Selected topics in identification, modelling and control: Progress report on research activities in the Mechanical Engineering Systems and Control Group. Vol. 5

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SELECTED TOPICS IN IDENTIFICATION,

MODELLING AND CONTROL

Progress Report on Research Activities in the

Mechanical Engineering Systems and Control Group

Edited by G.H. Bosgra and P.M.J. Van den Hof

Volume 5, December 1992

Mechanical Engineering Systems and Control Group

Delft University of Technology

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Published and Distributed by Delft University Press Stevinweg 1 2628 CN Delft The Netherlands Tel.: (0)15-783254 Telefax: (0)15-781661 By order of

Mechanical Engineering Systems and Control Group DelftUniversity of Technology

Mekelweg 2,2628 CD Delft The Netherlands

Tel.:+31-15-786400; Telefax:+31-15-784717

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Selected

Selected topics in identification, modelling and contral:

progress report on research activities in the mechanical engineering

systems and contra I group. - Delft:Mechanical Engineering Systems and Control Group, Delft University of Technology, Vol. 5-ed. by O.H. Bosgra and

P.M.J. Van den Hof.-ili. Met Iit.opg. ISBN 90-6275-834-7

SISO 656 UDC 531.7 + 681.5 NUGI 841

Cover design by Ruud Schrama

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Editorial

The present issue of Selected Topics in Identifica-tion, Modelling and Control is the fifth volume in the series, reporting on ongoing research in the Me-chanical Engineering Systems and Control Group at Delft University of Technology. We have eleven pa-pers on a variety of subjects that completely cover the subjects of Identification, Modelling and Con-trol.

In the area of Modelling, the issue of model uncer-tainty representation is consideredby Lambrechts

et

al. with a contribution aimed at obtaining struc-tured uncertainty models to be used in Jl-synthesis controller design. Wortelboer and Bosgra consider extensions to frequency weighted balancing model reduction. The modelling of a wind power gen-eration system is the subject of Van Baars and Bongers who present first results on the issue of experimental validation of their theoretical models. The real-time modelling of heat balances and the estimation of the relevant process variables on the basis of operational chemical process plant data is the subject of work reported by Baak et al. The field of Control is covered in contributions by Bongers and by Schrama et al. Both papers con-sider robust control issues with both uncertainty in plant and in controller, and derive new results with respect to robust stability. The interplay between the requirements of robust high-performance

con-trol and the achievements of error-analysis-directed system identification is shown in the contribution by Schrama and Van den Hof

In the area of System Identification we have two papers by Hakvoort ad dressing worst-case aspects of identification for two fundamentally different er-ror criteria. The issue of guaranteed error bounds in system identification results is discussed and worked out by De Vries and Van den Hof Finally, Heuberger et al. show the construction of orthogo-nal domains that may lead to advantageous sigorthogo-nal and system representations in system identification. The present issue contains results from applied projects and from theoretical studies. We appreci-ate the fact that results from collaborative projects performed in cooperation with external institutes and industrial research groups constitute a consid -erable part of this issue. Although most of the ma-terial presented here will eventually be published elsewhere in the open literature, we appreciate the efforts of our authors who in many cases have made available some of their most recent research results. We hope you enjoy the result.

ükko Bosgra Paul Van den Hof Editors

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@1992 Delft UniversityPress Se1ect edTopicsinIdentification,Modelling and Control Vol. 5, December1992

Pararnetric

uncertainty modeling using LFTs

Paul Lambrechtsl, Jan Terlouwl, Samir Bennani'' and Maarten Steinbuch] t Mechani cal Engineering Systems and Control Group, Delft University of Technology, Mekeluieq 2, 2628 CD Delft, Tlle Netherlands

§ Fac, of Aerospace Engineering, SectionStability and Coni rol, Delft University of Tech-nology, l\ïuyverweg1, 2629 IJS Delft, The Netherlands

UPhilips ResearcliLaboraiories, P.O.Box 80.000, 5600 JA Eindhoven, TheNetherlands

Abstract. In this paper a gen eral ap p roac h for modellingstructured real-valued para-metric perturbations is presented. It is based on a decomposition of perturbations into linearfractiona! transformation s (LFTs ), and is applicable to rational multi-dimensional (ND) polynomialper turbation sofent ries in sta te- space modeis. Modelreduction is used to reduce the size of the uncert ainty st ruct ure . The procedure will be applied for the uncertainty mod elling of an aircraft mod el dep ending on alt it ude and velocity (ftight envelope ).

1 Introduetion

In both robustness arialysis aIHI robust cont rol sys-tem design the conce pt of the struc t u reel sing ular value J.l as introduced by Doyle (1982) is ofgreat importance. lt allows a high degr ee of det ail in modelling the conditioris under which the cons id-ered con trol syst em should oper ate satisfactorily, both in thesenseof stabilityand perform an ce. The calcula t ion of ft for suc h models then results in a single number acting as an accu rate measurein in-dicating whether the beh avi our of the controlled systemis satisfactory or not. Therelevan ee ofusin g the st ruct ured singular val ue instead of measures that do not refiect the structural propert ies ofthe plan t uncert ain t ies,like the oo-norm orthe2-norm, can be found in liter a t ur e ; the la tter may lead to arbitrarily conse rvative statements when practical exam ples are cons ide red (see for instanee Bal as et al., 1990,Stein and Doyle , 1978 ,1991 , Doyl e et al., 1986, Skogest ad et al., 1988). In spite of this the use of J.l has been seriously hamp ereelby the con-siderable computational effort need ed for its ca lcu-lation with respect to a given uncertainty moelel. Recently elevelop ed methoels for calcula t ing close upper and lower bounds for the most gen era] cases (Fan anel Tits, 1986, 1991, Young et al., 1991)

now motivate the effort of modelling uncertainties in greateletail. Themain issue of this paper is to of-fer a com plete procedure for setting up the general st ruc t ure for the calcula t ion ofJ.l when uncertainties like real-valued parameter variations in state-space models and variations in operational conditions oe-cu r.

First,wewill givesomepreliminary results on the use of Linear Fractional Transformations (LFTs) and thei rimportan cefor uncertaintymodelling, fol-lowed by a definition of the structured singular value ft anel some relevan t uncertainty sets. Sec-tion 3 then will present a procedure for parametrie uncertain ty modelling based on astate-space model in whichuncertain entriesmay be given as rational mult i-d imensional polynomia l fun ctions of a set of parameters. The usefulness of this procedure for practi cal pro blems willbe demon strated by means of an extensive exam ple in sect ion 4, aftel' which some concl uding remarks follow in section 5.

2 Preliminaries

This section will review some of the properties of Linear Fractional Transformatiens (LFTs) and the structured singular valuefl along the lines of Doyle

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2.1 Definition of LFTs

and let .0._uE R(S

)ql

XPl and .0./ E R(S)q2 XP2 be a r-bitrary. We will then defin e the upper and lotoer LFT s as ope rators on .0.u and .0., respectively :

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with PMN:= PMI

+

PNI an el qMN:= qMl

+

qNI. Then the cascaele con neet ion obtained by setting lOM = ZN with r :=qAf 2= PN 2 (seefig.2) results in an upper LFT on .0.MNwith coefficient matrix: '1'0 demo ns t rate th is we will first look at the two most basicconnect ions betw een two LFTs: thecas -cade and parallel configur atio ns. Aft er that we will show a sim ple feedb ack configu rat ion for one LFT which can also be rew rit ten into thestandard form of fig. 1. These three configurat ions will play an impor t an t role in the a1gorithmic approach to un-certainty modellingwe will presen t in sect ion3.

Given matrices

M

and

N

par ti ti on ed as in eq.l: M E R(S)PM XqMl and NE R(S)PNXqN,with PAf:= Pll11

+

P1I12, qll1 :=qll1l+qIl12 , PN := PNI+PN 2 and

a

»

:= qNl

+

qN 2· Let Mand N be the coe fficient matrices of the up per LFTs on .0.M ER(s )qM lXPM I and .0.N E R(S)qNlXPNI resp ectively and defin e the com bined structure:

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Wewill conside r matriceswith ent riesthat are frac-tionsof polynomialsina corn plex- val ue dvaria bles; the space of all such real rati onal functions will be eleno t eelas R(s) ,M E R(s )PXqwill den ot e that M

is a p x q matrix with ent ries in R(s). Suppose a matrix M E R(s)is parti tione d as:

et al. (1987 , 1991). First we will give a defini-tionof upper an d lower LFTs and discus ssome im -por tan t possibili ti es of combining anel rearranging them . Next we will consieler the LF T conce pt as a fram ework for uncer t ai nt y modell ing and wit hin th is fram ework we will give a elefinition of Il and some releva nt uncertainty sets.

