Selected topics in identification, modelling and control: Progress report on research activities in the Mechanical Engineering Systems and Control Group. Vol. 7

-

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 d

en

Hof

Volume 7, December 1994

Mechanical Engineering Systems and Control Group

Delft University of Technology

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 Cont ro lGro up Delft University of Technology

Mekelweg 2,2628 CD Delft The Netherlands

Tel.: +3 1-15-786400; Telefax: +31-15-784717 email:arkesteijn@tudw03.tudelft.nl

CIP-GEGEVENS KONINKLIJKE BIBLIOT HEEK, DEN HAAG Selected

Selected topics in identification, modelling and control:

progress report on research activities in the mechanical engineering

systems and control group.-Delft:Mechanical Engineering Systems and Control Group, Del ft Universityof Technology,Vol. 7-ed.by O.H.Bosgra and

P.M.J.Van den Hof.-ill. Metlit.opg. ISBN 90-407-1077-5

SISO656 UDC 531.7+681.5NUGI841

Cover desig nbyRuudSch ram a

©1994 Copy rightDelftUniv ersityPress.All rights reserved.No part of this jou rna I may be reproduced, in any formor byany means, without written permission from the publisher.

Contents

Volume 7, December 1994

Robust performanceH2/Hoo optimal control M. Ste in buch and G.H.Bosgra

A unified approach to stability robustnessfor uncertainty descriptions based on fractional model representations

R.A . de Cal/ af on, P.M.J. Van den Hof and P.M.M. Bongers Solvab ility tests for the Lyapunov inequality

C.W. Schere r

An iter ative algorit hm for freque ncy- weighte d H2-nor m optim al redu ction and eente ring P.M.R. Wortelboer and G. H. Bosgra

Filtering and para met rization issuesin feed back relevant iden tification based on fractionalmodel representations

R.A . de Cal/ afon an dP.M. J. Van den Hof

1

9

17

23

35

Aninstrument al vari abiepro cedurefor theidentifica tion of probabilistic frequ ency responseuncertaintyregion s

R.G. Hakvoort and P. M.J. Van den Hof 45

Identifi cation and robust cont rol design of an industrialglass tube manufacturing process

D. de Roover and R. G. Hakvoort 53

Closed loop system identification of an industrial wind turbinesystem and a preliminary validation result

G.E. van Baars 63

Closed-loop identifi cationof a continuous crystaIIization process

R.A. Eek, J. Both and P.M.J. Van den Hof 71

Inner loop design and analysis for hydraulic actuators, with application to impedance con trol

J. Heintzeand A.J.J. van der Weiden 83

Stability analys is of a hydraulicservo-systemincluding transmission line effects

G. van Schothorst, P.C. Teerhuis and A.JJ van der Weid en 93

Zer o-ripple torque cont ro l in brushless DC motors

H. Huisman 101

Predictive servo control through optimal feedforward com pensat ion

search project isa collaborative effor t betw een three university researchgroupsof Delft University, indus-try and the national aerospace research laboratory NLR. Finally,the paper by Henk Huisman discuss-es a new approach to torque ripple minimization in brushless DC motors. In precision drive systerns such as video scanners, it is of utmostim por ta nceto have mass produced, cheap motors disp laying high servo controlperformance.

Robustcontrolissues are discussed in several paper-s. Some preliminary results on an uncompromised 11.2/1ioo optimal con trol problem are given in the

paper by Maarten Steinbuch and Okko Bosgra. As a contrib ution to the recent interest in LMI-based robusi controlproblem formulations, the paper by Gars ten Scherer has results on the solvability of Lya punov inequ al iti es. The issue of model reduc-tion for control design is considered in the paper by Pepijn Wortelboerand Okko Bosgra, where fre-quency weighted versions of the11.2optimal reduc-tion prob lem are studied. Some unifications of ro-buststability theoryfor fractionaluncertainty mod-elsare presen ted in thepaperby Raym on d de Galla-fon, Paul Van den Hofand Peter Bongers. Finally, in the pa per by Dick de Roover the ser vomechanis-m problem with fini te hori zon is cons idere d, usi ng an approach in which ideas of feedfor ward three-degrees-of-fr eedom cont ro l, predictive control, the int ern al model principle,an d rep et itive control, are com bine d .

In the present issu e there finally are two cont ribu-tions on the metho dology of syste m identific atio n. The pap er by Ramond de Gall afo n and Paul Van den Hofdiscusses a feed back relevan t identifi ca t ion approach using coprime fract ion al model represen-tations . The assessment of frequency domain un-certainty bounds fromsystem ident ifica ti on ex peri-ments is discussed in the paper by Richard Hakvoort and Paul Van den Hof

We wish to acknowledg e the important contribu-tions to the work in the group made by colleagues from industry. In this issue we have contributions by Maarten Steinbu ch and Pepijn Wortelboerfrom Philips Research, and by Pet er Bongers from U -nilev er Research.

Ifyou wish to react to any ofthe papers in this vol-ume, please do not hesitate to contact us.

Edit

orial

This is the sevent h volume in the seri es Se/eet ed Topics in Identificati on , M odelling, and Control re-por ting oncur rent resear ch proj ects in theM echani-calEngi nee ri ngSyste ms and Control Groupat Delft Univers ity of Technology, Delft , The Netherlands . This volume cont a ins 13 papers ,of which 6 address eng inee ring modellingand control problemsand the ot her 7 consider methodsand techniquesfor con trol design and system identi ficat ion. This reflects the gen eral pattern of activity in our group,where the research is both ad dressing theoretical issuesin con-trol theory and system ident ificat ion , as weilas con -trolenginee ringissues in thefieldsof process control and motion serv o systems .

An industrial glass tube manufacuring process is considered in the paper by Richard Hakvoort and Dick de Roover, where issues of system identification-based model uncertainty are studied in their relevanee to robust cont ro!. A full scale wind turbine power generatingsystem is considered by Gregor van Baars, where the validation of dy-nam ic system models on the basis of closed-Ioop system identification experiments is discussed. A la rge pilot-scale industrial crystallizeris considered in the paper by RobEek, Jaap Both and Paul Van den Hof The research repor ted hereis partof a col-la bora ti ve project,jointlywith theParticIeTechno l-ogy, Process Equip ment,andCrystallizationgro ups at Delft Univers ity, and co-sponso red by European and US industria l process industries. Here, syste m ident ificat ion experiments are used not only to ob-tain dynamic modelsfor control purposes, but the ex pe rim ent alevidence also reveals im por t ant infor-mation on the actual physical mechanisms taking placein a continuous crystallizer .

The paper by Hans Heintzeand Ton Van der Wei -den considers thefeedbackdesign of impedance con-trol for a hydraulically actuated robot. The project is part of a collaborativ e effort with industry to de-velop a brick-laying robot. The expe rime nt al work rep orted has been performed on a la rgescale proto-type3-DOF SCARArobot ava ila ble in our laborato-ry. The pape r by Ger ivan Schothorst. Piet Teerhuis and Ton van der Weidenconsiders a hydraulicservo system whichisdesigned as an act uat ion m echanis-m for the SI M ONA Flight Si m ulat or to be locat-ed at the Simona Resear ch Cente r operated by the Dep artment of Aerospace Engineering at Delft U-niversity. This simulator is a large 6-DOF Stewart platform,now under construction and plannedto be availab leby theen d of1995. The

re-Okko Bosgra Paul Van den Hof Editors

bosgra@tudw03.tudelft.nl vdhof@tudw03 .tudelft.nl

Selected Topicsin Identification,Modelling andCon t rol Vol. 7, December1994

Robust performance

H

_{2 /}

H

oo

optimal controll

Maarten Steinbuch § and ükko H. Bosgr-a!

