multivariable control systems

Multivariable Control Multivariable Control SystemsSystems

Ali Karimpour

Assistant Professor

Ferdowsi University of Mashhad

2

Ali Karimpour Sep 2009

Chapter 2Chapter 2

Introduction to Multivariable Control Topics to be covered include:

Multivariable Connections Multivariable Poles Multivariable Zeros Directions of Poles and Zeros Smith Form of a Polynomial Matrix Smith-McMillan Forms Matrix Fraction Description (MFD) Scaling Performance Specification Trade-offs in Frequency Domain Bandwidth and Crossover Frequency

3


Chapter 2

Multivariable Connections

• Cascade (series) interconnection of transfer matrices

• Parallel interconnection of transfer matrices

)(1 sG )(2 sG

)(su )(sy

)(1 sG

)(2 sG

)(su

)(sy

)()()()()()( 12 susGsusGsGsy

)()()())()(()( 21 susGsusGsGsy

)()()( 12 sGsGsG )()( 21 sGsG

Generally

)()()( 21 sGsGsG

4


Chapter 2


• Feedback interconnection of transfer matrices

)(1 sG )(2 sG

)(sy)(su

)()()()()( 12 sysusGsGsy

)()()()()())()(()( 121

12 susGsusGsGsGsGIsy

)()())()(()( 121

12 sGsGsGsGIsG

A useful relation in multivariable is push-through rule

12122

112 ))()()(()())()(( sGsGIsGsGsGsGI

Exercise 1: Proof the push-through rule

5


Chapter 2


MIMO rule: To derive the output of a system, start from the

output and write down the blocks as you meet them when

moving backward (against the signal flow) towards the input.

If you exit from a feedback loop then include a term

or according to the feedback sign where L is the

transfer function around that loop (evaluated against the

signal flow starting at the point of exit from the loop).

Parallel branches should be treated independently and their

contributions added together.

1 LI

1 LI

6


Chapter 2


Example 2-1: Derive the transfer function of the system shown in figure

211

221211 )( PKPIKPPz

7


Chapter 2

Multivariable Poles ( State Space Description )

Definition 2-1: 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,)( ),,,( DCBA

ip

)det()( AsIs

Thus the system’s poles are the roots of the characteristic polynomial

0)det()( 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.

8


Chapter 2

Multivariable Poles ( Transfer Function Description )

Theorem 2-1

The pole polynomial corresponding to a minimal realization of

a system with transfer function G(s) is the least common denominator

of all non-identically-zero minors of all orders of G(s).

A minor of a matrix is the determinant of the square matrix obtained by

deleting certain rows and/or columns of the matrix.

)(s

9


Chapter 2

Multivariable Poles ( Transfer Function Description )

Example 2-4

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

)1(0)2)(1(

)1)(2)(1(

1)(

2

ssssss

sss

ssssG

The non-identically-zero minors of order 1 are

2

1,

2

1,

1

1,

)2)(1(

1,

1

1

sssss

s

s

The non-identically-zero minor of order 2 are

2)2)(1(

)1(,

)2)(1(

1,

)2)(1(

2

ss

s

ssss

By considering all minors we find their least common denominator

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

10


Chapter 2

Multivariable Poles

Exercise 2: Consider the state space realizationDuCxy

BuAxx

000

000

0011

1021

100

111

010

321

0.108.06.7

00.36.18.0

002.24.0

004.08.2

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.

11


Chapter 2

Multivariable Zeros ( State Space Description )

DC

BAsIsP

yu

xsP )(,

0)(

DuCxy

BuAxx

Laplace transform

The zeros are then the values of s=z for which the polynomial system

matrix P(s) loses rank resulting in zero output for some nonzero input.

00

0,

II

DC

BAM 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.

12


Chapter 2


zerosmnExactlymCBrankandD

zerosCBrankmnmostAtD

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:

13


Chapter 2


zerosmnExactlymCBrankandD

zerosCBrankmnmostAtD

zerosDrankmnmostAtD

:)(0

)(2:0

)(:0

Example 2-5: Consider the state space realizationDuCxy

BuAxx

0014

1

0

0

560

100

010

DCBA

0000

0000

0010

0001

00

0

0014

1560

0100

0010

II

DC

BAM g

),( gIMeig

4z

10123 zerosofNumber

14


Chapter 2

Multivariable Zeros ( Transfer Function Description )

Definition 2-2

is a zero of G(s) if the rank of is less than the normal rank

of G(s). The zero polynomial is defined as .

