Multivariable Control Systemskarimpor.profcms.um.ac.ir/.../multivariable9-decoupling.pdfDr. Ali Karimpour Feb 2017 Lecture 9 2 Topics to be covered include: • Decoupling • Decoupling

Dr. Ali Karimpour Feb 2017

Lecture 9

Multivariable Control

Systems

Ali Karimpour

Associate Professor

Ferdowsi University of Mashhad

Lecture 9

References are appeared in the last slide.


Lecture 9

2

Topics to be covered include:

• Decoupling

• Decoupling by State Feedback

• Diagonal controller (decentralized control)

• Decoupling

• Decoupling by Transfer Matrix (Nyquist-array methods)

Multivariable Control System Design(Decoupling, Diagonal controller and Nyquist-array method)


Lecture 9

3

Introduction

DuCxy

BuAxx

DBAsICsG 1)()(

)()()()()(

)()()()()(

2221212

2121111

susgsusgsy

susgsusgsy

We see that every input controls more than one output and that every

output is controlled by more than one input.

Because of this phenomenon,which is called interaction, it is generally

very difficult to control a multivariable system.

Interaction


Lecture 9

4

Definition 9-1

A multivariable system is said to be decoupled if its transfer-function matrix is diagonal

and nonsingular.

A conceptually simple approach to multivariable control is given by a two-steps

procedure in which

1. We first design a compensator to deal with the interactions in G(s) and

2. Then design a diagonal controller using methods similar to those for SISO systems.

)()()( sWsGsG ss

)()()( sKsWsK ss)(sKs

Decoupling

Decoupling


Lecture 9

5


)()()( sWsGsG ss

Decoupling

Decoupling

2. Then design a diagonal controller using methods similar to those for SISO systems.


Lecture 9

6

Decoupling by State Feedback

110

2

1

5.11

5.1

110

2

)(

ss

sssG

Example 9-1 Derive a decoupling Transfer matrix for following system.

11

75.05.71

75.05.71

)(

s

ss

s

sWs

.....0

0.....)()( sWsG s

Let K=diag(0.01+0.01/s, 0.01+0.01/s)


Lecture 9

7

Decoupling

• Dynamic decoupling

• Steady-state decoupling

• Approximate decoupling at frequency ω0

s.frequencie allat diagonal is )(sGs


)()()( sWsGsG ss Decoupling

)()( choosecan we with exampleFor 1 sGsWIG ss

(s)l(s)GsKsWK(s)IslsK -

sss

1)()( have we)()(by Then

It usually refers to an inverse-based controller.

diagonal. is )0(sG

This may be obtained by selecting a constant pre compensator )0(1 GWs

.at possible as diagonal as is )(0

sGs

This is usually obtained by choosing a constant pre compensator1

0

GWs

)( ofion approximat real a is 00 jGG s for selection good a is frequency 0BW


Lecture 9

8

Decoupling

The idea of using a decoupling controller is appealing, but there are several difficulties.

a. We cannot in general choose Gs freely. For example, Ws(s) must not cancel any

RHP-zeros and RHP poles in G(s)

b. As we might expect, decoupling may be very sensitive to modeling errors and

uncertainties.

c. The requirement of decoupling may not be desirable for disturbance rejection.

One popular design method, which essentially yields a decoupling controller, is the

internal model control (IMC) approach (Morari and Zafiriou).

Another common strategy, which avoids most of the problems just mentioned, is to

use partial (one-way) decoupling where Gs(s) is upper or lower triangular.


Lecture 9

9

• Decoupling






Lecture 9

10


Decoupling of a control system in state space representation.

DuCxy

BuAxx

Let DBAsICsG 1)()( Suppose

diagonalsGsG 1)()( zeros RHP no is thereand 0|D| If

11111111 )()()( DBDCBDAsICDDBAsICsG

Otherwise (|D|=0):

)()()( tTrtKytu Static output feedback

Dynamic output feedback

)()()( tTrtKxtu Static state feedback


Lecture 9

11

Cxy

BuAxx

Let

)()()( Suppose 1 trtFxEtu

Cxy

rBExFBEAx

11 )(have Then we

The transfer function matrix is 111 )()( BEFBEAsICsG

We shall derive in the following the condition on G(s) under which the system can be

decoupled by state feedback.


Decoupling of a control system in state space representation.