(4) Fu(M ,.0.u ) . - M2á

M21(I - .0.uMIl )-I.6. uM1 2 F/(M ,.0.,) .- MIl

+

Md l - .0./i'v!n )- l.0. /M21

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o

Eith er LFT will be calleel uiell defin ed if t.he

con-cern ing inver se exists: elct (! - .0.u.MI1)

-#

0 ancl det(J - .0./M22 )

-#

O. The matri x M is some t imes

refcrred to as the coe fficicnt matrix of thc LFT. Note that if s is interpret eelas the Lapl ace vari-able , a matrix with ent ties in H(s) ca n be seen as a multivariable transfer function of ~. linear time invarian t finitc dirnen sion al syste m. In that case LFTs can be seenas ope rat.ions resul tingIrom Ieed -back st r uet u res as given in fig.1; eq.2 the n defines a closcd loop transfer functi on s from Wil! to ZII1 in both cases.

Fig. 2: Cascade connection of LFTs The par all el conneet ion obtained by setting WII1

=

WN and ZII1N

=

ZII1

+

ZN with 1' q:=qll12= qN 2 and 1'p: = PM2= PN2 (sec fig.3 ) also resu lt s in an up per LFT on .0.MN, th is time with coefficient matri x:

Fig. 1: Upperand lower LF T as feedback struct ure An important reason for usin g the conce ptof LFTs in linear syste ms theo ry is that lincar inte rconn ee-tions of LFTs can be rewritten as one sing le LFT. This implies that LFTs can be useel to sepa rate ly model specific details of the sys te rn under consid-eration after whic h a com p lete sys tem description can be obtained by working out all connect ions.

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Fig. 3: Pa rall el connectio n of LFTs

Fig. 4: An LF T in the feedback path Not e that we have conve n ie nt ly chose n thc inputs an d outp uts of both LFTs to be com pat ible, but that it is also possible to con nee t on ly part s of in-pu t and out p ut veetors by definin g a furth cr par ti-tion ingof M and N.

Next cons ide r the feedba ck configu rat ion given in figA with Mil square (PI =

qd

.

In th is case

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in which 8;1kj,i = l 7' den ote repeated seclar blo ck s and .6.;,i = 1

J

denote

J

ul!

blocks . Note that for .6.E

a

to be compatib le with a coefficient matrix accord ing to eq.l we must have for an up-per LF T

tz:

k;

=

PI

=

ql and for a lower LFT ~~:{k, = P2 = q2; extensions to the non-square case are reasonably straightforward. An often use-ful restrietion of theset a can be obtaincd by tak-ingbounds for theoo-norrns ofthe sub-blocks of .6., with the oo-norm of a matrix M E R(s)PXq_defined as:

it is thcn possibl e to specify a set of linea r mod-els rather than a single one. Especially if this set of models is closely related to physical properties of the system under consideration it thus provides a basis for non-conservative and trustworthy state-ments on rob us t ness of controlled systems in the fa ce of 'true ' uncertainties.

For many relevant choices of the subspace a it is possib le to determine whcther all modcls within the specified set are stabie by calculating a single non-conservat ivemeasure. This measure was intro-cluceclby Doyl e in 1982 as the structured singular val ue or J1 and is based on a blo ck -di agon al str uc-tureof

a:

a

= {cliag (81hl ,...,8rhr, .6.1, .. . ,.6.j): 8;E R(s),.6.; E R(s)kr+.xkr+i}

lV,l fN

z

the coefficient matrixofthe eq uivale nt LFT can he calculated as:

with ä den otin g the la rgest sing u lar value . Ass um-ing that sealing factors are incorporat ed in the co-efficient ma tri x of the LFT, .6. is an ele ment of a unit ball in

a

:

[ Mf11 Mf 12 ]

M

_f = Mf 21 Mf22 with: (6) 11lVl1100 :=supä(M(jw)) w . (9)

2.2 Uncertainty description s with LFTs and the structured singular value Clea rly, for this coefficient matrixto be weil defined we must have det(I

+

Mil)

=I

O. A more general form of th is structure is known as the 'Redheffer Star-P roduct' (Redheffer, 1959).

Anothermainadvantageofthe LFTconce pt is that it prov ides a fram ework for uncer t.ain ty mode ll ing.

Thc coefficien t ma trix can be seen as the part of a linear model that is assumed to be correct: the nomin al model then results as an LFT on .6.

=

O. Bytakin g A E

a

with

a

C R(s) agive nsubs pace,

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Ba

= {.6.

E

a :

11.6. /100 ::; l}

For the bloc k- d iagonal struct ure of eq.8 we can now define the structured singularvalue as foll ows:

1l.ó.(M) :=

(11) min{II.6.lIoo:.6. E a ,det (l - .6.M_{Il )} = O}-I unless no .6. E

a

makes 1- .6. 1\111singular in which case It.ó.(M ):= O.

Clearly It.ó.(M) det errnin es the smallest .6.E

a

for whichthe LFT und er cons ide rat ion isnolon ger weil defined, IfF(/I'I,0) and all.6.E

a

are stab ie t rans-fer funeti on matri ces ,we mayalso interpret th is.6. Definition 2.1 Given an upper LFT with coeffi-cieni matrix M, partitioned as in eq.l and given a compatible block-diagonal slruciure as in eq.8; 1l.ó.(M) is then dejin ed as:

(7) .- M21(I

+

Mil)-1!'v112

+

M22

M21(1

+

Mld-I

-(I

+

MII)- I/1112 (I

+

1\111)-1

1vl111 Mjl2 ' -Mf 2 1 . -M/ 22 .

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-wecan write th is as (15 ) ( 16) (14) A(p)x

+

B(p )u , C(p)x

+

D(p )u , y x

Now we would like to rewrite eq. 15 using an lIpper

LFT:

S(p)= M22

+

11-121(1 - .6.uMII) -I.6. uMI 2 (17) with .6.u E BaTT (eq.13) and the matrices

kin ,1\1₂₁,J1dll,/1112 indep enden t of .6.".

lf we cons ide r only the non-trivial case that

5i

i

0, i = 1 ...r we can the n define Pi: = 1/ 5i and rewri t e eq. 17 as:

With the (n

+

I)x (n

+

m) matr ix S(p) defined as:

S(p) := (A(P) B(P)) C(p) D(p ) the parameter vector p:

as the sm allest one suc h that F(M,.6.)is unstable .

Note thatlowerLFTs can easily be rewri t t en as

up-per LFT s such t.hat wecan use the samedefinit ion; Iurtherrnore wehavethat It isalso defined ifineq . l

we take P2 = q2 = O.

With this defin it ion we now have the possibil -ity to test the prop er ti es of a set of systems by constructi ng an approp riate LF T , norm ali zin g .6.

such that .6.E Baand finally de t erminingwheth er It

:s

1. For an overview of suc h tests in the g en-era l case of eq .8 we refer to DoyIe et al. (1991).

Fur th errnore , we will no t go into det ail on corn -put a ti onalissueswith respectto It butsim ply refer torecentdev elopments as rep orted by Fan and Tits (1986, 1991) an d Young et al. (1991). Wewillc

on-centrate on a furth er restri ct ed set of .6.s that di

-rectlyresul t sfrom real valued parameter va ria tion s

in state-space models as eens ielered in sect io n 3. For this purpose we define the set of .6.s that are

square and diagonal and consist only ofreal-v alue d

repeaiedscalar blocks:

In accordance wit h the genera lstruct.ureof eq.S we

eauusuallyassumethatsealingfactors arein corpo-ra t ed in the coefficient matrixand that .6. is Iurt her restricted to the bou nded set:

Note that we now ha ve tra ns formed the problem of

find ing an LFT represent ati on of eq. 14 to an N

O-realizatio n problern (see Bose (1982)) .

