Selected topics in identification, modelling and control: Progress report on research activities in the Mechanical Enginnering Systems and Control Group. Vol. 2

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 2, December 1990

Laboratory for Measurement and Control

Department of Mechanical Engineering and Marine Technology

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

Laboratory for Measurement and Control

Department of Mechanical Engineering and Marine Technology Delft University ofTechnology

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 control :

pragress report on research activities in the mechanical engineering

systems and contral graup.- Delft: Laboratory for Measurement and Contral, Department of Mechanical Engineering and Marine Technology,

Delft University ofTechnology, Vol. 2 - ed. by O.H. Bosgra and P.M.J.Van den Hof.- ill. Met lit.opg.

ISBN 90-6275-646-8

SISO 656 UDC 531.7+681.5 NUGI841

Cover design by Ruud Schrama

Con

tents

Volume 2, Decem ber 1990

On a new robust stability margin P.M.M. Bongers

Low order robust Hoocontroller synthesis P.M.M. Bongers and G.H. Bosgra

Necessary conditions for statie and fixed order dynamic mixed

H

2/Hoo optima! contro! M. Steinbuch and G.H. Bosgra

Subspace balancing - fast algorithms for balanced reduction of flexib!e mechanica!systems P. Wortelboer

Det ectionof spill-over critic al modes by eigenvalue tracking P. Wortelboer .

On system order and struetureindices of linear system sin po!ynomia! form P.M.J. Van den Hof

On rela xed delay structure condit ions in closed loop iden tifiability problem s P.M.J. Van den Hof, û.K. de Vries and P.Schoen

Open-loop identification of feedback cont rolled systems R.J.P.Schrama

Optima! experiment design for prediction error identification in view of feedback design R.G. Hakvoort

Cont ra ! of an evaporative ammonium su1fate crystallizer with fines remova! Sj. de Wolf, R.Koning and J. Jager

Modelling of a pasteurisation process

R.Lucassen, J. do Liuramenlo and A.J.J. van der Weiden

1 7 17 25 33 43 53 61 71 79

89

Editorial

It is with pleasure that we present the second is-sue of our bi-annual progress report on research activities in the Mechanical Engineering Systems and Control Group at Delft University. The edi -tion of the first volume, last april, yielded quite a number of positive reactions, both from the aca-demie and from the industrial world. It also gave rise to a number of questions as "Why are you do-ing this?" and "Now you probably do not have to publish any papers anymore in the regular maga-zines?" With respect to this first question we can formulate a twofold answer. Firstly this progress report provides a means for fast publication of re-cent research results; in this way it also serves as a medium that stimulates Ph.D.-students to for-mulate their -possibly preliminary- results in paper style. Secondly through this publication we hope to present an overview of the research activities in our group, in order to stimulate and probably also to initiate contacts with colleagues from academie and industrial research laboratories, as weil as with industries and governmental institutes.

In view of the second question raised above it should be stressed that this progress report does not keep us from publishing our research results in the regular international literature. In this re-spect the papers published in this magazine either have the status of preprints or extended versions of .papers being submitted for presentation at interna-tional symposia or for publication in internainterna-tional magazines, or they serve as concept papers that should be polished and finalized for presentation at international meetings.

This present issue contains eleven papers from different research topics being undertaken in our

group. Three papers written by Peter Bongers, Okko Bosgra and Maarten Steinbuch are situated in the area of robust control. Part of this research within our group is performed in collaboration with Philips Research Laboratory, Eindhoven. This also holds for the work of Pepijn Wortelboer who gives an account of the activities on large scale mod-elling and model reduction of mechanical systems. A system theoretic contribution in terms of bino-mial representations of linear systems is formulated in a paper written by Paul Van den Hof. Three papers show contributions in the area of (c1osed loop) system identification; they are authored by Paul Van den Hof, Douwe de Vries, Peter Schoen (in collaboration with the Laberatory for Thermal Power Engineering), Ruud Schrama and Richard Hakvoort. The last two papers of this issue rep-resent application-oriented projects. The paper by Sjoerd de Wolf, Robine Koning and Johan Jager .considers the modelling and control problem of a crystallization plant (in collaboration with the Laboratory for Process Equipment and a numbcr of chemical industrial companies). The paper by Ronaid Lucassen, Jan do Livramento and Ton van der Weiden discusses the modelling problem of a pasteurisation process in a Heineken brewary.

We would like to end this editorial by acknowl-edging the effort of Ruud Schrarna who did a per-fect job in preparing the appropriate ~TEX style files which give this issue its uniform layout.

Okko Bosgra Paul Van den Hof Editors

@1990 DeUtUniversit y Pre ss

On a new robust stability margin

Peter M.M. Bongers

Select ed Top ics in Iden tifi cati on ,Mod ellingand Control Vol. 2,December1990

Lab. Measurement and Control. Dept. Mechanical Eng. and Marin e Technology, Delft University of Technology, Mekelw eg2, 2628 CD Delft, The Nethe rlands.

Abstract. The aim of this paper is to derive a new robust stability margin. Known sufficient conditions for robust stability stated in gap-metriosense contain inherent con -servativeness in the formulation of the various steps. In this paper conservativeness in one of the steps is removed,resulting in a new robustness margin. Thekey issue is that more information of the specific controller is taken into consideration. The resulting robustness margin is less conservative than the margin in directed gap-metric sen se and is as easy to compute.

1

Introduction

In some recent papers ([4], [1]) a sufficient condi -tion for robust stability has been stated in the gap -metric.

In the gap-m etri c robustness the nominal plant is factorized in normalized coprime factors. The differen cebetween a perturbed plant and the nom-inalplant is described by perturbations on the nor -malized coprime factors of the nominal plant. Ro-bustn essof theclosed loop for a class of perturbed plants is guarante ed if thenorm ofthecoprime p er-turba tions is small enough. The maximum a llow-abIe norm ofthe pert ur bat ionsis determi nedby the infinity norm of the closed loop, hence only cru de information ab out the applied controller is taken into conside ra t ion .

The main idea behind the new and less conse r-vative robust ness mar gin to'be considered in this pap er is to factorize a given cont roller (that s tabi-lizes the nom inal plant) and use this fact ori zation as abasis to determine a factorization thenomina! plant.

Thedifference between a perturbed plant and thc nomina! plant is now describedbyperturbations on the coprime factors of the nomina! plant which in-cludes detailed information about the applied con-troller. Robustness of the closed loop for a class of perturbed plants is now guaranteed if the norm of

the coprime perturbations is less than one.

The !ayout of this pap er is asfollows: after some pre!iminaries,stabilityof a nominal closedloop sys-tem is discusscd in scction 3. The usual concept of stability in gap-metric sense is given in section 4. Reduction of conservativeness introduced by the gap-metric is discus sed in section 5 and is illus-trated by an example.

