Multivariable Control SystemsMultivariable Zeros Element zeros Decoupling zeros (input and output)...

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

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

Lecture 4

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

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.

Lecture 4

Multivariable Poles (Though Rosenbrock’s System Matrix)

)()()()(

)(PsWsRsQsP

Thus the system’s poles are the roots of the following system

0)()( sPs

DuCxyBuAxx

DBCAsI

0)( AsIs 0)()( AsIsPs

So they are compatible.

Lecture 4

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.

Lecture 4

Example 4-1

)1)(1()1)(1()2)(1()1(0)2)(1(

)1)(2)(1(1)(

sssssssss

The non-identically-zero minors of order 1 are

)2)(1(1,

ssssss

The non-identically-zero minor of order 2 are

2)2)(1()1(,

)2)(1(1,

)2)(1(2

By considering all minors we find their least common denominator

)1()2)(1()( 2 ssss

Multivariable Poles (Thrugh Transfer Function Description )

Lecture 4

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.

Lecture 4

Multivariable Poles

Exercise 4-1: Consider the state space realizationDuCxyBuAxx

000000

00111021

100111010321

0.108.06.700.36.18.0002.24.0004.08.2

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.

Lecture 4

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

Lecture 4

Multivariable Zeros

Different definition of multivariable zeros

• Element zeros.

• Decoupling zeros (input and output).

• Transmission zeros or blocking zeros.

• System zeros.

• Invariant zeros.

Lecture 4

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

Lecture 4

Multivariable Zeros (Through State Space Description )

BAsIsP

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.

Then the zeros are found as non trivial solutions of

This is solved as a generalized eigenvalue problem.

Rosenbrock system matrix

Lecture 4

DCBAsI

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

Invariant zeros

)(),(min)(result that s of values thoseare zerosInvariant

CnBnsP

)( of minorsgreatest all of gcdOr sP

Lecture 4

Multivariable Zeros

Element zeros

501010011

4000030000200001

)4)(3)(2)(1()4)(3)(2()(

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)

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)

Lecture 4

1210112

300020005

s= -2 is output decoupling zero (o.d.z)s= -5 is input decoupling zero (i.d.z)

s= -4 is transmission or blocking zeroInvariant zeros

s= -4, s=-5 are invariant zeros.

Element zeros

s= -4, -4

System zeross= -4, s=-5, s=-2 are system zeros.

1210112300

1002000005

Multivariable Zeros

Lecture 4

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

Lecture 4

17zerosmnExactlymCBrankandDzerosCBrankmnmostAtD

zerosDrankmnmostAtD

:)(0)(2:0

For square systems with m=p inputs and outputs and n states, limits on the number of transmission zeros are:

Multivariable Zeros

Lecture 4

zerosmnExactlymCBrankandDzerosCBrankmnmostAtD

zerosDrankmnmostAtD

:)(0)(2:0

Example 4-2: Consider the state space realizationDuCxyBuAxx

0014100

560100010

0000010000100001

00141560

01000010

),( gIMeig

10123 zerosofNumber

Lecture 4

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 )

Lecture 4

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

Lecture 4

Example 4-3

)1)(1()1)(1()2)(1()1(0)2)(1(

)1)(2)(1(1)(

sssssssss

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

The greatest common divisor of all the numerators of all order-2 minors is

1)( ssz

Lecture 4

Multivariable Zeros

Exercise 4-3: Consider the state space realizationDuCxyBuAxx

000000

00111021

100111010321

0.108.06.700.36.18.0002.24.0004.08.2

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.

Lecture 4

Transmission zero assignment

Lecture 4

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

Lecture 4

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

Hppp pqAqptAt

ppp qBuCty

Lecture 4

Example4-4:

)1(25.4

It has a zero at z = 4 and a pole at p = -2 . H

653.0757.0757.0653.0

271.000475.1

588.0809.0809.0588.0

45.442

51)3()( 0

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

)1(25.441

2.18.0

)( which does not contain any component of the input signal e4t

Lecture 4

Example4-5:

)1(25.4

It has a zero at z = 4 and a pole at p = -2 .