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0]

]

- I - !"'[II /1112 PTI k ,. (1:3) BaTT := {diag (8,h" ... ,b,-h,.): 8;E [-1,

+

1]}

3 P

a ra m e t rie uncertainty

modelling

3.2 Existence of a sol ut ion

Using a cons t ruct ive algorithm we are now able to

prove the followin g theerem.

In this pa ragrap h we will consider the pro b lein of state-space mode ls wit h pa ram e t ri e uncer t. ain t y

occurring as real ra ti onat ND- po ly nomia ls. This

genera lizes earl ier results in parametrie uncertainty

mode lling as given by Mo rt o n and McAfoos (198.5)

and Steinbuch et al. (1991, 1992). In section 3.1 the prob1emwill be Ior m u la t ed , wh ich tUrIJS out to be an NO-real izat ion problem. Section:3.2 discu sses the existence of a solution and section :3.3 prov ides

an algo r it hmfor solving the realizat ion pro ble m by construct ing an appro p riate LFT .

3.1 Transformation of a state-space model

to an LFT

Cons ide ra vect or p= (PI,...,]JT )EW containingl' bounded scala r paramet er s. Let the model of the

per turbed syst em be givcn as a st a tc- space reali za

-tien in which theentries of the matri cesdep end on

Theorem 3.1

;\ solu ii on to ihe probl em o] tran sJor m inga

state -s pace model

with pura m et ri e uncert.ainlsj lo an LFT exisls

ij

ihe sl ate -s pace matrices can begiven as real rationel ND-polynomials in the parameters.

Proof:

Rea l ratio nal varyin g entr iesina state-spacemodel

can be describ ed as LFT s individ ually. Based on the prop erties of the int er connecti on of LF T s, treated insubsec tio n2. 1,these individ ual LFT s can bc collected in one LFTafte r wa r ds. For details on

the algorithm sec section 3.3. 0

Minimality of the obtain ed LFT ca n not be gua r-anteed since it is not straightforward to gen eralize

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to ND-systems. By for instanee Hinamoto (1980), Kung and Le vi(1977) and Roesser (1975) 20c

oun-ter pa rts of these notions are conside re d , leading to thedefinitionof local and global cont rolla b ility and observability. As is weil known,for 1Dsystems the minimality ofastate-sp ac edescriptionis equivalent

to the property that such a realization is c ontrol-lableand observable. By Kung and Levi (1977) it is

shown by means of an exa m ple that global observ-ability and cont rolla b ility does not imply minimal

-ity. However,by removinglocallyunobservableand

uncontrollable perturbations the dimensions of the obtained LFT can be reduced substantially. This

app roach has been implementeel in the algorithm

give n in the next section .

3.3 A procedure for the transformation

I ₆ 2 61 l

L

-I

_J

[

I

J

0 Pol 5 2 0 Po251

-

1 0 f . -

-

1 0 I -61 0 0 62

-

-0 0 51 52 1 0 0

-

0 1 0 _{: .}

-Fig. 5: The varying terms 111 the numerat or nij

written as LFTs

61 0 0 0

o

62 0 0 ~_~

o

0 61 0

o

0 0 62 1. Sealing the varying parameters

Lower an d upper boun d veetors for the p a-rame te r vector P can be dcterm ined, den ot ed

resp ectively as pand ij: IO\\' defin e Po = (p

+

p)/2,

8 =- (iJ - p)/ 2, Ó = (ÓI " .ór ) ,

ó7

E [-1,+ 1], suc h th~"i Pi = Poi + 8iÓj for

i = 1· · ·1'. Substi tu ti on ofthisresult in eq . 14

then gives scaled NO-poly nom ia l exp ress lons

for all vary ing numer a tors and den omin ator s.

For instan ce , su p pose a nu merato r is give n as

njj(Ph P2) = PIP2 , with PI = Pol + 81ÓI and P2= P02+ 82Ó2. Thcn the scaled numerat or is

o

0 0 0

o

0 0 0

o

0 0 0 1 0 0 0 1 1 0 1 Po2 5 1 Pol52 5152

o

PoIPo2 ~ _

2. Individual varying terms as LFTs The varying parts of a nurnerator or d

enomi-nator consist of a number of termsthat can be

written as sepe ra te LFTs. For exa m ple, the scaled numerator njj(151,152) given above has

th ree varying terms resu lt ing in three LFTs (see fig.5). Of course the sarne ca n be done with the varying terms in denominators.

3. Numerators of varying entries

Using the fact that two parallel LFTs form

again an LFT (see eq.5) the addition of all terms in each nurnerator ca n again be written

as an LFT. Sinceweuseupper LFTs,thenom

-inal (constant) part of each nurnerator results

in a feedthrough term, which ca n be in

corpo-rated in the M22 term of the comb ineel LFT. The resultingLFT giving thenurneratornjj of

eq . 19 is depi ct ed in fig.6.

njj(ÓI,Ó2) =

PoIP02+ P02Sl Ó1 + Pol 8 2Ó2 ₊ 81 82ÓIÓ2 ( 19)

Fig.6: The numer ator njj given as a single LFT

4. Denominators of varying entries

A similar procedure is followed for the denom-inators, However, there is a slight eliffer ence.

In fact we are interested in an LFT-d escription

of the inverse of the NO-polynomial d

enomi-nator. This inver secan bethought of as a feed-back structure as in figA, with the d

enomina-tor minus 1 in the feedback path. Accoreling to eq.6 th is structure can then be rewritten as an LFT under thecondition that the term Ml l+1 isinvertible. This corresponels with the restri

c-tion that the norninal parts of all the varying denominators of a state-space model must be

unequal to zero.

5. Combining numerators and denomina-tors of individual entries

Cascade con ne et ion of the LFTs of each numerator-denorninator pair found in the pr

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6. Combining all varying entries

We now hav e a comp lete descrip ti on ofallu

n-cert a in ent riesspecified in eq.14 in the form of LFTs. Combining the LFTs for the A, B,

C

an d Dma tricessep ar a t elyby mean s ofeq .5 r e-su lts in fig.7 that can bercwri ttenasone single LFT with ~ = diag(~ A ,~B,~c ,~D )'

procedure can be perform ed inter acti vely. In the next section an example is give n.

4 U ncertainty modelling for the

Phugoid

approximation of the

DHC2-Beaver aircraft.

Fig. 7: LFT description of st ate-s pace param e tri e

uncertainties

7. Transformation to the real-repeated blockstrueture

~ ca n now be rearan ged into the real-va lue d rep eated scala r block struct ure of eq. 12 by

mean s of inter changing rows and colum ns of

the LFT. Note that due to step 1 the ent riesof

~aree1ements of[- 1, 1],suc h that ~ E BA"T'

(20

)

[

u(

_{O( t)}t) ]

1]

[u(t)]

O(t)

A [ 0

[

1~(t)]O(t) y(t )

The airc raft mod el cons ielereel inthisexam ple is the linear approxirn ati on moelel of the phugoid motion anel ca n be give nin state-s pace for m as:

4.1 Modelling the phugoid motion

The design exa m p le is bas ed on the variat ions of aerodynarnic coefficients and relative mas s within the flight envelope, as they ap pear in the phugoiel

approximation of the Beav er airc raft. The phugoid

motion is a low frequen cy badly damped os

cilla-tory effect appearing in forward velocity and alti

-tude of the aircraft. For good aircraft design it is important that the effcct of the phugoid mot ion is

minimizeelto ertsure sa t isfact ory handling an d fl y-ingquality, especially under instrumentflight rules.