2

Preliminaries

Algebraic Structure

Definition 2.1 The us ed olqebraic si ru ciure of rings was introduced in

([2

), [7

}):

:F:=

{a/b

I

a EH,

s

«

H\Ol, a quotientfield ojH.

9

:= {a/b

I

a EH, bE I\Ol, a (not necessar -ily commutative) ring with iden ti ty H := asubri ng of

9

with ide ntity

I := {g E

9

I

g-1 E

g}

,

the set of mult iplica-tive unit s of

9

J

:= {h EH

I

h- 1EH}, theset of multiplica-tive units of H

In the seque! of this paper we wil! identify the ring H with lRHool the spaceof continuous time fi -nite dimensional real rational proper stab!e transfer

-P(I

+

CP) - I ] (I

+

CP)-I functions, therefore all stabie transfer functions will

have entries belonging to'H. With a slightabuse of notation we will denote P EFmxn as P EF.

Rem a r k 2.2 The transfer funetion P E F ean haveunstablepoles and ean havepoles andor zeros on the imaginary axis.

ping the external inputs

(

el ,

e

2)

onto the outputs (UI,

U2)

is given by:

H(P,C) = [_IC

~]-I

=

[I -

P(I

+

CP) - lC (I

+

CP)- IC M* M

+

N* N

=

'I (M M*

+

N N*

=

I)

Faetorizations

Definition 2.3 {7}

normalized coprimeness In addition th e pair M,N(M, N) is called normalized right (left)

coprime (nrefor nlcf) if it is coprime and:

T(P,C) = [ -::]

(XM

+

YNt

l

[y X]

(1)

Using the coprime representation the closed loop structure of Fig.1 can be redrawn as in Fig.2 with

ç

=

Xe2

+

Yel'

Stability of the closed loop, i.e, the controller C internally stabilizes the plant P, is guaranteed if and only if H(P,G)E'H. Define:

T(P,G) = H(P,G) -

[

~ ~

]

= [ - : ]

(I

+

GPt1 [G I] the n by definit ionof'H,B(P,G) E 'H <=?T(P,G) E 'H. Now let P = N M-1_{with(N, M) a rcf of Pand} let C =

X-I

Y

wit h

CV, X)

a lef of C then:

(NU

+

MV

= I) (P = M-IN)

UN+VM = I

P = NM-1

fractions

A

plant P E

F

has a right (lef t)

[rac

-tional

repres entatio n if thereexist N, M(N,M)E 'H such that P = NM- I(= M-1N).

coprimeness Furt hermore we say that th e pair M,N(M,N) is 7'ight (lef t) coprime (rcf orlcf) ifthere ezisis U,V(U,V)

E'H

such that :

Meyer and Fran klin [6] have presen ted a state -space repre sent ation of normalized coprime f actor-izati ons.

+ UI

-

X-I

->;

-y

f+-=-3

Closed loop stability

In thispaper we will study the closed loop stability according to Fig.1, where we assume that a stabi -lizing controller G has been designed for the plant P.The closedloop transferfunctionH(P, G)

map-+ p

Fig. 2: Closed loop structure with coprime repre-sentation

Lemma 3.1 Let P E F be given as P = NM-1 with

(N, M)

a re] of Pand let the controlle rCE

F

begiv en asC= X-I

Y

with

(Y,

X) a lcfofC. Then stability of the clos ed loop is equi valen t to:

Fig. 1: Closed loop st ructure

Remark 3.2 A-:-1 is the transfer funet ion from

ç

to 7J. Thus stabili t y of the closed loop is equivalent to stability of the tran sf erfunct ion s[rom

ç

to 7J and from 7J to

ç.

C

+ Pro of: Vidyasagar et al.

[7]

o

2

-For robust stability it is essential that the closed loop transfer function remains stabIe for plants

Pt:.

close to P. Usually the controller is designed with knowledge ofP only. In a more precise formulation it is usefull to deterrnine the largest class 5 such that:

Lemma 4.2 Given the plant and controller of Theerem

4.1

then the minimum achievable 11

[

~~]

1100

equals the directed gap [rom P to

r;

denoted as

8(P, Pt:.)

r;

E5 :=

{Pt:.

E

F

IT(Pt:.,

C) E

'H}

with C designed on

P

such that

T(P,

C) E'H.

4

Robust

metric

Stability

.

In

gap-Proof: The perturbed plant can be factorized as

Ps

=

(Nt:.Q)(MD.Q)-1

with

(ND., MD.)

and

Q

E'H,

Then the perturbations are written as:

[

~~

] = [

~:

] - [

~:

]

Q

and a lower bound on the coprime perturbations is: For a given controller such that

T(P,

C) E

'H

we

de-termine in this section the class

5

g in a gap-metric sense.

11 [

~~]

1100=

J~L

11 [

~:

] - [

~:

]

Q

1100

Theorem 4.1 Let the nominal plant P E

F

be

given as

P

=

NnM;:1

with

(Nn, Mn)

a nrc] of

P

and let a

Pt:.

E5g be given as:

Pt:.

=

Nt:.Mó.1

= tN; - ~N)(Mn

-

~A1)-1 (3) and let the controller

C

E F internally stabilizing P be given as

C

=

X-lV

with

(X, V)

E

'H

such that:

([ X

Y] [

~:

])

= I (4)

Then a sufficient condition [or stability of the per-tutbed system

Pt:.

is given by:

This is the expression of the directed gap from P

to

Ps

as stated in Georgiou

[3].

0

The class

5

g can now be formulated in terms of a gap-metric boundary:

5

g

=

{Pt:.EFIT(P,C)E'Hand

8(P, Pt:.)

11 T(P, C) 1100<

I}

Remark 4.3 Clover and McFarlane

[5J

derived a controller synthesis method such that

11 T(P, C) 1100

is minimized over all admissable controllers.

Using this and (4) we get as necessary and sufficient condition for stability:

Proof: According to (3), [

~~]

can be written

[

~M]

1

11

~N

1100< I1 [T(P,

C)]

1100

N

ew

stability margin

and let

Pt:.

E

F

be given as:

5

Theorem 5.1 Let the controllerCEF internally stabilizing

P

be given as

C

=

X;:

1

V

_n

with

(Y

_{n, Xn)}

nlcf and let P E F be given as P = N M-1 _with (N, M) a re] of P such that:

(5)

[

~~

] = [

~:

] - [

~:

]

as:

By exploiting the nref of Pand (4) in (1) we have:

11

[X

V]

110011 [

~~]

1100< 1

(6)

Since

[X

y], [

~~

]

E

'H

by definition, a suffi-cient condition for robust stability is:

11 [T(P, C)] 1100=11

[X

V]

1100

Hence Theorem 4.1 is proved. o

Then a sufficient condition [or stability of the per

-turbed system

Pt:.

is given by:

[

~M ]

11

~N 1100<

1

Pro of: The stability condition for the perturbed closed loop can be writen as:

[ - -]

[~M]

Again applying the small gain theorem wehave a

suffieient condition of stability:

11

[X

n

Y

n

]11

co11 [

~

~

] llco< 1

Using the fact that 11

[X

_n

Y

_{n ]} llco= 1by d

efini-tion of normalized left eoprimeness, the theorem is

proved. 0

Remark 5.2 The fact that

[X

_{n Yn] is co-inn}er

implies thatihe maximum singular value is constant over all[requencies. Thus taking the infinity norm does not introduce conservativeness.