§Philips Research Laboratories,Prof. Holstlaan4,5656AA Eindhoven ,TheNetherlands . ttMechanical Engineering Syst em sandContral Group,

Delft Uni versi ty of Technology,Mek elweg 2,2628 CDDelft,The Netherlands.

Abstract . This pap er considers therobust performance mixedH

2

/H

00 optimal cont rol probl em formulation. An ex plicit param etrization of H00 norm bounded , causal, r eal-rational uncert ain ties is used , based on LMIs. This leads to a const rained H2 optimal cont ro l prob lem. In case the uncertainty can be considered to lieon its bound,a new

par am etri zationfor lossless bounded real functions can be used. Using this param

etriza-tion ,it ispossibl etoformulatean unconstrain edoptimization problemfor thesolution of

the robu st performance mixed H2/ Hoo optimal control problem. The theory is applied to a Comp act Discrobust control problem .

Keyword s. H2/H00 control;optimalcontrol;robust control; compact disc player; uncer-tainty;opt im iza t ion .

1

Introduction

In the sixties the opt im al LQG control approach provid ed a systemati c solution to the control d

e-sign problem of multivariable systems , see for

in-stance Anderson and Moore (1971), Kwakernaak and Sivan (1972) and Levin e and Athans (1970).

Despi te of itsrelevan eein theformulationof p erfor-mance req uir em en ts,LQoptimal control was shown

to possess na guara nteed robustness margins if ap -plied in conjunction with an observer or Kalman filter, see Doyle and Stein (1981). The robustness issue has attracted a lot ofatte nt ion during the las t decade, resul t ing in the developme nt of H00 co n-trol theory, see for instanee Fra ncis (1987), Doyle and Glover (1988), Maciejows ki (1989) and Doyle et al. (1989). Robust cont rol based on the Hoo norm of a weighted closed-Ioop transfer fun ct ion is capableof handling the design problem for systems with uncert ainties. However , the use ofone single

number, albeitfrequency dependent, to address r o-bustness and perform an cefor multivariabl esyst ems

is rather restrictive. The design problem of stating lThis paper ispresentedat the 33rd IEEECon ferenceon Decision and Con trol, Orlando , FL, USA , December14-1 6, 1994. Copyrig htof this paper remainswith IEEE .

performance and robustness objectives in both the H2 and Hoo framework has recently been stated by several authors, see Bernstein and Haddad (1989),

Doyle et al. (1989), Mustafa (1989), Rotea and Khargonekar (1991), Scherer (1995), Steinbuch and Bosgra (1991a,1991b),Stoorvogel (1993),Paganini

et al. (1994), Yeh et al. (1992),Zhou et al. (1994).

In general one can define the problem as follows.

Suppose the plant is given by its transfer function G(s)with threesets of inputs and outputs.

W2 Z2 Wl G(s) Zl U y

I

_F(s)

I

I

I

The manipulated vari abl es are u, the measured outp ut s are y. The signa l sets (Wl, Zl ) are r e-lated to H2 or LQ type of performan ce crite ria,

whereas (W2, Z2) are rela ted to Hoo norm r equire-ments. Doyle and Zhou (1989) and Zhou et al. (1994) posed theprobl em Zl = Z2, Wl =FW2and gave a veryniceinterpret ation for an induced semi-norm on the transfer functi on. Their work showed that the H

2

/

H00 cont ro l probl em definiti on can bestated precisely in termsof signalsets . Bernstein and Had-dad (1989) have considered thedual mixed H2 /Hoo

probl em for the situation where Wl

=

W2 ,Zl =F Z2· They used a performan ce crite rion relating w(

=

Wl

=

W2) with Zl, under the const raint of an H00 norm bound on the tran sfer function from w to Z2.

They deriv ed necessar y condit ions forredu ced order

controllers and necessa ry and sufficient condit ions for the fullorde r case. Must afa (1989) showed that

if Wl

=

W2 and Zl

=

Z2 the auxilia ry performan ce

index of Bernst ein and Hadda d can be interpret ed

nicely as an entropyexp ression, yielding the cent ra l

H00 controller for the full orde r case.

In all these cases, the probl em solved does not add ress the true robust perform an ce control prob

-lem relevan t in practi cal applications : only a very

restri cted class of problem sdohave thesame inputs (or dua lly outpu ts) for the H2 performance m ea-sure s as for the H00 norm- bounded uncertain ti es.

The mixed H2/ Hoo problem with minimizing the transfer functi on in the H2 sense from Wl to Zl, while constraining the H00 norm of the transfer function from W2 to Z2 to some bound is unsolved . However , this problem does not address robust H2

performance.

In contrast to the aforem entioned results, in this paper a robust performance mixed H

2/

H00optimal control problem (or 'robust H2 problem ' as called

by Stoorvogel (1993) and Paganini et al. (1994)) is

considered including a parametrizationof theworst

case H00norm boundeduncertaintyrelatingthesig -nals W2 and Z2 (Steinbuch and Bosgra (1991b)),see Fig. 2.

Inthisproblem ,theworst- casenorm-bounded

un-certainty represents unmod eled dynamicsof thesy s-tem for which we are abIe to formul ate both their

magni tude bound and the ir structural inter conn ec-tion with the system dynamics.

The robusi performan ce mixed H2/Hoo control

probl em is to minimize the H2 norm of the tr

ans-fer functi on from Wl to Zl usin gthe feedback Kts), while maximizingthe H2 norm of thesam etransfer

functi on over the allowableuncertainties:

sup min11TW1 - Z1(K,L),)

11

2

(1) 11.::\11""<1'K( s )

In the liter ature there exist vari ous formulations

for performance optimization problem s involving

worst-case disturban ces , resulting in a minimax

formulati on , e.g. Mills and Bryson (1994) and Sweriduk and Calise (1993).

From an application poin t of view this rolrust p

er-formancemixed H2/H_oo cont rol problem add resses many design problems in which H2 performan ce is

theoptimizationcrite rion, su bject to (un)s t ructu red H00 norrn-bounded uncertainties. This problem is

hard to solve, but becau se of the possibl e impli -cat ions for use in practi ce, it seems worthwhile to

further investigate the problem . Not e also the in-teresting work of Paganini et al. (1994) in which

the behavioral framework is used to asym p to tica lly

calculate the worst-case uncertainty for the robust

performan ce

lh/ H

00 cont rol probl em. In thispap er

we will explore the validity and usefuln ess of this

mixed H dH00 probl em formul ation using num eri-cal too!s. In earl ier work we considered the calc u-lati on of the worst- case uncer t ain ty, see Stein bu ch and Bosgr a (1994). In this paper we will exte nd the calcula tionofthe wor st case uncer t ain ty and in-vesti gate the related control design . A numeri cal

approa ch will be presented and applied to a mixed H2/Hoo Com pac t Disc control prob lem.

I

_L),(s)

_I

I

Z2 W2 Wl G(s) Zl U y

I

_K(s)

I

I

I

Fig.2: Robust performan ce H2 /Hoo cont rol

prob-Iem.