Also for cont rolle r design it is important to find

an accur a te ch a racter iza t ion of this effect. We will ther efor e start with thc definition ofthe analyt ica l phugoid modelin which st a bility derivati vesared e-fined tha t have becn quantified accu ra te ly over the

whole flight envelo pe by Tjee and Mulder (1988). Inour exa m ple weareint er est eelin modellingp

a-ram et er variationof the airc raft in cr uise flight co n-diti on s over the ent ire fligh t envelope . Theflight e

n-velopc represen t s a set of Ilyin g cond it ions, in term s of velocity and alti tude, under which the aircr aft ca n operate . The goal of this excrc ise isto obtain an aircraft model that accura.tely represents all fiight

co nd it io ris that may occurandth a t may beused for

stab ility anel performan ce arialysis and also ca n be

useel in robust cont roller synt hes is. We will show that once the variations are exp licit ly defined, the

model can be writ.ten as an LFT such that c alcu-lation of thestructureel sing ula r value may provide

a measure for the unwanted effect of the phugoid

motion .

u

8. Reducing the dimension of ~

The resulting LFT may now be rcduced in di

-mension ; if possible the individual repeated

blocks are replaced hy smaller blocles . 'I'he re

-duetion procedure isstartcelby sepe rat ing the

first repeated block from the deri ved LFT re -sult ing in a su bsyste m which lia s a st ate-s pace form.Theremainingpart of~may then bein

-terpretedas an uncertain tyblock act ing on th is su bsyste m . Subsequen tl y the uncontrollabl e and unobservable per turba t.ionsof the subsys -tem may be removeel using a standard (I J)) reduction technique . Rewritin g the LFT suc h that the next rep eated block is sepa ra tc d then

allows to pe rform this reduction ste p for all blocks. Although a mini mal reali zation ca n not always be obtained by t.his procedure, many exa m p les have shown tha t. an exte ns ive reduc

-tion of dimensionscan be achieved.

u

With these steps we now have an LFT description

which is equivalent to the state-space systcm of

eq .14. These steps have been implem enteel within the environment of PC Matlab such that the entire

wit.h:

._ [

21'~~)

C

CX

u

(p

,

V)

-go/V]

(13)

The state vector x = (u,0) represe n t s the l ongi-tudinal component of the velocity vector (u) and

the pitch angle (0). The in pu t vector 10 is added to demonstrate the possibilityof mod ell in g the e

f-fect of, for instanee, air turbulenceon the phu goi d motion. As a measure of the effect of the phugoid motion we assume that the pit ch ang le 0 can be measured . The ter ms

Cx.

and

Cz;

rep resent the stablity der ivatives which are known in terms of

al-titude (air dens ity p)an d veloc ity V. The accelle

r-ation of gra vity is given as 90 = 9.80 665 m/ s2 an d

the factor Jlc= p~c represents the relativeaircr aft mass with m denoting the nominal aicraft mass , S the wing area allel ë the mean aero dy n amic ch ord

of the wing profil e. The air density p is assurn ed to dep end on alt it ude h acco reling to the Standarel Atm os p here model:

Polyn o mialfit of the Iligh t envelopeas afuncti on ofó_r

.' ,---=---~

maximal order of 2are give n by: Cx ; = 5.95e- 02 b;b; - 6.37e-0 2 b; by

+

1.38e-02 b;

+

3.97e-02 bx

8;

- 3.99c-0 2 bxby - 7.95e-04 _bx - 9.9 1e- 02

8;

+

1.23e-Ol by- 0.14 (24) Cz• 1.02e-Ol b;8; - 2.90e -018;8y

+

1.83e-0 1

8;

+

1.0ge-OI 8x8; - 2.41 e-Ol 8xby

+

1.91e-0 2 bx - 2.37e-Ol b;

+

7.68e-Ol 8y - 1.43 and plot t ed in fig.9. Finall y the rel a ti ve airc raft

0.'

-

0.'

- I

~oL..:__--~~---_:_---~---....J

Fig. 8: Flight env elo pe and its ap p ro x imat ion

v

(22) ,\ = -0.0065 K/ m , [ Ta ]

~+

1

p= Po Ta+,\h with Ta

=

288.15 K, Ra

=

8314.32 J/K·k m ol and Ma= 28.9644 kg/k mo!.

4.2 Fitting the stability derivatives

1'0 obtain a paramet rie elescription ofthe stab ility

derivat ives C

_{x •}

and Cz; as a functi on of p anel V

we willuse a 2dime ns ion al pol yn omi al fittin g pr o-ced ure . Dat a regard ing the stab ilit y derivat ivesfor

several com b inat ions of panel V within the f1ight envclo pe is avail abl e from Tjee and Mulder (1988).

Alt hough the area of the flight enve lope is not square it can be approximated by means of p

oly-nornia lfit s ofthe nom inalvalue and deviati on ofV as a funct ion ofp. Bot h thc ap p roxi mat io n of the

area and the surfaces defined by

Cx.

and

Cz;

can then be givcn as a fu net ion of two new parameters bxand by varying between -1 and 1. For the a

p-proximation of the area, secend order pol yn omi al fits have been deter.milled res ul ti ng in :

h p V 4000

s.

+

6000 [ft] - 1.25e-Ol bx

+

1.03

[k

g/ m:']

- 3.57b;by

+

3..57b; - 3.50 brby

+

3.50 _bx

+

12.3 by

+

47.7 [mis] (23 )

Fig. 9: Fitteel su rfacesof Cx ; and Cz;

The flight envelo pe thus described is visu alizeel in

fig.8 as a function of bx' Polynomial fits for

e

x.

an d C

_z

•

within this area an d with term s havinga

mass ca n be detcrmin ed as:

(14)

4.3 Results

o

;

= [0 0 0 0 0 0

I

0 00 0 0 00

I

0 ( (34)

Using theprocedureofsection 3 we arriveat an up-per LFT description ofthe state-space modelgiven

in eq.20 with thirteen uncertainty inputs and out

-put s

1tÓ%6 X = Ax

+

[B_Ó1 B_Ó2

I

Bu ] 1tÓy l

(26)

To dem onstrate the possibili ty of usin g the st ruc-tureelsingular value 11 as a measure for thc worst

case influen ce of disturbance input w on the phugoid motion, we will calcu late Jl for the in

ter-conneetion structure given in eq.26 . The struct ure ofthe uncertain tyblock~forthis LFT canbe given according to eq .8 as À = diag(óxI6, óyI7, ów) wit h

óx, óythe real paramet er varia ti on s as defined

be-fore and Ów a complex per turba ti on to exp ress our elemand to restri et the phugoid moti on. We used a

prelirnin aryrelease of theMUSYN tooibox to ca lcu-lateItfor this mixeelreal -rep eated com plex probl em

resulting in fig.lO. Our cho iceofdistur bance input

YÓy7 Y YÓ%6 llÓ%6 YÓyl = Cx

+

[

D

_Ó_' DÓ2

I

Du ] llÓyl o-1.33e-02 0 2.05e-Ol-8. 0 ; e - 0 2-6.15 e - 0 1 lO" - -_. w [rad/sj Real ft for the Phugoid mot ion of the Beaver aircraft 40 JI 35

I

30 25 20 15 10 5 0 10 - 1 Fig. 10:

Upperand lower bou nds of realIJ.

5 Conclusions

'w and ou t put Y was ar bit rary and just to d emon-st rate the procedure. The value ofltof 38.6 can be

interpreted as the maxim al amplificat ion occur ring In the tra ns fer functi on matri x Irorn w to Yunder the worst case cond it ions within theflight envelope

and with the worst case disturban ce w (a sine of

frequ en cy 0.27 rad / s l oA cho iceof inputs and

out-puts bas ed on a physical interpreta ti on of dist ur-bances and the desir ed su ppress ion of the phugoid

mot ion is cur re nt ly a research topic at the facul ty

of aerospace enginee ring.