The perturbed plant ean be factorized as PI:). =

(NI:).Q)(MI:).Qt1 _{with Q E 'H, then the coprime}

perturbation are written as:

[

~~

] = [

~

] - [

~:

]

Q

An uppper bound on all allowable perturbations is described by:

11 [

~~

] llco=

J~~

11 [

~

] - [

~:

]

Q

llco

(7)

The class Sb ean be written as:

Sb = {PI:).

I

T(P,C) E 'Hand

j~~

I1 ([

~:

] - .[

~:

]

Q)

A-I llco<

I}

To prove that this stability margin is less eons

erva-tive than the gap-metrie margin we state the main

result of this paper:

Theorem 5.3 TheclassSb is larqer than or equal

io theclassSg

Before proving this theor em we first need some

lemm a 's.

Lemma 5.4 L_{et (Nn, Mn)} be a nrcf of Pand let

(Yn

,

X

n

)

bea nrc] of

C,

then: with:

Proof: Pre-multiplyT(P,C) by [

-N: M:]

and

post-multiply T(P,C) by

[1~]

,

both norm pr

e-serving bydefinition of nrcf and nIcf, then:

IIT(P,C)llco = 11

ix;«;

+

YnNntlL

=

IIA-

1

L

o

Lemma 5.5 Let ihe plantP be described byits nrc]

(Nn,M n), then an expression of the re] (N,M) IS

given byNnA-1,MnA- 1 with:

Pro of: Sinee the cont roller will stabilize the p

er-turbed plant the functi on AE

.J

hence :

o

Proof of Theorem 5.3: Using Lemma 5.5in (7)

an upper bound on the per turbations ean be writ

-ten as:

11 [ élM ] Ilco= inf 11 [ Mn] A-I - [ MI:). ] Q lico

s«

QE'H

n;

NI:).

IC we define a new freedom

Q

= QA-l the upper

bound on the eoprimeper turbati ons ean bewritten

as:

11 [élM] llco= [nf 11 ([ M n] _ [ Nh]Q)A - l ll

s

»

QE'H N n NI:).

Then an estimate of this boun d is given by a

weightedgap-metriedist an ee: S(P ,PI:).)11 A-I lico

Henee

s,

ç

Sband the theor em is proved . 0

Corollary 5.6 Since /IT(P,C) /Ico = IIA-111co and

IIT(P,C)lIco ~ 1 this new stability margin equals

the gap-m etri c if

A

is an al matrix, witha areal

number.

Example

For sim plieity this example contains sing le loop

syste ms . All the necessar y calulations are i

mple-ment ed in PC-M ATL AB.

The nominal closed loop consists of a fourt h order plan t Peontrolled by a second order

con-troller C, their resp ecti ve Bod epl ot s are given in

fig. 3. Normalized coprime fact ors of Pand C

are used to compute A, A-I and IIA-lil_co' The

Bodeplot of A is given in fig. 4. The area below

IIA-lll:,l = IIT(P,C)II :,l in Fig.4 is allowed for c

o-prime perturbations if the robustness is measured in gap-rnetric sense. Whereas the dashed area can also be exploited by the eoprime perturbations if the new robustness margin is used.

Fig.4: Bode plot A

Fig. 3: Bodeplot plant P and cont roller

C

[1] Bongers, P.M. M., O.H. Bosgra (1990), Low order rob ust Hoo controller synthesis, To ap-pear at the 29t h _{Conf. Decision and Control,} Hawaii,USA.

[2] Desoer, C.A., R-W. Liu, J. Murray, R. Seaks (1980), Feedback System Design: The F rac-tional Representation Approach to Ana lysis and Synthesis, IEE E Trans. Autom at. Conir., AC-25,399-412.

[3] Georgiou, T.T.(1988), On the computation of the gap metric,Syst.Control Lett., vo111, 253-257.

[4] Georgiou, T.T, M.C. Smith (1990), Optimal robustness in the Gap metric, IEEE Trans. Automat. Contr., AC- 35,673-686.

[5]

McFarlane, D.C., K. Glover (1989), Robust controller design using normalized coprime factor plant descriptions, Lecture Notes in Control and Information Sciences,vo1.138. [6] Meyer, D.G., G.F.Franklin (1987), A connec

-tion between normalized coprime factorization and linear quadratic optimal regulator theory, IEEE Trans. Automat. Contr., AC- 32, 1041-1047.

[7] Vidyasagar, M., H. Schneider, B.A. Francis (1982), Aigebraic and Topological Aspects of Feedback Stabilization, IEEE Trans.Automat. Conir., AC- 27, 880-894.

References

margin. Thereby for a larger class we have guar-anteed a robustness margin. The new robustness margin is as easy to compute as the gap-metric ro-bustness margin. In an example the difference be-tween both margins is illustrated.

10'

1~1 10' frequcnoy fJh)

Next we construct constructfactors (MA,

NA)

as:

[

~:

] = [

~

] A-I

+

72 [

~

]

.

such that PA= NAM;;/ E Sb and near the boundary of Sb. By Theorem 5.1 T(PA ,C) E1{.

The gap-metric distance g(P,PA)= .45, accord-ing to Theorem 4.1the stability of the closed loop is guaranteed in gap-metrio sense if g(P, PA)

<

IIA-III:,l

=

0.138.

Hence PA E Sbbut PA

ti.

S9thus the new sta-bility margin is less conservative than gap-metric robustness. lil' lOl plant

-

10'

]

amtroller :g, E I~I

.

I~' I~' 1~1 10' la' I~' frequcnoyfJhJ

6

Conclusions

In this pap er a newrobustn ess margin hasbeen de-rived. Com paredtothe gap -met ric robust ness, the factorization of the applied controller is explicite ly taken into account. Hen ce one aspect ofconse rva-tiveness is reduced by introducing this robust ness

Selected Topics in Identification, Modelling and Control Vol. 2, December 1990

Low order robust

H

_oo

controller synthesis

Peter M.M. Bongers, Okko H. Bosgra t

Lab. Measurement and Control, Dept. Mechanical Eng. and Marine Technology, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands.