Lecture 9

12

Theorem 9-1 A system represented by

with the transfer function matrix G(s) can be decoupled by state feedback of the form

Cxy

BuAxx

)()()( 1 trtFxEtu

if and only if the constant matrix E is nonsingular.

)(

0

0

lim

.

.

1

2

1

sG

s

s

E

E

E

Epd

d

s

p

Proof: See “Linear system theory and design” Chi-Tsong Chen

md

d

new

s

s

sG

0

0

)(

1

Furthermore the new system is in the form:

pd

p

d

d

AC

AC

AC

F

.

.

2

1

2

1



Lecture 9

13

Example 9-1 Use state feedback to decouple the following system.

xyuxx

110

001

00

10

01

6116

100

010

Solution: Transfer function of the system is

65

6

65

6

6116

6

6116

116

)()(

22

2323

2

1

ss

s

ss

sss

s

sss

ss

BAsICsG

The differences in degree of the first row of G(s) are 1 and 2, hence d1=1 and

]01[6116

6

6116

116lim

2323

2

1

sss

s

sss

sssE

s

The differences in degree of the second row of G(s) are 2 and 1, hence d2=1 and

]10[65

6

65

6lim

222

ss

s

sssE

s



Lecture 9

14

Now E is unitary matrix and clearly nonsingular so decoupling by state feedback is

possible and

10

01E

Solution (continue):

5116

010

2

1

2

1

d

d

AC

ACF

)()(

5116

010

10

01)()()( 1 trtxtrtFxEtu

The decoupled system is

xCxy

rxrBExFBEAx

110

001

00

10

01

6116

6116

000

)( 11

Exercise 9-1: Derive the corresponding decoupled transfer function matrix.



Lecture 9

15


Example 9-2 Use state feedback to decouple the following system. The system is related

to a continuous chemical process with two reactor connected in series [Luyben ,1997]

2 0 1 0.0025 1 0,

2 4 0 0.0025 0 1x x u y x

1

1 0.00252 2

2 0.0025( 2)( 4) ( 2)

( ) ( ) ( ) s s

s s s

G s c sI A B D G s

So E is non-singular

1 0.0025lim [ ] [1 0.0025]1

2 2min(1,1) 11

min(2,1) 12 2 0.0025lim [ ] [0 0.0025]2

( 2)( 4) 2

1 0.00252 2

2 0.0025( 2)( 4) ( 2)

( )

E ss ssd

dE s

s s ss

s s

s s s

G s

1 0.0025( ) 2

0 0.0025E rank E


Lecture 9

16


1

2

1

1 1

1

22

1 0 2 0

2 4[0 1]

d

d

C A C A AF

AC AC A

1 1( )x A BE F x BE r

y Cx

1 1

2 0 1 0.0025 1 0.0025 2 0 1 0.0025 1 0.0025( )

2 4 0 0.0025 0 0.0025 2 4 0 0.0025 0 0.0025

1 0

0 1

x x r

y x

0 0 1 0

0 0 0 1 0 0( ) , ( ) ( ) 0

0 01 0

0 1

x x r

G s rank A rank

y x


Lecture 9

17


Now the MIMO system is decoupled and G(s) is diagonal, we can design two SISO

controller for the diagonalized G(s), using state feedback again we have :

1 1 1

1 10 01 0 1 0( ) ( )

0 1 1 0 1 10 0

s sG s C sI A BE F BE

s s

1 1 1( ) ( )G s C sI A BE F BE


Lecture 9

18


System model with state feedback decoupler


Lecture 9

19


Diagonalized system model with state feedback controller plus a gain to reduce Ess


Lecture 9

20


0 5 10 15 20 25 30-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Am

plitu

de (

mol*

A/ft3

)

Step responsse for closed loop system (Controller designed for decoupled system)

r1

y1

r2

y2

Applying the designed controller to the system result in the following response to specified

reference input r1,r2


Lecture 9

21

Property of Decoupling by State Feedback

1- All poles of decoupled are on origin.

3- No transmission zero in decoupled system.

4- Transmission zero of the system are deleted .

5- Unstable transmission zero is the main limitation of method.

2- Decoupled system is:

ndd

decouple ssdiagsG

...,,)( 1


Lecture 9

22

• Decoupling






Lecture 9

23

Diagonal controller (decentralized control)

Another simple approach to multivariable controller design is to use a diagonal or

block diagonal controller K(s). This is often referred to as decentralized control.