Where is the number of finite zeros of G(s).

iz )( izG

)()( 1 ini zssz z

zn

15


Chapter 2


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 2-2

16


Chapter 2


Example 2-9

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

)1(0)2)(1(

)1)(2)(1(

1)(

2

ssssss

sss

ssssG

according to example 2-4 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(22

2

22

sss

s

sss

ss

sss

ss

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

1)( ssz

17


Chapter 2

Multivariable Poles

Exercise 3: Consider the state space realizationDuCxy

BuAxx

000

000

0011

1021

100

111

010

321

0.108.06.7

00.36.18.0

002.24.0

004.08.2

DCBA

a- Find the 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 zeros of the system through its transfer function.

18


Chapter 2

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

is input zero direction and is output zero directionzu zy

We usually normalize the direction vectors to have unit length

12zu 1

2zy

19


Chapter 2


Pole directions:

is input pole direction and is output pole directionpupy

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

20


Chapter 2


Example:

)1(25.4

41

2

1)(

s

s

ssG

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

H

GsG

653.0757.0

757.0653.0

271.00

0475.1

588.0809.0

809.0588.0

45.4

42

5

1)3()( 0

H

GzG

6.08.0

8.06.0

00

050.1

55.083.0

83.055.0

65.4

43

6

1)4()(

zu

Input zero direction

zy

Output zero direction

21


Chapter 2


Example:

)1(25.4

41

2

1)(

s

s

ssG

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

)2()( GpG

)3(25.4

431)2()(

GpG

pu

Input pole direction

py

Output pole direction

??!!

H

G

6.08.0

8.06.0

00.00

09010

55.083.0

83.055.0)001.02(

001.0 examplefor Let

22


Chapter 2

Smith Form of a Polynomial Matrix

)(sSuppose that is a polynomial matrix.

Smith form of is denoted by , and it is a pseudo diagonal

in the following form

)(s )(ss

00

0)()(

ss ds

s

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

)(si )(1 siis a factor of and 1

)(

i

ii s

23


Chapter 2


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

)(si )(1 siis a factor of and 1

)(

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}

………………………………………………………..

………………………………………………………..

24


Chapter 2


The three elementary operations for a polynomial matrix are used to find Smith form.

• Multiplying a row or column by a constant;

• Interchanging two rows or two columns; and

• Adding a polynomial multiple of a row or column to another row or column.

)().....()()()()().......()( 121122 sRsRsRssLsLsLs nns

)()()()( sRssLss

25


Chapter 2


Example 2-11

2

1)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: Derive R(s) and L(s) that convert to )(s )(ss

26


Chapter 2

Smith-McMillan Forms

Theorem 2-3 (Smith-McMillan form)

)()(

1)( s

sDsG

G

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 DG(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

00

0)(

00

0)(

)(

1)(

)(

1)(

~ sMs

sDs

sDsG ds

Gs

G

27


Chapter 2


Theorem 2-3 (Smith-McMillan form)

)()(

1)( s

sDsG

G

00

0)(

00

0)(

)(

1)(

)(

1)(

~ sMs

sDs

sDsG ds

Gs

G

)(

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

)(

)(,

)(

)()(

2

2

1

1

s

s

s

s

s

sdiagsM

r

r

)()()()(~

sRsGsLsG )(~

)(~

)(~

)( sRsGsLsG

11 )()(~

,)()(~ sRsRsLsL

and unimodular are )( and )(~

,)(,)(~

matrices The sRsRsLsL

28


Chapter 2


Example 2-13

)2)(1(2

1

)1(

2)1(

1

)2)(1(

4

)(

sss

ssssG

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

)2(4)(,)(

)(

1)(

sssD

s

sss

sDsG G

G

)3)(1(0

01)(

ssss

2

30

0)2)(1(

1

)(

)()(

~

s

s

ss

sD

ssG

G

s

29


Chapter 2

Matrix Fraction Description (MFD)

function transferSISOan is 65

1)(Let

2

ss

ssg

12 651)( can write We

ssssg

polynomial polynomial

This is a Right Matrix Fraction Description (RMFD)

165)( writealsocan We12

ssssg

polynomial polynomial

This is a Left Matrix Fraction Description (LMFD)

30


Chapter 2


A model structure that is related to the Smith-McMillan form is matrix fraction description (MFD).