2

Pr

e lim in a rie s

Given a stabiestrictly proper transfer function ma-trixG(s) with statespace realizationC(sI - A)-lB,

then thefollowing performancemeasures can bed

(i) FTX

+

XF

+ [

X G HT

1

a:

1 [

C

:

X ]

<

0 Definition2.1 TheH2 norm of atransferfunct ion

Gts )is defined as:

(

1

J

OO

)

1/2

11 C 112

=

271" -

00

t7-(CT (- jw )C(jw))dw (2)

untli tr(. ) the trace mat rix operat or.

(ii)

.J

>

0 with

.J

= [

I _JT ]

-J

I (10) (11) ( 12) The2-norm ean be eomp uted with Lyapunov e

qua-tions:

whereS is the eont ro lla bilit yGram ian and P is the observa bilit y Gram ia nsolving:

AT p + P A +CTC = 0

(4

)

Proof: Follows direet1y from the Bounded Real Lemmaas formulatedin Pet ersen et al. (1991). 0 In the sequel wewill denotetheset of all transfer functions .6.(s) with 11.6.

11

00

<

1 asV.

This result direetly leads to the fol1owin g parametrization whieh eha racte rizes all real rati o-nal causal stable transfer functions .6.(8)of order n having 11 .6.

11

₀₀

<

1:

1. Choose J sueh tha t Definition 2.2 The H

00

norm of a transfer fun

c-tion C(s) is defi n ed.as:

.J

>

0 (13)

11 C

1100

=

maxë (C(jw)) wER

untliü the maximum singular value.

(5) 2. Let C and H be matrices of appropriate di-mensions eontaining free parameters, and let F

=

F.

+

Fk with

F.

=

Ft

and Fk

=

-Fr ,sueh tha t

3

Parametrization

of

H

oo

norm

bounded transfer funetions

(14) and

3.1 Inequality formulation

Theorem 3.1 Let (F ,C,H,J) be an asymptoti-call yst abie min imalrealizat ion of thetransfer [utic-tion .6.(s )

=

H(sI- F)-lC

+

J. Then thefol/owing sta te me nts are equiv alent.

In Steinbueh and Bosgr a (1991b) a parametriz ati on wasint roducedfor stab lestrict1y prop er Hoo n

orm-bounded uneer t ainty mod eIs. In this paper wewill extend this parametrization to proper H

00

norm-boun deduneertainties using Linear Matrix In equal-iti es(LMIs),seeSection3.1. Basedon numerical e x-peri enees with this parametrization it seems worth-whileto ex ploit thebound ary eonditions, i.e. when the worst- cas eperturbation is losslessbounded real , see Section 3.2.

. 1

1=1,2 "" ''2n

(15)

(16) for neven , and

3.2 Lossless bounded real formulation When the inequality const raints stated in thepr evi-ous sectionareall activethentheworst-case p ertur-bation is lossless bounded real (all-pass property). This means that the perturbation is on its bound at all frequ eneies and for all its singular values. In order to investigatethemixedH

2/

H

00

problemalso for this case, wedevelop a suitable parametrization . Let us first eonsider losslesspositive real and lossless bounded real transfer funetions, see also Anderson and Vongpanitlerd (1973).

for n uneven.

These equat ions follow from thetheorem aboveby . selecting a coordinate fram e sueh that X

=

l and by utilizing an additional orthogonal transformation that brings Fk into modal from.

11

.6. 11

₀₀

< 1

(6)

3X

>

0 sueh that (7)

[FTX+XF

XC HT]

CTX - I JT _<0 (8) H J - I 3X

>

0 sueh that (9) 1. 2. 3.

Proof: foll ows by direct application of D

efini-ti on 3.2. 0

Lemma 3.4 Let.6.(s )

=

(I - f(s»)(I + f(s »)- l , then

.6.(s ) is lossless bound ed realiffI'(s ) is /oss/ess posi-tivereal.

(1

8)

Ax

+

Bçui;

+

B2lU2 Cl x + DI2lU2 C2x

+

D2llUi x

r

_.6.(s)

1

I

r

lU2 z2 lUl M(s) zl

Fig.3: H2 worst case uncertainty.

Consi de r Fig. 3. The noise dist urbances lUI ha ve unit noise int ensi ty, and the uncertainty .6.(s) r ep-resen ts unstru ctured H00 norm bou nde d pert u rba-tions of the nominal closed-loo p system (M(s) i n-cludesG(s )and /( s»).

4

Worst case perturbations

and let .6.(s )

=

H(sI - F)-IG+ J. The n the c losed-loop syste m is:

In cont ras t to Stoorvogel (1993), causality of the perturbation is assume d as an implicit and n eces-sary ingredi ent to pose the tr ue probl em. Thus we consider a setV ofreal-ration al ca usal stable t rans-fer function matrices .6.(s ) sat isfying 11.6. 1100

<

IJ

andVa if.6.T (- s ).6.(s)= J. As performanceindi ca-tor we use the varianee of the sign al Zl, i.e. the H2 norm of the closed- loo p transfer funct ion from lUl to Zl. Let Mt.s) have the statespace reali za ti on

(17

)

F - flT(1

+

J) - Ifl

-V2

fl

T

₍

₁

₊

_J)-I

V2(

1

+

J)-lfl (I - J)(I + J)- I F G H J Lemma 3.5 Let I'(s )

=

fl(sl - F)-l(; + J, with F

+

FT

=

0 ,

o

=

flT and

J

+

JT

=

0, with

F E IRn x n and

J

E IRm xm, and with fl and (;

of com pat ible dimensions. Then the real matrices F, fl and

J

parametrizes all /oss/ess positive real

transferfunctions I' wit h st ate dimension n.

Lemma 3.6 Let .6.(s )

=

(I - f(s»)(I + f(s») -l with

r(s)

=

fl(sl - F)-l(;

+

J,

with F, fl and

J

as dejined in the previ ous lemm a. Then a stat e space

realization for .6.(s)

=

H(sl - F)-IG

+

J is given

by: Proof: .6.T(-s ).6.(s )

=

(I+ rT(_ s »)-l(1- rT(- s »)(1 - f(s»)(1+ f(S»)-1= (I + rT(_s »)-l(1+ rT (-s )f(s»)(1+ f(S»)-1

=

(I + rT(- s »)-l(l+rT (-s»)(I+f(s»)(I+f(s»)-l

=

I, and conversely. 0

Defini tion 3.2 The realrationa / fun cti onI'(s ),sE <C, is loss/ess positive rea/iff(s)

+

rT (- s )= O. Definition3.3 The real rationol fun ct ion .6.(s),

s E <C, is loss/ess bounded rea/ if .6.T (-s).6.(s)

=

I.

The set of allsuch .6.(s ) is den oi edVa. or max tr[Cf[C]S (21) (F. ,F k, G, H ,J ) sup I1 T

_{W' _}

_Z

_'(

.6.)11₂

=

suptr[Cf [C]S (20) 6EV 6EV

(

19)

( A+ B2JC2 GC2

(

s

,

+

GBD2JD21) 21 lUl ( Cl+ D12JC 2 D12H ) ( ; )

(

~)

=

denoted as [A], [B],

[

Cl.

Notice th at D 12JD2l

=

0 for 11TW1 - Z1(.6.)112

<

00.

Let the system

(19)

be stabie and consider the

consirainedoptimization problem

A nd this isa param et rization[or allst able loss/ess

bounded real.6.(s ).

