Spécialité: Mathématiques Option: Théorie Mathématique du ... · Ahmed BARIGOU Jury de soutenance TERBECHE Mekki Président Professeur Université d’ORAN MESSIRDI Bekkai Directeur

UNIVERSITÉ D'ORANFaculté des Sciences

Département de Mathématiques___________________________________________________________________________

THÈSEprésentée pour l'obtention du diplôme de

Doctorat d'État

Spécialité: MathématiquesOption: Théorie Mathématique du Contrôle

CONTRIBUTIONÀ L'OBSERVABILITÉ DES SYSTÈMES LINÉAIRES PERTURBÉS

Par

Ahmed BARIGOU

Jury de soutenanceTERBECHE Mekki Président Professeur Université d’ORAN

MESSIRDI Bekkai Directeur de thèse Professeur Université d’ORAN

BEKKAR Mohamed Examinateur Professeur Université d’ORAN

BOUAGGADA Djillali Examinateur Maître de Conférences -A- Université de Mostaganem

TALHAOUI Abdallah Examinateur Maître de Conférences -A- E.N.S.E.T. d’ORAN

Année universitaire 2010-2011

In the Name of Allah, the Beneficent, the Merciful

DedicationTo my family in supporting me

Praise be to Allah

RÉSUMÉ DE THÈSEContribution à l’observabilité des systèmes linéaires perturbés

Mots-clefs : Opérateur borné, système linéaire, perturbation Bochner-intégrable,

espace de Hilbert, espace de Banach, observabilité idéale, observabilité au sens de

Kalman, invariance conditionnelle, invariance contrôlée, Factorisation opérationnelle.

Le concept d' "observabilité idéale" des systèmes perturbés reste un sujet

d'actualité malgré tous les développements dédiés à cette problématique depuis les

années 1970.

Ce travail de thèse est axé sur l'extension des principaux résultats obtenus

jusqu'ici dans nR aux espaces de dimension infinie, et notamment aux espaces de

Hilbert et de Banach.

Les systèmes linéaires autonomes soumis à une perturbation inconnue supposée

intégrable au sens de Bochner sont considérés dans leur cadre abstrait, et sont donc

susceptibles de donner lieu à de multiples applications.

Les résultats que nous obtenons tournent principalement autour des voies et

moyens permettant d'accéder malgré cette perturbation à la reconstruction de l'état de

tels systèmes, à partir de la seule connaissance d'une observation recueillie à leur

sortie sur un intervalle de temps fini.

Nous avons développé ces résultats d'une façon pratiquement intégrale dans le

cas d'espaces de Hilbert à travers deux approches distinctes mais complémentaires,

l'une étant axée sur les projecteurs orthogonaux au chapitre 2, et l'autre au chapitre 3

sur le concept d'observabilité directionnelle moyennant une factorisation d'opérateurs

inspirée d'un grand principe d'analyse fonctionnelle de Douglas. Ce sont là entre autres

les contenus faisant l'objet notamment des théorèmes 1, 2, et 3 de cette thèse.

Le quatrième et dernier chapitre se veut une synthèse géométrique démontrant le

lien entre les deux approches précitées. Plus précisément, le théorème 4 localise le

sous espace idéalement observable au sein d'un semi treillis inférieur

conditionnellement invariant, alors que le théorème fondamental 5 identifie le sous

espace en question à l'élément minimal de ce semi treillis.

SUMMARYContribution to the Observability of Disturbed Linear Systems

Keywords: Bounded operator, linear system, Bochner-integrable disturbance, Hilbert

space, Banach space, ideal observability, Kalman observability, conditioned invariance,

controlled invariance, operational factorization.

The concept of ideal observability of the disturbed systems remains a subject of

topicality despite the developments dedicated to this problem since the years 1970.

This work of thesis is centered on the extension of the principal results obtained up

to now in nR to spaces of infinite dimensions, and in particular to Hilbert and Banach

spaces.

The autonomous linear systems subjected to an integrable unknown disturbance

within the meaning of Bochner are considered within their abstracted framework, and

are thus likely to give place to multiple applications.

The results which we obtain turn mainly around the ways and means making it

possible to reach in spite of this disturbance the rebuilding of the state of such systems,

starting from the only knowledge of an observation collected at their output on an

interval of finished time.

We developed these results in a practically integral way in the case of Hilbert

spaces through two distinct but complementary approaches, one being centered on the

orthogonal projectors in chapter 2, and the other in chapter 3 on the concept of

directional observability realising a factorization of operators inspired of a great principle

of functional analysis, the Douglas one. It is there inter alia the contents being the

object in particular theorems 1, 2, and 3 of this thesis.

The fourth and final chapter is intended to be a geometrical synthesis showing the

connection between the two above mentioned approaches. More precisely, theorem 4

locates under space ideally observable within a semi lattice lower conditionally invariant,

whereas the fundamental theorem 5 identifies under space in question with the minimal

element this semi lattice.

Contents

Acknowledgements v

Chapter 1. Introduction to the ideal observability problem 1

1. The state of the art, symbols and notations 12. Contents and organization of this thesis 4

Chapter 2. An ortho-projectional approach of the problem 91. Statement of the ideal observability problem 102. Identi�cation of the ideal observability subspace 113. Consequences 164. Example of an ideally observable system 215. Conclusion 23

Chapter 3. A directional approach of the problem 251. Preliminaries 262. A criterion for non-ideal observability 293. On the lack of ideal observability by a scalar output 344. Conclusion 37

Chapter 4. On the invariance of the ideal observability

subspace 391. Preliminaries 392. Invariance connections between Kalman and ideal

observability 423. A semilattice of Kalman conditioned invariants 43

iii

iv CONTENTS

4. Conclusions and �nal remarks 47

Appendix A. Some extensions of the directional approach toBanach space 51

1. Introduction 512. The ideal observability subspace 523. On the lack of ideal observability 554. Conclusion 57

Appendix. Bibliography 59

Acknowledgements

I would like to express my gratitude to my supervisor, Pr MES-SIRDI, for his guidance, and above all his friendship.

My great and sincere appreciation goes to Pr TERBECHE, thepresident of my thesis committee, and to all the members of the boardof examiners: Pr BEKKAR, Dr BOUAGGADA and Dr TALHAOUI.

Ahmed BARIGOUOctober 2010.

v

CHAPTER 1

Introduction to the ideal observability

problem

1. The state of the art, symbols and notations

This thesis is mainly devoted to the ideal observability of ab-stract linear systems with bounded operators. The contributionsbrought in this context may be considered as the extensions to Hilbertspace of the fundamental results developed up to now in the �nitedimensional spaces.

The systems considered all along this work will be supposed tobe governed in their phase space by the triple1 (A;B;H) accordingto the equations

(1.1) x0(t) = Ax(t) +Bu(t); t � 0;

(1.2) y(t) = Hx(t); t 2 [0; T ] :

Unless otherwise stated, the following assumptions (A1)-(A4)will be made partially or totally throughout the three following chap-ters.

(A1) The (unknown) function u(:) is locally integrable in thesense of Bochner2, i.e. u(:) 2 L1loc([0;+1[ ;U);

(A2) The spaces of phase, output and disturbance, noted X, Yand U are real separable Hilbert spaces;

1To be more concise in notation, the system (1:1) (1:2) is often simply

called the three-map system or the triple (A;B;H):2See [28] for more details on this type of integrability.

1

2 1. INTRODUCTION TO THE IDEAL OBSERVABILITY PROBLEM

(A3) The operators A, B and H are linear and continuous, i.e.A 2 L(X;X) � L(X), B 2 L(U;X) and H 2 L(X;Y ),with H non invertible.

(A4) Where necessary, the following additional assumption willbe also taken into account concerning the transmission mech-anism3 B:

ImB = ImB; with E + ImB = E + ImB

for any closed subspace E � X:

The notations A�, ImA, kerA will concern respectively the ad-joint, the range and the null subspace of the operator A. We shalluse sometimes y(:) instead of fy(t); t 2 [0; T ]g, "a.e." instead of "al-most every where", and "i¤" for "if and only if". The symbol Sp(:)will stand for the linear span of (:); and for any subspace E, E will bethe closure of E ; while the scalar product "< z; E >= 0" will mean"< z; e >= 0 for all e 2 E".