Abstract. In this paper robust stability of the closed loop behaviour of plants is in-vestigated. It is shown that when the distance between the nominal plant model and the boundary of the nominal plant envelope is measured in the gap metric, a sufficient condition on the closed loop stability can be given. Low order controllers are derived using either model reduction or controller reduction techniques. It is shown that using controller design followed by controller reduction techniques the size of the plant envelope is larger than if one is using plant model reduction followed by controller design.

1

Introduction

The problem of robust stability of closed loop plants has received a considcrable amount of attention in recent years. Using Hoc based controllers, bounds on the nominal plant envelope can be given such that every plant within the envelope is stabilizcd by the controller. Two common envelope descrip-tions are defined by an additive transfer function or a multiplicative transfer function. These envelope descriptions are studied in [6],[2] and necessary and sufficient conditions for robust stability are given for all plants in the envelope description.

The sufficient conditions for robust stability im-ply a test on an Hoc norm of a certain transfer func-ti on and hence the existence of a robust stabilizing controller can be determined by iterative Hoc mini-mization techniques as for example described in [5).

A more general plant envelope description can be found using the algebraïc control theory [23), [21), [22], [4)). The associated plant envelope de-scription [27) has been studied by Vidyasagar and Kimura [25) in terms of stabIe factor uncertainty and Graph metric uncertainty. In this envelope de-scription the various plants within the envelope (eg.

tThe original version of this paper was presented at the 29t h _{IEEE Conf. Decision and Control, December 5-7,1990,}

lIonolulu, Hl, USA. Copyright ofthis paper remains with the IEEE

perturbed plants) may have a different number of unstable poles. The bound defining the envelope considered in the Graph-mctric is also independent of the factorization of the various plants (i.e. thc nominal plant as weil as the perturbed plant).

Glover and McFarlane [11) have shown that if a normalized left coprime representation of the plant is used, a Hoc controller of the same order as the plant can be calculated and a robustness margin

in terms of stabIe factor uncertainty can be given without any iterative calculations.

In th is paper we show that Hoc controllers based on a norrnalized right coprime plant description in-duce a stability margin in the gap metric. Since the gap metric distance is smaller than or equal to the graph metric distance [20) the plant envelope stabilized by the controller is larger.

Anderson and Liu [1], McFarlane et al. [15] ob-tain low order controllers by reducing the designed high order controllers or design the controllers on reduced plant modeIs. They use balance and trun-cate [19] on the stabIe factor plant description to obtain low order modeIs.

In th is paper the control synthesis problem is rewritten such that the control design is a onc block Nehari problem and the reduction is a Hankel norm reduction problern [10]. As a result a tighter bound on theHoc norm is cstablished allowing robust sta-bilization of a larger plant envelope by the low order

controller.

2

Problem formulation

Feedback Configuration: The plant

P

will be controlled by the controller C in the feedback

con-figuration as given in Fig.1

p

r---,

Ä I I I I I I I I I I u

+ -

-}

y L ~

Fig.2: Perturbed system representation

Fig. 1: Feedback configuration

The closed loop transfer function H(P,C)

map-ping the inputs

(eI,e2)

onto the outputs

(U1,U2)

is:

T(P,C) = H(P,C) -

[~ ~]

= T(P,C)

or more Pt>'s. If the distance between the nominal

and the perturbed plant model is measured in the

gap-metric, denoted as ó(P, Pt», both models can

have a different number of unstable poles and dif-ferent factorizations and yet fit in a theory of

ro-bust stability. Vidyasagar [24] proved that if the

coprime factors of the nominal plant and the per-turbed plant are converging, their closed loop be-haviours converge too. With this important result we are able to measure the distance between two open loop plants and relate it to the closed loop

behaviour of both plants. Also the fact that the

real plant lies within the plant envelope has impor-tant implications about its closed loop behaviour.

A sufficient condition to ensure robust stability of the feedback configuration is given by:

IIT(P,

C)lloo

ó(P,

r;

<

1 (2)

Vidyasagar and Kimura [25] proved the same ro-bustness result for the graph-metric distance be-tween the nominal and the perturbed plant model. The distance measured in the graph-rnetric is larger than or equal to the distance measured in the gap metric, so we have a stronger robustness result. (1) -P(I

+

CP)-l ] (I

+

CP)-l

[

-C I

l P]

-1 [ 1 - P(I

+

CP)-lC (I

+

CP)-lC H(P,C)

=

with the restrietion on C th at det(I

+

CP) =1= O.

The feedback configuration of Fig. lis called inter-nally stabie if and only if the four transfer functions in (2) are stable, Note that stability of three trans-fer functions in (2) does not imply stability of the fourth transfer function [3].

Instead ofH(P, C) we will study the closed loop

transfer function :

Robust Stability: In general if a perfect model

of the plant

P

is available, it will be non-Iinear and

of extremely high order. In engineering practice

the plant will be described by a high order lin

-ear model. This nominal model is an

approxima-tion of the real plant. The discrepancy between

the nominal model and the plant is approximated by an uncertainty model. The feedback loop wil! be called robustly stabie if the transfer function

T(P,C) remains stabie for all plant variations

de-scribed by the uncertainty model. In this paper

we will study the coprime factor uncertainties, the

nominal plant model P

=

N M-1 _{is perturbed to}

Ps

=

(N +!:i.N)(M +!:i.Mt1_{as depicted in Fig.2.}

Itis assumed that the real plant lies in the plant

envelope, therefore it can be bounded by Pand one

Low Order Controllers: Using the Hoo

con-trol design method Glover and McFarlane [11] have

found a controller

C

such th at IIT(P,

C)ll

_oo

is

min-imized over the class of all stabilizing controllers.

They showed that a controller with the same order of the nominal plant model give rise to the

mini-mum achievable IIT(P,

C)ll

oo'

Due to the fact that

the norninal plant model is in general of a high or-der this will result in a high oror-der controller. For practical implementation the controller should have areasonabie low order which is frequently much lower than the nominal model. Anderson and Liu [1] presented three ways to obtain the low order controllers: (i) reduction of the nominal model fol-lowed by controller design on the reduced model; (ii) controller design on the nominal model followed

P(s) or P if there is na confusion. With a slight abuse of notation a transfer function is given by: P(s) :=D

+

C(sI - A)-IB := [ sI_-;A

ff]

AT denotes the transpose of A and P*(s) de-.notes pT₍

_-s).

_{For minimal plants P(}

_s)

_{E 1{,}

(P( s) E 1{l.) the controllability and observability

Grammians Wc and Wo, respectively are the pas-itive definite symmetrie solutions of the following Lyapunov equations:

by controller reduction; (iii) direct design of the low order controller. McFarlane et al.[15] presented ap-proach (i) and (ii) together with an upper bound on performance deterioration, meaning an increase of IIT(P,

C)lloo'

In combination with eqn

(2)

it can be seen that under performance deterioration due to controller reduction the amount of allowable uncer-tainty is decreased.