Theorem 3.7 Defin e the matrices0 and if; as up-per triangu/ar realmatrices, with zero on their di-agonal, and with appropriate dimensions, such that

F

=

0 - OT, and

J

=

if; - if;T , then the triple (0,

if;, fl) parametrizes all stabie lossless bounded rea/ transfer functions .6.(s)

=

H(sI - F)-lG

+

J, with

H,F,G and J dejin ed by(17).

Proof: follows by some matrix manipula ti ons and

is omitted her e. 0

Since ma tri ces

F

and J are defined as s kew-sym met rie (see Lemm a 3.5) ,we further reduce the number of free variables and end this section with the main result.

withF., Fç ;C,H,J according to (13)-(16),and with the state-varianee matrixS

=

sr

the solution to:

and let ~(s)

=

H(sI - F)-IC

+

J,then the p er-turbed system is given by:

[AJS

+

S[Af

+

[B][Bf

=

0 (22) In case of ~ being lossless bounded real , i.e.

~ EVa the optimization problem can be

reformu-lated as an unconstrainedoptimization problem: _with y Apxp

+

B plWI

+

Bpu

c

.;», +

Dplu Cpx p

+

DlpWI (27)

(23)

max tr[Cf[CJs (8,,p,H )

with S the solution to (22), and with (F,C,H, J) defined by (17), where

ft'

=

0 - OT,

J

=

</J - </JT.

5

Numerical exploration of the

con-trol design problem

In the preceding section we have considered worst-case per turbations ~(s) for the fixed system Mte ).

As Mt.s) includes the feedb ack cont roller J«s) , we have to investiga te how ~( s) and Kt.s )are related .

Recall the set V of allstabIe norm-bounded un-certainties. Let ~k (S) EV be the worst- case dis tur-bance, i.e. a disturb an ce attaining

X Ax

+

B1WI

+

B2W2

+

B 3u ZI CIx+ D12W2

+

D 13U

(26)

Z2 C 2x

+

D21WI + D23U y C3x

+

D3IWI

+

D32W2 su p 11TWt-Z t( ](' ~) 11₂

(24)

6E"D

for a certain feedb a ck cont roller Kt;s ). Althou gh the underlying probl em may show a much more com plicatedst ruct ure, we assume herethat (at least locally) there exists a unique maximizing ~k(S).

Nowconsider all such~k(S) , and determinethef eed-backlaw JC thatwouldbe H2-optimalincase~k(S) indeed would qualify as worst-case uncertainty. By assigning an H2-optimal K" to each ~k(S), we it er-ate over ~k(S) until it satisfies the conditions for a worst-case disturbance, i.e. the gradients with resp ect to its paramet ers are zero for a fixed H

2-optim al](*.

Using the parametrization of the worst case un-certaint y we are now able to rewritethe control d e-sign problem in a numerically tractabIe way. Con-side r Fig. 2,with the noise disturbances WI having unit noise intensity and uncertainty ~(s) E V or

~(s) EVaresp ectively. The design problem is

(A; - CJIDpIBJ)X

+

X(A p -

BpD;ICpt)-XBpBJX

+

CJI(I - DpID;I)CpI

=

0(29)

(A p - BpIDîpCp)Y

+

Y(A; CJ DIpBJI) -YCJCpY

+

Bpl(I - DîpDIP)BJI

=

0(30) and the H₂ optimal control law u

=

J(*(s)y is defined by

DIp

= (

D31

+

D32JD2I )

Notice that D I2JD2I

=

0 for 11_TWt-Zt(~) 11₂

<

00 and that D32JD23

=

0 is assumed for simplicity. Thefeedback cont rollerKts)connects the measured variables y with the inputs u.

Let the system (27) be stabIe and assume for the moment that the uncertainty paramet ers (F,C,H,J) are fixed. Then the optimization prob-lem

min 11 Twt_zt(u

=

J«s)y)

Ib

(28)

K

can be solved as a standard H2 or LQG type of problem:

(25) sup min 11T_Wt-Zt(](,~) 112

6E"D(. )K(. )

Fig . 5: Configuration of the CD MIMO control loop . Xrad xJoe f-L---.-o--.--spot L . . - _ - ' trac k- - -o--o-l

the controller uses position-error information pro-vided by four photo diodes. As input to thesystem the controller generates contro! curre nts to the r a-dial and focus actuator, which both are p ermanent-magnet/ coil systems.

In Fig.5 a block-diagram of the control loop is shown .Thedifference between theradia l(Xrad) and vert ical (Xjoc)spot positionand the referen ce track is detected by an optica ! piek-up which generates a radia !erro r signa l (erad) and a focus error sig-na! (ejoc)' A controller J(( s) feeds the system with the currents iradand ijoc,see also Steinbuch ei al.

(1994).

In thenumericalexperimentsa MIMO 2x2 mode! (G)of orde r 10 is used . We will consider two types of uncertainties . The first is a mu!tiplicativeoutput uncertainty. The uncertainty ~ is 50

%

.

The noise disturbances w] acting on the multivariablecontrol

loop enter the loop at the input of the system and measurement noise act on the feed ba ck loop. The performance measures ZI are the inpu t to the plant

and the outp ut ofthe plant ('spot'). Our interest is in how the uncert ainty ~( s) can disturb the H2 p

er-formance from w ] to Z], and how the H2 optima! controller Kts) counteracts this. By using a g en-eral purpose optimization program the results are obtained, using the formulation of the previous s ec-tion. The inputs w] and W2 have been scaled such

that the nominal performance equals 1 and such that " ~ 1100 = 1 corresponds to 50%modelunc er-tainty.The results are presented in the tablebelow, and have been obtained with a first order (2 x 2) uncertainty; see Steinbuch and Bosgra (1994) for investigations with respect to the number of states in a worst-case ~. The results obtained using the LMIformul at ion showed that alltheboundary con-ditionsweremet, andafterapplicationof thelossless bounded real formulati on it appeared that identical solut ionswere obtained. Itsho uldbe noted thatthis resul t holds for this example. Itisnot clearwhet her more generalstateme nts can be mad e. This is s ub-ject of future resear ch.

Theresults aresum m arized in Tabl e1. In the Ta-bie the column'nom ina l,J(n om ,~

=

0'den ot es the valu eof the objec t ive functi on for the unperturbed max 11 Tw,_z,(u

=

J(*(s)y) 112 (33)

(8, q" H )

with J(* the so!ution to (29)-(3 1).

The optim ization prob!em inclu din g the un cer-tainty ~ E V can now be form u!ate d as a co n-sirai nedopt im iza t ion prob! em over a standard H2

optima!cont ro!prob!em:

6

E

xample:

Compact Disc player

ro-bust control problem

with IC the so!ut ion to (29) - (31) , and F.,Fk,G,H, J according to (13)- (16) .

Ir ~ E Va the optim iza tio n prob!em can be f or-mulated as an unconstrainedopt im ization prob!em:

Fig.4: Schematic view ofa rotatingarm Compact Disc mechanism. max 11 Tw,_ z,(u

=

IC (s )y) 112 (32) (F.,Fk,G, H,J) V (A p- Bp(BJ X

+

DJ]CP]»v

+

(YCJ

+

Bp]D[p)Y (31) u.