(28) (27) (32 ) (29) (:30) o-1. 4 8 e-Ol 1.81e-01 0 o 0 o 0 o 0 o 0 -3. 0 5 e- 0 2 -5.88e-03 -lA 8 e- 0 2 5.0 8 e - 0 2 -4. 1 1e-03-IAQe-OI -1.8 5 e-0 2 0 -1. 48 e-0 2 0 -3.87e-03 0 J.OOe-D3 0 o 1 o 0 0 0 0 0 -7. 34 e- 0 2 0 l.DSe-OI 0 0 0 o-1.2 2 e- Ol 0 0 0 0 -7.14 e-Ol 0 0 0 0 0 o 2.37e-Ol 0 0 0 0 o 0 0 5.57('-01 0 0 o 0 0 0 5.20(' Ol 0

-1. 42 1e-0 2 5.2ge-02 2.03e- 02 1.90e-OI l.iJe-OI 2.IGe-OI 1.23e-01 1.9 8 e - 0 2 -1.ï5e-OI-5. 0 0 e - 0 2 1.12e-OI 1.3 6 e-Ol -3.3 74 e -Ol 4.9 6 e-0 3 4.8 2 e - Ol -2. 6 2 e-0 2 3.36e-02 4.04 e-0 2 o 8.0 Se - 0 2 0 4.l.S ~-O I-3.20t'-Ol -2. 81~- OI o 3.05~-02 0-4.24~-0 1-2.8i ~-01-2.7 B ~ - 01 o-6.45 ~ - 0 2 0 r.r se-or 2.16e-02 2.55e-01 BÓ, = [

-1.0 1 8 e-Ol 8.25e -03 J.45e - Ol 1.8 ge-02 9.92e -03 2.oie- 0 2]

o-1. 2 2 e-Ol 0-4 .4 7e- 01-1.5 4e- Ol-2.ï4e- Ol B [-1. ]79.61e-Ol 1.94e-020 0

00 ]

62

==

3.0 4 3.70e-Ol 6.68 e - 0 300 00 A - [-3.47e -02 -2. 0 6 e-OI ] - 3.5 4 e-Ol 0 with :

c

=

0 0 0 0 0 0 0

-1. 0 2e- 0 2 8.77e-02-2.41e-01 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-1.33e-Ol -a.ss- 01 1.5 0 e-Ol 2.01e-Ol 1.50!.>e-Ol 0 0

1.69~-01 4.0 6 ~-0 2 1.0 1e-Ol 3.56 ~-0 1-7..Sg e-02 0 0

5.64e-02 1.25e-Ol -2.14e-Ol 1.21 e-01 -3.41~-02 0 0

3.27e-02 1.0Ie-Ol 3.53e-02-4.9 5 e-0 2-9.58~ -02 8.18~-02 8.6!.><, -02

-1.80e-02 -2.1ge-01-7.88e-02 3.4ge-02-1.65~-023.34e-Ol-2.12 e-0 2 -3.86e-02 -4.12('-02-1.34e- 02-4.40e-02-1.5 7 ~ - 012.64~-01 2.72t'-01

2.24e-02-1.3 9 ~ - 01-5.14~-02 1.62e-01 8.3&~ 02 2.62t'-0 3-1. 4 ge Ol

(33)

The developmen t of meth od s for analys is and d e-sign bas edon thestr uct uredsing u larvalueJlcauses

an incr easin gdem a nd for the const r uctionof

accu-rate uncer t ainty modelsin the form of LFTs. Us

u-ally the knowled ge conce rn ing unce rtainty in math -em at ical models of physical sys tems is available in terru sof paramet er variaticris. In st ate-space m od-els this often appear s as variations ofent ries that

(15)

can be approximated accu rately by mean s of r a-tios of ND-poly no m ia ls in ind ep e nd en t variab les wh ich have a phys ical int cr pre t a ti o n. Inth is pape r

an algo rithm is presen t ed wh ich is used to tra

ns-form a state-space mod el wit h this type of para-metric uncert ain ty to an LFT descript ion wit h a

real-repeated perturbation matr ix. Alt ho ug h thc

dimension ofthispcrturbat ion matr ix may in itially bc very high , a red u ction proced ure is pro posed

that usu all y decreases it sig n ifica n t ly. Ilowever , thisproceduredoes not guarantcemin imalityof the resul tin g structure. The proposed proced ure has been ap pl ied to the uncer t ainty mod ellin g for the

phugoid app rox imat io n of the DI-IC2 -B eav e r a

ir-craft resultin g in an LFT description allo wing the analysisoftheinfluen ce ofdis t ur bancesove rtheen -tire fligh t envelope. The pro cedure hasbeen i mple-mented in Mat Lab , such that unce rta inty mod els can be set up in an int er a cti ve user-fri endl y man-ner.

Ackn

ow

ledgements

Maarten Steinbuch want s to tha n k Ma th ew Newl in

and Pet er Youn g , both fro m Ca liforn ia !ns ti tu t e of Technology, for pro vidin g a tes t vers io n ofthe ir I

t-arialysis software for real and com plex structures. The auth ors also wish to than k Paula Roch a from Delft Unive rsi ty of Techn ology Iac. of Math ern

at-ics for her hel p in ga in ing a be tter unelerst.aneling of development s in the area.of 2D and ND syste m

rep rese ntat io ns.

This research was spo uso re d by Philips Research

Laboratori es in Eindhov en , Delft. University of

Tech no logy and Nat io nal Aer osp ac e Laberat ory

(NLR) in Amst erdam.

References

Bal as G.J ., Packard A., Doyl eJ.C. (1990 ). Theory and applications of ro b ust mu lt ivariable cont ro l.

in Hoo and It Short Course. i\lusyn inc., Delf t,

june 25-28.

Bose N.K. (1982). Applied niullidimen sional sys

-tem th cory. Van Nostrand Reinh eld Co., N.Y .,

Doyle J.C . (1982). Analysis of feed back systems with structured uncer t a in ti es . lEE Proe.• Pari

D, Con tro l Theor y and Applical ious , vol.129,

no.6(nov), pp.242-250 .

Doyl e J.C ., Packard A. (198 7). Uncer tain mul

ti-varia b ie systems from a state space pe rsp ect ive.

in Proc. of the Americau Con trol Confe rence,

pp.2147- 21 52.

DoyleJ.C. ,Pa cka rd A.,Zhou K.(1991). Revi ew of LFTs, LMIs an d lt. in Proc. ofili e IEEE Co/!

-[erence on Decision and Control.pp.1227-1232.

Doy le J.C . , Len z K., Pa cka rd A. (1986) . Design examp les using Il-synthesis: spaceshutt le la t eral axis FCS during reen try. in Proc. of the IEEE

Conference on Decision and Control. pp.22

18-2223.

Fan M.K.H., Tits A.L. (1986). Characterization

and efficie nt computat ion of the structured sin-gular value . IEEETrans. on Automatic Contro l.

vol.AC-3 i, no.S, pp.734-743.

Fa n M.K .H. , Tits A.L., Doyl eJ.C . (199 1). Rob ust

-ness in the presence ofmixed parametric uncer-tainty and unm od ell ed dynam ics. IEEE Trans. on Automat ic Control, vol A C-36, no.1, pp. 25-38.

Hinam ot o T. (1980). Reali za ti on of a state-space

model from 20 inp u t outp ut map. IEEE Trans.

on Circuits(j' Systems, vol.CAS-27,no.1,

pp.36-44.

Kun g S., Levi B.C . (1977). New results in 20 sys-tem theory, part II:2D state-space models rea

li-sarionsand the notions ofcontrollability,

observ-ability and min imality. in Proc. of the IEEE,

vol.65, no.6, pp.94.5-961.

Mor t on B.G., McAfoos R.M. (1985). A ft-t est for robust ness arialysisofa real-para m e ter var iation problem. in Proc. of the A merican Contro l Con

-ference, pp.l3.5-138 .

Reelheffer R. (1959 ). Inequ ali ti es for a matri x Ri

c-cat ieq uat ion. JournalofMathematics and M e-chan ics, vol.8, no.3.

Roesser R.E. (1975) . A discr ete st ate-s p ace model

Ior linear imageproces sing. IEEE Trans. on

Au-lomedie Con trol vol AC-20 , no.1 , pp.1-10.

Skogest aelS.,Mor ari M.,DoyleJ.C. (1988). Robust cont ro l of ill-condi ti oned plan t s: high -purity di

s-tillat ion. IEEE Trans. on Auto mat ic Cont rol.

vol.AC-33. no.12, pp. 1092- 1105.

Stein G., Doyl e .J.C. (1978). Singu lar values and

feed ba ck: designexamp les . in Proc. 16th Annual

Aller/on Conf. on Communication, Control and

Compuiaiion, Univ. of Illin ois,pp.460-471.

SteinG., DoyleJ.C . (1991). Beyond singu lar valu es

and loop shapes. Joum al of Guidance, vol.14, no.1, pp .5-lG.

Steinbuc h

M. ,

Terlouw J.C., Bosgr a

O.H

.,

Smit

S.G . (1992). Uncertainty modelling and st

ruc-tureelsingu lar value cornp utat ion app lied to an

electromechan ica l system. lEE Proc., Part D, Control Theor y and Applicaiions, vol.139, no.S, pp .301 -30 7.

Steinb uc h

M

.,

Terlo uw J.C., Bosgr a

O.H

.