In this paper the trade-off between the order of the controller and the allowable plant perturbation will be investigated. Itwill be shown that the con-troller design on the nominal model followed by controller reduction implies a larger bound on the plant envelope than reduction of the nominal model followed by controller design on the reduced model.

AWc

+

WcA T

+

BBT

=

0 ATWo+WoA+CTC=O ( -AWc-WcAT +BBT=O) -ATWo - WoA

+

CTC

=

0 (3)

3

Preliminaries

Proposition 3.1 ([lB), [13}, [19}) Let either P(s) E1{ or P(s) E 1{l., and suppose that P(s) has a minimal state-spaee representation:

StabIe multivariable linear systems can be studied by considering them as transfer function matrices having all entries belonging to the ring 1{. More-over, in many cases (e.g. convolution operators) the ring 1{ is commutative and is an integral do-main (i.e. 1{ has no divisors of zero). We consider the class of possibly unstable multivariable systems as transfer function matrices whose entries are el-ements of the quotient field

F

of 1{. Throughout th is paper we use

([26], [4]):

F := {a/b 1a E 1{,

s

«

1{\O}, a quotient field

of1{

o

:= {a/b

I

a E1{, bE

I\O},

a not necessarily commutative) ring with identity

1{ := a subring of

0

with identity

I :=

{g

E

0

I

g-I E

O},

the set of multiplica-tive units of

0

:J

:= {h E1{ 1h-I Efi}, the set of

multiplica-tive units of1{

Note that:

:J

CIC 1{

c

c

c

F

With a slight abuse of notation systems P E Fmxn are denoted as P E F. In the se-quel of this paper we will study real rational

fi-nite dimensional continuous linear time invariant systems. For this reason we choose to identify the ring

0

with IRL

_oo,

the space of proper real-rational functions with no poles on the extended imaginary axis. On IRL

_oo

we use the norm 11.1100:

Ilf(s)lloo

= suPwä[f(jw)]

The subring 1{ is then identified with IRH

_oo

the subspace of IRL

_oo

of systems with no poles in the closed right half plane, and analogously

1{l. is identified with IRH;;'. The following nota-tion is used. We will denote transfer funcnota-tions as

P = [

sI_~/

I

~

]

then there exists a non singular state transforma-tion matrix T sueh that the plant Pb has a realiza-tion

= [ sI - T-IAT

I

T-IB ]

Pb CT D

andWc, Wo as symmetrie positive definite solutions to the appropriate Lyapunov equations

(3)

with the property:

Then the plant Pb (s) is ealled a balaneed realization

of P(s).

Proposition 3.2 Let either P( s) E 1{ or P(s) E

'H",

The Hankel singular values of P(s) are:

with Wc, Wo the symmetrie positive definite so-lutions to the appropriate Lyapunov equations

(3). The Hankel norm of P(s) is defined as 11 P(s)

IIH=

af!

Definition 3.3

([26], [12])

fractions: A 'plant P E F has a right (left) fraetional representation if there exist

N,M(N,M) E 1{ sueh that P = NM-I ₍₌

coprimeness: The pair M,N(M, N) is called right (lef!) coprim e (ref or lcf) if ih ere exists

U, V(U, V)

E'H

suc h that:

VN +VM= I

P = NM-I

(NÜ +ÛV

= I)

(P = Û - IN)

4

Perturbation bounds

Let the nominal plant be given by

P

E:Fand let

a plant in the envelo pe be given by Pe:. E:F. The boundary ofthe nominalplantenvelope containing

all plant s to be stabilized by the cont roller C EF

.is given by:

normalized coprimen ess: In addition ihe pair M,N(M,N) is called normalized right (lef t)

copri m e (nrcf or nlcf) if they are coprime and: M* M

+

N* N = 1 (M M*

+

N N* = I)

Meyer and Franklin

[17]

have presented a state

-space representation of norrnalized coprime factor-izations.

Definition 3.4

[16]

Letthe pair (N, M) E 1{ be

a nrcf of P E :F, then the Graph Hankel singular values of Pare defined as:

with

uF

defined as in Proposition 3.2.

Definition 3.5

([24],

[7], [9]) Let Pi E:Fadmit a rcf (Ni,Mi) E'H such that (Ni, Mi) is normalized.

The distance between two plants PI and P2 in the

sense of ihe gap metric is definedas:

with:

Moreover :

The next theorem gives a sufficient condition on robust stability for all plant s Pe:. in theclass V.

Theorem 4.1 Let C E:F be a nominal controller

stabilizing the nominal plant P E :F, then

C

s iabi-lizes all Pe:. in the class V ij:

e

IIT(P,

C)lloo

<

1

To prove theorem 4.1 we first need a lemma. In this lemma a simple expression for the Hoo norm of

the closed loop transfer function T(P,C) is given .

It shows that the norm ofT(P,C) equalsthe norm of a specific lef of the stabili zing controller.

Lemma 4.2 Let P E:Fadmit a nrcf (N ,M) E1{ and let C E :F be such that C st abilizes P, an C admits a lcf given by CY,X) E'H such that X M + YN = I, then:

IIT(P

,

C)lloo =

II[f

X]L

Pro of: The closed loop transfer function

T( P,

C) can be written as:

T(P,C) T(P,C)

= [ - : ]

(I

+

C

P) - l

[

C

1

J

=

[

-:]

(YM +X N t l

[X

Y J

then :

Remark 3.6 Given twoplunis PI and P2 th at d

if-[er in relevant propert ies, then mimizat ion over Q may be expected to lead to Q = 0, since as the[a

c-torizat ion is normaliz ed, thegap betw een thes e two

plants equals one.

Let PI be given and let P2 describe ihe same

plant, in a different factorization, then the transfer

function Q is the unitary freedom in the normal-ized factorization: Q*M* MQ

+

Q*N* NQ = 1 or Q*Q = 1 leading to a gap which equals zero.

II

T_{(P, C)lloo =}

II[

-:

]

_[

_{X Y}

J

L

Now

(N, M)

is a nrcf of

P

,

so

11 [ - : ]L

thus:

1,

(6)

5

Controller

characterization

(X(6.M

+

MQ)

+

Y(6.N

+

NQ) E.J (9)

Thus (10) provides not only closed loop stability for one perturbed plant but for all plants in the enve-lope in the class V.

o

Remark 4.4 Glover and McFarlane [11] proved stability[or a class of stabiefactor perturbations on the normalized left coprimefactors of the nominal plant. The restriction of Q to be Q = 1 and the

factorization of (M2,N2 ) as right coprime factors defines the allowable stabiefactor perturbation on the normalized right coprime factors. /lence sta-biefactor perturbations effeclivel y allowing an even

sm aller classV than thegraph-metric.