=

- (BJ X

+

DJ]CP])v

Both the radi a! and the verti cal (focus) position of the laser spot, relative to the track of the disc, have to be con t ro lled active!y. To accom plish this , In FigA a schematic viewof a Compact Disc mec h-anism is shown. The mechanism is composed of a turn-tab!e DC-motor,and a ba!ancedradia!arm for track-following. An optica l elem ent is mou nted at the end of theradia!arm. A diode!ocated in thisel -eme nt generates a laser beamthat passesthroug h a seriesof optica!!enses to givea spot on the inform a-tion !ayer of the discoAn objective lens,suspended by two paralIe! !eaf springs, can be actuated ver ti-cally for focussing.

11T_{w ,}_ _{z ,} 112 nominal:J(n om , .6.

=

0 1.00 nominal: J(* , .6.

=

0 1.06 perturb ed : J(* , .6.

_k

1.38 per turbed: J(n om , .6.

_k

1.48

Table 1: H2performan cefor CD player with output

uncer t ain ty

case with the st anda rd H2 opt im al control [(nom .

The second colum n 'no m inal, J(*, .6.

=

0' means the robust performan ce opt im al controller solving (32) or (33) and with which the H2 perform an ceis

evaluated wit hout a perturbation . The third co l-um n 'pe rt urbe d ,[(*, .6.

_k'

means therobust p

erfor-man ce optimal cont roller solving (32) or (33) and with which the H2 performance is evaluated with the worst case perturbation , i.e. th is number isthe valueof (32) or (33). Finally ,thelast column 'pe r-tur be d ,J(n om ,.6.

_k'

meansthenominal performance optim alcont roller [(n om for which the H2 p erfor-man ceisevaluate d for theworst case perturbation, i.e. this nu mber is the value of (32) or (33) with J(

=

J(n om fixed.

The results shown in theTable indicate that the worst case varianeeof theperformance variables Zl increases with a factor 1.48 if thenominal optimal H2 cont roller is used. In caseof the optimized con-troller for theuncertainty,thenominal performance degrades with 6%. The worst-case performance is 1.38,which is about 10% better than with the nom-inal feedback . In the following figure the nominal and perturbed transfer functions are shown from the input i-: «to the radial spot position.

Disc player MIMO control loop, but now with an uncertainty affecting the resonance at 860 Hz. The uncertainty perturbs especially the dampingof the resonance,with an amountalmostequa l to 100%I. Again,the noisedisturbances Wl acting on the mul-tivariablecontrolloopente r theloop at the input of thesystem and measurem ent noise act on the f eed-back loop. The performance measures ZI are the input to theplantand theoutput of theplant . The results of the ca lcula t ions are presented in Tabl e 2.

I1Tw , _

z

,

112 nominal: J(n om, .6.

=

0 1.00 nominal: J(* ,.6.

=

0 1.01 perturbed : IC ,.6.

_k

1.01 perturbed : [{n om ,Ó

_k

17.90

Table2: H2 performancefor CD player with r eso-nanceuncertainty

The varianeeof the performance variables Zl in -creases with a factor of almost 18(!) if thenominal optimal H2 cont roller is used , while in case of the

optimized controller for the uncertainty,the robust performance is only 1

%

less than the nominal p er-formance. Analysis of the worst-case uncertainty revealed that it is equal to a proportional term (1) driving the dam ping of the system close to zero. The H2 /H00 optimal controller counteracts this ef-fectively. In the following figure the nominal and perturbed transfer functions areshown.

no~-:per ... . '!e!~!_ ~~.. __..~.. .__ IO-' ~ - -__ !oom_per ~\.opf.J)Cr

.,.

~nol 10.2 10-) oom "'W'" ,.•• • -,'.;\..... ~..'.'.....-. 10-) IO~ frequency(Hz ) 10' 10' Irequency(Hz)

10' _{Fig.7: Nominal (-) and worst case perturbed (--,} ... ) transfer function from WI to ZI

Fig. 6: Nominal (-) and worst case perturb ed (- -, ... ) transfer function from Wl to Zl

Thesecond exam ple considers thesameCompact

1Although this ca n be described as a pararnetri c un-certainty (real-valued) it is treated here as a complex uncertainty.

Conclusions

A new formulation of the robust performance

Hz!H00 opt im al con trol problem has been pro-posed in this paper , and an ex plicit parametriza-tion for a worst-casenorrn-bounded uncertainty has been used ,yieldingan (un)constrainedoptimization problem . A Compact Discsystem with an unstruc

-tured uncertainty has been discussed. Using a nu-merical algorithmwe haveshown th at it is possible to calculate worst case uncertainties and to rob us-tify the controllerto counteract with the worst case perturbation. The robust performance obtained in these examples shows that it is worthwhile to fur

-ther explore the theory of the robust performance mixed H2/H_oo control problem .

References

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Francis, B. (1987). A Course in Hoo Control The

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Levin e, W.S. and M. Athans (1970). On the de-termination of the optimalconst ant outp ut f eed-backgains forlinear multivariablesystems. IE E E Trans. Autom. Control. AC- 15,44-48.

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00

)

controller. 1. Guidance Contra ! and Dynamics, 17, 153-160.

Mustafa, D. (1989). Relations betw een maximum ent ropy/H00 control and com bined H00/LQG control. Syst. Contra!Leii., 12, 193-203.

Paganini, F., R. D'Andrea and J.C . Doyle (1994).

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1994 Amer. Contra!Conf., pp. 2782-2786.

Petersen , LR., B.D .O. Anderson and E.A.

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Rotea, M.A. and P.P . Khargonekar (1991). H2

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n.;

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Steinbu ch, M. and O.H. Bosgra (1991a) . Neces-sary conditions for stat ic and fixed order dy-namicmixedH2/Hooopt imalcont rol. Proc. 1991

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Steinbu ch, M. and O.H. Bosgr a (l991b) . Robust performance in Hz] H00 opti m al cont rol. Proc.

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O.H. Bosgra (1994). Multivariable cont rol of a compact disc player using DSPs. Proc. 1994 Amer. Control Conf., pp. 2434-2438.

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458-462.

Yeh, H.H.,S.S.Banda and B.C.Chang (1992). Nec-essary and sufficient conditions for mixed H2 and Hoo optimalcontrol. IEEE Trans. Autom.

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@Delft Universi ty Press Selected TopicsinIdentification,Modelling and Control Vol. 7, December1994

A unified approach to stability robustness for

uncertainty descriptions based on fractional model

representations

Raymond de Callafonl, Paul Van den Hof and Peter Bongerst

Mechanical Engineering Systems and Control Group

Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands.

Abstract. The powerful standard representation for uncertainty descriptions in a basic perturbation model as introduced by Doyle can be used to attain necessary and sufficient conditions for stability robustness within various uncertainty descriptions. In this paper these results are employed to formulate necessary and sufficient conditions for stability robustness of several uncertainty sets based on simpIe additive coprime factor uncertainty, gap-metric uncertainty as weil as the recently introduced and less conservative A-gap

uncertainty.

Keywords. Stability robustness,coprime factorizations, gap-metric

1

Introduction

In a model-based control design paradigm, the de-sign is basedon a (necessarily) approximative model

F

of a plant to be controlled. An apparently

suc-cessful control design leads to a controllerC, having somedesired closed loop propertiesfor thefeedback

cont rolled model

F,

but due to the possible mis-match betw een the actual plant Poand the model

F

,

averificationof thesedesir edclosed loop prop er-ties is preferr ed before implem entingthe controller

Con the act ua l plant Po. In this paper thedi scus-sionisdirected towardstheverificationof oneof the

most importantclosed loop properties: stability. To evaluatest a bility when the cont roller C is being applied to the plan t Po, a characte rizat ion of the discrepancy betw een theplant Poand the mod el

F

can be used (Doyle, 1982; Francis, 1987; Doyle et

al., 1992). Since the real plant Pois unknown , the

discrepancy in general ischaracte rized by aso called

uncertainty set , denot ed with P. Typically an

un-cert ainty set

P

is defined by the (nominal) model

P,

which is found by physical modelling or id

entifi-lThe work of Raymond de Callafonis sponsored by the Dutch Systems and Control Theory Network.