Clearly, this works well if G(s) is close to diagonal, because then the plant to be

controlled is essentially a collection of independent sub plants, and each element in

K(s) may be designed independently.

However, if off diagonal elements in G(s) are large, then the performance with

decentralized diagonal control may be poor because no attempt is made to counteract

the interactions.


Lecture 9

24

The design of decentralized control systems involves some steps:

3- The design (tuning) of each controller ki(s)


1- How many control loops is necessary?

loop-assignment problem or input-output pairing

2- The choice of pairings (control configuration selection)


Lecture 9

25

Input-Output Pairing

Definition of RGA (Relative Gain Array)

Physical Meaning of RGA: Let

TGGGRGAG )()(

ijijij hg /

2221212

2121111

ugugy

ugugy

jkuj

ijiij

k

u

y uyg

,0

or 0inputsother if and between relation

222121

2121111

0 ugug

ugugy

1

22

212 u

g

gu

ikyj

ijiij

k

u

y uyh

,0

or 0outputsother if and between relation

Relative gain?

1111

22

21

12111)( uhu

g

gggy


Lecture 9

26


Let

1

1)( TGGG

2221212

2121111

ugugy

ugugy

λ=1 Open loop and closed loop gains are the same,

so interactions has no effect.

λ=0 g11=0 so u1 has no effect on y1.

0<λ Closing second loop, no sign change the gain between y1 and u1.

λ<0 Closing second loop leads to changing the sign of the gain between

y1 and u1.(Very Bad)

1_ To avoid instability caused by interactions in the crossover region one should

prefer pairings for which the RGA matrix in this frequency range is close to identity.

2_ To avoid instability caused by interactions at low frequencies one should avoid

pairings with negative steady state RGA elements.

In this section we provide two useful rules for pairing inputs and outputs.


Lecture 9

27


RGA property:

1- It is independent of input and output scaling.

2- Its rows and columns sum to 1.

3- The RGA is identity matrix if G is upper or lower triangular.

4- Plant with large RGA elements are ill conditioned.

5- Suppose G(s) has no zeros or poles at s=0. If λij() and λ(0) exist

and have different signs then one of the following must be true.

* G(s) has an RHP zeros. * Gij(s) has an RHP zeros.

* gij(s) has an RHP zeros.

6- If gijgij(1-1/λij) then the perturbed system is singular.

7- Changing two columns/rows of G leads to same changes to its RGA


Lecture 9

28


Example 9-3 Select suitable pairing for the

following blending system. (ω is output

flow and x is the composition and defined as

percent of of ωA to total flow)

Solution:

BA

A

BA

ww

wx

www

B

A

w

w

w

x

w

xx

w

0

0

0

0111

RGA of the system is

00

00

1

1

xx

xx

If x0=0.1

1.09.0

9.01.0AB wxww &

If x0=0.9

9.01.0

1.09.0BA wxww &


Lecture 9

29


Example 9-4 Select suitable pairing for the following plant

8.14.01.18

7.04.85.15

4.16.52.10

)0(G

Solution: RGA of the system is

98.107.09.0

43.037.094.0

41.145.196.0

)0(


Lecture 9

30


Combination of SVD, C.N. and RGA can help in this matter.

1- How many control loops is necessary?

2- The choice of pairings (control configuration selection)


Lecture 9

31


3

2

1

3

2

1

3

2

1

)0(

020.095.090.0

008.095.052.0

006.09.048.0

u

u

u

G

u

u

u

y

y

y

Example 9-5 Determine the preferred multiloop control strategy for a process with

the following steady-state gain matrix, which has been scaled by dividing the process

variables by their maximum values.

Solution:

01.037.065.0

56.079.036.0

45.016.071.0

99.002.001.0

01.005.099.0

02.099.005.0

01.000

014.10

0062.1

01.083.056.0

68.041.060.0

73.038.057.0

)0(G

233211 && uyuyuy

132231 && uyuyuy

162..

NC


Lecture 9

32


99.002.001.0

01.005.099.0

02.099.005.0

01.000

014.10

0062.1

01.083.056.0

68.041.060.0

73.038.057.0

)0(G 162..