Remark 4.3 Vidyasagar and Kimure [25] proved a similar statement on robustness. However they measured the distance between the nominal plant and any periurbed plant in the graph-metric. The restrietion on Q that Q E

.J

and

IIQli",

~ 1 as fac-torization freedom in (10) defines the graph-metric distance between the two plants, thereby allowinga smaller classV.

I1 [

~~

] IL

=

J~~

I1 [

~:

]

Q - [

~:

]

L

(10) In (5) and (8) the freedom of choosing the transfer function Q to fit the nominal plant and the per-turbed plants as close as possible can be used to minimize the norm of theerror realization:

Since (X MQ

+

YNQ )= Japplicationof thesmall gain theorem and lemma 4.2 proves part

(b).

By proving part (a) and (b) it is shown th at a controller

C stabilizes all plants P_tlin the envelope ofP if: Simple manipulations show that (9) cart be written as:

T(Ptl, C) is stable if and only if:

T(Ptl,C)

= [

-~QQ

] (Y NtlQ+X MtlQtl [X Y] is stabIe if and only if(X MtlQ+Y NtlQ) E

.J

which is equivalent with [X(M +6.M)+Y(N+6.N)] E

.J

.

Without loss of generality the controller factoriza-tion (Q X, QY) can be chosen such that (Q X M + QYN) = J. Hence:

[X(M

+

6.M)

+

Y(N

+

6.N)] E

.J

if and only if

(J

+

[X Y] [

~~

])

E

.J

then

rtr:

C) EH. By applying lemma 4.2 to (7)

part (a) is proved .

part (b): Let

P

have a rcf (N Q, MQ) E

H

with Q E Hand factorize the perturbed plant

Ptl

as:

P«

= NtlMó.1 ₌

(6.N

+

NQ)(6.M

+

MQ)-I The

uncertainty can be written as:

The small gain theorem applied on (6) leads to the condition that if:

with:1I [

~~

]

11

",

<

E The closed loop transfer function of the perturbed plant controlled by the nominal controller:

T(P,C) = [ - : : ] (Y N

+

XM)- I [X Y ]

A per turbed plant P_tl E

:F

satisfying (4) can be

factorized as:

with _{(Ntl, Mtl)} E te a nrcf of

Ps

,

Q E H allow-ing (NtlQ, MtlQ) to be an arbitrary rf of Pand (6.N,6.M) the coprime factor uncertainty. This uncertainty can be written as:

Proof of theorem 4.1: : The proof is split into two parts. In part (a) we show robustness for S(P,Ptl )

s

E,in part (b) robustness for S(Ptl,P)

~

e.

part (a): Given (QX,QY) E H a lef of C and

(N, M) E1ia nrcf of P,then T(P,C)can be writ -ten as:

[

~~

] = [

~

] Q- [

~:

]

(8)

The controller used in the feedbackconfiguration of Fig. 1 must satisfy two objectives: First it has to

The controller design procedure implied by this the-orem is remarkably simpie: solve a one-block Ne-hari problem as in Lemma 5.3in order to obtain a controllerwhich stabilizesthe closed loop andrnax -imizes its robustness in view of Theerem 4.1. The minimal achievable valueof Q in the one block

Ne-hari problem is well known.

Lemma 5.3 Let R be [M" N" ], (R E 'Hl. by dejinition) with McMillan degreen and let Q E 'H

with McMillan degreen, then: stabilizethe closedloop. Thiscan beform alized as

to find a controller C EF based on a nominal plant model P E F such that the closed loop mapping T(P,C) E'H. Secondly it has to stabilize all plant models P~ ED. In view of theorem 4.1 this poses an Hoo minimization problem on the closed'loop T(P,C) : infcE.r- IIT(P,C)lIoo

By definitions of the ring 'H, T(P,C) E 'H if and only if H(P,C) E

'H;

thus without loss of generality we may inspeet T(P,C) on stability. If T(P,C) E'H we say that C stabilizes P. The sta-bility ofT(P,C) can be checked by inspecting each of thefour transfer functions . However as stated in theorem 5.1 it is possible to check the stability of

T(P,C)by ins pecting jus t one trans fer function. Proof: [8] p.68

o

Theorem 5.1 Let P E F have a re](N,M) and

let C E F have a re] (X,

Y)

and a lef

(Y;

X), then thefollowing statements are equivalent:

(a) C stabilizes P

(b)

[

~

- ;

]

E.1

(c)

(XM

+

YN

)

E.1

The minimal achievable Hoo bound on the closed loop T(P,C) can be stated in terms of the Graph Hankel singular values.

Corollary 5.4 For a given plant P E

F

with its nrcf (N,M) E 'H, controlled by a controller C E

F such that

C

stabilizes P, theH00 bound on the closedloop transfer funetion isgiven by:

IIT(P,C)lloo ~, Proof: (a) {:}(b) is proved by in [25]

(a) {:}

(c)

T(P,C) = T(P,C) with: 1 , ~ - - - ; = = = = = == =

)(1-

I1

[M" N"]

1111) can be written as: Hen ce T(P, C) E 'H iff

(X

M

+

YN

)

E .1, smce

[

-:

]

,

[

X Y

]

E'H by definit ion . 0 McFarl ane [14] has shown the existence of a con-trollerC such that the closedloop is internally sta-bieandits infinity norm is minimized. For our pur-poses we formulate this result as follows.

Theorem 5.2 Let(N, M) E'Hbea nrcf of P(s) E

F and let

(Y, X)

be a lef of an admissible controller C(s) EF, then:

IIT(P,C)lIoo ~,

if and only if:

I1 [

~

]

"

+

[X Y]

L

~

)(1 -

,-2)

= Q

Proof: The rcf version ofTheorem4.2 in [14] 0

Remark 5.5 Notethat the open loop plant directly detemi nes the maximal achievable closed loop ro-busin ess margin.

6

Low order controllers

It isdesirabie to find a controllerof possibly low or -der. In fact a controller design theory should indi-cate those situationsin which a low order controller is not able to satisfy the performance requirements. Although in the previous section no attention was paid to the order of the controller it is desired to obtain an r th order controller for an nth order plant

P, with r

<

n. In th is section we will show the common properties and the differences of the two indirect ways to obtain low order controllers: (i) model reduction of the plant and cont roller design on the reduced model; (ii) cont roller desing on the nominal model and controller reduction. In thepr e-vious section the con trol problem was reduced to determining:

with P

=

N M-I _{E :F,(M, N) E 'H a nrcf and}

C

=

X-I

Y

E :F,

(X,

Y)

E'Ha lef. For an nth order plant model Glover and McFarlane [11] showed the existence of an nth order controller.