Shortly speaking, our aim will be the reconstructibility of thestate trajectory x(:) of (1:1) on the observation time interval [0; T ] ;for given A; B; and H; and solely from the knowledge of the outputy(:) generated according to (1:2) under the action of the disturbanceu(:):

Due to L.S. Pontriaguine4, the original statement of the idealobservability problem for the �nite5 dimensional system (1:1) (1:2)has been handled under miscellaneous viewpoints and by means of awide variety of mathematical tools and arguments.

The �rst and most important attempt in our opinion, and whichdates back to 1969, is due to Basile and Marro through their paperon the "observability of linear time invariant systems with unknowninputs" [8]. Besides, it�s worth noting that this contribution camejust as a simple and greatly successful application of their earlier

3The assumption (A4) will be mainly used in Chapter 2. Such assumptionis automatically veri�ed in the practical case where B is of �nite rank.

4as mentioned in [19]5where X = Rn; Y = Rq ; U = Rp, with properly sized real matrices A, B,

H and u(:) locally Lebesgue integrable.

1. THE STATE OF THE ART, SYMBOLS AND NOTATIONS 3

extended invariance concepts at the origin of the �nite dimensionalGeometric Approach6 [7].

However, the "ideal"7 terminology was initially introduced in1970 by Nikol�skii [19] who de�ned the �nite dimensional process(1:1) (1:2) as an ideally observable system i¤ its initial state x(0)can be determined solely from the output y(:):

This "ideal" concept was revisited later on by Marinich [18],Ivanov [15] and others with somewhat di¤erent problem statements.In this context, the "relative ideal observability" in the sense ofMarinich is concerned in �ne with the determination of the whole8

trajectory x(:):Yet, despite these apparent di¤erences in the problem state-

ments, we proved that the algorithms obtained in these cases areall equivalent [3]. In particular, the criterion giving the rebuildingof x(0) within the meaning of Nikol�skii joins that which ensures therebuilding of all the state x(:) within the meaning of Marinich. But,a posteriori, it is not surprising to realize that the process makingit possible to recover the initial state x(0) of an autonomous systemis enough to largely recover all the state trajectory x(:) of such asystem. The same remarks still hold [3] when we compare the al-gorithms of Nikol�skii and Marinich with that of Basile and Marro[8].

So, all along this work, we have been inspired by the fact thatall our comparisons and other auxiliary results obtained in X = Rn

have been achieved with no misleading mathematics owing to the

6The Geometric Approach is based on the extension of some classical invari-ance concepts. Its fundamental tools have been �rstly introduced in the state

space X = Rn under the denominations of controlled invariance, conditioned

invariance, and (A;B) invariance. The �rst two dual concepts are due toBasile and Marro.

7as well as the "perfect" terminology used in [21].8i.e. not only x(0); as for Nikols�kii, but the determination of x(t) or Gx(t)

for all t 2 [0; T ] :


use of the principal tools highlighted higher within the geometric ap-proach of multivariable control systems. Consequently, for In�niteDimensional Linear Control Systems, some suitable invariance ex-tensions and other generalizations have been done purposely in analmost natural way.

2. Contents and organization of this thesis

The presentation of this thesis is organized as follows.

Chapter 1: The present introduction.Chapter 2: The Ortho-Projectional Approach followed in thischapter is a somewhat straightforward extension to Hilbertspaces of the �nite dimensional "geometric approach" de-veloped in [7, 8]. Some basic concepts and tools are in-troduced rapidly and as far as needed for the desired iden-ti�cation of the ideal observability subspace. In the maintheorem of this chapter, the ideal observability subspace ofthe system (1:1) (1:2) is determined by an explicit expres-sion which constitutes an e¢cient algorithm leading to thereconstruction of the system trajectory x(:): This is doneindependently of the observation time, and solely by meansof the system parameters A; B, and H: Besides, the ex-ample of an ideally observable system, coming at the endof the chapter, shows how such a reconstruction presentsthe state vector x(t) as the sum of a Fourier series in anorthonormal basis of the Hilbert phase space X : Amongthe consequences collected in several corollaries and otherremarks, we state a criterion for relative ideal observabil-ity which allows to get as particular cases the criterion ofobservability within the meaning of Kalman [24], as wellas the equivalence of the fundamental �nished dimensional

2. CONTENTS AND ORGANIZATION OF THIS THESIS 5

results of Marinich and Basile & Marro9. Finally, and con-trary to what happens in X = Rn; our unexpected resultin Corollary 4 states the absence of ideal observability inin�nite dimensions when the output is minimal and thusunidimensional10.

Chapter 3: Di¤erently from what precedes, The Ideal Di-rectional Approach of this chapter is rather centered on thecharacterization of the lack of ideal observability [4]. Inde-pendently of the Ortho-Projectional Approach of Chapter2, a criterion generalizing Nikol�skii�s theorem11 is estab-lished under assumptions (A1)-(A3), with the additionalassumption (A4) which represents a realistic12 restriction onthe transmission mechanism B: This is done in Theorem 2.Consequently, Corollary 7 describes the ideal observabilitysubspace as the intersection of a "family" of Kalman ob-servability subspaces, and states a somewhat new criterionfor the ideal observability of the system (1:1) (1:2). Also,and contrary to what occurs in �nite dimension, such a de-scription will con�rm, as mentioned in the preceding chap-ter as well as in [5, 6], that the notion of ideal observabilitydoes not have signi�cance in the Hilbert space X when His reduced in assumption (A3) to a simple functional onX : This is done in Theorem 3. It should be remarkablethat we present the latter result as an independent and selfcontained problem, with much more details on the exact �-nite dimensional state trajectory part recoverable from theunidimensional output.

9The connection between the principal results of [8] and [18] was already es-

tablished directly in [3]. But here this connection comes as a simple consequenceof Theorem 1.

10so that H is reduced to a simple (linear continuous) functional on the

Hilbert X :11the main theorem exposed (with no available demonstration) in [19].12but not useless since, in control problems practice, a �nite number of

sensors is often taken in many situations.


Chapter 4: This ultimate chapter is an attempt for a geo-metrical13 synthesis of the two approaches developed inChapters 2-3. The purpose is to link their fundamental re-sults, with a special look for a simple formulation connect-ing the Main Theorem 1 of Chapter 2 with the Main Theo-rem 2 of Chapter 3. The description of some deeply sharedinvariance and conditioned invariance properties betweenKalman and ideal observability is summarized through thepresentation of a semilattice F constituted with a collec-tion of special Kalman observable subspaces, and which isin fact a semi lattice of closed conditioned-invariants underthe pair (A�; kerB�): These considerations lead to Theo-rem 4, and �nally to the main Theorem 5, where a newconstruction process describes the ideally observable sub-space = as exactly the in�mal element of F ; and so that= proves to be the least (A�; kerB�) conditioned invariantcontaining the initially (given) observable subspace ImH�:

Appendix A: This appendix is given just to show that someresults obtained in Hilbert space may extend to Banachspace, while other results still need to be renewed withsome suitable adaptations14. In fact, it should be remem-bered that, in the case of Hilbert spaces, the transition15

from X � to X occurs via Riesz theorem; and this is whythe subspace = of ideal observability could be (and was in-deed) shown in Chapter 2 as the greatest subspace of X onwhich the orthogonal projection of the state trajectory x(:)

13As noticed above, here we are concerned by the so called "geometric

approach of control problems" initiated mainly by Basile and Marro, and byWonham and Morse. Recall that the concepts of "conditioned invariance" and

"controlled invariance" are due to Basile and Marro [7]. Similarly, the conceptof (A,B) invariance has been introduced and widely developed by Wonham and

Morse [25, 26, 27].14and thus with some additional constraints on the assumption (A1)-(A4).15whence the identi�cation between the two spaces and all the advantages

of this identi�cation.

2. CONTENTS AND ORGANIZATION OF THIS THESIS 7

can be uniquely determined on [0; T ] from the known out-put y(:): But for a Banach space X , the subspace = shouldbe naturally localized in the dual space X � provided withthe weak-(*) topology16 � (X �;X ) : So, we must deal withthe loss of the fundamental tools of the hilbertian struc-ture, and hence the loss of crucial advantages17. In fact,it is common that some complementation problems arisingin Banach space are due (not only to the in�nite dimen-sion, but also and more) especially to the lack of an innerproduct18. This is among the reasons why Douglas theoremdoes not extend to Banach space19, and this is consequentlywhy the eventuality of any extension for the main theoremof Chapter 3 remains under investigation. Nevertheless, thematter presented in this appendix reconsiders, within thetopology � (X �;X ) ; the concept of ideally observable direc-tion20 to establish the weak-(*) closedness and the condi-tioned invariance of = under the couple (A�; kerB�) and togive a necessary condition for ideal observability. Besides,likewise the end of Chapter 3, the question of non-ideal ob-servability under a scalar output is treated in detail as aself contained and independent problem.