NC

Determine three control loop is not suitable so:

55.0133..,

55.072..,

38184..,

,

1132

1131

1121

21

NCuu

NCuu

NCuu

andyy

46.1139..,

64.069..,

36.051.1..,

,

1132

1131

1121

31

NCuu

NCuu

NCuu

andyy

71.068..,

25.3338..,

37.045.1..,

,

1132

1131

1121

32

NCuu

NCuu

NCuu

andyy

233211 && uyuyuy 132231 && uyuyuy

1321 & uyuy

1322 & uyuy


Lecture 9

33

The RGA based techniques have many important advantages, such as very simple in

calculation as it only uses process steady-state gain matrix and scaling independent.

Moreover, using steady-state gain alone may result in incorrect interaction measures and

consequently loop pairing decisions, since no dynamic information of the process is

taken into consideration.

Many improved approaches, RGA-like, have been proposed and described in all

process control textbooks, for defining different measures of dynamic loop

interactions.

[1] D.Q. Mayne, “The design of linear multivariable systems,” Automatica, vol. 9, no. 2, pp.201–207, Mar. 1973.

Relative Omega Array (ROmA),

[2] ARGA Loop Pairing Criteria for Multivariable Systems

A. Balestrino, E. Crisostomi, A. Landi, and A. Menicagli ,2008

Absolute Relative Gain Array (ARGA),

Relative Normalized Gain Array (RNGA),

[3] RNGA based control system configuration for multivariable processesMao-Jun He, Wen-Jian Cai *, Wei Ni, Li-Hua XieJournal of Process Control 19 (2009) 1036–1042



Lecture 9

Next example, for which the RGA based loop pairing criterion gives an

inaccurate interaction assessment, are employed to demonstrate the

effectiveness of the proposed interaction measure and loop pairing criterion.

Example 9-6:

Consider the two-input two-output process:

26


RGA=Diagonal pairing

RNGA =Off-diagonal pairing

To illustrate the validity of above results, decentralized controllers

for both diagonal and off-diagonal pairings are designed respectively based on

the IMC-PID controller tuning rules.

To evaluate the output control performance, we consider a unit step set-point

Change of all control loops one-by-one and the integral square error (ISE) is

used to evaluate the control performance.


Lecture 9

35


The simulation results and ISE values are given in Figure. The results show

that the off-diagonal pairing gives better overall control system performance.

off-diagonal

diagonal


Lecture 9

36

• Decoupling






Lecture 9

37

Nyquist-array methods

Nyquist-array methods ( Compensator structure )

We shall again assume that the plant's transfer function is square.

Suppose that a compensator is rational, invertible, and has all its poles and zeros in the

left half-plane (including the origin).

Theorem 9-2

Let K(s) be square, rational and invertible, and have all its poles and zeros in the open

left half-plane. Then

abccba KsKsKsKsKsKKsK )()()( and )()()(

aa KK and are permutation matrices, in other words, reorder the outputs or inputs,

)( and )( sKsK bb are products of elementary matrices

)( and )( sKsK cc are diagonal matrices, with rational, non-zero, and with poles and

zeros in the open left half-plane only.


Lecture 9

38


This theorem implies that compensator design can be split into two stages.

• In the first stage are used to make the return ratio diagonally dominant.)( and sKK ba

• The second stage begins when dominance has been achieved and consists of

designing a set of separate SISO compensators one for each loop.

At this stage no attention is paid to the remaining interactions in the system,

Except that Gershgorin bands of the return ratio replace SISO Nyquist loci.


Lecture 9

39


Design can be pursued using either

• Direct Nyquist Arrays (DNA)

)()()()(0 sKsKKsGsQ cba

• Inverse Nyquist Arrays (INA)

Since Kc(s) post-multiplies the other transfer functions, and is diagonal, hence

column dominance of the G(s)KaKb(s) is not destroyed by Kc(s).

)()()()( 11111

0 sGKsKsKsQ abc

Since pre-multiplies the other transfer functions, and is diagonal,

hence row dominance of the is not destroyed by .

)(1 sKc

)()( 111 sGKsK ab

)(1 sKc

Therefore, when working with INA it is usual to try to achieve row dominance.

Therefore, when working with DNA it is usual to try to achieve column dominance.

)(sGaK)(sKb)(sKc

)(1 sG )(1

sKb

)(

1sKc

1

aK

There is the Ostrowski bands, which allows the designer to take into account the

effects of interactions between loops.


Lecture 9

40


The direct Nyquist-array (DNA) method

The DNA has several advantages.

1- The designed compensator does not need to be inverted, and one consequence of this

is that any structure imposed by the designer, such as setting certain elements to zero,

is retained.