(1991) . Hobustn ess arialys is for reaIand complex p

(16)

lil Proc. of the A in eri cati Control Co nfe rence, Boston, pp.556-561.

Tjee R.T.H, Mulder J.A . (1988 ). Stabili ty and cont ro l derivatives of the De Havillancl DHC-2

'BEAVER' aircraft. in Techn. Univ. De"!! , R

e-port LR-556.

Young P.M. , Newlin M.P., Doyl e J.C . (199 1). 1I analysis with real parametri e uncertainty. in Proc. of the IE E E Conferen ce on Decision atul Control .pp.1251-1256.

(17)

@1992 Delft Uni versity Press _S_{elec ted}_Top_{ics in Id}_{entification ,}_M_{od elling a}_{n d}_Co_{n t ro l}

Vol. 5,December1992

R

obustness

of feedback systems under simultaneous

plant and controller perturbati

on

Peter M.M. Bongers

Mechanical Engineering Syst ems and Control Group

Delft University of Techno/agy, Meke/weg2, 2628 CD De/ft, TheNetlie rl atids .

Abstract. The aimof thispap er is to deriv e a newrobust stability margin for simulta-neou s perturbat.ionof plan t and cont rolle r which is less conservative than thegap-metric robu st ness. Known sufficient conel it ions for robust stability stateel in the gap-met rio

contain inheren t conse rvat ive ness in the formu lation ofthe vari ou s steps . In this paper conservative ness in one of the steps is removeel, resulti ngin a new an el less conse rvat ive robu stn ess margin . The key issueis that mor einfor m at ion ofthenominal feeelbacks ys-temis taken int o considcration. The im provement ofthe new robustn ess mar ginwill be

illustrated by an example.

Keywor els. robu st ness, sirnultaneous perturbations, coprime Iactorizations, gap-metric

1 Introduction

A perfect mod el ofthe real pla nt, if it is available, will in gene ra l he non -lin car and ofext re me ly high

order. In enginee ring practice the plant will be a p-proxirn ated by a low oreler linea r moel el. The di s-crepancy between the nomina! moelel and the plant

is the n elescribeelby asetofplan t uncert .aintym oel-els. In the next ste pa cont rolle r will he synt hes ized insucha way that it robnstlystabili zes thenorninal modeland the set of plantuncertain tymoelels,fora

pre-sp ecifieelperforman ce; Meth od s to elesign such

robust cont rolle rs are for exam p legive n in (Doyle

et al.(1989), McFarianeanel Glover (1989), Bonger s

and Bosgra (1990)) .

When dealing with industri al processes the

con-trolcom pu ters are calcula t ing in finiteworel length arit hmet ics or even integer arit hmet ics, while for the cont roller limiteeltime anel space on the com -puter is available, Therefore the implem enteel c on-troller is only an approximation of the elesign eel cont roller . The discr epancy between the design eel cont roller and the implcm ented controller can be

elescribeel by a set ofcont roller uncertaintymoelels.

Thefeeelback con nee t ion of plant moelel and c

on-tra iler will be called robustly stable ifthe feedb ack

syste m rem ain s stabie for all plant va riat ions d e-scrihe el by theset of plant uncert ain ty moelels anel

allcont rollervariat ions elescribeelby the setof con-troller uncert ain ty mod eis.

In sorne recent pa pers by Georgiou and Smith ( 1990a), Bonger s and Bosgr a (1990) a sufficient condit ion for robu st stability of a closeelloop

sys-tem has been stateel for plan t perturbations m ea-sureel in the gap-metr ic. In Georgiou and Smith (1990b) gap -m etri orobustness under simultaneous

plant and controller perturbations has been s

tud-ieel. In the ga p-me t ric robustnessthe nominalplant (controller)isfactorized in normalized coprimef

ac-tors. The differ en ce between a perturbed plant

(controller) and the nominal plant (controller) is described by perturbations on the normalized co

-primefactors of thenominal plant (controller). Ro

-bustnessof the closed loop for a class of perturbed

plantsand cont rolle rs is guaranteed if the norm of the perturbations on the normaliz ed coprime fac

-torsissm allenongh. Themaximum allowablenorm of the perturb ati on s is det ermined by the infinity norm of the feedback syst em. This means that in the gap- me t ric robustness only cru de information

(18)

Defi n iti on of distance measures

M* M

+

N*N

=

I (M M*

+ IVIV*

=

1)

with M*= MT₍

_{-8) .}

The gap metric distance (El-Sakka ry (1985),

Zames and El-Sakkary (1980)) 8(Pt,

P

_{2 )} between

the two plants is defined as

Thepair M,N(M,

IV)

is anormalized right (left) coprimefraetionalrepresentation(nrefornlef)

if itis a coprimefraetionalrepreseniaiionand:

max{d(P

I ,

P

2 ),

d(P

2,

Pd}

QE1t.\~~I

=~1

I

1[

~:

]

-

[

~:

]

QIL

d(P

I , P2 ) i(PI ,P2 )

P = NM-t ₍₌

_M-

t

_IV)

The pair M,N(M,

IV)

is right (left) coprime[ra

c-tional represeniaiion (reforlef) if it is a right (left) fraetional representation and ihere exists U,V(Ü,V) E H such thai:

UN

+

VM

=

I

(IVÜ

+ MV

=

1)

Suppose PI, P2 are two plants with nref

(NI,Md,(N2 ,Nh) respecti vely, and suppose C is a co nt rolle r such that T(PI,C)E H with

(

X, Y

)

a nlefof C.

The graph metricdistance (Vidyasagar (1984) )

d(PI,P_{2 )} between the two plants is defined as

Defini tion 2.1 ( Vidya sagar (1982))

A plant P EF has a right (left) fraetional repr

e-sentation if there exist N,

M(IV

,

M) EH such that

about the norninal feedback system is taken into

considera t ion .

The main idea behind the new and lessconserva

-tive robustness margin,to be conside re d in this pa-per ,is to take into account moreinformation about

the nominal feedback system. Onecan think of this inforrnation as refinement of the infinity norm to a

frequency dependent maximumsing u la r value, and

the directionalityof the feedbackloops in mu

ltivari-ablesystem s.

In order to takeaccount of the c!osed loop cha

r-actcrist ics a norrnalizedcoprimefactoriza tionof the norninal controller (plant)is used to define aspecific cop rime factorization of the norninal plant (con

-trolIer).

The difference between a sim ult a ne o usly per

-turbed plant and controller

(Pt>.,

C

t>.)

and thenom

-inal plant and controller (P,C) is now described

by perturbations on the specificcopr ime factors of

the nominal plant and nomin al cont ro lle r which

in-clu des detailed information about the nomin a] c

on-tro llerand plant resp ecti vely.

It will be shown that th is ne w rob ustn essmargin allows a larger c!ass of co pr im e fa ct or pla nt and

cont ro lle r perturba tions than allowed in the ga p-metri c.

The layoutof th is paperis as follows: afte l'some preliminaries in Section 2, stab ility of a nominal

c!osed loop system is discu ssed in section 3. Then

the new robustness margin will be derived in S

ec-tion 4, whereit willbe demon strateelthat this mar

-gin is less conservative tban thc gap-metric. Tbe

wbole procedure ~ill be illust r a t ed by an example in Section 5 foll owed by tbe conclusions

The A-gap margin 8A(Pt,P2 ) between the two plants is defined as

where A =

[X

y]

[

~:

].

Note that the nref

of P, is shaped with A- t to account for the closed

loop operation of tbe plant . Ifthereis no controller,

A-t willbe Mt-t, as can be checked easily, and the A-gap will be:

max{l(PI,

P

2 ) ,l(P2 ,

Pd}

J~L

11 [

~:

] [

~:

]

Qll

oo

8(PI,

P

2 )

l(PI,P2)

2 Preliminaries

In this note we adopt tbe ring theoreticsettingof Desoer et al.(1980) and Vidyasagar et al.(1982) to

st u dy stabie muItivariabIe Iinear systems by con-sidering them as transfer function matrices hav

-ing all ent r ies belonging to the ring 1{, In this note we will identify the ring H witb IRHoo, the

spa ce of stabie reaI rational finite dimensional

lin-ear time-invariant continuous-time systems. We

cons ide r the class of possibly non-proper and/or unst able muItivariablesystemsas transfer function matrices whose en tries are elem ents of the quo

-tientfield

F

of

H

(

F

:=

{a/b

I

a E

H

,

bE

H

\ O}).