16See for instance [23] and Tome 2 of [16].17Such as orthogonality and re�exivity.18For instance, a subspace (even closed) need not to be complemented in a

Banach space [23]. See also footnote 17.19See [13] for other possible extensions concerning adjoint operators in Ba-

nach space.20reconsidered in Chapter 3, but formerly introduced in [6].

CHAPTER 2

An ortho-projectional approach of the

problem

In accordance with the introductory chapter of this thesis, let usrecall that we are concerned with the abstract linear systems in theform

(0.1) x0(t) = Ax(t) +Bu(t); t � 0;

(0.2) y(t) = Hx(t); t 2 [0; T ] ;

where x(t) is the state vector, and y(t) the output observed on a�nite interval of time [0; T ] under the action of a disturbance u(:).Unless otherwise stated, the following assumptions (A1)-(A4) will bemade partially or totally throughout this chapter:

(A1) The (unknown) function u(:) is assumed to be locally inte-grable in the sense of Bochner, i.e. u(:) 2 L1loc([0;+1[ ;U);

(A2) The spaces of phase, output and disturbance, noted X , Yand U are real separable Hilbert spaces;

(A3) The operators A, B and H are linear and continuous, i.e.A 2 L(X ;X ) � L(X ), B 2 L(U ;X ) and H 2 L(X ;Y),with H non invertible.Where necessary, the following additional assumption willbe made:

(A4) The transmission mechanism B is a closed range operator(i.e. ImB = ImB) such that E + ImB = E + ImB for anyclosed subspace E � X :

9

10 2. AN ORTHO-PROJECTIONAL APPROACH OF THE PROBLEM

The notations A�, ImA, kerA will concern respectively the ad-joint, the range and the null subspace of the operator A. We shall usesometimes y(:) instead of fy(t); t 2 [0; T ]g, "a.e." instead of "almostevery where", and "i¤" for "if and only if". The symbol Sp(:) willstand for the linear span of (:); and for any subspace E , E will be theclosure of E ; while the scalar product " < z; E >= 0 " will mean "< z; e >= 0 for all e 2 E ".

1. Statement of the ideal observability problem

Using the known equation of motion (0:1) with its given coe¢-cients A, B and H, and assuming that (0:2) has been measured for(0:1) under the action of the perturbation u(:), the ideal observabilityproblem is to rebuild the trajectory x(:) on the interval [0; T ] solelyfrom the available information y(:), for every possible y(:).

It�s worth of note that the ideal observability of the couple(0:1) (0:2) includes the systems in the form:

x0(t) = Ax(t) +B1u1(t) +B2u2(t);

y(t) = Hx(t), t 2 [0; T ] ;

where u1(:) is a control known to the observer, while u2(:) is a pertur-bation on the system. The known input part u1(:) and the unknowninput part u2(:) can be reduced to the case of completely unknownu(:). It is su¢cient to take

B1u1(:) +B2u2(:) := Bu(:);

with

B := [B1; B2] and u(:) := [u1(:); u2(:)]T:

The approach followed in the sequel is the extension to Hilbertspaces1 of the �nite dimensional "geometric approach" developed in[7, 8].

1Our fundamental results in this context have been published in [5]. Thissection recalls the background of such generalizations.

2. IDENTIFICATION OF THE IDEAL OBSERVABILITY SUBSPACE 11

Under the assumptions (A1)-(A3), and starting from the initialinformation (0:2) ; one would like to get the greatest possible orthog-onal projection of the state x(t) in accordance with the followingde�nitions.

Definition 1. The ideal observability subspace =; associated tothe system (0:1) (0:2) ; is the greatest (in the sense of subspaceinclusion) closed subspace of X where the orthogonal projection ofthe state vector can be uniquely recognized along the interval [0; T ]from the output y(:) under the action of the perturbation u(:):

Definition 2. For G 2 L(X ; T ); where T is any separableHilbert space, the system (0:1) (0:2) will be said G-ideally observ-able if the vector Gx(t) can be uniquely determined for all t 2 [0; T ]from the output y(:) under the action of the perturbation u(:):

From De�nitions 1 and 2, it�s natural to state the following def-inition for the crucial case where T = X and G = IdX .

Definition 3. The system (0:1) (0:2) is called ideally observ-able i¤ = = X :

2. Identi�cation of the ideal observability subspace

In the present context, the reconstructibility of Gx(:) or x(:) isobviously related to the knowledge of the subspace =: A formulationof such a subspace will be given after the following lemma (whichmay be seen as an extension of the "matrix pseudo-inverse") [5].

Lemma 1. For any D 2 L(V;W) and any v 2 V; where V andW are Hilbert spaces, the two following statements are equivalent:

(a) It�s possible to recognize the vector Dv;(b) It�s possible to determine the orthogonal projection of the

vector v on the subspace ImD�.

Proof. The Hilbert space V can be written as

V = V1 � V2; V1 := ImD�; V2 := kerD;


so that any v 2 V takes the unique form

v = v1 + v2; v1 2 V1; v2 2 V2:

Let us show that (a) =) (b); meaning that the knowledge of Dvimplies the determination of v1; where v1 is the orthogonal projectionof v on the subspace ImD�:

If bD denotes the restriction of D to the subspace V1; i.e. bD :=DjV1 ; then

ker bD = f0g and Im bD = ImD;

which makes bD invertible, with an inverse operator bD 1 (a non nec-essarily continuous left inverse ) de�ned on ImD and taking its values

in V1: Consequently (since Dv2 = 0; or since D(v1 + v2) = bDv1):bD 1Dv = bD 1D(v1 + v2) = bD 1Dv1 = v1 .

Reciprocally, the proof of (b) =) (a) is immediate. In fact, theknowledge of v1 allows to obtain Dv as follows:

Dv = D(v1 + v2) = Dv1 (since v2 2 kerD).

�

Theorem 1 (the Main Theorem). The ideal observability sub-space of the system (0:1) (0:2) is determined by the expression

= =

1[

i=0

Xi = Sp�Xi; i 2 N

;

where

X0 = ImH�;

Xi+1 = X0 +A�(Xi \ kerB

�); i 2 N;

and where Sp(:) stands for the closure of the linear span of the se-

quence�Xi1i=0

:

Proof of the Main Theorem. We shall prove successivelythe three following statements:


(S1) It�s possible to identify the orthogonal projection2 of thevector x(t) on the (closed) subspace Xi; for all i 2 N;

(S2) The strong limit (denoted P ) of the sequence of projectorsfPig

1i=0 is exactly the orthogonal projector on the subspace

=;(S3) The subspace = is maximal (i.e. the greatest subspace in

the sense of De�nition 1).PROOF OF (S1). Writing

z0(t) = y(t)

and

Z0 := H;

the given output (0:2) becomes

(2.1) z0(t) = Z0x(t); t 2 [0; T ] :

According to Lemma 1, the equation (2:1) allows to recog-nize along [0; T ] the orthogonal projection P0x(t) of x(t) onthe subspace X0 = ImZ�0 : Besides, we get by di¤erentiationof (2:1)

(2.2) z00(t) = Z0Ax(t) + Z0Bu(t); a.e. on [0; T ] :

If Q1 denotes the orthogonal projector on the subspace

(ImZ0B)?= kerB�Z�0 ;

then one gets from (2:2)

(2.3) Q1z00(t) = Q1Z0Ax(t); t 2 [0; T ] :

So, we can write (2:1) and (2:3) simultaneously in the form

(2.4) z1(t) = Z1x(t); t 2 [0; T ]

with

z1(t) =

�z0(t)Q1z

00(t)

�;

2denoted Pix(t); where Pi stands for the orthogonal projector associated to

the subspace Xi:


and

Z1 =

�Z0

Q1Z0A

�2 L(X ;Y � Y)

de�ned for all x 2 X by the expression

Z1x =

�Z0x

Q1Z0Ax

�:

So, once more from Lemma 1 (with Z1 instead of D), theequation (2:4) allows to recognize along [0; T ] the orthogo-nal projection of x(t) on the subspace ImZ�1 ; where Z

�1 2

L (Y � Y;X ) is the adjoint of Z1 de�ned for all (y1; y2) 2 Y2

by the expression

Z�1

�y1y2

�=�Z�0 A�Z�0Q

�1

� � y1y2

�;

i.e.:

Z�1

�y1y2

�= Z�0y1 +A

�Z�0Q�1y2:

It remains to establish the equality X1 = ImZ�1 so as toprove that this projection coincides indeed with P1x(t):But, since Q1 is self-adjoint, we get

ImQ1 = ImQ�1 = kerB

�Z�0 ;

and consequently

ImZ�0Q�1 = Z

�0 ImQ

�1 = ImZ

�0 \ kerB

�;

whenceImZ�0Q

�1 = X0 \ kerB

�;

and

ImA�Z�0Q�1 = A

� ImZ�0Q�1 = A

�(X0 \ kerB�):

ThusImZ�1 = ImZ

�0 + ImA

�Z�0Q�1

i.e.