It also gives the designer more freedom in the choice of compensator, since its inverse

does not need to be realizable

2- The plant need not be square, since it need not have an inverse.

3- The inverse compensator designed by the INA method has right half-plane zeros,

particularly if it is designed semi-automatically by one of the methods described in

the next section.

However, there is no tool, such as the Ostrowski bands, which allows the designer

to take into account the effects of interactions between loops.


Lecture 9

41


Achieving diagonal dominance

• Cut and try

• Perron-Frobenius theory

• Pseudo-diagonalization

It is based on some straightforward transformation.

Perron-Frobenius theory allows us to check whether a plant can be

made diagonally dominant by input and output scaling

A way of automatically generating compensators with a more general

structure.


Lecture 9

42


Achieving diagonal dominance ( Cut and try )

It is sometimes possible to examine the display of a Nyquist array and observe that

some straightforward transformation will achieve diagonal dominance.

2323

223

1

23)(

s

s

s

ss

s

s

s

sG

Let

Its inverse is

ss

sssG

2

1)(ˆ

This is clearly neither row dominant nor column dominant anywhere on the Nyquist

contour.

ss

sssGKa

2

1

01

10)(ˆˆ

01

10ˆ

aK

1

2

ss

ss

This is clearly row dominant and column dominant anywhere on the Nyquist contour.


Lecture 9

43



ss

sssGKa

2

1

01

10)(ˆˆ

01

10ˆ

aK

1

2

ss

ss

So that the compensated plant is represented by

23

2

23

2323

1

)(

s

s

s

ss

s

s

s

KsG a

01

10aK

Physically, this corresponds to nothing more than a re-ordering of the inputs (or a re-

assignment of inputs to outputs.

In this artificial example both the direct and the inverse array have been made equally

dominant, and the compensation required in each case is the same.

This is not usually true. It is quite possible for a particular compensator to make the

direct array dominant, but not the inverse array, and vice versa.


Lecture 9

44



The ‘elementary matrices’ are supposed to represent simple transformations devised

by the designer.

In practice, it is rarely possible to make much progress by relying on being able to find

such transformations by ad hoc means.

10

21)(

ssKb

An alternative strategy is to try to diagonalize a system at one frequency, and hope

that the effect will be sufficiently beneficial over a wide range of frequencies.

If the system has no poles at the origin, then K=G-1(s) is a realizable (because constant)

compensator.

Fortunately, we already have an algorithm (ALIGN algorithm) for performing the

required approximation on other frequencies.


Lecture 9

45



A plant has the inverse transfer function

2

115

3101

2

)(ˆ2

2

s

sss

ss

ss

sG

5.01

32)0(G

15.0

5.125.0)0(ˆ GKb

It is not diagonally dominant at low frequencies, since

The compensator

IGK b )0(ˆˆ

sasssGKb

44

18

14

29

)(ˆˆ

gives

which is obviously diagonally dominant, and it gives column, but not row, dominance at

high frequencies, since


Lecture 9

46


Achieving diagonal dominance ( Perron-Frobenius theory )

Perron-Frobenius theory allows us to check whether a plant can be made diagonally

dominant by input and output scaling.

Theorem 9-3 (Mees, 1981):

If G is square and primitive, then there exist a diagonal matrix S such that

is diagonally dominate, if and only if

1~ SGSG

(I) 21

diagp GG

If (I) satisfied then X which achieve diagonal dominance is

nsssdiagS ,,..., 21

Where λp Perron-Frobenius eigenvalue and (s1, s2, … ,sn)T is left Perron-Frobenius

eigenvector of .1

diagGG

)(sG1S S


Lecture 9

47



Output scaling is physically impossible, since the meaningful plant outputs

(which are variables such as velocity, or thickness of steel strip) cannot be affected by

mathematical operations.

But we can use

But we must be wary of falling into the trap of believing that this return ratio tells us

anything about interaction at the plant output.

The output variables may be interacting with each other to a considerable extent, and

this interaction may be being hidden by the measurement scaling S.


Lecture 9

48



Fortunately, the Perron-Frobenius theory gives useful results, even if only pre-

compensation (input scaling) is allowed.