Let the nth order plant Pn E :Fbe given by its

nrcf(Nn ,Mn ) and let th~ rt~order controller CrE

:Fbe given by by its lef(Yr, Xr ),then the controller

synthesis problem of inf

_{c E}

.1"IIT(Pn, Cr

)1100

is:

11[

M ]* [ - - ] 11

inf n + X; Y,.

(Yr,Xr)E'H N;

00

plant reduction and controller

design

Equation (12) can be rewritten as:

(12)

with ap(Pn) the graph Hankel singular values in de-creasing order.

Pro of: Apply Hankel norm reduction on the nrcf

of then th order nominal plant to obtain a (r + 1)th .order ref of the reduced order plant. Using Glover

(10]

the Hankel norm model reduction of the nIh order plant model to an (r +

1)lh

order reduced model induces an Loo error bound smaller than 2:i:r+2 ap(Pn(s)). The first (r+1) ap(Pn(s)) re-main the same.

Thus the Nehari approximation of an (r + l)th order reduced plant model results in an rt h order controller with a distance of

af(P

n ) . Using these

results in Theorem 5.2 proves the lemma. 0

(14) inf 11[Mn]* + [.Kr

f

r ]11 (13)

(Yr,Xr)E'H N;

00

=

(Jr~l)

I1 [

~:

r-[

~:::

r

+

[

~:::

r

+

[.Kr

~]

1100

The intention of model reduction on the nominal plant can be seen in the first two terms of the right hand side of (13), controller design on the reduced model can be seen in the last two terms (rhs) of (13). Using the triangle inequality (13) leads to:

I1[ M ]* [ - - ]11 inf n + X; Y,. (YroXr)E'H N;

00

~

(Nr+l,il}!+tlE'H11 [

~:

] * - [

~:::

] *L

+

(Yro~!)E'H

I1 [

~:::

]*

+

[.Kr

~]

L

The first term (rhs) of (14) is a Hankel norm model reduction problem and can be solved using the for-mulas of Glover [10], the second term (rhs) is a one block Nehàri problem as in Lemma 5.3. This is a Nehari problem on the reduced order plant model. Based on these observations an upper bound on the left hand side of (14) can be estimated in terms of the graph Hankel singular values. Thus an upper bound on _IIT(Pn,

Cr)lloo

can be given.

Lemma 6.1 Let P;

=

NM-I _{denote an nth} or-der plant controlled by an rth order controller Cr=

X;l"Yr,

then an upper bound on T(Pn,Cr) isgiven by:

The first term (rhs) of (14) can be seen as the dif-ference between a nominal plant model and a per-turbed model. Then by applying the same argu-ments as in section 4 the low order control stabilizes all perturbed models PI),. if:

Corollary 6.2 Based on (15) we can state that it ispossible to trade-off a lower order of the controller by robustness of the closed loop against plant per-turbations.

controller design and controller

reduction

Equation (12) can be rewritten as:

(Yrot~E'H

I1 [

~:

].

+

[.Kr

~]

L

(16)

= )nf ,,[

~n

] . -

[X Y]

+

(YroX r) n

[X Y]

+

[.Kr

~]

1100

The intention of controller design on the nominal plant can be seen in the first two terms (rhs) of

(16),

controller reduction can be seen in the last two terms (rhs] of (16). Using the triangle inequality (16) leads to:

in which the first term (rhs) is a one-block Nehari problem on the full order model and the second term (rhs) is a Hankel norm model reduction prob-lem on the controller.

Lemma 6.3 Let the graph Hankel singular values of the controller

C

n - 1 be given by aF(Cn-I), then

an upper bound of T( P, Cr) is given ay:

Proof: The Nehari approximation of an nIh order

plant model results in an (n - 1)lh order controller with a distance of

af(P

n ) . Apply Hankel norm

re-duetion on the rcf of the

(n -

1)lh order controller to obtain r1h order rcf of the reduced controller.

Using Glover (1984) the Hankel norm model re-duetion of the

(n -

1)lh order model to an

r

1h

or-der model induces an L_{oo error bound smaller than}

L:i=r+2 aF(Cn-1(s)). Using these results in

Theo-rem 5.2 proves the lemma. 0

Theore m 6.4 Let an nIh order plant P E :F be given by its nrcf(Nn,Mn) and an rtk,

(r

<

n), or-der controller with lef

(.KT!

Y;.)

such the closed loop is maximum robust or that:

_inf

II[

~n

],.

+

[X

r

Y;.

lil

(19)

(Yr.Xr)E1t' n 00

then if a con troller is designed using the controller design and cont ro ller redu cti on approach an equal or larger robustn ess margin is created tha n a c on-trollerdesigned us ing thecontroller design and con-troller reduetion approach .

Proof: In the Neh a ri approximation of an nIh or

-der unstable model by an (n - 1

)th

order stabie model the bound is determined only by the first singular value. The remaining singular values of the

(n -

1

)th

order approximation are smaller or equal to the singularvalues of the nIh order model.

Hence the bound of (18) isin general smaller than the bound of(15),and the theorem is proven . 0

7

Conclusions

In this paper we proved stabilityfor an envelope of plants. This envelope is bounded by a gap metric distance to the nominal plant. Compared to both the stabie factor uncertainty and the graph metric uncertainty we have increased the size of the al

-lowable plant envelope. Itis shown that a trade-off

can be made between the order of the controller and the size of the nominal plant envelope. Alow order controller induces a smaller bound on the envelope than a full order controller. We have,also shown that plant model reduction followed by controller

.design induces a smaller bound on the plant enve-lope than controller design followed by controller reduction.

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@1990DelftUn ivers ity Pres. _{Select ed Top ics in Identification ,ModeIlingand Control} Vol. 2, December1990

Necessary conditions for statie and fixed order

dynamic mixed

H

₂/

H

oo

optima1 contro1

Maarten Steinbuch tand ükko H. Bosgraf

tPhi/ips Research Laboratcries, P. a.Box 80. 000, 5600 JA Eindhoven , The Netherlands

§Lab. Measurem ent and Conirol, Dept. Mechanica/Eng. and Marine Technoloog,

Delft University of Technoloq ç, Mekelw eg2, 2628 CD Delft, The Neth erlands.

Abstract. Thispaper considers thedesign ofst at ie and fixed order dynamic controllers based on two dual mixedH2 /H00 performancecriteria. Necessary conditions for static and fixed order

dynamic output feedback laws are derived for the minimum of an H2 norm control problem,

under the restrietion of an Hoo norm bound on a transfer function. The design method is

illustrated with an example.