(2.5) ImZ�1 = X0 +A�(X0 \ kerB

�) = X1:


Now, by induction, let us suppose that we have got thefollowing equation (2:6) which makes it possible to recog-nize the projection Pix(t) of the state x(t) on the subspaceXi = ImZ�i

(2.6) zi(t) = Zix(t); t 2 [0; T ] ;

with

zi(t) =

�zi 1(t)Qiz

0i 1(t)

�;

and

Zi =

�Zi 1

QiZi 1A

�:

Then, by di¤erentiation of (2:6) ; and by projection of theobtained resulting equation on the subspace

(ImZiB)?= kerB�Z�i

we get

(2.7) Qi+1z0i(t) = Qi+1ZiAx(t); t 2 [0; T ] :

Thus, by gathering (2:6) and (2:7) ; we can write

(2.8) zi+1(t) = Zi+1x(t); t 2 [0; T ] ;

where

zi+1(t) =

�zi(t)

Qi+1z0i(t)

�;

and

Zi+1 =

�Zi

Qi+1ZiA

�:

Finally, if we take into account the obvious inclusions

A�(Xi \ kerB�) � A�(Xi+1 \ kerB

�) for all i 2 N;

then the relation Xi+1 = ImZ�i+1 can be derived in thesame way which led us to the equality (2:5). So, the pro-jection Pi+1x(t) of the state x(t) on the subspace Xi+1 isactually determined by (2:8) according to Lemma 1 (withZi+1 instead of D).


PROOF OF (S2). We notice that

Xi � Xi+1 for all i 2 N;

which gives a non decreasing sequence of subspaces�Xi1i=0

:

Consequently, the increasing sequence of associated projec-tors fPig

1i=0 admits a strong limit P; where P is also an

orthogonal projector. Besides, this is exactly the orthogo-nal projector3 on the subspace

= =

1[

i=0

Xi

(i.e. Px = x for all x 2 = and Px = 0 for all x 2 =?):PROOF OF (S3). Let us suppose that the subspace = =1[

i=0

Xi is not maximal, and recall that the solution x(t) of

(1:1) can be uniquely written as

x(t) = x1(t) + x2(t)

with x1(t) 2 = and x2(t) 2 =?: Now, because =? � kerH

(since ImH� � =); our supposition means that the compo-nent x2(t) a¤ects the output y(t) and thus could be recog-nized outside = (i.e. in =?): But this contradicts the factthat y(t) = Hx(t) = Hx1(t) (since x2(t) 2 =

? � kerH):

�

3. Consequences

We obtain in this paragraph a criterion of G ideal observability,and consequently a criterion of ideal observability. We also get asparticular cases the criterion of observability within the meaning ofKalman [24], as well as the results concerning spaces of �nished di-mensions established in [8] and [18]. An interesting result is given by

3See [2] and Tome 1 of [16]

3. CONSEQUENCES 17

Corollary 4: the absence of ideal observability in in�nite dimensionwhen the output is unidimensional.

Corollary 1. The system (0:1) (0:2) is G ideally observablei¤

(3.1) ImG� � =

where

(3.2) = = Sp�Xi; i 2 N

with

(3.3) X0 = ImH� and Xi+1 = X0 +A

�(Xi \ kerB�) for all i 2 N:

Proof. We �rst notice that ImG� � = is equivalent to ImG� �= since = is closed. Now, if the system is G ideally observable, thenthe output y(:) allows the identi�cation of Gx(:) and consequentlythe identi�cation of the projection of x(:) on the subspace ImG�

(according to Lemma 1). But = is the greatest subspace includingsuch a projection, whence (3:1). Conversely, if ImG� � =; then theprojection of x(:) on ImG� is known, and so is Gx(:); whence theG ideal observability of (0:1) (0:2) : �

Corollary 2. Under assumption (3:3) ; the system (0:1) (0:2)will be ideally observable i¤

(3.4) = = Sp�Xi; i 2 N

= X :

Proof. It is enough to make G = IdX in Corollary 1. By theway, this corollary comes also and immediately from Theorem 1 andDe�nition 3. �

Corollary 3. The system (0:1) (0:2) is Kalman-observable4

i¤

(3.5) SpnImA�

i

H�; i 2 No= X :

4i.e. observable in the sense of Kalman[24]


Proof. In the sense of Kalman5, the function u(:) is not a dis-turbance, but a control known to the observer; which makes it pos-sible to put B = 0 without loss of generality[24]. Thus, with B = 0(i.e. kerB� = X ) in Corollary 2, we get from the expression (3:3)

Xi = X0 +A�Xi 1 = ImH

� + ImA�H� + :::+ ImA�i+1

H�;

with X0 = ImH�; whence the expression (3:5) : �

Corollary 4. If the output space is taken as Y = R (with anin�nite dimensional phase Hilbert space X ), then the system (0:1) (0:2) is ideally observable i¤ the two following conditions are bothveri�ed:

(3.6) SpnA�

i

h; i 2 No= X ;

(3.7) B = 0;

where h 2 X stands for the vector representation6 of the functionalH : X !R:

Proof. Suppose �rst that the system is ideally observable (i.e.= = X ), and let us derive (3:6) and (3:7) : According to (3:4) with(3:3) ; one can see that

X0 = ImH� = Sp fhg ;

and

X1 = X0 +A�(X0 \ kerB

�)

= Sp fhg+A�(Sp fhg \ kerB�)

= Sp fhg+A�Sp fhg ;

where there must7 be

Sp fhg \ kerB� = Sp fhg :

5See [24]6rising from Riesz representation theorem.7If not, there would be Sp fhg \ kerB� = f0g ; Xi = X0 = Sp fhg for

all i 2 N; and hence = reduced to = = Sp fhg ; which contradicts the madeassumption = = X :

3. CONSEQUENCES 19

Thus, up to now we have got

X1 = Sp fhg+A�Sp fhg = Sp fh;A�hg ;

with B�h = 08: For the remainder of the proof, one can easily showby induction that:

Xi = Spnh;A�h; :::; A�

i

ho;

B�A�j

h = 0; j = 0; 1; :::; i 1; i = 1; 2; ::: :

Thus, we have

X = = = Sp fXi; i 2 Ng

= Spnh;A�h; :::; A�

i

h; i 2 No

= Spnh;A�h;A�

2

h; :::o;

whence (3:6) : Finally, the ful�lled condition (3:6) with the relations

B�A�i

h = 0 for all i 2 N lead together to B� = 0; and hence to(3:7).