§Now with UnileverResearch, Technical Application Unit, Vla ardingen ,The Netherlands.

cation techniques, and some bounded 'area ' around it (Doyle et al., 1992). The uncertainty set Pitself

reflects all possible perturbations of the (nominal) model

F

that may occur. Some typical examplesof commonly used unstructured uncertainty sets that are norm-bounded by using an Hoo-norm can for example be found in Doyle and Stein (l981),Doyle et al.(1992):

• Additive uncertaintyset:

PA(F,V,W,r):= {P

I

P=F+.6.. A, with

IIV

LlAWil00 ::;

r}.

(1) • Multiplicative (out put) uncertaintyset :

PM(P, V,

w

,

r) := {P

I

P =

[I

+

LlM]F ,

with IIVLlMWlloo ::;

r}

.

(2)

• Additive (right) coprime fractional uncertainty

set, where

F

has been factorized first in a right

- - - - 1

coprimefactorization P := N D and theset

is defined as

PCF(JÎl,ÎJ,VD, VN, W,r):=

{P

I

P= [JÎl

+

LlN][ÎJ+ LlD]-1, with

Clearly, by defining the uncertainty set in such a way that at least the plant Po E P, stability ra-bustness results for the set P will refiect sufficient cond it ions under which the plant Powill be stabi-lized by C,(Doyl e et al., 1992). Inthis persp ective, special attention will be given in this paper to an uncertain ty set PcF which ischa rac te rized by

per-turbations on an additivecoprimefactor description of the nominal model

P,

like in (3). The sp ecif-ie app lica t ion of the uncertainty set description of

(3) will be motivated by the favourable properties it has over a standard additive (1)or multiplicative

(2) uncertainty set description.

Stability robustness results for uncertaintysets em-ployin gweighted and unstructured additive pertur-ba tion s on a cop rime factorization ,gap-meti cbased uncert ain ty set s and the recently introduced A-gap uncer t ai nty sets will be sho wn to be closely related to each othe r . The cont ribu t ion of this paper is in the unified treatment of the sit uat ions of the dif-feren t uncer t ainty sets, by em ploy ing the weigh tin g an d the factoriz ation used in an uncertainty PCF

as given in (3). While stability robustness results for uncertainty set susin g addit ive perturbations on norm aliz ed(Jeft) coprime fact oriz ations (Gloverand McFarlan e,1989) andgap-met ric based uncertainty sets (Georg iou and Smit h, 1990a) have separately been derived before, thispaper amp lifiestheir rela-tion, as weil as the extension to a less conservative A-gap (Bongers, 1991; Bonge rs, 1994) uncer t ainty set descript ion.

The out line of this pap er will be as follows . In

section 2 som eprelimin arynotation s an d definitions will be given, while in section 3 the basic stability robustn ess result usin g the powerful perturbation mod el (Doyle , 1982) will be sum m a rized . This per-tu rba t ion mod el gives rise to an unified approach to handle stability robustness for various uncertainty descriptions , including additive weighted perturba-tions on acoprime fa ctorization. Section4contains the results ofap plying this unified approachto ad-ditive uncertainty descriptions on fractional model representations likein (3) and favourable properties areilluminated. Thelink with gap and A-gap based stability robustness results is discussed in sections 5 and 6, the latter one being less conservat ive than the former one , as sho wn in secti on 7. The paper ends with some conclud ing remarks .

2

Preliminaries

Through out th is pap er , the feedback configur at ion of a plant Pand a cont roller C is denoted by T(P,C) and defined by the feedback conneetion st ruct ure depicted in figure 1.

P and Care represented by real rational (discret e

1'1 U

I

P

I

+'

+

Î

I

I

I

- _{1' 2}

Ye

_I

C

r

Ue ~+

Fig. 1: Feedback conneetio nstruct ure T(P,C)ofa plant Pand a controller C

time) transfer function matrices, and it will be as-sumed that the feedback conneeti on is well-posed, i.e. th at det[I

+

CP]=1= O.

Infigure 1 the signals u and y refiect resp ectiveiy the inputs and out puts of the plant P. The signals

U_e and Ye are resp ectively the inputs and out p uts of the cont roller Cp, and 1'1 and 1"2 are external

referen ce signais . The feedb ack system T(P,C) is defined to be internallystabie if the mappingfrom COI(1'2, 1"1) to col(ue,u) is BIBü stabi e , i.e. if the corresp onding transfer function is in IR7-l

_{oo ,}

bein g the I-Iardy space of real rational transfer function matri ceswith bounded 7-l

_oo-n

orm (Franc is, 1987) :

II

Glloo :=

sup o-{G(ei w)} (4)

wE[D, ")

with 0- the maxim um sing ula rvalue.

Thedyn ami csoftl~eclosed loop systemT(P ,C) will also be describ ed by the mapping from eol( r2, 1"t}

to col( y, u) which is given by the transfer functi on

matrixT(P,C):

T(P, C) :=

[~]

[I+CP]-l

[

C

I] . (5)

Using the theory of fractional representations , as e.g. presented in Vidyasagar (1985), a plant P is

expressed as a ratio of two stabie transfer functions

N and D. For two transfer functions N,DE IR7-loo

the pair(N,D) is called right coprimeover IR7-loo if there exist X, Y E IR7-loo such that X N

+

Y D

=

J.

Thepair(N, D) is a right coprimefactorization (ref) of P if additionally det{D} =1= 0 and P

=

N D-1

. A

right coprimefactorization (N,D) is called normal-ized (nref) if it satisfies N* N

+

D*D

=

I,wher e * denotes com plex conjugate transp ose. For (normal-ized) leftcoprimefactorizations (lef) dual definition-s exist . With resp ect to internal stability of the feedback system T(P,C) the following lemmawill be used .

Lemma 2.1 LetP have ref(N,

D)

atul Icf(D,N),

and let C have re] (Ne,De) and lef(De,Ne) . The

t. thefeedback systemT(P,C) given in figure 1is

internallystabie

H. T(P,C) EIR1l00 1 with T(P,C) defined in (5). m. A- I EIR1looJ with A:=

[

'D

e N

e

1[

~

]

IV. Ä-I EIR1l001 with Ä:=

[D N

1

[~:]

Proof: iÇ:} ii: Fram figure 1 and lemma 2.1it can

be seen that

~

~

-~

d z rl U

Y

P

r-

_Y

r

+

-

_r2 Yc C Ue ~+ d z

3

Stability robustness

in standard

form

IGiven two topologies Ol andO2 ,Ol is said to be weaker

than 02if Ol is a subcollection of 02, see also Vidyasagar

et al. (1982)

Definition 2.2 Considerthetwo plants PI,P2 with

nrcf's(NI,DI) ofPI and (N2,D2 ) ofP2 . Then the

gap between PI andP2 is expressed by

(7)

(6) U Ue M IIMll~lIoo

<

1

Fig. 2: (a) Feedback conneetion structure of a

perturbed plant Pt>. and a controller C;

(b) general feedback conneetion structure

K(M,~).