• Right matrix fraction description (RMFD)

• Left matrix fraction description (LMFD)

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

31


Chapter 2


or)()()(~ 1 sDsNsG )()()(

~ 1 sNsDsG

We know that

)(~

)(~

)(~

)( sRsGsLsG )(~

)()()(~ 1 sRsDsNsL 1)()( sGsG DN 1)()()()(

~ sDsRsNsL

This is known as a right matrix fraction description (RMFD) where

)()()(,)()(~

)( sDsRsGsNsLsG DN

32


Chapter 2


or)()()(~ 1 sDsNsG )()()(

~ 1 sNsDsG

We know that

)(~

)(~

)(~

)( sRsGsLsG )(~

)()()(~ 1 sRsNsDsL )()( 1 sGsG ND

)(~

)()()( 1 sRsNsLsD

This is known as a left matrix fraction description (LMFD) where

)(~

)()(,)()()( sRsNsGsLsDsG ND

33


Chapter 2


We can show that the MFD is not unique, because, for any nonsingular

mm )(smatrix we can write G(s) as:

111 )()()()()()()()()( ssGssGsGsssGsG DNDN

Where is said to be a right common factor. When the only right

common factors of and is unimodular matrix, then, we

Say that the RMFD is irreducible.

)(s

)(sGN )(sGD

)(),( sGsG DN

It is easy to see that when a RMFD is irreducible, then

zssGsGzs N at rank losses )( ifonly and if )( of zero a is

(s)Gφ(s)

zssGsGps

D

D

det is G(s) of polynomial pole that themeans This

at singular is )( ifonly and if )( of pole a is

34


Chapter 2


Example 2-16

)1(

42

)1(

2

)2)(1(

82

)2)(1(

4

)2)(1(

1

)2)(1(

1

)(22

s

s

s

s

ss

ss

ss

ss

ssss

sG

30

11

00

1

20

0)2)(1(

1

114

014

001

)(~

)(~

)(~

)(2

2

s

s

ss

s

sssRsGsLsG

1

2

2

1

2

2

3

10

3

1)1)(2(

24

24

01

30

11

10

0)1)(2(

00

20

01

114

014

001

s

sss

ss

ssss

sss

s

ss

)(sGN )(sGDRMFD:

35


Chapter 2


Example 2-16

)1(

42

)1(

2

)2)(1(

82

)2)(1(

4

)2)(1(

1

)2)(1(

1

)(22

s

s

s

s

ss

ss

ss

ss

ssss

sG

30

11

00

1

20

0)2)(1(

1

114

014

001

)(~

)(~

)(~

)(2

2

s

s

ss

s

sssRsGsLsG

)(sGN)(sGD

LMFD:

00

)2(30

11

11

01)4)(1(

00)1)(2(

30

11

00

20

01

100

010

00)1)(2(

114

014

0011

2

1

2

2 s

s

ssss

ss

ss

ss

s

ss

36


Chapter 2

Scaling

ryedGuGy d ˆˆˆ;ˆˆˆˆˆ

rDreDeyDyuDudDd eeeud ˆ,ˆ,ˆ,ˆ,ˆ 11111

rDyDeDdDGuDGyD eeeddee ;

ddedue DGDGDGDG ˆ,ˆ 11 ryedGGuy d ;

37


Chapter 2

Performance Specification

• Nominal stability NS: The system is stable with no model uncertainty.

• Nominal Performance NP: The system satisfies the performance specifications with no model uncertainty.

• Robust stability RS: The system is stable for all perturbed plants about the nominal model up to the worst case model uncertainty.

• Robust performance RP: The system satisfies the performance specifications for all perturbed plants about the nominal model up to the worst case model uncertainty.

38


Chapter 2

Performance Specification

• Time Domain Performance

• Frequency Domain Performance

39


Chapter 2

Time Domain Performance

The objective of this section is to discuss ways of evaluating closed loop performance.

Although closed loop stability is an important issue,

the real objective of control is to improve performance, that is, to make the output y(t) behave in a more desirable manner.

Actually, the possibility of inducing instability is one of the disadvantages of feedback control which has to be traded off

against performance improvement.

40


Chapter 2


Step response of a system

Rise time, tr , the time it takes for the output to first reach 90% of its final value, which is usually required to be small.

Settling time, ts , the time after which the output remains within5% ( or 2%) of its final value, which is usually required to be small.

Overshoot, P.O, the peak value divided by the final value, whichshould typically be less than 20% or less.

41


Chapter 2



Decay ratio, the ratio of the second and first peaks, which should typically be 0.3 or less.

Steady state offset, ess, the difference between the final value and the desired final value, which is usually required to be small.