Conversely, if (3:6) and (3:7) are assumed, then (3:6) means thatthe system is Kalman observable, and (3:7) speci�es the absence ofany perturbation. Hence, the system is ideally observable. �

Remark 1. Corollary 4 states that there can never be any idealobservability9 with a scalar output under the action of a non-nullperturbation. Yet, this situation may fail10 whenever Y = Rn withn > 1:

Corollary 5. Let X = Rn; U = Rm and Y = Rk; with suitablysized matrices A; B and H: Then the ideal observability subspaceassociated to the system (0:1) (0:2) is given by the expression

= = Xn 1

8since Sp fhg \ kerB� = Sp fhg :9i.e. the state trajectory x(:) cannot be entirely reconstructed.10A counterexample is given by Example 1 at the end of this chapter.


where

X0 = ImH�

and

Xi = X0 +A� (Xi 1 \ kerB

�) for i = 1; 2; :::n 1:

Proof. The ideal observability subspace is closed and can bewritten as = = Sp fXi; i 2 Ng. Besides, as in [7], it�s clear that ifXi = Xi+1 for some i; then Xi+j = Xi for any j 2 N: Now sincedimX = n and rankH � 1; we get necessarily Xn 1+j = Xn 1 forj 2 N: Finally, since Xi � Xi+1 for all i 2 N; we get = = Xn 1: �

Corollary 6. Let X = Rn; U = Rm and Y = Rk; with suitablysized matrices A; B; H and G: Then the system (0:1) (0:2) isG ideally observable i¤

(3.8) Sp [G�] � Sp [H�; A�C�1 ; :::; A�C�k ] ;

where Sp [:] denotes the linear span of the column vectors in [:] ; thematrices Ci being de�ned by the following statements:

(3.9) Sp [C�i ] � Sp�H�; A�C�1 ; :::; A

�C�i 1�; C0 = 0;

(3.10) CiB = 0;

(3.11) rank [H�; A�C�1 ; :::; A�C�i ] = rankH +

iX

j=0

rankCj;

under the additional condition that (3:12) =) (3:13) ; where (3:12)and (3:13) are de�ned by:

(3.12) fq 2 Sp [H�; A�C�1 ; :::; A�C�k ] \ kerB

�g

(3.13) fA�q 2 Sp [H�; A�C�1 ; :::; A�C�k ]g :

Proof. Let =0 = Sp [H�; A�C�1 ; :::; A

�C�k ] : According to Corol-laries 1 and 5, it will be su¢cient to prove that =0 = Xn 1: But thecondition (3:12) =) (3:13) means that

(3.14) A�(=0 \ kerB�) � =0:

4. EXAMPLE OF AN IDEALLY OBSERVABLE SYSTEM 21

In addition, if we take into account the relations (3:9) ; (3:10) and(3:11) ; =0 appears obviously as the minimal

11 subspace of Rn whichcontains ImH� and which ful�lls the invariance condition (3:14) :Hence12 =0 = Xn 1: �

Remark 2. Corollary 6 states a deep connection between theapproaches developed separately in [8] and [18] about the ideal ob-servability problem in �nite dimensional spaces.

4. Example of an ideally observable system

Let us consider the system (0:1) (0:2) with the following para-meters:

X = l2 =

(x = (xi)

1i=1 ; xi 2 R;

1X

i=1

x2i <1

);

A : x = (x1; x2; :::) 7 ! Ax = (x2; x3; :::) ;

H =

�1 0 0 :::

0 1 0 :::

�;

B� =�1 0 0 :::

�:

If feig1i=1 denotes the natural basis

13 of X =l2; the shift operatorA 2 L (X ) gives

Ae1 = 0 and Aei = ei 1 for i = 2; 3; ::::

A simple calculus gives similar expressions for A�; and precisely:

A�ei = ei+1 for i = 1; 2; ::::

In addition,

ImB = Sp fe1g

11as in [7].12=0 is the least (A�; kerB�) conditioned invariant subspace containing

ImH�; which implies =0 = Xn 1:13The canonical base feig

1

i=1 ; where ei = [0; 0; :::; 1; 0; 0; :::] with 1 at posi-

tion i:


with

kerB� = Sp fe2; e3;:::g ;

and by a simple computing of H�; we get

ImH� = Sp fe1; e2g :

Finally, we obtain successively:

X0 = ImH� = Sp fe1; e2g ;

X1 = X0 +A� (X0 \ kerB

�) = Sp fe1; e2; e3g :

and similarly:

Xi = Sp fej ; j = 1; 2; :::; i+ 2g :

So, it�s now evident that

= = Sp�Xi; i 2 N

= Sp fXi; i 2 Ng = Sp fe1; e2; :::g = l2:

Hence, our system is ideally observable. Besides, we can give in thissituation the explicit expression of the state x(t) by means of theobserved output y(t): In fact, if y1(t) and y2(t) represent the twocomponents of y(t); then it�s clear that

y1(t) =< x(t); e1 >

and

y(i)2 =< x(t); ei+2 > for i = 0; 1; :::;

where < : ; : > denotes the scalar product in l2; and y(i)2 (t) stands

for the derivative of order i of the function y(t) for t 2 [0; T ] : Thus,the vector x(t) can be written as the sum of the Fourier series

x(t) = y1(t)e1 +1X

i=0

y(i)2 (t)ei+2; t 2 [0; T ] :

Remark 3. In the above example, it is remarkable that the per-turbation u(t) itself can be determined as

u(t) = y01(t) y2(t) a.e. in [O; T ] :

This is due to the fact that kerB = 0 (which makes B left invert-ible). In general, it�s only possible to determine Bu(t) as Bu(t) =

5. CONCLUSION 23

x0(t) Ax(t) whenever the ideal observability is already ful�lled. Theprojection of u(t) on the subspace kerB cannot be recognized fromthe output of the system.

5. Conclusion

We gave here above an explicit expression of the ideal observabil-ity subspace, independently of the observation time, and solely bymeans of the system parameters A;B, and H. Just like the observ-ability in the sense of Kalman, this independence with respect to theinterval of observation was foreseeable for an autonomous system.

The hilbertian phase space X being assumed separable, the prop-erty of ideal observability allows the rebuilding of the state x(t) inthe form of a decomposition in Fourier series along an orthonormalbasis.

However, the assumption of separability of X is not necessary inthe majority of the elaborated results, apart from certain particularsituations14 non detailed in this chapter.

Besides, it is worth of note that the results established in thischapter (in the context of the approach followed here by orthogo-nal projectors) generalize most of the main known results in �nitedimensional spaces.

14For instance, if H is of �nite rank (say Y = Rk), it is clear that such anassumption of separability will have to be required in Corollaries 2 and 3.

CHAPTER 3

A directional approach of the problem

This chapter will be mainly devoted to the lack of ideal observ-ability, and more precisely to the characterization of this absence.

We shall deal as in Chapter 2 with abstract linear systems inHilbert phase space X , with bounded operators and under the actionof a disturbance u(:) generating a measured output y(:) on some �niteinterval of time [0; T ].

However, unlike the preceding chapter, the present approach willbe based on the notion of ideally observable directions �rstly intro-duced in [6], and recently much more developed in [4].

In this new context, we shall give a criterion for non-ideal ob-servability. Such a characterization will be established by means ofa range inclusion of operators according to a well known Douglas1

theorem [12].Meanwhile, a complementary insight is brought (directly or in-

directly) into the ideal observability subspace = and it�s orthogonalcomplement =?, with a prelude for the next Chapter 4 where somefundamental invariance properties will be highlighted.

Here are now some preliminaries needed in the following para-graphs.

1See also [1] on Douglas equations, and [13] for an extension of Douglastheorem to Banach space (but only for adjoint operators!).

25

26 3. A DIRECTIONAL APPROACH OF THE PROBLEM

1. Preliminaries

1.1. Some initial concepts. Once more, let us consider thesystem:

(1.1) x0(t) = Ax(t) +Bu(t); t � 0;

(1.2) y(t) = Hx(t); t 2 [0; T ] ;

under the same assumptions (A1)-(A4) of Chapter 1.As introduced in the particular context of [6], and more recently

in [4], the statement of the ideal observability problem will be basedon the following de�nition.

Definition 4. The vector � 2 X will be called an ideally observ-able direction for the system (1:1) (1:2) i¤ the following implicationis true for any perturbation u(:) 2 L1loc([0;+1[ ;U) :

fy(t; x0; u(:)) = 0; for all t 2 [0; T ] g

=) f< x(t; x0; u(:)); � >= 0; for all t 2 [0; T ]g ;

or (brie�y) if and only if:

(1.3) y(:) � 0 =) < x(:); � >� 0:

In this de�nition, x(t) = x(t; x0; u(:)) stands for the solution2

of equation (1:1) generated by the perturbation u(:) with the initialcondition x0 := x(0), whereas y(t) := y(t; x0; u(:)) is the correspond-ing output.

The formulation (1:3) means that the trajectory x(:) can berecognized along the direction � from the knowledge of y(:) on [0; T ].

Definition 5. The ideal observability subspace = � X for thesystem (1:1) (1:2) is the set of all the directions � 2 X verifyingthe implication (1:3) :

2Such a solution is known to be

x(t) := x (t; x0; u(:)) = S(t)x0 +

tZ

0

S(t �)Bu(�)d�

with S(t) = exp(At):

1. PRELIMINARIES 27

Definition 6. The system (1:1) (1:2) will be said ideally ob-servable i¤ = = X .