(b) !C(M,~) is BIBO stablefor all~ with 11~1100

::;

"y _{if and only if}

It is weilknown that thefeedbacksystem depictedin

Figure 2(a) can be recasted into the situation given

in Figure 2(b), where the uncertainty ~ has been

pulled out (Doyle et al., 1992) using the artificial

signals d and z. Provided that the transfer M is

BIEO stabie, intern al stability of the configuration

in Figure2(a) isequivalent to BIBO stability of the

feedbacksystemK(M,~)in Figure2(b),wher eM

ll

will denote the map from from d onto Z only.

The small gain theorem can now be applied to

char-acterize stability results for the conneetionstructure

of Figure 2(b) and has been summarized in the

fol-lowing lemma.

Lemma 3.1 Let the stabie transfer functions

M,~ E IR1loo construct a feedback connection

K(M,~). Then

(a) A sufficient condition for BIBO stability of

K(M,~) is given by

max{5(PI,P2 ) ,5(P2,PI)}, with

QEiR~oo

11 [

~:

] - [

~~

]

Qll

oo

5(PI,P2)

5(Pi ,Pj)

Fractionalrepresentations havea close relation with

approxim at ion in the graph topology. The graph

topo logy is the weakest top ology' in which a vari

a-tion of the eleme nts of a stabi e feedback c

onfigura-tion aro un d their nominal values,preserv esstability

of that closed loop system (Vidyasagar et al.,1982).

The graph topology is known to be induced by s

ev-era l metri cs, as e.g. the graph metric introduced

in Vidyasagar (1984) or the gap metric introduced

in Zames and EI-Sakkary(1980),being expressed in

the following way (Georgiou, 1988).

For describingthestability robustness of several

un-cert ainty sets based on fractional model

representa-tions, the standard results on stability robustness

for a general situation as depicted in Figure 2 will be used .

Henc e T(P,C) E IR1ioo if and anly if H(P,C) E

IR1i_oo which is equ ivalent to T(P, C) being int

er-nall ystabie .

i Ç:} iii, iv: See Vidyasagar (1985), Bangers (1994)

+

with A

=

[beD

+

JÎleJÎl] and (De,Ne)a lef ofC. As the weightingfunctions arestabieand stably inv ert-ible, and A-1 is stabie according to Lemma 2.1 it

follows that M1 1 is stabie. Application of Lemma

3.1then proves the result. 0

Fig. 3: Additiv e pert ur bations on a righ t coprime fact oriz ati on

+

.6. r------ --- --- --- -- ----, I I , I I I I I I I I I I I I I l _

Denoting col(d1 ,d2 )

=

.6.z with the appropriate

signals defined as indicated in figure 3, the co r-responding stabie transfer functi on Ml l satisfying

z

=

Ml1col(d 1,d2) can be written as

In the next section it will be shown how these r e-sults can be exploited to deriv estability robustness conditions for gap-metri cuncertaintysets as weIlas for uncertaintysets basedon further generalizations of the gap-metric . To this end , the result on the e -quivalencebetwe en several formulations of thesame uncertainty sets wiII be presented first.

Theresult of secti on 3onstability robustness can be applied to various Hoo-norm bounded uncertainty sets byrewritingthe uncertaintydescription into the basi c perturbation modelIC(M, .6.). In this section th is is done for uncer t ain ty sets based on coprime fact or uncertainties.

A crucia l aspect in the result of lemma3.1 is the assum pt ion that .6. E lR1ioo. In case of an add i-tive (1) or multiplicative (2) uncertainty set in the basic per turbation mod el , this assumption implies the condit ion that the locati onof all unstable poles of Pare assume d to be fixed. Additive per turba-tions on coprime factoriza tio ns (3) are mor e fl exi-bie and allowchanges in both the num be r and the locat ions of poles and zeros anywhere in (f; (Chen and Desoer, 1982). Mor eover , fractional r epresen-tations havea close rela tion with approx imationin the graph topology. Dist an ce measure s (or metrics) like thegrap h and gap metricgiven in defini tion 2.2 ind uce this same grap h topo logy and can also be used to evaluatestability robust ness proper ti es ofa closed loop system (Vidyasagar, 1984 ; EI-Sakk ar y, 1985; Vidyasagarand Kirnura ,1985).

First ly, the uncer t ain ty set PCF (3) ,as menti oned in the introdu cti on ,will be discussed.

Corollary 4.1 Consider a plant

ft

with rcf( JÎl ,D),

st abilized by a given controller C, and consider ihe

uncertainty set

PCF( JÎl ,D, VD,VN,W,,) :=

{P 1P

=

[JÎl

+

.6.N][D

+

.6. D]-l, with A proof of this lemma can befound in Doyle et al.

(1992) and Maciejowski (1989) .

4

Stability robustness for

uncertain-ty descriptions based on fractional

model representations

for stabie and st ably invertibl e filters VD, VN, W. Theti ihefeedback system T(P,C) is internally sta-biefor all P EPCF if and only if

,

<

IIW-1 [D

+

CJÎl]- l

[I

C

]

[V~l V~

l]

1[1

Proof: Definin g

.6. :=

[~D ~N] [~~

]

W, such that 11.6.1100 ::; " (9) the basic perturbation structure of the uncertain-ty set PCF ca n be written into a form that cor -resp ond s to IC (M,.6.) for a specific form of MIl.

Proposition 4.2

Theuncertaintyset PCF(JÎl,D,VD,VN,W,,) as d

e-finedin Corollary

4.1

can alternativelybewrittenin thefollowing equivalent forms:

(a) PCF(JÎl,D, VN, VD, W,,)

=

{P

I

(b) PCF(N, ÎJ, VN,VD,W,I)

=

{P

I

P

=

N nD;;,1_{, (Nn, D n) nrcf, and there}

exists aQE IR1ioo such that

(a) PCF(N,ÎJ,VN, VD,

w

,

I)

=

Pg(P, I);

(b) For1

<

1,Pg(p ,

I

)

=

P~(p,

I)·

Proof: Part (a). According to Proposition 4.2(b)

and taking into account thespecificchoieeofw

eight-ing functions in the lemma,it follows that PCF(N ,ÎJ,VN,VD,W'/)

=

{P 1P

=

NnD;;,l, (Nn,D n) nrc], and there exists a

Q

E IR1i_oo sueh that

Proof: Part (a) follows by simpIe calculation.The

proof of part (b) is more involved. Inthis derivation the freedom in (11)is used to denote:

(NW

+

V;Vl!::iN)

=

NnQ (ÎJW

+

VD1!::iD)

=

DnQ

with (Nn,Dn) a nrcfand

Q

E IR1ioo. Such factors

ean always be found . It follows then that

Since

(N

,

ÎJ) is chosen to bea nrcfof Pit is straight-forward to verify that PCF

=

Pg.

Part (b) is proven in Georgiou and Smith (1990a).

The restrietion to 1

<

1 is ca used by the fact that

these sets with 1

>

1 can not be stabilized by a

single controller. 0

whieh proves the resul t . Note that the factor Q

cance ls in the representation of P. 0

!::iN VN[N nQ- NW] !::iD VD [Dn Q - ÎJW].

for whieh the following relation with the coprime

factor uncertainty sets can be shown, as presented

befor e.

