1

Control Engineering Group

Automatic Control & Computer Science DepartmentPublic University of Navarra

31006 Pamplona, Spain

http://www.ayc.unavarra.es/ceg/home.htm

2

Head of the Group: Prof.Dr. Mario García-Sanz

Doctors: Dr. Igor Egaña Santamarina

Dr. Montserrat Gil Martínez

Dr. Juan Carlos Guillén

Dr. Xabier Ostolaza Zamora

PhD. Students: Marta Barreras Ana Huarte

Alejandro Asenjo Javier Castillejo

Juan Antonio Osés Pablo Vital

Javier Villanueva Juan José Martín

Manuel Motilva Irene Eguinoa

Europe

Pamplona (Spain)

Control Engineering GroupAutomatic Control & Computer Science DepartmentPublic University of Navarra, 31006 Pamplona, Spain

http://www.ayc.unavarra.es/ceg/home.htm

3

With Prof. Isaac Horowitz, October 2000

Control Engineering GroupAutomatic Control & Computer Science DepartmentPublic University of Navarra, 31006 Pamplona, Spain

http://www.ayc.unavarra.es/ceg/home.htm

Basic Research on QFT Robust Control. +Applied Research Projects with Industry.

In the last 5 years: - More than 50 papers (Journals and Conf.) on QFT. - 7 Patents. - 5 Ph.D. Dissertations on QFT. - 6 Research Projects with Industry - 3 Research Projects supported by Spanish Gov. - 10 Ph.D Students working on QFT.

Guest Editor, Robust Frequency Domain Special Issue,Int. J. Robust Non-linear Control, Wiley, 2003.

Organizer of the 5th International Symposium onQuantitative Feedback Theory and Robust FrequencyDomain Methods, Pamplona, August 2001.

NATO Lecturer. (USA, Italy, Portugal, etc)5th QFT Symp., Pamplona, August 2001

M.TorresCompany

PublicUniversityof Navarra UR

CEIT

Aerospace Division

BoeingAirbus - A380 - Eurofighter - EurocopterNorthrop Gr.British Aerosp...

ControlEngineering

Group

Applied Research Projects

- Wastewater Treatment Plants- Drainage and Sewage Water Networks

Energy Division

Applied Research Projects- Wind Turbines Design- Aerospace Machinery- Paper Machinery

QFT Research- Fundamentals- MIMO Processes- Time-delay Systems- Distributed Plants- Non-linear Controllers

Paper Division

Tetra PakGeorgia Pacific

International Paper...

5

EXTERNAL DISTURBANCE REJECTION INUNCERTAIN MIMO SYSTEMSWITH QFT NON-DIAGONAL

CONTROLLERS

M. García-Sanz1, M. Barreras1, I. Egaña1, C.H. Houpis2

€

1Automatic Control and Computer Science Department,Public University of Navarra, 31006 Pamplona, SPAIN.

2Air Force Institute of Technology, Air Force ResearchLaboratory, Wright-Patterson AFB, Ohio, U.S.A.

http://www.ayc.unavarra.es/ceg/home.htmEmail: mgsanz@unavarra.es

Authors wish to gratefully appreciate the support given by the Spanish “Ministerio de Ciencia yTecnología” (MCyT) under grants CICYT DPI’2000-0785 and DPI’2003-08580-C02-01.

6

• Disturbance attenuation in multivariable systems is often an importantand difficult problem to deal with.

• This work circumvents rejection of external disturbances in MIMOsystems with plant model uncertainties.

• The Quantitative Feedback Theory (QFT) is applied to design fullypopulated matrix controllers to attenuate the effect of externaldisturbances at plant input and plant output.

1.- Introduction

7

• The study starts with some previous ideas– suggested by Garcia-Sanz and Egaña (2002)– about the design of non-diagonal QFT controllers to reduce the loop

coupling in tracking problems,García-Sanz, M. and I. Egaña (2002). Quantitative Non-diagonalController Design for Multivariable Systems with Uncertainty.International Journal of Robust and Non-Linear Control. Nº 12, pp. 321-333. John Wiley & Sons.

• and considers the approach– introduced by Houpis and Rassmussen (1999)– about MIMO systems with external disturbances.

Houpis, C.H. and S. J. Rasmussen (1999). Quantitative Feedback TheoryFundamentals and Applications. Marcel Dekker, New York, ISBN: 0-8247-7872-3.