)2()( GpG

)3(25.4431)2()(

Input pole direction

Output pole direction

6.08.08.06.0

00.0009010

55.083.083.055.0

)001.02(

001.0 examplefor Let

Lecture 4

One Property of Zero Direction

DCBAsI

System zero

rank. losses )( that of valueThe sPs

Thus we have a zero at s=z:

DCBAzI

DuCxyBuAxx

Now show that the output of following system is zero for all t

ztzz eutuxx )(,)0(

0,0)( tallforty

Lecture 4

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.

Lecture 4

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

)(,,........)(,)()( Where 21 sssdiags rds

)(si )(1 siis a factor of and1

Lecture 4

)(,,........)(,)()( Where 21 sssdiags rds

)(si )(1 siis a factor of and1

1 gcd {all monic minors of degree 1}

2 gcd {all monic minors of degree 2}

r gcd {all monic minors of degree r}

………………………………………………………..

Lecture 4

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

Lecture 4

Example 4-6

21)2(2

)2(4)(

10 1}1,2,2,1gcd{1 ss

)3)(1(}34gcd{ 22 ssss

s )3)(1()(1

22 sss

)3)(1(0

Exercise 4-4: Derive R(s) and L(s) that convert Π(s)to Πs(s)

Lecture 4

Theorem 4-4 (Smith-McMillan form)

1)( ssd

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

)(1)(~ sMs

sdsG ds

Lecture 4

Theorem 4-4 (Smith-McMillan form)

1)( ssd

)(1)(~ sMs

sdsG ds

)()(,,........

)()()(

ssdiagsM

)()()()(~ sRsGsLsG )(~)(~)(~)( sRsGsLsG

11 )()(~,)()(~ sRsRsLsL

and unimodular are )( and )(~,)(,)(~ matrices The sRsRsLsL

Lecture 4

Example 4-7

)2)(1(21

)2)(1(4

)2)(1()(,5.0)2(2

)2(4)(,)(

)3)(1(0

0)2)(1(

Lecture 4

Example 4-7

824824

)2)(1(1)(

ssssss

824824

sssssss

)4(30)4(30

00)4(30

11)( 2

00)4(30

01)( 2

)4(001

)()( 24 sss s )()()()()()( 4312 sRsRssLsLss

)(~)(~)(~)(~)()(~)(

1)( sRsGsLsRssLsd

0)2)(1(

Lecture 4

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.

Lecture 4

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

Lecture 4

Example 4-8

)2)(1(82

)2)(1(4

)2)(1(1

0)2)(1(

114014001

)(~)(~)(~)(2

ssssRsGsLsG

31)1)(2(

100)1)(2(

114014001

)(sNR )(sDRRMFD:

Smith-McMillan form

Lecture 4

Example 4-8

)2)(1(82

)2)(1(4

)2)(1(1

0)2)(1(

114014001

)(~)(~)(~)(2

ssssRsGsLsG

)(sN L)( sD L

00)2(30

1101)4)(1(00)1)(2(

10001000)1)(2(

114014001 1

ssssss

Smith-McMillan form

Lecture 4

Transmission zero assignment Transmission zero assignment

Lecture 4

Zeros position depends on the location of sensors and actuators.

Theorem4-5: Transmission zeros cannot be assigned by state feedback.

DCBAsI

sP )(by defined is System

DDKCBBKAsI

sPf )( :isfeedback with System

DCBAsI

DDKCBBKAsI

)()( sPsPf

Theorem4-6: Transmission zeros cannot be assigned by output feedback.

Changing zero position can affect the output response.

Lecture 4

What about following structures?

CxyBuAxx

Theorem4-7: Transmission zeros cannot be assigned by series compensation.

CBDAsIBHGFsI

uDHwuuGFww

sKˆˆ

)()()( sPsPsPs

CBDAsIBHGFsI

controllersystemnew zzz

Lecture 4

CxyurBAxx

Theorem4-8: Transmission zeros cannot be assigned by dynamic feedback.

CBBDCAsIBH

GCFsIsPf

DyHwuGyFww

)()( sPFsIsPf

IBDIBHGFsI

CBBDCAsIBH

GCFsIsPf

systemcontrollernew zpz

Lecture 4

Example 4-9: Suppose

sssG zeroion transmissa is 1s

Now let:

Feed forward controller

zeroion transmissanot is 1s

Lecture 4

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.

Lecture 4

Exercises

Exercise 4-5: Consider following system.

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.

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.

Lecture 4

Exercises(Continue)

Exercise 4-7: Consider following transfer function.

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.

Lecture 4

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

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