Lemma 5.1 Let a plant Pand a controller

C constitute an internally stab/e feedback sys-tem T

(P,

C). Consider the uncertainty set PCF(N ,ÎJ,VN, VD,W'/) under the additiona/ con-ditions th at

(N

,

ÎJ) is a nrcf of P, and VD

=

J, VN= J, and W = J. Then

(16) 1

<

IIT(P,

C)II

;;:,I .

Lemma5.1 relates the set defined by a gap metri c

bound with theset ofcop rime factor purturbation s

by a special choi ee of the weigh ting functions VD,

VN ,Wand the coprime fact oriza t ion(N ,ÎJ) of t he model

P.

This gives rise to an unified approach to handlesets of plants that are bounded by a ga p

metric ,and the stability robustness result for these

sets follows now directly from Corollary 4.1.

Corollary 5.2

Consi der the situativn of Lemma 5.1 with 1

<

1. Then for each of the three sets of plants PcF, Pg and P~, T(P,C) is internallç stable for all PEP if and only if

Note that the result of this corollary is not new. It was shown already in Georgiou and Smith (1990a), where a complete proof of the stability robustness

result is given. It has been shown here that the stability robustness results in the standard form has simply be exploited, as formulated in section

section 3. Restricting attention to the situation that 1

<

1is natural, as Bode's sensitivity integral shows that IIT(P,C)lIoo

>

1, (Maciejowski, 1989), show-ing that stability robustness can only be achieved for sets with1

<

1.

Proof: The proof follows simply by substituting

the specific weightings in the result of Corollary 4.1,

employing the fact that premultiplication of thee

x--T -T

pression within the norm by [N D

jT

leaves the

norm invariant, due to the normalization of the ref.

o

(13) (14)

{P

I

b(P,

P) ::; I}

{P 16(P,P) ::;/}, Pg(P,I) P~(P,

I)

5

Stability robustness based on

dis-tance measures

In this sect ion stability robustness results for g

ap-metric uncertainty sets are formulated. The main

result of this section is not new, but al ready proven

sepa ra te ly in Georgiou and Smith (1990a). The

close relation of the stability robustness result here with the formulation in the previous seetion

eon-cerninggen eral coprimefactor uncertainty sets will

beilluminated .This relation will be employed in the

next section to formulate similar results for

uncer-tainty sets based on theso-called A-gap,as recently introduced in Bongers (1991) and Bongers (1994). The following uncertaintysets are being considered

Finall y it sho uld be noted that the gap and graph metri c are induced by the same topo logy and are

uniformly equiva lent (Georgi ou , 1988;Packard and Helwig , 1989 ). Therefor e stability robustness in the gra ph metricyields a similar result as mentioned in

corollary 5.2.

6

Stability robustness in the A-gap

The resuItsobtainedin theprevioussection for gap-based stability robustness can be further extended

for uncertaintysetsbasedon the recently introduced A-gap, (Bongers, 1991;Bongers, 1994).

Definition 6.1 Let two plants

Pi

,

P2 have nrcJ's (NI ,DI), (N2 ,D2

2

re~pectively. Let C be a

con-troller with nlcf(De,Ne) such that T(Pl ,C) is in-ternally siable. Then theA-gap between the plonis

Pl, P2 is definedto be expreseed by

unth. A

=

[DeDl

+

NeNd .

The differen ce between g(P1,P2) and 8t.(P1,P2) is

the addit iona lshaping of the nrcf of PI with A-I int o arcf(N,

iJ)

.

In this wayÄ:= DeÏJ+NeN = I,

with

N

=

NIA-1,

IJ

=

DIA-1, which is used

to conside r the close d loop ope ra tio n of PI in

-duced by the controller C bein g em ployed. This

mak es the distance betw een PI and P2 dep end en-t on the nrcf of the cont roller C. Note that the

distan ce measure 5A(Pl,P2) is not a metric since 8t.(Pl,P2)

-I-

8t.(P2 ,Pi) due to theinfluenceofthe

cont ro ller C (Bon gers ,1994).

Accordingly , an uncertainty set bas ed on A-g ap

un-certainty can bedefined as:

PgJP , , ):= {P

I

8t.(p ,P):Ç, }.

Thisuncer tainty set can alsobeshown to be e

quiv-alent to an uncertainty set of coprime factor un

cer-tainties , provided appropriate weighting functions

are chosen .

Proof: The proof of (a) is straight forwa rd, along

the same lines as the proof of Lemm a 5.1(a ). R

e-sult (b) thenfollow sdirectlyfromCoro lla ry4.1,e m-ploying the fact that A[b

+

CN]- l [I C]

=

[DeNe]

having an co-nor m of 1 due to the fact tha t it is a

normalized left cop rime factorization. 0

As said before,in case of the A-gap,the uncertainty

set defined accordingly considers perturbations of the nominal plant

P

that are controller dep end ent.

The introduetion of weightings in the gap met

-ric has also been studied in Gedd es and Postleth

-waite (1992), Georgiou and Smith (1990b)or Qui and Davidson (1992). In Gedd es and Post!ethwait

-e (1992) a multiplicativeuncertainty description on thenrcf

(N,

0)

of themodel

P

is being used ,leading

to an uncertainty structure ~ having a diagonalf

or-m. Dueto thediagonal form only necessaryands uf-ficient conditions based on the st ru ct u red singular value J.l{.} ca n be obtained. The weightings in the

weighted gap of Gecirgiou and Smith (199Db)have

to be defined a posteriori which makes the cho ice of the weightingfunctions , to access robustness i

s-sues on the basis of a weighted gap, not a trivial

task. Informationon the si ze ofthe coprime fact or

perturbationscan be used in the weighted pointwise gap metricdefined in Qui and Davidson (1992) ,but st ill an efficientcom pu ta t ional method for pointwise gap metricis not availa bleyet.

7

Conservatism issues

All stability robustness results in this paper refiect

necessary and sufficient con dit ions of an un

certain-ty set to be stabilized by a single cont roller. As such no conservatism is introduced in the test for

checking stability robustness itself. However, for a single given cont roller, differ ent of such un cert.ain-ty sets contain a different portion of the set of all

syste ms that is stabilized by the cont roller. In th is

persp ectivethe conce ptofconservatism is an intrin -sic property of the uncertaintyset beingused. As a

result an uncertaintyset will be called morecons

er-vative than another if one controller stabilizes both sets ,while the former set is contained in thelatter.

Theorem 7.1 (Bongers, 1991) Consider a plant Pand a stabilizing controller C with nlcf (De,Ne).

Consider thefollowing two uncertaintysetsresulting

[rom ihe stability robusin ess resulis in the previous

sections:

Lemma 6.2 Let a plant Pand a controllerC with

nlcf (b; Ne) cons tit uie an internally stabie f

eed-back syst em T(P, C). Consider the uncertainty set

PcF( N ,D,VN, VD,

W

,,)

uiuler the additional c on-ditions that (N,D) is a nrcf of P, and VD

=

I, VN

=

I, and W

=

A-I with A

=

[DeD

+

NeN]. Tlten

(a) PCF(N,D,VN, VD,W, ,)

=

PgJP, ,);

(b) T(P,C) is internally stabie [or allP EPCF if and only if,

<

1.

S6(P,C) SgJP,C) then {u P6(P,b), b

<

IIT(P,

C)II~;;n

{U

PgJP, c), c

<

I}

(17)