Theorem 9-4

If G is square and primitive, then there exist a diagonal matrix K(s) such that

is diagonally row dominate, if and only if

)()(~

sKsGG

(I) 21 GGabs diagp

If (I) satisfied then K(s) which achieve diagonal dominance is

)(,,...)(,)()( 21 skskskdiagsK n

Where λp Perron-Frobenius eigenvalue and (k1(s), k2(s), … ,kn(s))T is right Perron-

Frobenius eigenvector of .1 GGabs diag

)(sG)(sK


Lecture 9

49



Everything else remains the same, except that if a dynamic compensator is used then

the elements of compensator must be chosen to have realizable inverses.

A drawback of using diagonal compensators

• Row dominance when using the DNA method

• Column dominance when using the INA method

This is exactly the opposite of what we would like, since further diagonal compensation,

for the purpose of ‘loop shaping’, may destroy the dominance which has been achieved.


Lecture 9

50


Example 9-7 Consider the transfer function

12

2

10010

1

15

1

)5)(1(

4

)(

2 sss

s

sss

s

sG

10-2

10-1

100

101

102

0

0.5

1

1.5

2

2.5

3

3.5

4

Frequency (rad/sec)

Gain

(dB

)

The Perron-Frobenius eigenvalue of

)(ˆ)(ˆ 1 jGjGabs diag

is:

Its value is smaller than 4 dB ( i.e. λp<2 )

at all frequencies, so it is possible to obtain

column dominance by using a diagonal

compensator.



Lecture 9

51


The second element of the Perron-Frobenius left eigenvector, when the first element

is fixed at 1 is:

10-2

10-1

100

101

102

-10

-5

0

5

10

15

20

25

30

Frequency (rad/sec)

Gain

(dB

)

31.0

898.4438.0)(ˆ

2

s

ssk

This matches the variation of the

eigenvector very well.

The inverse compensator

)(ˆ,1)(ˆ 2 skdiagsK

Leads to column dominant.



Lecture 9

52



-20 -10 0 10 20-20

-10

0

10

20

Re

Im

-20 -10 0 10 20-20

-10

0

10

20

Re

Im

-20 -10 0 10 20-20

-10

0

10

20

Re

Im

-20 -10 0 10 20-20

-10

0

10

20

Re

Im

GK ˆˆNvquist array of


Lecture 9

53


Achieving diagonal dominance ( Pseudo-diagonalization )

This can be done by choosing some measure of diagonal dominance, some compensator

structure, and then optimizing the measure of dominance over this structure.

We shall use the term pseudo-diagonalization for any such scheme, although the

term is often reserved for the particular scheme proposed by Hawkins (1972).

Hawkins assumed that inverse arrays are to be used, but his method can be applied

equally well to direct arrays.

If we have a plant G(s) and a constant compensator K, with Q(s)=G(s)K, then

Inividual elements of Q are given by

j

T

iij kjgjq )()(

N

k ji

jk

T

ik

N

k ji

kijkj kjgpjqpJ1

2

1

2

)()(

Hawkins proposed to minimize

subject to the constraint1jk

jth column of K

Otherwise it leads to kj=0

ith row of G


Lecture 9

54



N

k ji

jk

T

ik

N

k ji

kijkj kjgpjqpJ1

2

1

2

)()(

Hawkins proposed to minimize

subject to the constraint1jk

Hawkins method may not prevent from being made small, as well as the

off-diagonal elements, so that diagonal dominance may not be obtained.

)( kjj jq

Suppose, however

N

k

kjjk

N

k ji

kijk

j

jqp

jqp

J

1

2

1

2

)(

)(

2

2

)(

)(

k

jk

T

jk

k ji

jk

T

ik

j

kjgp

kjgp

J

The solution is given by the multi frequency ALIGN algorithm. (Maciejowski (1989)).


Lecture 9

55



Ford and Daly (1979) have extended this approach to dynamic compensators:

)(),...,(),...,()( 1 sksksksK mj

skksk jjj ...)( 0

j

T

iij jjq )()(

)(....)()()(

jgjjgjjgjT

i

T

i

T

i

T

i

j

j

j

k

k

...

0

k

jk

T

jk

k ji

jk

T

ik

j

jp

jp

J2

2

)(

)(

Vector

ijth element of G(s)K(s) is :

ith row of G


Lecture 9

56



A realizable compensator is therefore obtained by dividing Kj(s) by any polynomial

of degree β (or greater).

Pseudo-diagonalization can be applied to either direct or inverse Nyquist arrays.