42


Chapter 2


Excess variation, the total variation (TV) divided by the overall change at steady state, which should be as close to 1 as possible.

The total variation is the total movement of the output as shown.

0/ variationExcess vTVvTVi

i


43


Chapter 2


• IAE : Integral absolute error deIAE

0

)(

• ISE : Integral squared error deISE 2

0)(

• ITSE : Integral time weighted squared error deITSE 2

0)(

• ITAE : Integral time weighted absolute error deITSE )(0

44


Chapter 2

Frequency Domain Performance

Let L(s) denote the loop transfer function of a system which is closed-loop stable under negative feedback.

Bode plot of )( jL

)(

1

180jLGM

180)( 180 jL

1)( cjL

180)( cjLPM

cPM /max

45


Chapter 2


Let L(s) denote the loop transfer function of a system which is closed-loop stable under negative feedback.

)(

1

180jLGM

180)( 180 jL

1)( cjL

180)( cjLPM

cPM /max

Nyquist plot of )( jL

46


Chapter 2


Stability margins are measures of how close a stable closed-loop system is to instability.

From the above arguments we see that the GM and PM provide stabilitymargins for gain and delay uncertainty.

More generally, to maintain closed-loop stability, the Nyquist stability condition tells us that the number of encirclements of the critical point -1 by must not change. )( jL

Thus the actual closest distance to -1 is a measure of stability

47


Chapter 2


One degree-of-freedom configuration

nGKIGKdGGKIrGKIGKsy d111 )()()()(

Complementary sensitivity function )(sT

Sensitivity function )(sS

The maximum peaks of the sensitivity and complementary sensitivity functions are defined as

)(max)(max

jTMjSM Ts

48


Chapter 2


11 )()()()( jLIjKjGIjS

)(max

jSM s

Thus one may also view Ms as a robustness measure.

sM

1

49


Chapter 2


There is a close relationship between MS and MT and the GM and PM.

][1

2

1sin2;

11 rad

MMPM

M

MGM

sss

s

][1

2

1sin2;

11 1 rad

MMPM

MGM

TTT

For example, with MS = 2 we are guaranteed GM > 2 and PM >29˚.

For example, with MT = 2 we are guaranteed GM > 1.5 and PM >29˚.

50


Chapter 2

One degree-of-freedom configuration

nGKIGKdGGKIrGKIGKsy d111 )()()()(

Complementary sensitivity function )(sT

Sensitivity function )(sS

Trade-offs in Frequency Domain

)(sT

IsSsT )()(

TndSGSrrye d KSndKSGKSru d

51


Chapter 2

Trade-offs in Frequency Domain

IsSsT )()(

TndSGSrrye d KSndKSGKSru d

1)()( sLIsS

• Performance, good disturbance rejection LITS or or 0

• Performance, good command following LITS or or 0

• Mitigation of measurement noise on output 0or or 0 LIST

• Small magnitude of input signals 0Tor 0or 0 LK

• Physical controller must be strictly proper 0Tor 0or 0 LK

• Nominal stability (stable plant) small be L

• Stabilization of unstable plant ITL or large be

52


Chapter 2

Bandwidth and Crossover Frequency

Definition 2-3

below. from db 3- crossesfirst )S(j

wherefrequency theis , ,bandwidth The B

Definition 2-4

above. from db 3- crosses )T(jat which

frequency highest theis , ,bandwidth The BT

Definition 2-5

above. from db 0 crosses )L(j where

frequency theis , ,frequency crossover gain The c

100

101

102

103

-60

-40

-20

0

20

40

60Singular Values

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

L

T

S

53


Chapter 2


Specifically, for systems with PM < 90˚ (most practical systems) we have

BTcB

In conclusion ωB ( which is defined in terms of S ) and also ωc ( in terms of L ) are good indicators of closed-loop performance,

while ωBT ( in terms of T ) may be misleading in some cases.

100

101

102

103

-60

-40

-20

0

20

40

60Singular Values

Frequency (rad/sec)

Sin

gula

r V

alue

s (d

B)

L

T

S

54


Chapter 2


Example 2-17 : Comparison of and as indicators of performance. BTB

1,1.0;1

1;

)2(Let

z

szs

zsT

zss

zsL

036.0B054.0c 1BT 1.0 zero RHPan is There z

55


Chapter 2


Example 2-17 : Comparison of and as indicators of performance. BTB

1,1.0;1

1;

)2(Let

z

szs

zsT

zss

zsL

036.0B054.0c 1BT 1.0 zero RHPan is There z

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