The set of multiplicative units ofH is defined as:

.J

:=

{h

E

H

I

h- I E H}. In the sequel systems

(19)

3 Closed loop stability

In this paper we will study the closed loop stabi

l-ity according to Fig. 1, where we assume that a stabilizing controller

C

has been designed for the nominalplant P.

el + u _p Y

+

C -₊ e2

Fig. 1: Closed loop structure

The closed loop transfer function T( P,C) map

-ping the external inputs (el,e2)onto the outputs (u, y) is given by:

T ( P,C) = [

~

] (1

+

CP)- I

[

J

C]

For bounded exogeneous inputs (el,e2),stability

of the closed loop , i.e. the controller C internally

stabilizes the plant P, is guara nteed ifanelonly if T(P,C) E 1{. Now let P = NM-I _wi_th _(N,_M₎ _a refof P an d let C =

X-

I

Y

with

(

X,

Y)

a lefof C then:

Theorem 3.1 (Vidyasagar et al.(1982))

Let

P

E :F

b

e

given as

P

=

N M-

I _wi_th

₍

_{N, M)}

a refof Pand let the con tro lle r C E:F be given as C

=

X-

Iy

with

(X,Y)

a lefofC or as C

=

YX

-

I

with (Y,X) a ref of C. The n the following st ate-ments are equivalent :

1. The closed loop is stabie

2. ( [

X y

] [

~

])

= 1\E

J

3. U:=

[

~

- : ] E

J

For robuststability it is essent ia lthat the closed loop transfer functi on rem ain s sta blefor plants P6

"close to" P cont rolledby cont rolle rsC6 "closeto"

C. Usually the cont ro llerCiselesign ed with kn owl-edge of

P

only.

4 Main result

In this section a sufficient condition for feedback

stability uneler simultaneous plant and controller

perturbations is presented. As a by-p roduct we have sufficient conditions for only plant pe

rturba-tions or only controllerperturbations.

Next it will be shown that this robustness margin

is less conservative than a simila r margin baseel on the gap-metricdistance between the nom inalplant

and perturbeel plant summed wit h the gap-met ric

elistance between the nominal controlle r and the

pertur be d controller, as elerived by Georgiou and

Sm it h (1990b) .

This will immediately imply that the gap-met rio

robustness mar gin is also less conservat ive than a marginbased on graph-metricelistances, asder ived by Viel yasagar and Kim ura(1986)

Fina lly it will be shown unde r what conelitions the new robu st ness marginequalsthe gap-metricbaseel

robustness margin.

Theorem 4.1 Suppose T(P,C) is siable, and that

both plant and controller are perturbed to P6,C6 respectiveiy. Then all pairs (P6 ,C6 ) form a stabie

closed loop T(P6,C6 ), provided:

Proof: Let

(Nt

M

), (Y,_

Xl

be nref of

P

,

C, resp ec-tive lyand let(M, N), (X, Y)benlefofP,C,resp

ec-tively. Then frorn stabilit y of T(P,C) the matrix

[M-Y]

U

=

N

X E

J.

Denote

U

=

[M -Y]

[1\-1 _0 ] o

N

X

0 1\-1 wh ere 1\

=

[

X y

] [

~

]

,

À

=

[1\1

N

] [

~

] .

ThenU;;I can be partitioneelas

U-I = [

X_

~

]

o -N M

Now let (N_{6 ,}M_{6 ) ,}(Y_{6 ,}X₆ ) be refof P6 ,C6 , r e-spectively . Accor din g to Theorem 3.1 T(P6 ,C6) E

'Hifand only if

Select real number s

s;

>

8A

(

P

,

P

6 ),8c

>

8

A

(C,Ct>,)

(20)

Now if

11 (1 -

U~U; II")Q

<

1 the n (accord ing to Lemma A.I) U~ E.:J and thereby the pair(P~, C~) is stabIe. U;- U~

=

[A B],

where

Now assembling the pieces, usin g the definiti on of the gap-metric and Lem m a A.3 we have that

ó/\(P,P~ ) S;

sçr,

P~)IIT(P,

C)lIoo

A = [[

~

] A- I - [

~:

] ] B

= [[ -;

,

.

]

A

- I - [ - : : ] ]

[

A

B]

U

o -I A

[

je

)

/]

+

B

[-

N

Û J

Along the same lines we have

Then usi ng the fact that 111'(

C,

P)

1

00

111'(P,

C) 1

1

00

(Georgiouand Smith (1990b)) we have that

8 /\

(P,P~)

+

8 /\

(C, C

~) S; (8(P,P~)

+

si

c,

C~))

II

T(P, C

)ll

_oo

5 Example

In this section the applicat ion of the presented 1'0

-bustness margin will be illustrated using an exam-ple. For simp licity only SISO systems are

consid-ercd, which implies that the improvement of the new robustness marg in by taking int o account di-rectionalityof the feedback loops can not be demon-strated .

In Fig. 2 the frequency respon ses of both the nomi nal plan t model

P

of order 5 and a per tu rb ed plan t mod el P~ are sho wn .

Using the control design method described in Bon ger s and Bosgr a (1990) a cont rolle r C of order 3 has been designed on P suc h that IIT ( P,

C)

ll

_oo

is minimized. In Fig. 3 the frequen cy resp on ses The transfer function A-I can be seen as a weighting function on the gap be t ween the nom -inal pla nt an d the pert urbed pla nt . Only when

A = al, with a E IR, the extraction of A-I will not introduce conservatism . In that case the gap-met ri c robust ness isnot more conservat ivethanthe new robustness margin.

It can be show n (Bongers (1992)) that a spe-cificchoiceof the controllerorder in a lRI-I oc-nor m design based on normalize d coprime fact or iza t ions will lead to A

=

al.

However in gene ral the ma xi m um sing ula r value of 1\ will be frequ en cy dep end en t. For mu lt iva ria ble cont ro l design s the sing ular valu es ofAwill in gen-era l not be eq ual, which means tha t in A also d i-recti on ality will be presen t .

The presented robustness mar gin tak es both the featuresof directionality and frequen cydep enden cy int o account. This im pli es that the present ed 1'0

-bustnessma rgin haspracti cal ben efits compare d to the gap-me tric robustness mar gin.

o which proves the theo rem

Theorem 4.3 Suppose 1'(P,C) is stable, then a sufficient eondition [or ó/\(P,P~)

+

ó/\(C,C~)

<

is !Jiven in the gap-metrie by

ó/\(P,

P~

)

J~L

I

1 [

~

]

1\- I - [

~:

]

-l

=

J~~II([

~

] [

~:]

0 )

A-111

oo

<

J~~II(

[~]

-

[

~:

]

O) ll

oo

II A- 11I

oo

ó(P,P~ )

+

sic,

C~)

<

IIT(P,

Cl

11 :.

1

and

Corollary 4.2 Suppose 1'(P,

Cl

is stabie. There holds Proof:

II[

A B ]

U;

llloo

<

I

IA

[

je

f

'

Jll

_oo

+

1

B

[

-N

i

!

J

1

100 <

s,

+

Óc if Ó/\(P,P~)

<

1 then T(P~,C) EH

In the next theor em it will he shown that a s uf-ficient cond it ion for stab ility accord ing to Th

eo-rem 4. 1 ca n be stated in term s ofgap- met r ic di s-ta nce . Ther eb y we will show that Theor em 4.1 is a gene ralizat ion of the gap-metric ro bu s tn ess

The robust ness resul ts of (Bongers (1991

l,

Schra ma et al.( 1992)) are a specia l case of Theo

-rem4. 1, which can be seen in the next corollary. Now ifóp

+

s

,

<

1 the n U~ E

J

which proves that T(P~,C~) EH,which cornpletes the proof. 0

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of both thc designed cont roller C and a perturbed cont roller Có are show n.

Ifthe robu st ness is measured in the gap-metric, the closcd loop system T(Pó ,Có ) (Theor em 4.3)

rem ai ns stabIe providcd 10' <; 10' _-₁₀₀ 10' ·200 10° -300 u 10,'

~

"0, ~ -400 Ö. ₁₀_" ., ~

-a.