Lemma 2. Under assumptions (A1)-(A3), the ideal observabilitysubspace = is closed; it contains ImH� and veri�es the conditioned-invariance property:

(1.4) A�(= \ kerB�) � =:

The orthogonal complement =? (which is closed and contained inkerH) veri�es the dual controlled-invariance property3:

(1.5) A=? � =? + ImB:

Proof. It�s clear that = is subspace. Moreover, for z 2 X ; if Fzdenotes the functional de�ned on X by Fz(�) :=< z; � >; then

= = \z2E

kerFz; with E := fx(t) 2 X : Hx(t) = 0; t 2 [0; T ]g :

Thus, = is closed as the intersection of the kernels of continuousfunctionals.

It�s also trivial that

fy(t) = Hx(t) =< x(t); ImH� >� 0g

=) f< x(t); � >� 0 for any � 2 ImH�g ;

and thus ImH� � = .For the inclusion A�(= \ kerB�) � =, it�s su¢cient to establish

the implication:

(1.6) y(t) � 0 =)< x(t); A�(= \ kerB�) >� 0:

By De�nition 5 of =, and by continuity of the scalar product, weget by successive di¤erentiations the following implications:

y(t) � 0 =)< x(t);= >� 0

=) < x0(t);= >= 0; a.e. in [0; T ] ;

whence:< Ax(t) +Bu(t);= >= 0; a.e. in [0; T ] :

3=? + ImB = =? + ImB under assumtion (A4).


Therefore:

< x(t); A�= > + < u(t); B�= >= 0; a.e. in [0; T ] ;

and hence < x(t); A�(= \ kerB�) >� 0; which proves (1:6).To �nish the proof, notice that the inclusion (1:5) derives from

(1:4) by taking the orthogonal complement in both sides: �

Remark 4. The part of Lemma 2 concerning = remains valid inan arbitrary Banach space X as shown in Appendix A. The subspace= is then in the dual space X � provided with the weak-(*) topology�(X �;X ).

Remark 5. De�nition 6 means that the system (1:1) (1:2) isideally observable i¤ 4:

y(:) � 0 =) x(:) � 0

for any perturbation u(:) 2 L1loc([0;+1[ ;U):

Remark 6. It is remarkable that the sequence of subspaces fXig ;produced in Chapter 2 and recalled below in expression (1:7) ; canalso be derived directly from (1:3) by successive di¤erentiations andby continuity of the scalar product. Remember that the subspace =is described in the preceding chapter as the greatest subspace of X ,on which the orthogonal projection of the state function x(:) can berecognized on [0; T ] uniquely from the output y(:). Under the crucialassumption (A4), and especially with of a �nite dimensional rangedoperator B; the expression of = still remains ( a fortiori) determinedby the following closed linear span of an increasing5 sequence of sub-spaces fXig

1i=0:

(1.7) = = Sp�Xi, i 2 N

;

with

X0 = ImH�;Xi+1 = Xi +A

�(Xi \ kerB�); i 2 N:

4Since the implication (1:3) must be ful�lled for all � 2 X :5i.e. such that Xi � Xi+1 for all i 2 N:

2. A CRITERION FOR NON-IDEAL OBSERVABILITY 29

When = = X , which means =? = f0g, we get exactly the situationwhere the determination of the whole state x(t) can be done on theinterval [0; T ]. As explicitly obtained in the example of Chapter 2,such a determination may be understood as a representation by aFourier series in some orthonormal basis of the separable Hilbertspace X .

1.2. Some connections between Kalman and ideal ob-

servability subspaces. Let us recall initially (with more detailsthan done in Chapter 2) that when u(:) is not a disturbance, buta control at the observer�s disposal, then the concept of ideal ob-servability is reduced to the well known Kalman observability. Thesystem (1:1) (1:2) is in fact known to be Kalman observable i¤ oneof the two equivalent statements is true:

(1.8) K(H) := Sp(ImH�; ImA�H�; ImA�2H�; :::) = X ;

(1.9) \i2N

kerHAi = f0g :

It�s worth of note that the Kalman observability subspace K(H)in (1:8) is the least subspace which contains ImH� with the invari-ance property A�K(H) � K(H):

The last two relations (1:8) and (1:9) show that Kalman ob-servability does not depend on B; which allows to consider B = 0without loss of generality as far as u(:) remains a known function.

Therefore, the ideal observability of the system (1:1) (1:2) im-plies obviously it�s Kalman observability. Of course, the conversedoes not hold in general.

2. A criterion for non-ideal observability

The aim of this section concerns the connection between the idealobservability subspace of the perturbed system (1:1) (1:2) and it�sKalman observability subspaces when the function u(:) takes thefeedback forms u(t) =Mx(t) as M describes the space L(X ;U):


As a generalization of the �rst main result6 given in [19], thenext Theorem 2 states a criterion for the lack of ideal observabilityof (1:1) (1:2) : Its forthcoming proof will be based on the followingDouglas theorem [12] on the range inclusion of operators on Hilbertspace.

Lemma 3 (Douglas theorem). Let H1; H2 and H3 be Hilbertspaces. If E 2 L (H1;H3) and F 2 L (H2;H3) ; then the followingare equivalent7:

(i) ImE � ImF;(ii) There exists (a linear and bounded operator) C 2 L (H1;H2)

such that E = FC,(iii) There is a positive number � such that EE� � �FF �:

Theorem 2 (the Main Theorem). Under assumptions (A1)-(A4), the two following statements are equivalent:

(i) The system (1:1) (1:2) is not ideally observable;(ii) There exists a bounded linear operator M 2 L(X ;U ) for

which The system

(2.1) x0(t) = (A+BM)x(t)

(2.2) y(t) = Hx(t); t 2 [0; T ]

is not Kalman-observable.

Proof of the Main Theorem. The implication (ii) =) (i)is immediate.

In fact, by contradiction, if the system (1:1) (1:2) were ideallyobservable, and thus with an unknown function u(t), then it wouldbe a fortiori Kalman observable with any feedback u(t) =Mx(t):

The implication (i) =) (ii) will be proved by invoking Douglastheorem.

6It is unfortunately given in [19] with no available demonstration (nor in-

dication on the way to such a result).7We are only concerned with the equivalence between the two statements

(i) and (ii). See[1] for more applications.


Let us suppose that (1:1) (1:2) is not ideally observable, mean-ing =? 6= f0g, and let us then prove the existence of an operator Mfor which the system (2:1) (2:2) can not be Kalman observable on[0; T ]. Precisely, in accordance with the criterion (1:9) ; we will showthat:

(2.3) \i2N

kerH(A+BM)i 6= f0g :

If P denotes the orthogonal projector on the subspace =?; then theinclusion (1:5) in Lemma 2 becomes

(2.4) AP (=?) � P (=?) + ImB;

orImAP � ImP + ImB:

Thus:

(2.5) ImAP � Im [P;B] ;

with AP 2 L(X ;X ); and where [P;B] 2 L(X�U;X ) is de�ned by

[P;B]

�x

u

�= Px+Bu for all (x; u) 2 X � U :

Now, according to Douglas theorem, the inclusion (2:5) ensures theexistence of an operator C 2 L(X ;X�U) such that

Cx = (C1x;C2x) for all x 2 X ;

and for which we get:

(2.6) AP = [P;B]C = PC1 +BC2:

The relation (2:6) gives

APx = PC1x+BC2x for every x 2 X ;

and particularly:

(2.7) PC1x = (A BC2)x 2 =?; for all x 2 =?:

So, with =? � kerH, we get from (2:7):

(2.8) H (A BC2)x = 0; for all x 2 =?:


Now, since =? 6= f0g, we can take an element z0 6= 0 in =? for which

Hz0 = 0: Besides, we can write from (2:8):

(2.9) H(A BC2)z0 = 0:

Similarly, since

z1 := PC1z0 = (A BC2)z0 2 =?

in accordance with (2:7), then once more from (2:8) :

H (A BC2) z1 = 0;

which gives in return:

(2.10) H(A BC2)z1 = H(A BC2)2z0 = 0:

Following steps (2:9) and (2:10) ; a simple induction leads to theresulting equalities

�H(A BC2)

iz0 = 0; i 2 N;

which mean that

z0 2 \i2N

kerH(A+BM)i;

and this is the desired result (2:3) with M = C2. �

Remark 7. An equivalent version of the Main Theorem (Theo-rem 2) is: " Under assumptions (A1)-(A4), the system (1:1) (1:2)is ideally observable i¤ the system (2:1) (2:2) is Kalman observablefor every M 2 L(X;U) ": In other words, Theorem 2 signi�es thatthe ideal observability of (1:1) (1:2) is ful�lled if and only if

KM (H) = X for every M 2 L(X ;U);

where the subspace KM (H) is de�ned in accordance with (1:8) by:

(2.11) KM (H) := Sp�Im((A+BM)�)iH�; i 2 N

:

Besides, in the situation where there is no ideal observability for(1:1) (1:2), theorem 2 ensures the existence of an analytical con-trol u(t) = M0x(t) for which there is no Kalman observability for


(2:1) (2:2) : By means of such a control, the trajectory of the sys-tem starting from a point in the subspace =? can be maintained8 in=? itself, and remains invisible to the observer.

