)(~)( sRsGsLsG )(~)()()(~ 1 sRsDsNsL 1)()( sDsN RR 1)()()()(~ sDsRsNsL
)(~)(~)(~)( sRsGsLsG )(~)()()(~ 1 sRsNsDsL )()( 1 sNsD LL )(~)()()( 1 sRsNsLsD
It is easy to see that when a RMFD is irreducible, then
zsssGzs R at rank losses (s)Nor)(N ifonly and if )( of zeroion transmissa is L
(s)Gφ(s)pssDsDsGps
Ddet is G(s) of polynomial pole that themeans This at singular is )(or )( ifonly and if )( of pole a is RL
Dr. Ali Karimpour Feb 2014
Lecture 4
42
Example 4-8
)1(42
)1(2
)2)(1(82
)2)(1(4
)2)(1(1
)2)(1(1
)(22
ss
ss
ssss
ssss
ssss
sG
3011
00
120
0)2)(1(
1
114014001
)(~)(~)(~)(2
2
ss
ss
ssssRsGsLsG
1
2
21
2
2
310
31)1)(2(
2424
01
3011
100)1)(2(
0020
01
114014001
s
sss
sssss
sss
ss
ss
)(sNR )(sDRRMFD:
Matrix Fraction Description&
Smith-McMillan form
Dr. Ali Karimpour Feb 2014
Lecture 4
43
Example 4-8
)1(42
)1(2
)2)(1(82
)2)(1(4
)2)(1(1
)2)(1(1
)(22
ss
ss
ssss
ssss
ssss
sG
3011
00
120
0)2)(1(
1
114014001
)(~)(~)(~)(2
2
ss
ss
ssssRsGsLsG
)(sN L)( sD L
LMFD:
00)2(30
11
1101)4)(1(00)1)(2(
3011
0020
01
10001000)1)(2(
114014001 1
2
1
2
2 ss
ssssss
ssss
sss
Matrix Fraction Description&
Smith-McMillan form
Dr. Ali Karimpour Feb 2014
Lecture 4
44
Poles and Zeros in Multivariable Systems
Topics to be covered include:
Multivariable Poles and Zeros
Direction of Poles and Zeros
Smith-McMillan Forms
Matrix Fraction Description (MFD) and Smith-McMillan Forms
Transmission zero assignment Transmission zero assignment
Dr. Ali Karimpour Feb 2014
Lecture 4
45
Transmission zero assignment
Zeros position depends on the location of sensors and actuators.
Theorem4-5: Transmission zeros cannot be assigned by state feedback.
Proof
DCBAsI
sP )(by defined is System
DDKCBBKAsI
sPf )( :isfeedback with System
IK
IsP
IKI
DCBAsI
DDKCBBKAsI
sPf
0)(
0)(
)()( sPsPf
Theorem4-6: Transmission zeros cannot be assigned by output feedback.
Changing zero position can affect the output response.
Dr. Ali Karimpour Feb 2014
Lecture 4
46
What about following structures?
CxyBuAxx
sG
)(
Theorem4-7: Transmission zeros cannot be assigned by series compensation.
00
0)(
CBDAsIBHGFsI
sPs
uDHwuuGFww
sKˆˆ
)(
)()()( sPsPsPs
DHI
GFsI
CBAsI
I
CBDAsIBHGFsI
sPs
000
0
000
00
00
0)(
Transmission zero assignment
controllersystemnew zzz
Dr. Ali Karimpour Feb 2014
Lecture 4
47
What about following structures?
CxyurBAxx
sG)(
)(
Theorem4-8: Transmission zeros cannot be assigned by dynamic feedback.
00
0)(
CBBDCAsIBH
GCFsIsPf
DyHwuGyFww
sK
)(
)()( sPFsIsPf
000
00
00
0
00
0)(
CBAsI
I
IBDIBHGFsI
CBBDCAsIBH
GCFsIsPf
Transmission zero assignment
systemcontrollernew zpz
Dr. Ali Karimpour Feb 2014
Lecture 4
48
What about following structures?
Transmission zero assignment
Example 4-9: Suppose
11
11
32
11
)(
ss
sssG zeroion transmissa is 1s
Now let:
1110
)(sK
12
12
31
11
)()(
ss
ss
ss
ssKsG
Feed forward controller
zeroion transmissanot is 1s
Dr. Ali Karimpour Feb 2014
Lecture 4
49
Transmission zero assignment
For zero assignment procedure see following papers.
R.V Patel and P. Misra “Transmission zero assignment in linear multivariable systems” ACC/WM5 1992.
A. Khaki Sedigh “Transmission zero assignment for linear Multivariable plans” 10th IASTED International Symposium, 1991.
Dr. Ali Karimpour Feb 2014
Lecture 4
Exercises
50
Exercise 4-5: Consider following system.
xy
uxx
001001
213102131
a) Find the SMM form of the system system.b) Find the pole and zero polynomial of the system.c) Find the RMFD and LMFD of the system.
Exercise 4-6: Consider following transfer function.
2
422
1
)(sss
sGa) Find the SMM form of the system system.b) Find the pole and zero polynomial of the system.c) Find the RMFD and LMFD of the system.
Dr. Ali Karimpour Feb 2014
Lecture 4
Exercises(Continue)
51
Exercise 4-7: Consider following transfer function.
124
23
31
42
42
21
)(
ss
ss
ss
ss
ssG
a) Find the SMM form of the system.b) Find the pole and zero polynomial of the system.c) Find the RMFD and LMFD of the system.
Dr. Ali Karimpour Feb 2014
Lecture 4
52
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