-500 10') 10· -600 10" ·700 '-._0'.... 10· -800 10° 10' 10' 10" 10" 10' 10' 10' 10" 10"

Irequen cy[rad/st Ireq uency[rad/sj

Fig.

2:

Frequencyresponse P

(

-)

,

P

ó (- -)

ScP, Pó )

+

S(C, Có ) ~ "T( P,C)";:'1

Thegap between the nominal plan t and perturbed

plant is:

ScP, Pó ) = 0.19,

and the gap bet ween thenominalcont roller and the perturbed cont roller is:

S(C,Có ) = 0.14

The nom ina l plant ,controller pair imply a robust-ness mar gin of:

IIT

(

P, C)11:;,1 = 0.1 It is obvious that even the individual perturbations do not sat

-isfy the robustn essmar gin, therefore stabilityof the perturb ed feedback syste m can not be guaranteed. Next the refinem ent of the new robustness mar-gin will be shown. The improvement of the new robust ness mar gin, by taking into account the fre

-quen cydep enden cy ofthe feedback system, is ilIus-trat ed in Fig. 4.

Fig.3: Frequ cn cy respon se C (- ) ,Có (- -)

10' 10'

10' Irequency [radIs]

10" Fig. 4: Frequency resp onse

a

(A )

(

-)

"T(P,C)II:;,I (- -) 0.5 0.45 0.4 0.35 u 0.3 "0 .è ö. ~ 015 0.2 0.15 0.1 0.05 10"

Suppose, for a moment, only plant p erturba-tions are present. The n stability ofthe closed loop in gap-metric sense is gua ranteed provided that

S(P,Pó ) is smallerthan the dash ed linein Fig.4.

Therefinemen t towards thcnew robustnessmar

-gin can be seen as follow s: The frequency where

thelarg est differ cn cebetw een Pand Pó liesis not

taken into account in SCP, Pó ) , it is in SA(P,

Ps).

Let

Q

be theoptimal solut ion in Theorem 4.3,then

Irequen cy[rad/si ·200L....~~~~~... 10' 10" 10° 10' 10' 100 .. so

c

-:

0 '" u ~ _-so ~

-a.

-100 ·ISO Irequency[nul,,!

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T(Pb" C) is stabIe if

6 Conclusions

Bon ger s P.M.M. , Faet oriz ation al Approach to

Ho-bust Conirol: Application to a Flexi ble Wind Turbine.PhD Thesis , Delft Unive rs ityof Tech-nology, The Netherlands, In preparat ion, . Bongers P.M. M. (1992). On a new robust stability

margin. in Recent Advances in Mathematical

Theory of Syst ems, Control. Neiuiorks and Sig-nal Processing, Proc. of theInt. Symp.

MTNS-91, H. Kim ur a , S. Kodarna (Eds.), 377-382.

Bongers P.M .M ., O.H. Bosgra (1990) . Low order

H_oocontrollersynthesis. Proc. 29th Conf. D

esi-cion and Control.Hawaii, USA, 194-199. DesoerC.A., R.W . Liu,J. Murray, R.Saeks (1980).

Feedback systems design: the fractional repre-sentatio n approach to arialys is and synthesis.

IEEE Trans. Automat. Conir., AC-25, 399-412.

Doyl e J.C ., K. Glover, P.P. Khargonekar, B.A. Fran cis (1989). State-space solutio ns to stan-dard Hz and Hoo problems. IE E E Trans. Au-tomat. Conir., AC-34 ,831-847.

El-Sakkary A.K. (1985). The gap metric: robust-ness ofstabilization offeedback systems.IEE E

Trans. Automat. Contr., AC - 3 0,240-247. Georg iou T.T., M.M.Smith (1990). Optirnal

ro-bustness in the gap metric. IEE E Trans. Au-tom at. Conir., AC-35 ,673-686.

Georg iou T.T. , M.C . Smith (1990). Robust con

-trol of feed b ack systems with combined plant and cont ro ller uncer t a inty. Proc. A mer. Conir. Conj., 2009-201 3 .

McFari an e D.C., K. Glover (1989). Robust

Con-troll er Des ign Using Normalize d Cop rime

Fac-tor Plant Des cript ions. Lecture Notes 111 Cont rol an d Infor m a ti on Sciences, vo1.1 38, Springer Verl ag ,Berlin , German y.

Schrama R.J.P., P.M.M . Bon ger s , D.K. de Vries (1992).Assessment of robu st stability from ex-per imental dat a. Proc. Amer. Conir. Cotij.,

References

The derivation of a new rob ust stabilitymargin for simultaneous perturbation of plant and controller

has been presented. It has been shown that this margin is less conservative than similarrobustness margins stated in the gap-metric or graph-metric. The im provemen t ofthe new marginliesin the fact tha t fre que ncy de pende ncy ofthe feedb ack sys~em and directionali t y of the feed ba ck loops are taken int o account. The applicat ion of this robustness margin has been illustr a t ed by an example.

freque ncy[rad/sj

.sero' 10" 10" 10I 10' frequency [rad/s j

and thelam b da- gap between thenorninal controller and the perturbed controller is

lO"

r

~..._---._

···

----

~

···· ··· ··

_..

· ···

· _·

. I

10. 1

~. ",-_.-

-It can be seen easily seen that robu st stability of the per turb ed feed b a ck system is gua ra nteed by the new robu stness marg in.

For com ple teness the Irequen cy response of thc (2,2)-elemen t of the feedb a ck syste m T(P,C) IS given in Fig. 5.

10')

Fig. 5: Frequen cy response P(l

+

Cp)-I C (- ) ,

Pb,(I

+

Cb,Pb,)-I Cb, (--)

Thcla mb da-gap betweenthc norninalplantand the pcrturbed plant is

IS smalle r than

o-(A)

,

then T(Pb"

C)

IS st a bIe. The re by the area of allowable P~s is exten de d t o-wards the solid curve in Fig. 4.

When the stability robustn essis measured in the new robustness margin (Theorem 4.1), the per -tur be d closed loop T(Pb" Cb,) remains stab ie pro -vided :

Next if the differ en ce between P and Pb" define d as

(23)

Chicago, Illinois, USA, 286-290.

Vidyasagar M. (1984). Thegraph metri cfor unsta -bIe plants and robustness est irnates for f eed-back stability. IEEE Trans. Automat. Contr., AC-29,403-418.

Vidyasagar M., H. Kimura (1986). Robust con-trollers for uncertain linear multivariable sys-tems. Automaiica, 22,85-94.

Vidyasagar M., H. Schneider,B.A. Francis (1982). Algebraic and topological aspects of feedback stabilization. IEEE Trans. Auiouiai. Contr.,

AC-27, 880-893.

Zames G., A.K. EI-Sakkary (1980). Unstable sys-tems and feedback: the gap melric. Proc. 18th Allerton Conf., 380-38.5.

A

Pr

oofs

Lemma A.I Givena transferfunet ion H EH. If

l

il

-

Hl

loo

<

1 then H E

J.

Proof: For an arbitrary fun ction F E H, a su ffi-cie nt con d it ion for

I - F

to ha ve a stab ie inver se is give n by the small ga in cond it io n 1/

F

II'X)

<

I.

Define

H

:=

I

-

F

and the lemm a is proveel. 0 Theorem A.2 Let(N, M)bearcf of P and(Y, X)

bearcf of C, then T(P,C) E H ijand only ij

[

M-

Y]

U= IV X E

J

.

Proof: Vidy asagar an d Kimura(1986 ), lemm a .5.1 o Lemma A.3 Let (N,M),(Y,X) be nrcf of P,C. respeetive/y and iet

(Nt,N)

,

(.':J')

benlcf of P,C, . i D

:ft

A -_

[x

-

r ,,;. ] [ iNH ] ,A --respeetwey. eme a I a

[Nt N]

[

~

]

,

then I/T(P,

C)ll

oo IIT(C,

P)ll

_oo

Pr o of: Using the normalized coprime factoriza-tions for P,C,T( P,C) can beex p resse d as

The fact that [

~

]

is a nrcf , [.); }/ ] is a nlcf

and

(

XM+ YN) -I

= />.-1 proves the firsl part.

Selected topics in identification, modelling and control: Progress report on research activities in the Mechanical Engineering Systems and Control Group. Vol. 5

-

-

-.3/

j .

t