The following corollaries state a connection between the subspaceof ideal observability = and the set of Kalman observability subspaces

fKM (H); M 2 L(X ;U)g :

The equality \KM (H)M2L(X ;U)

= X is an additional criterion for the ideal

observability of the perturbed system (1:1) (1:2) :

Corollary 7. Under assumptions of Theorem 2, the ideal ob-servability subspace for the perturbed system (1:1) (1:2) veri�es theequality

= = \KM (H)M2L(X ;U)

;

where KM (H) denotes the Kalman-observability subspace de�ned by(2:11).

Proof. The proof of = �

\KM (H)M2L(X ;U)

!derives simply from

Theorem 2.In fact, let � 2 = , so that (1:3) holds true for every possible

u(:) 2 L1loc([0;+1[ ;U), and in particular for every u(t) = Mx(t).Thus, � is a Kalman observable direction for the system (2:1) (2:2),meaning that � 2 KM (H) for all M 2 L(X ;U).

Conversely, if � 2 \KM (H)M2L(X ;U)

, then it is Kalman observable with

u(t) =Mx(t) for all M 2 L(X ;U), and thus � 2 = from Theorem 2;

whence

\KM (H)M2L(X ;U)

!� =. �

8i.e. the trajectory is said to be " controlled on =? " by means of thefunction u(t) =M0x(t):


Corollary 8. Under assumptions of Theorem 2, the perturbedsystem (1:1) (1:2) is ideally observable if and only if

\KM (H)M2L(X ;U)

= X :

Proof. By De�nition 6, the system (1:1) (1:2) is ideally ob-servable i¤ = = X , and, from Corollary 7, it is so i¤ = = \KM (H)

M2L(X ;U)

=

X . �

3. On the lack of ideal observability by a scalar output

Under assumptions (A1)-(A3), with no restriction on the opera-tor B;we consider in this section the particular situation where theoperator H is a simple functional on X .

In this case, Y = R and the system (1:1) (1:2) is observed bymeans of a scalar output y(t) = Hx(t) � < x(t); h > on [0; T ], whereh 2 X is the Riesz representation of H.

When established in Chapter 2-Corollary 4, the lack of ideal ob-servability by means of a scalar output was derived as a simple con-sequence of the main ideal observability subspace expression (1:7) :

But now, according to our main theorem above, this lack of ob-servability can be handled directly9 and by other arguments. Merely,it will be treated in the next theorem in the form of an independentand much more complete and signi�cant result, with a forthcomingremark on its right appreciation.

Theorem 3. Under assumptions (A1)-(A3), with no need for(A4), the system (1:1) (1:2) with a scalar output is ideally ob-servable if and only if the two following conditions (C1)-(C2) aresimultaneously checked:

(C1) = = Sp(h;A�h;A�2h; :::) = X ;

(C2) B = 0:

9i.e. not as a consequence of some proposition, but as an independant resultas done in Theorem 2.

3. ON THE LACK OF IDEAL OBSERVABILITY BY A SCALAR OUTPUT 35

If B 6= 0, the ideal observability subspace = is reduced to the �nitedimensional subspace given by the expression:

(3.1) = = Sp(h;A�h;A�2h; :::; A�k0h);

where k0 is the smallest integer for which

HAk0B 6= 0; with HAiB = 0 for i = 0; 1; :::; k0 1:

Proof. Notice that condition (C1) means1\i=0kerHAi = f0g :

If the two conditions are checked, (C1) means that (1:1) (1:2) isKalman observable; and (C2) means the absence of any disturbance.Thus, the system is ideally observable.

Reciprocally, ideal observability implies condition (C1).For (C2), we shall prove by contradiction that, if B 6= 0, then

condition (C1) can�t be achieved, and = is reduced to the �nite di-mensional subspace in (3:1). So, let us suppose B 6= 0, which impliesthe existence of k 2 N such that HAkB 6= 0 (if not there would be

ImB �1\i=0kerHAi = f0g, whence B = 0). Now, let us take the

smallest integer k0 for which

HAk0B 6= 0 with HAiB = 0; i = 0; 1; :::; k0 1;

and chooseu0 2 U such that HA

k0Bu0 6= 0R:

Let us consider the control u(:) expressed by:

(3.2) u(t) =M0x(t) := HAk0+1x(t)

HAk0Bu0u0

�= u0

HAk0+1x(t)

HAk0Bu0

�:

For such a control, where

x 2 X 7 !M0x := u0HAk0+1x

HAk0Bu0

means clearly that M0 2 L(X ;U), the system (1:1) (1:2) with ascalar output becomes:

(3.3) x0(t) = (A+BM0)x(t)

(3.4) y(t) = Hx(t) =< x(t); h >; t 2 [0; T ] :


Taking into account the condition HAiB = 0; 0 � i < k0, we getsuccessively:

H(A+BM0) = HA+HBM0 = HA;

H(A+BM0)2 = HA(A+BM0) = HA

2 +HABM0 = HA2; :::;

H(A+BM0)k0 = HAk0 1(A+BM0) = HA

k0 :

Besides, by de�nition of M0; the relation

HAk0+1x+HAk0BM0x = 0

is true for all x 2 X ; which gives

H(A+BM0)k0+1 = HAk0+1 +HAk0BM0 = 0;

and leads to

H(A+BM0)k0+p = 0 for all p � 1:

Consequently, according to (2:11), the Kalman observability sub-space for (3:3) (3:4) takes the form:

(3.5) KM0(h) = Sp(h;A�h;A�2h; :::; A�k0h);

and thus, condition (C1) fails. Finally, = � KM0(h) from Corollary

7; and KM0(h) � = derives from (1:3) with HAiB = 0; 0 � i < k0

by successive di¤erentiations (as done in the proof of Lemma 2). �

Remark 8. Unlike the short version of Chapter 2-Corollary 4,Theorem 3 above gives an explicit description of the ideal observabil-ity subspace =, showing that = is �nite dimensional whenever thedisturbance acting on the system is not null. In fact, if B 6= 0, then=? is of �nite codimension, and the state x(:) can thus only be recog-nized along k0+1 directions. In this situation, if the Main Theorem2 insures the existence of a control which enables to maintain thetrajectory in the invisible subspace =?; then it is remarkable that theproof of Theorem 3 has the advantage to give concretely and explicitlyan example of such a control in the form (3:2). However, Theorem3 should not be considered as a simple particular case of the MainTheorem 2. Indeed, besides the fact that there is no restriction onB; it should be rather considered as a degenerated situation, since,

4. CONCLUSION 37

in addition, Theorem 3 fails for Y = Rm with m > 1. The exam-ple given for X = l2 in Chapter 2 proves curiously that two outputs(say Y = R2) may well be su¢cient for the ideal observability of thesystem (1:1) (1:2).

4. Conclusion

Taking into account the fact that much has been written onsimilar subjects in in�nite dimensional spaces, we have tried, forthe sake of completeness, to make as well as possible a self-su¢cingchapter.

The reader is only referred to some selected works (regardless oftheir priority among too many others in literature) for their director indirect relationship with this paper.

For instance, [10], [14] and [22] deal with di¤erent observabilitymatters, and for excellent expositions of di¤erent subjects related tothis paper, we can�t forget neither [11] for the in�nite-dimensionaltheory nor [9] and [27] for the �nite dimensional aspects of control.