8

• Consider the n x n linear multivariable system shown in Fig.1.

2.- Rejection of External Disturbances at Plant Input

r = 0 -

’ d i

P di

d i

G

u v + y

d 0

+

’ d 0

P d 0

P + + Fig. 1 General structure of a MIMO

system with plant outputand input external disturbances.

• P is the plant, P є P and P is a set of possible plants due to uncertainty,• G is the full populated matrix controller,

• Pdi and Pdo are the plant input and output disturbance transfer functions.• di’ and do’ are the plant input and output external disturbances respectively.

=

=

nnnn

n

n

nnnn

n

n

g...gg............

g...ggg...gg

G;

p...pp............

p...ppp...pp

P

21

22221

11211

21

22221

11211

9

• denoting the external disturbance at plant input by

• then the output due to the external disturbance at plant input is,

• and the closed loop transfer matrix TY/di that relates the disturbance atplant input to the output y becomes,

'idii dPd =

( ) '//

1ididiYidiYi dPddy TTPGPI ==+= −

( ) PGPIT 1/

−+=diY

(1)

(2)

r = 0

-

’ d i

P di

d i

G

u v + y

d 0

+

’ d 0

P d 0

P + +

10

• Multiplying Eq.(2) by (I + P G),• pre-multiplying by• and denoting

where Λ is the diagonal part and B the balance of P-1 respectivelyand Gd is the diagonal part and Gb the balance of G respectively

• and rearranging it, yields the next expression

TY/di = ( I + Λ-1 Gd ) -1Λ-1 – ( I + Λ-1 Gd ) -1 Λ-1 [( B + Gb ) TY/di ]

1−P[ ]

bd

ij

GGGBPpP

+=

+Λ===− **1

(7)

[ ]

+

=+Λ===

0...p...0...

p...0

p000...000p

p*n

*n

*nn

*

1

111*ij

*1- BPP

+

=+=

0...g...0...

g...0

g000...000g

n

n

nn 1

111

bd GGG

11

the only part which has anon-diagonal structure. Itwill be call the couplingmatrix Cdi

• By inspecting Equation (7) two different terms can be found:

TY/di = ( I + Λ-1 Gd ) -1Λ-1 – ( I + Λ-1 Gd ) -1 Λ-1 [( B + Gb ) TY/di ] (7)

gii *

1

iip

ui0 yi

-

∑=

=n

i

dicd1

jijdi-j

* 1 ii p g ii

0 y i

-

u i

di i

1 ≤ i ≤ n

A diagonal termTY/di-d = ( I + Λ-1 Gd ) -1 Λ-1

(8) A non-diagonal termTY/di-b = ( I + Λ-1 Gd )-1 Λ-1 [(B+ Gb )TY/di](10)

12

This coupling matrix is essential to analyse the reduction of the cross-coupling effects.

Cdi = ( B + Gb ) TY/di

Each element of Cdi obeys,

where is the delta of Kronecker.

The influence of the non-diagonal elements gik (i ≠ j) on the cross-couplingelements described in Eq.(15) is difficult to analyse directly because of thecomplexity of the expression.

For this reason, one hypothesis and two simplifications are stated in orderto make the quantification of coupling effects easier.

)δ(1t)gp(c ikkjik

m

1k

*ikij −+=∑

=−di

(11)

(15)

≠⇔==⇔=

=ik0δik1δ

δki

kiki

13

Hypothesis H1: The diagonal elements tjj in Eq. (15) are assumed to bemuch larger than the non-diagonal ones tkj,

( ) ( ) jkforgptgpt ik*ikkjij

*ijjj ≠+>>+

Simplification S1: Applying Hypothesis H1, Eq. (15) can be rewritten as,

ji;)gp(tc ij*ijjjij ≠+=−di

Simplification S2: The elements tjj can be replaced using the expressionobtained from the equivalent system in Eq. (8).

1*jjjj

1*jj

jj1

−

−

+=

pg

pt

14

Due to the previous considerations the coupling effect cdi-ij can be defined as,

ji;)(

)(

jj*jj

ij*ij ≠

+

+=− gp

gpc ijdi (19)

In order to find out the optimum non-diagonal controller,• Eq. (19) is made equal to zero,• and a nominal plant that minimises the maximum non-parametric uncertainty radii in Eq. (20) is chosen,*

ijp∆N*

ijoptij pg −= (21)

N*ijp ij∆

Note, that every uncertain plant is represented by the following family,

{ } ( ) *ijijij

N*ij

*ij ppp ∆≤∆≤∆+= 01

for i,j=1,…, n

Where is the nominal plant and thenon parametric uncertainty radii.

