Multivariable Control Systems - Personal Datakarimpor.profcms.um.ac.ir/imagesm/354/stories/mul_con/... · 2014-02-13 · Dr. Ali Karimpour Feb 2014 Lecture 9 6 Decoupling The idea

Multivariable Control Multivariable Control SystemsSystems

Ali KarimpourAssociate Professor

Ferdowsi University of Mashhad

Lecture 9

References are appeared in the last slide.

Dr. Ali Karimpour Feb 2014

Lecture 9

2

Decoupling Control

Topics to be covered include:

• Decoupling

• Decoupling by State Feedback

• Diagonal controller (decentralized control)

• Decoupling


Lecture 9

3

Introduction

CxyBuAxx

BAsICsG 1)()(

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

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

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

2211

22221212

12121111

susgsusgsusgsy

susgsusgsusgsysusgsusgsusgsy

pppppp

pp

pp

We see that every input controls more than one output and that everyoutput is controlled by more than one input.

Because of this phenomenon, which is called interaction, it is generallyvery difficult to control a multivariable system.


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)(sK s

Decoupling

Decoupling


Lecture 9

5

Decoupling

• Dynamic decoupling

• Steady-state decoupling

• Approximate decoupling at frequency ω0

s.frequencie allat diagonal is )(sGs

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

)()( choosecan we with exampleFor 1 sGsWIG ss

(s)l(s)GK(s)IslsK -s

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

possible. as diagonal as is )( 0jGs

This is usually obtained by choosing a constant pre compensator 10 GWs

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


Lecture 9

6

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

7

Pre and post compensators and the SVD controller

The pre compensator approach may be extended by introducing a post compensator

)()()()( sWsGsWsG ssps

The overall controller is then

)()()()( sWsKsWsK spss


Lecture 9

8

Decoupling Control

• Decoupling


• Diagonal controller (decentralized control)



Lecture 9

9

Decoupling by State Feedback

In this section we consider the decoupling of a control system in state spacerepresentation.

DuCxyBuAxx

Let DBAsICsG 1)()( Suppose

diagonalsGsG 1)()(

0|D| ifThen

11111111 )()()( DBDCBDAsICDDBAsICsG

But in the case of |D|=0

)()()( tTrtKytu Static output feedback

Dynamic output feedback

)()()( tTrtKxtu Static state feedback


Lecture 9

10

Decoupling through state feedback

CxyBuAxx

Let

)()()( Suppose 1 trtFxEtu

CxyrBExFBEAx

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.



Lecture 9

11

Theorem 9-1 A system represented by

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

BuAxx

)()()( 1 trtFxEtu

if and only if the constant matrix E is nonsingular.

)(0

0lim

.

.

12

1

sGs

s

E

EE

Epd

d

s

p

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

md

d

new

s

ssG

0

0)(

1

Furthermore the new system is in the form:

pdp

d

d

AC

ACAC

F..

2

1

2

1



Lecture 9

12

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

xyuxx

110001

001001

6116100010

Solution: Transfer function of the system is

656

656

61166

6116116

)()(

22

2323

2

1

sss

ss

ssss

sssss

BAsICsG

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

]01[6116

66116

116lim 2323

2

1

ssss

ssssssE

s

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

]10[65

665

6lim 222

ss

sss

sEs



Lecture 9

13

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

1001

ESolution (continue):

5116010

2

1

2

1d

d

ACAC

F

)()(

5116010

1001

)()()( 1 trtxtrtFxEtu

The decoupled system is

xCxy

rxrBExFBEAx

110001

001001

61166116000

)( 11

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



Lecture 9

14

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:

ndddecouple ssdiagsG ...,,)( 1


Lecture 9

15

Decoupling Control

• Decoupling


• Diagonal controller (decentralized control)• Diagonal controller (decentralized control)


Lecture 9

16

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

17

The design of decentralized control systems involves two steps:

1_ The choice of pairings (control configuration selection)

2_ The design (tuning) of each controller ki(s)



Lecture 9

18

Input-Output Pairing

Definition of RGA (Relative Gain Array)

Physical Meaning of RGA: Let

TGGGRGAG )()(

ijijij hg /2221212

2121111

ugugyugugy

jkuj

ijiij

kuy uyg

,0

or 0inputsother if and between relation

222121

2121111

0 ugugugugy

122

212 u

ggu

ikyj

ijiij

kuy uyh

,0

or 0outputsother if and between relation

Relative gain?

122

2112111 )( u

ggggy


Lecture 9

19


Let

11

)( TGGG2221212

2121111

ugugyugugy

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 leads to 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

20


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

21


Example 9-2 Select suitable pairing for the following plant

8.14.01.187.04.85.154.16.52.10

)0(G

Solution: RGA of the system is

98.107.09.043.037.094.041.145.196.0

)0(


Lecture 9

22

The RGA based techniques have many important advantages, such as very simple incalculation as it only uses process steady-state gain matrix and scaling independent.Moreover, using steady-state gain alone may result in incorrect interaction measures andconsequently loop pairing decisions, since no dynamic information of the process istaken into consideration.

Many improved approaches, RGA-like, have been proposed and described in allprocess control textbooks, for defining different measures of dynamic loopinteractions.

[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 SystemsA. 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 9-3:Consider the two-input two-output process:

26


RGA=Diagonal pairingRNGA =Off-diagonal pairing

To illustrate the validity of above results, decentralized controllersfor 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-pointChange of all control loops one-by-one and the integral square error (ISE) is used to evaluate the control performance.


Lecture 9

24


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

25

Exercise 9-3: Use state feedback to decouple the following system and put thepoles of new system on s=-3.

xyuxx

110001

001001

6116100010

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

xyuxx

10000010

00110110

0100000000010000

Exercises


Lecture 9

26

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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Multivariable Control Systems - Personal Datakarimpor.profcms.um.ac.ir/imagesm/354/stories/mul_con/... · 2014-02-13 · Dr. Ali Karimpour Feb 2014 Lecture 9 6 Decoupling The idea