Keywords. Hd Hoo control, min max optimization,optimal control,fixedordercontrol,robust

control, static feedback

1

Introduetion

In the sixties the optima! LQG control approach pro

-vided a systematic solution to the control design problem of multivariable systems, see (Anderson and Moore,1971;Kwakernaak and Sivan, 1972; Levin eand Athans, 1970). Despite of its relevance in the formu-lation of performance requirements , LQ optimal con-trol was shown not to possess guaranteed robustness margins if applied in conjuction with an observer or Kalman filter (Doyle and Stein, 1981). The robust

-ness issue has attract ed a lot of attention during the

last decade, resulting in the development of Hoo con

-trol theory (Doyle and Glover , 1988; Doyle and

oth-ers, 1989; Francis, 1987; Maciejowski, 1989). Robust control based on the II00 norm of a weighted clo

sed-loop tran sfer functi on iscap abl eof handlin g the design

problem for syste ms with uncertain ties. Itis based on

determining a controller for the worst-case uncertainty

and for the worst-case performance specificat ion. A

I-though this approach is very attract ive to ensure

sta-bility of the per turbed closed-Ioo psys te m ,it isnot al

-ways usefulforperform an cerequirements. This hasled

to the design problem ofstat ing perform an ce and ro-bustness objectives in both theH2 and II00 fram ework

(Berns te inand IIaddad ,1989;Doyle,Zhou and Bod

en-heimer, 1989; Mustafa, 1989; Rotea and Khargonekar,

1990). In generalone can define the problem as follows. Suppose the plant is given by its transfer futietion G(s)

with three types of inputsand outputs. The manipu-lated variables are denoted by u, the measured outputs are denoted by y. The sign als (Wl,ZI ) are related to

H2 type of performance crite ria, whereas (W2 , Z2) are

related to H00 norm requirements.

W2 Z2 Wl G(s) ZI U y

I

F(s)

I

I

I

Fig. 1: Standard H2/1100 control problem.

IfW2

=

0 and Z2isnottakeninto considerat ion, Fig.1

shows the configuration for a stan dard H2 (LQG) d

e-sign problem of finding a controller that mini mizes the 2-norm of the tra nsfer function from the whitenoisein

-puts Wl tothe performance measuresZ! . On the other

hand if w!

=

0and z! is not taken into account, then Fig.1showsthe configuration for ast anda rdH00design

problem of finding a controller F(s) such that the

00-norm of thetransferfunction from the L2-boundeds

ig-nalsW2 to the performancemeasures Z2 is bounded to

somerealnumber. Ifall signals are takeninto account,

Consider the following state-space realization Proofsee (Willems, 1971).

following performance measures can be defined.

(4) (3) 11 G(s)1100

<

1

x

₌

Ax

+

Blwl

+

B2W2

+

Bu ZI Clx

+

DI3U (7) Z2

=

C 2x

+

D23U y Cx (i) I1G(s)1100

=

maxa(G(jw)) wER

with

a

themaximum singular value.

Lemma 2.3 Let (A, C) be observable and (A,B) eon-trol/able. Then thefol/owing statementsare equivalen t:

where S

=

ST ~ 0 is the controllability Gramian and P

=

P" ~0 is the observability Gramian:

AS+SAT+BBT=O ATp+PA+CTC=O

(ii) 3P

=

p T

>

0 satisfyingthe Riecati equa tion

ATp+PA+I-2PBBTp+C TC=0 (5)

sueh that A

+

1-2BBT P is stable.

(iii) 3S

=

ST

>

0 satisfying the Riccati equation

AS

+

SAT

+

1-2S CTC S

+

BBT

=

0 (6)

sueh that A

+

1- 2SCT C is stable.

(

1 1 00 . )1/2

11 G(s)

Ib

= -

tr(GT(-jw)G(jw))dw (1)

211" -00

The Hoo norm of a transfer function can be calcu -lated using the relation between the singular values of

G(jw) and the eigenvalues of the Hamiltonian matrix

related to (5) or (6) (Bruinsma and Steinbuch, 1990). Definition 2.2 The //00 norm of a transfer function

G(s) is definedas:

with tr(.) the traee matrix operator.

The 2-norm can be computed as:

Definition 2.1 The H2 norm of a transfer funetion

G(s) is defined as:

problem is to find a controller F(s) such that the 2 -norm from WI to ZI is minimized, under the constraint of a bound on the oe-norm of the transfer function from

W2 to Z2. However, in th is case a worst-case distur-bance W2 'does not affect the H 2 performance meaning that this problem addresses a robust stability nominal

H2 performance kind of problem. There is not yet a general solution for this type of mixed.H2/Hoo prob-Iem. Rotea and Khargonekar (1990) derived conditions for which it is solvable in the state-feedbackcase.

Robust performance mixed H2/Hoo can be intro -duced in some sense by taking the veetors WI and W2 as identieal or, by duality,ZI and Z2' For these types of problems an upper bound for the pure H2 performance can be derived while constraining the oo-norrn of the other transfer function. This has been the approach of Bernstein and Haddad (1989) and of Doyle, Zhou and Bodenheimer (1989). Doyle, Zhou and Bod en-heimer (1989) and Zhou and othe rs(199U) have posed the problem ZI

=

Z2, WI

=f.

W2and have given a very nice interpretation for an induced semi-norm on the transfer functions. Their work showed that the pr ob-lem can bestated preciselyin ter ms ofsignalsets, and they have derived necessary and sufficient conditions for the full-order feedback case. Bernstein and Had -dad (1989) have considered the mixed H2 /Hoo prob

-lem for the situation where WI

=

W2,ZI

=f.

Z2. They used a performance cri terion relating W (= WI = W2)

with ZI ,under the constraint of an Hoo norm bound on the transfer function from W to Z2. They have derived necessary conditions for fixed order controllers and nee-essary and sufficient conditions for the full order case. Mustafa (1989) has shown that if WI

=

W2 and ZI

=

Z2

the auxiliary performance index of Bernstein and Had-dad can be given the nice interpretation as the result of minimizing an entropy expression, yielding the central

H00 controller for the full order case.

In this paper we will consider both the problem definition of Doyle, Zhou and Bodenheimer (1989): ZI

=

Z2,WI

=f.

W2,as well as the case of Bernstein and Haddad (1989): ZI

=f.

Z2, WI

=

W2. The contributions of th is paper are: (i) we will extend the results given by Doyle, Zhou and Bodenheimer (1989) to the fixed order feedback case, (i i) we will give an al ternati ve def -inition of the problem posed by Bernstein and Haddad (1989), using a worst-case output injection interpreta -tion of Z2, (iii) we will explore the duality between the two problems, (iv) it will be shown that the necessary conditions for optimality for fixed-erder controllers are a special case of those for statie output feedback of an enlarged system and (v) the design method will be il-lustrated with a numerical example.

2

Norms and Notations

Given a stabie strictiy proper transfer function matrix G(s) with state space realization C(sI - A)-IB the

In this model we assume that there is no direct feedthrough from.the disturbance vectorWI to the out-puts y and ZI,in order to assure a bounded H2 norm

from WI to ZI. For simplicity we assume also that there is no feedthrough from the disturbance vectorW2 to y