(20) *ijp∆

N *ijp

*ijp∆

dB

φ

N *ijp

15

Effect ofoptimum

non-diagonalcontroller

The minimum achievable coupling effects can be computed substituting theoptimum controller, Eq. (21), in the coupling expression of Eq. (19) andtaking into account the uncertainty radii of Eq. (20).

ijjj

N*jjjj

*Nij

ijgij ∆∆(1 gp)

pc OPT

ijg ++=

=

In the same manner, the maximum coupling effect without any non-diagonal controller -pure diagonal controller cases- can be computedsubstituting gij=0 in Eq. (19),

( )ijjj

N*jjjj

*Nij

0ijgij ∆1∆(1

+++

== gp)

pc

n1,...,ji,for∆∆0∆∆0 jjjjijij =≤≤≤≤ ,p,p **

16

• Consider the n x n linear multivariable system shown in Fig.1.

3.- Rejection of External Disturbances at Plant Output

Fig. 1 General structure of a MIMOsystem with plant outputand input external disturbances.

• P is the plant, P є P and P is a set of possible plants due to uncertainty,• G is the full populated matrix controller,

• Pdi and Pdo are the plant input and output disturbance transfer functions.• di’ and do’ are the plant input and output external disturbances respectively.

=

=

nnnn

n

n

nnnn

n

n

g...gg............

g...ggg...gg

G;

p...pp............

p...ppp...pp

P

21

22221

11211

21

22221

11211

r = 0 -

’ d i

P di

d i

G

u v + y

d 0

+

’ d 0

P d 0

P + +

17

• denoting the external disturbance at plant output by

• then the output due to the external disturbance at plant output is,

• and the closed loop transfer matrix TY/do that relates the disturbance atplant output to the output y becomes,

(24)

(25)

'oodo dPd =

( ) 'ododo/Yodo/Yo dPddy TTGPI ==+= −1

( ) 1−+= GPIT do/Y

r = 0 -

’ d i

P di

d i

G

u v + y

d 0

+

’ d 0

P d 0

P + +

18

• Multiplying Eq.(25) by (I + P G),• pre-multiplying by• and denoting

where Λ is the diagonal part and B the balance of P-1 respectivelyand Gd is the diagonal part and Gb the balance of G respectively

• and rearranging it, yields the next expression

TY/do = ( I + Λ-1 G )-1 + ( I + Λ-1 Gd )-1 Λ-1 [ B – (B + Gb TY/do) ]

1−P[ ]

bd

ij

GGGBPpP

+=

+Λ===− **1

(26)

[ ]

+

=+Λ===

0...p...0...

p...0

p000...000p

p*n

*n

*nn

*

1

111*ij

*1- BPP

+

=+=

0...g...0...

g...0

g000...000g

n

n

nn 1

111

bd GGG

19

the only part which has anon-diagonal structure. Itwill be call the couplingmatrix Cdo

• By inspecting Equation (26) two different terms can be found:

(26)

A diagonal termTY/do-d = ( I + Λ-1 Gd ) -1

(27) A non-diagonal termTY/do-b = ( I + Λ-1 Gd )-1 Λ-1 [ B – (B + GbTY/do)](29)

TY/do = ( I + Λ-1 Gd )-1 + ( I + Λ-1 Gd )-1 Λ-1 [ B – (B + Gb TY/do) ]

1 ≤ i ≤ n

* 1 ii p

g ii

0 y i -

u i

do i

gii *

1 ii p

u i 0 y i -

∑==

n i do c d

1 jij - do j

20

This coupling matrix is essential to analyse the reduction of the cross-coupling effects.

Cdo = B – (B + Gb TY/do)

Each element of Cdoobeys,

where is the delta of Kronecker.

The influence of the non-diagonal elements gik (i ≠ j) on the cross-couplingelements described in Eq.(32) is difficult to analyse directly because of thecomplexity of the expression.

For this reason, one hypothesis and two simplifications are stated in orderto make the quantification of coupling effects easier.