But a practical difficulty arises if a dynamic compensator is found for an inverse array:

its inverse needs to be realizable.

In particular, it can be viewed as an extension of the ALIGN algorithm, and can

therefore be applied in the context of the characteristic-locus method.

Pseudo-diagonalization have described in the context of Nyquist array methods,

it can clearly be applied whenever approximate inverses of frequency responses are

required.


Lecture 9

57

Design example

0732.005750.1

6650.104190.4

000

00000.11200.0

000

,

6859.00532.102909.00

0130.18556.000485.00

00000.1000

0705.001712.00538.00

0000.101320.100

BA

000

000

000

,

00100

00010

00001

DC

the model has three inputs, three outputs and five states.

Example 9-8: Consider the aircraft model AIRC described in the following

state-space model.

DuCxy

BuAxx



Lecture 9

58

Design example

• We shall attempt to achieve a bandwidth of about l0 rad/sec for each loop.

THE SPECIFICATION

• Little interaction between outputs.

• Good damping of step responses and zero steady-state error in the face of step

demands or disturbances.

• We assume a one-degree-of-freedom control structure.



Lecture 9

59

Design example

PROPERTIES OF THE PLANT

• The time responses of the plant to unit step signals on inputs 1 and 2 exhibit very

severe interaction between outputs.

• The poles of the plant (eigenvalues of A) are

jj 1826.00176.0,03.178.0,0

so the system is stable (but not asymptotically stable).

• Thus this plant has no finite zeros, and we do not expect any limitations on

performance to be imposed by zeros. since

the number of finite zeros of the plant can be at most

)(2 CBrankmn 013.25



Lecture 9

60


Design example Obtaining column dominance

10-3

10-2

10-1

100

101

102

-100

-50

0

50

100

150

Frequency (rad/sec)

Gai

n (d

B)

Column dominances of the plant: column 1 (solid curve), column 2 (dashed curve)

and column 3 (dotted curve).

)(

)(

jg

jg

jj

ji

ijcolumn dominance measure

column 1


Lecture 9

61


We use pseudo-diagonalization (the algorithm of Ford and Daly (1979)) to obtain

column dominance, and apply it to one column at a time.

For this purpose, and in the rest of this design example, all frequency responses are

evaluated at a set of 50 frequency points, equally spaced on a logarithmic scale

between 0.001 and 100 rad/sec, except for a greater density of points in the region

of 0.18 rad/sec .

Applying pseudo-diagonahzation to the first column and optimizing over constant

compensator elements only, with uniform weighting on all frequencies, was not

successful.

Diagonal dominance was improved at low frequencies, where it was not needed, but

remained almost unchanged at high frequencies.

Frequencies above 0.1rad/sec were therefore weighted 10 times as much as lower

frequencies, but dominance was still not achieved above l rad/sec.



Lecture 9

62


This produced virtually perfect diagonal dominance, with a dominance measure less

than 10-6 at all frequencies.

33

54

43

1

1031.11005.4

1011.31083.1

1078.91052.1

)(

s

s

s

sk

The lower degree of dominance may lead a simpler compensator structure. Only one

element of the column may need to be dynamic, for example.

For the second column, it was again necessary to optimize over a first order dynamic

structure,

2

32

2

2

1040.30

1050.41074.5

1022.10

)(

s

s

s

sk

Again, almost perfect dominance was obtained everywhere.



Lecture 9

63


The third column proved to be the most difficult to compensate.

Optimizing over a first-order structure with high weighting on low frequencies lead to:

Reducing the weighting on the low frequencies gave no benefit at higher frequencies.



Lecture 9

64


So optimization over a second-order structure was attempted, with uniform weighting

at all frequencies.2

2313033 )( skskksk

This achieved dominance, except

in a narrow range of frequencies

near of frequencies near 0.2 rad/s.

The weighting on frequencies

between 0.1 and 1 rad/sec was

therefore increased to 10 times

as much as on other frequencies.



Lecture 9

65


1221

2223

1222

3

1052.11008.31086.2

1034.31058.21007.5

1014.11078.91021.4

)(

ss

ss

ss

sk

The design obtained for the third column of the compensator is

)(),(),()( 321 sksksksKb

10-3

10-2

10-1

100

101

102

-140

-120

-100

-80

-60

-40

-20

0

20

Frequency (rad/sec)

Gain

(dB

)



Lecture 9

66


Design example Loop compensation

The first two columns are so dominant that the Gershgorin circles, superimposed on the

(1,1) and (2,2) loci, cannot be distinguished from the loci themselves, and the

Compensation of the first two loops is no different from compensation of a SISO system.