(31)

(32)

≠⇔==⇔=

=ik0δik1δ

δki

kiki

)δ(1)()δ(1 ikkjik

m

1k

*ikijij −+−−= ∑

=− tgppc *

ijdo

21

N*ijp ij∆

Note, that every uncertain plant is represented by the following family,

{ } ( ) *ijijij

N*ij

*ij ppp ∆≤∆≤∆+= 01

for i,j=1,…, n

Where is the nominal plant and thenon parametric uncertainty radii.

(20) *ijp∆

N *ijp

*ijp∆

dB

φ

N *ijp

Applying again Hypothesis H1 and Simplifications S1 and S2, the finalexpression of the coupling effect can be written as,

(33)ji;)(

)(

jj*jj

ij*ij

*jj*

ijij ≠+

+−=− gp

gpppcdo

In order to find out the optimum non-diagonal controller,• Eq. (33) is made equal to zero,• and a nominal plant that minimises the maximum non-parametric uncertainty radii in Eq. (20) is chosen,*

ijp∆

(34)N*jj

N*ij

jjoptij

p

pgg =

22

Effect ofoptimum

non-diagonalcontroller

The minimum achievable coupling effects can be computed substituting theoptimum controller, Eq. (34), in the coupling expression of Eq. (33) andtaking into account the uncertainty radii of Eq. (20).

( ) ( )jjijjj

N*jjjj

*Nij

ijgij ∆∆∆1

−++

== gp

pc OPT

ijg

In the same manner, the maximum coupling effect without any non-diagonal controller -pure diagonal controller cases- can be computedsubstituting gij=0 in Eq. (33),

n1,...,ji,for∆∆0∆∆0 jjjjijij =≤≤≤≤ ,p,p **

( ) ( )ijjj

N*jjjj

*Nij

0ij ∆1∆1

+++

== gp

pc

ijg

23

4.- Design Methodology

Composed of n stages, as many as loops, performs the following steps for everycolumn of the matrix controller G.

⇒⇒

⇒

=

nnnk2n1n

knkk2k1k

n2k22221

n1k11211

2n1n

2k1k

2221

1211

1n

1k

21

11

g...g...gg

.........

g...g...gg

.........

g...g...gg

g...g...gg

...

0...0...gg

.........

0...0...gg

.........

0...0...gg

0...0...gg

0...0...0g

.........

0...0...0g

.........

0...0...0g

0...0...0g

G

Step 1 Step 2 Step n

• Stability: It is necessary and sufficient that the plant of each successive loop is stabilised(Chait and Yaniv, 1991)

• Before starting the sequential procedure, analyse the effect of interactions in the systemand identify input-output pairings using the Relative Gain Array, (Bristol, 1966).

• Afterwards, rearrange matrix P* so that ( )-1 has the smallest phase margin frequency,( )-1 the next smallest phase margin frequency, and so on, (Houpis, Rasmussen, 1999).

*p11*p22

24

Step B:

Design the (n-1) non-diagonal elements gik (i≠k, i = 1,2,...n) of the kth

controller column, minimising the coupling cdi-ik or cdo-ik described in Eqs.(19), (33), and applying the optimum non-diagonal controller equationsEq. (21) and Eq. (34) respectively.

Step A:

Design the diagonal element of the controller gkk for the inverse of equivalentplant described in Eq.(37), using a standard QFT loop-shaping method,

[ ] [ ][ ] ( )[ ]( ) [ ] [ ]( )

[ ] ( )( )[ ]

[ ] )(

gp

gpgppp

k

kk

37 k;i 1k

11-i1-i1k*

1)-(i1)-(i

1i)1(i1k*

i1)-(i11-ii1k*

1)-(ii1k

*iik

e*ii

** PP =≥

+

++−=

=

−−

−−−−−−

25

5.- Example

Set of plants P due to uncertainty

105τ 2272k22

83τ 211511k21

128τ 12-1.8-2.2k12

30.5τ 1130.5k11

τ maxτ minτKmaxkmink

++

++=

2

1

22

22

21

21

12

12

11

11

2

1

11

11uu

kk

kk

yy

ττ

ττ

Coefficients of parametric uncertainties

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

d2

g12

g21

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

d2

g12

g21

p11

p22

Desired performance specifications:• Robust Stability:

• Robust Output Disturbance Rejection

• Reduction of coupling effect as much aspossible

2,12.11 i

i =≤+

iL

L

2,1,/5010)(

)( =<ω+

≤ israds

ssdsy

i

i

26

Relative Gain Array

127325 plants generated due to uncertainty:

λ11є [0 , 0.1) for 22220 plants

λ11є [0.1 , 0.2) for 39985 plants

λ11є [0.2 , 0.3) for 35695 plants

λ11є [0.3 , 0.4) for 23155 plants

λ11є [0.4 , 0.5] for 6270 plants

0 500 1000 1500 2000 25000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Value s of landa11

Numbe r of sample s

land

a11

0.005.00

10.0015.0020.0025.0030.0035.00

%

0.4-0.5

Values of landa

Values of landa. Percentage.

0.3-0.40.2-0.30.1-0.20-0.10.005.00

10.0015.0020.0025.0030.0035.00

%

0.4-0.5

Values of landa

Values of landa. Percentage.

0.3-0.40.2-0.30.1-0.20-0.1

27

DESIGN PROCEDURE (0 >λ11 >0.5)

• Step A-1: Standard QFT loop-shaping for*11

1p

-100 -80 -60 -40 -20 0-50

-40

-30

-20

-10

0

10

20

30

0.1

0.5

1

2

5

10

20

50

100

Phase (deg)

Gai

n (d

B)

Templates

0.10.5125102050100

22770 points

28

DESIGN PROCEDURE (0 >λ11 >0.5)

• Step A-1: Standard QFT loop-shaping for*11

1p

-100 -80 -60 -40 -20-50

-40

-30

-20

-10

0

10

20

30

0.1

0.5

1

2

5

10

20

50

100

Phase (deg)

Gai

n (d

B)

Te mplate s

0.10.5125102050100

900 points

29

ss 3022718s7550g 211 +

+=

Design of g11

dd

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g22r2 Y2

1

d2

g12

g21

p11

p22

e11

r2

Y1

p12

p21

g11

22-Y2

1

d2

g12

g21

p11

p22

dd

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g22r2 Y2

1

d2

g12

g21

p11

p22

e11

r2

Y1

p12

p21

g11

22-Y2

1

d2

g12

g21

p11

p22

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g22r2 Y2

1

d2

g12

g21

p11

p22

e11

r2

Y1

p12

p21

g11

22-Y2

1

d2

g12

g21

p11

p22

-360 -315 -270 -225 -180 -135 -90 -45 0

-60

-40

-20

0

20

40

60

Open-Loop P has e (deg)

Ope

n-Lo

op G

ain

(dB

)

0.10.5125102050100

30

• Step B-1: Optimum Non-diagonalcontroller for the first column

sssss

302907313591057088751g 23

2

21 ++−−−=

Step A-2: Once the first column has been designed, the equivalent plant of the second channel is calculated:

[ ] [ ] [ ] [ ]( ) [ ] [ ]( )[ ] [ ]1111

*11

1121*121211

*21

1*222

e*22 gp

gpgppp

+

++−=

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g22r2 Y2

d1

d2

g12

g21

p11

p22

e11

r2

Y1

p12

p21

g11

22-Y2

d1

d2

g12

g21

p11

p22

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g22r2 Y2

d1

d2

g12

g21

p11

p22

e11

r2

Y1

p12

p21

g11

22-Y2

d1

d2

g12

g21

p11

p22

31

Standard QFT loop-shaping for *e22

1p

-180 -135 -90 -45 0-60

-50

-40

-30

-20

-10

0

10

20

30

0.1

0.5

1

2

5

10

20

50

100

Phas e (deg)

Gai

n (d

B)

Te mplate s

0.10.5125102050100

22842 points

32

1314 points

-180 -135 -90 -45 0-60

-50

-40

-30

-20

-10

0

10

20

30

0.1

0.5

1

2

5

10

20

50

100

Phas e (deg)

Gai

n (d

B)

Te mplate s

0.10.5125102050100

Standard QFT loop-shaping for *e22

1p

33

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g22r2 Y2

d1

d2

g12

g21

p11

p22

e11

r2

Y1

p12

p21

g11

22-Y2

d1

d2

g12

g21

p11

p22

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g22r2 Y2

d1

d2

g12

g21

p11

p22

e11

r2

Y1

p12

p21

g11

22-Y2

d1

d2

g12

g21

p11

p22ssss

17020277168498101207s26745g 23

32

22 +++⋅+=

Design of g22

-360 -315 -270 -225 -180 -135 -90 -45 0

-80

-60

-40

-20

0

20

40

Open-Loop P has e (deg)

Ope

n-Lo

op G

ain

(dB

)

0.10.5125102050100

34

• Step B-2: Optimum Non-diagonal controller for the second column

sssssss14187.170471.277

18725108.1104.37430g 234

5253

12 ++++⋅+⋅+=

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g222 Y2

d 1

d 2

g12

g21

p11

p22

e11

2

1

p12

p21

g11

22-2

d

dg12

g21

p11

p22

+

+

- e1

e2

r1

r2

Y1

p12

p21

g11

g222 Y2

d 1

d 2

g12

g21

p11

p22

e11

2

1

p12

p21

g11

22-2

d

dg12

g21

p11

p22

35

• Disturbance d1 at plant output

Response y1 Response y2

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

g12

g21

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

g12

g21

RESULTS

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 50.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Time (se conds)

y1

DIAGONAL CONTROLLER

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

0.75

0.8

0.85

0.9

0.95

1

1.05

Time (se conds)

y2

DIAGONAL CONTROLLERDIAGONAL CONTROLLER DIAGONAL CONTROLLER

36

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 50.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Time (se conds)

y1

NON-DIAGONAL CONTROLLER

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

0.75

0.8

0.85

0.9

0.95

1

1.05

Time (se conds)

y2

NON-DIAGONAL CONTROLLERNON DIAGONAL CONTROLLER NON DIAGONAL CONTROLLER

• Disturbance d1 at plant output

Response y1 Response y2

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

g12

g21

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d1

p11

p22

g12

g21

RESULTS

37

RESULTS

• Disturbance d2 at plant output

Response y1 Response y2

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

g12

g21

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

g12

g21

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

p11

p22

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

1

1.025

1.05

1.075

1.1

Time (se conds)

y1

DIAGONAL CONTROLLER

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 50.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Time (se conds)

y2

DIAGONAL CONTROLLERDIAGONAL CONTROLLERDIAGONAL CONTROLLER

38

RESULTS

• Disturbance d2 at plant output

Response y1 Response y2

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

g12

g21

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

g12

g21

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

p11

p22

+

+

-

-

e1

e2

r1

r2

Y1

p12

p21

g11

g22

+

+

-

-r2 Y2

d2

p11

p22

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5

1

1.025

1.05

1.075

1.1

Time (se conds)

y1

NON DIAGONAL CONTROLLER

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 50.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

Time (se conds)

y2

NON DIAGONAL CONTROLLERNON DIAGONAL CONTROLLER NON DIAGONAL CONTROLLER

39

6.- Conclusions

• A QFT methodology

to design fully populated matrix controllers

to solve the MIMO external disturbance rejection problem at both,plant input and output,

and in the presence of model plant uncertainty, was presented.

• The definition of a specific coupling matrix

allows the statement of the controller design methodology.

• An example has been also presented

to improve the understanding of the methodology.

40

7.- Current Work• Applying the new methodology to control an industrial furnace.

A 7x7 MIMO System

Power: 900 kW

Length: 37.60 m

For Wind Turbine Blades Polymerisation

4.8 m

Group 1

37.600 m

7.8 m5 m5 m5 m5 m

Group 2 Group 3 Group 4 Group 5 Group 6 Group 7

5 m

Industrial furnace for compositemanufacturing located at M.Torres

(Spain)

41

EXTERNAL DISTURBANCE REJECTION INUNCERTAIN MIMO SYSTEMSWITH QFT NON-DIAGONAL

CONTROLLERS

Authors wish to gratefully appreciate the support given by the Spanish “Ministerio de Ciencia yTecnología” (MCyT) under grants CICYT DPI’2000-0785 and DPI’2003-08580-C02-01.

M. García-Sanz1, M. Barreras1, I. Egaña1, C.H. Houpis2

€

1Automatic Control and Computer Science Department,Public University of Navarra, 31006 Pamplona, SPAIN.

2Air Force Institute of Technology, Air Force ResearchLaboratory, Wright-Patterson AFB, Ohio, U.S.A.

http://www.ayc.unavarra.es/ceg/home.htmEmail: mgsanz@unavarra.es