?00

0?0

003980

)(sKc

10-3

10-2

10-1

100

101

102

-100

-80

-60

-40

-20

0

20

40

60

80

Frequency (rad/sec)

Gain

(dB

)

Response of (1, l) element with and without compensation.

72)log(20 k


Lecture 9

67



The frequency response of the (2,2) element, shown as a Nyquist plot in Figure, is

essentially constant at -25dB (=0.0562) at all frequencies.

Response of (2,2) element before compensation.

?00

0174

0

003980

)(s

sKc


Lecture 9

68



The response of the (3,3) element is shown as a Nyquist plot with its Gershgorin band

(computed column-wise), and its magnitude is shown in Bode form in Figures

Response of (3, 3) element before compensation.

10-3

10-2

10-1

100

101

102

-14

-13

-12

-11

-10

-9

-8

-7

Frequency (rad/sec)

Gain

(dB

)

Gain of (3, 3) element before compensation.


Lecture 9

69



Suitable compensation of this element is obtained by first changing its sign, so that the

locus starts and ends on the positive real axis, and then inserting an integrator with

enough gain to add about l0 dB at l0rad/sec.

s

ssK c

6.3100

0174

0

003980

)(

10-3

10-2

10-1

100

101

102

-40

-20

0

20

40

60

80

Frequency (rad/sec)

Gain

(dB

)

Gain of (3, 3) element after compensation.


Lecture 9

70



For this element the thickness of the Gershgorin band is significant, so we should check

that the band does not overlap 1 in order to be sure that our inference of closed-loop

stability is correct.

Response of (3,3) element after compensation,

with Gershgorin band, on a Nichols chart.


Lecture 9

71



The compensation is not yet finished, because the product Kb(s)Kc(s) is not realizable:

each element in the first and third columns has one more zero than poles.

REATTZATTON OF THE COMPENSATOR


Lecture 9

72


Design example Analysis of the design

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Re

Im

M=3 dB-circle

Characteristic loci, with 3 dB M-circle.


Lecture 9

73



10-3

10-2

10-1

100

101

102

-40

-35

-30

-25

-20

-15

-10

-5

0

5

Frequency (rad/sec)

Gain

(dB

)

Largest and smallest closed-loop singular values (principal gains).


Lecture 9

74



0 0.5 1 1.5 2 2.5 3-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

Am

plit

ude

Closed-loop step responses to step demand on output 1 (solidcurves),

output 2 (dashed curves) and output 3 (dash-dotted curves).


Lecture 9

75



0 1 2 3 4 5 6 7 8 9 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Impulse Response

Time (sec)

Am

plit

ude

Response to impulse disturbance on plant input 1.


Lecture 9

76

Exercise 9-3: Use state feedback to decouple the following system and put the

poles of new system on s=-3.

xyuxx

110

001

00

10

01

6116

100

010

Exercise 9-2: Decouple following system and find the decoupled transfer function.

xyuxx

1000

0010

00

11

01

10

0100

0000

0001

0000

Exercises

Exercise 9-1: Mentioned in the lecture.


Lecture 9

77

Exercises

Exercise 9-4: RGA of a 4 input, 4 output system is:

919.1900.0030.2215.0

910.1270.0314.3135.0

154.1286.0429.0011.0

164.0080.0150.0931.0

Suggest suitable pairing.


Lecture 9

78

Exercises

Exercise 9-5: Consider following system.(Final)

21

54

01

)(

s

sssP

a) Show that P(s) is not diagonal column dominant.

b) Show that L can convert P(s) to diagonal column dominance.

12

01L

c) Derive controller such that its steady state error to step input is zero and open loop

bandwidth of each canal be 10 rad/sec.

s

Controllerans 520

010:

d) Derive step response.


Lecture 9

79

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

• تحليل و طراحی سيستم های چند متغيره، دکتر علی خاکی صديق

• Process Dynamics and Control, Seborg, Edgar, Mellichamp and Doyle, 2011.

Documents

Multivariable Control Systemskarimpor.profcms.um.ac.ir/.../multivariable9-decoupling.pdfDr. Ali Karimpour Feb 2017 Lecture 9 2 Topics to be covered include: • Decoupling • Decoupling