Concerning the concept of ideal observability for the system(1:1) (1:2) in �nite dimensional spaces (say X = Rn) as in [8,18, 19], somewhat di¤erent (but �nally equivalent) de�nitions areused here and there according to the approach followed to solve thequestion. In this chapter, the statement of the problem, based on De-�nition 4, leads to the obtained results practically without recourseto any complete explicit expression of the subspace=. In fact, acomplementary insight is brought into the ideal = and non-ideal =?

observability subspaces for perturbed systems governed in Hilbertspace by (1:1) (1:2). Under assumptions (A1)-(A4), the lack ofideal observability is stated by means of an "operational" approach,with a range inclusion of operators (2:5) instead of the "geometric"subspace inclusion (1:5). The given results show also that the (posi-tive) size of the time interval does not play any role.


Furthermore, under some additional10 (more restrictive) assump-tions and adaptations, a nonnegligible part of these results may berenewed almost just as they are in Banach space.

10For instance, a �nite rank for B should be helpful for some beginningattempts. See also Appendix A.

CHAPTER 4

On the invariance of the ideal

observability subspace

This ultimate chapter is an attempt for a geometrical1 synthesisof the two approaches developed in Chapters 2 and 3. Our desire isto carry out such a synthesis as far as possible through some sim-ple formulations linking the principal results of these two chapters,among which namely the Main Theorems 1 and 2.

1. Preliminaries

We shall deal once more with the ideal observability subspace ofthe abstract systems governed according to the model under consid-eration until now, namely:

(1.1) x0(t) = Ax(t) +Bu(t); t � 0;

(1.2) y(t) = Hx(t); t 2 [0; T ] ;

and with the same assumptions (A1)-(A4) which have been taken allalong Chapters 2 and 3.

1Recall that we are here concerned by the so called "geometric approach ofcontrol problems" initiated mainly by Basile and Marro, and by Wonham and

Morse. The concepts of "conditioned invariance" and "controlled invariance" are

due to Basile and Marro[7]. Similarly, the concept of (A,B) invariance has beenintroduced and widely developed by Wonham and Morse[27].

39

40 4. ON THE INVARIANCE OF THE IDEAL OBSERVABILITY SUBSPACE

Meanwhile, a great attention will be also devoted to the set ofall the Kalman observability subspaces associated to the systems inthe form

(1.3) x0(t) = (A+BM)x(t); t � 0;

(1.4) y(t) = Hx(t); t 2 [0; T ] ;

where M describes the whole space L(X ;U):Yet, let us �rst recall from Chapters 2 and 3 the following list of

some basic informations concerning the ideal observability subspace= associated to (1:1) (1:2) and the Kalman observability subspacesKM (H) associated to (1:3) (1:4).

(a) The expression of = by means of the ortho-projectional ap-proach has been given as2:

(1.5) = = Sp fXi, i 2 Ng ;

where fXig1i=0 is an increasing set of encased

3 subspacessuch that:

X0 = ImH�; Xi+1 = Xi +A

�(Xi \ kerB�); i 2 N:

(b) The ideal observability subspace = is closed; it containsImH� and veri�es the (A�; kerB�) conditioned-invarianceproperty4:

(1.6) A�(= \ kerB�) � =:

The orthogonal complement =? (which is closed and con-tained in kerH) veri�es the dual (A; ImB) controlled-invariance property:

(1.7) A=? � =? + ImB:

2[Chapter 2-Theorem 1] :3i.e. Xi � Xi+1 for all i 2 N:4[Chapter 3-Lemma 2] :

1. PRELIMINARIES 41

(c) The ideal observability subspace for the perturbed system(1:1) (1:2) veri�es the equality5

(1.8) = = \KM (H)M2L(X ;U)

;

where KM (H) denotes the Kalman-observability subspacede�ned by the following (1:9):

(1.9) KM (H) = Sp�Im((A+BM)�)iH�; i 2 N:

(d) The system (1:1) (1:2) is ideally observable i¤6 the system(1:3) (1:4) is Kalman observable for every M 2 L(X ;U);which means (for instance and inter alia) that

[= = X ]() [KM (H) = X for all M 2 L(X ;U)] :

To end these preliminaries, let us introduce a useful result whichhelps to convert a geometric inclusion into an operational inclusion.The question (already encountered in Chapter 3) is actually how tohandle a range inclusion of operators instead of a geometric subspaceinclusion.

Lemma 4. Under assumptions (A1)-(A4), the following state-ments are equivalent for any closed subspace E � X :

(1.10) A�(E \ kerB�) � E ;

(1.11) AE? � E? + ImB;

(1.12) (A+BF ) E? � E? for some F 2 L(X ;U):

Proof. The equivalence (1:10)() (1:11) is straightforward bytaking the orthogonal complement in both sides of (1:10) or of (1:11)with assumption (A4).

The equivalence (1:11)() (1:12) is the corollary result (with Einstead of =) of the arguments developed via Douglas Theorem along

5[Chapter 3-Corollary 7] :6[Chapter 3-Remark 7] :


the demonstration of the Main Theorem of Chapter 3 (Theorem 2).�

2. Invariance connections between Kalman and ideal

observability

As announced at the beginning of the above section, we shallread out a geometrical connection development for the main resultsprecedingly presented through the approach by projectors as well asthrough the directional approach of the ideal observability concept.This reading will mainly result in a complementary description ofthe construction process of the ideal observability subspace = for thesystem (1:1) (1:2) : Such a process will be achieved by recoursingto the well known Kalman observability subspaces KM (H) attachedto the systems (1:3) (1:4) :

So, in addition to the expression (1:8) of the ideally subspace =by means of the Kalman observable subspaces KM (H); the followingresults point out some fundamental invariance properties shared bythe two kind of subspaces. The strongest results will follow in theforthcoming Theorem 4 and Theorem 5.

Lemma 5. For any system (1:3) (1:4) ; the Kalman observabil-ity subspace de�ned by the expression (1:9) veri�es the conditioned-invariance property:

(2.1) A�(KM (H) \ kerB�) � KM (H); for all M 2 L(X ;U):

Proof. For the proof of (2:1) ; we recall that KM (H) is the leastsubspace7 such that

(2.2) ImH� � KM (H);with (A+BM)�KM (H) � KM (H):

So, if � 2 KM (H) \ kerB�; then

(A+BM)�� 2 KM (H) with B

�� = 0;

and thus

(2.3) (A+BM)�� = A�� +M�B�� = A�� 2 KM (H);

7This is obvious from (1:9) : See also [24].

3. A SEMILATTICE OF KALMAN CONDITIONED INVARIANTS 43

whence (2:1) : �

The ideal observability subspace =; which veri�es the alreadyproved conditioned invariance property (1:6) ; has in addition thefollowing in�mal subspace property.

3. A semilattice of Kalman conditioned invariants

Let us associate to the perturbed system (1:1) (1:2) all thefree systems (1:3) (1:4) obtained with all analytical controls in theform u(t) = Mx(t) for t 2 [0; T ] when M describes the whole spaceL(X ;U):

Doing this, we associate to the (closed) ideal observability sub-space = of the system (1:1) (1:2) the collection F made up of allthe (closed) Kalman observability subspaces KM (H) of (1:3) (1:4);i.e. :

(3.1) F := fKM (H) j M 2 L(X ;U) g

where

(3.2) KM (H) = SpnIm [(A+BM)�]

iH�; i 2 N

o:

In fact, such a collection F of special Kalman subspaces will beproved in the sequel to be a semi-lattice of conditioned-invariantsubspaces8 under the pair (A�; kerB�), with an essential role for it�sin�mal9 element inf

M2L(X ;U)F :

Meanwhile, the next result is an important completion to thepreceding Lemma 5. In fact, Lemma 5 insures that the elements ofF de�ned by (3:1) verify the conditioned invariance (2:1) ; but thequestion is whether the converse is true. In other words, the point

8See [7, 9] for more details on this type of invariants.9As it will be proved in the sequel, it�s fundamental to notice that F con-

tains it�s greatest lower bound (and so it�s in�mal element), i.e. infM2L(X ;U)

F =

\

M2L(X ;U)

KH(M) 2 F :


is whether the collection F includes actually all the (A�; kerB�)-conditioned invariants containing ImH�:

Lemma 6. Considering the collection F de�ned by (3:1) ; thefollowing statements hold true in the phase space X :

(a) Every element KM (H) 2 F is an (A�; kerB�) conditioned-invariant subspace containing ImH� according to the inclusion

A�(KM (H) \ kerB�) � KM (H):

(b) Conversely, if E � X is a closed subspace which containsImH� and such that A�(E \ kerB�) � E ; then E 2 F .

Proof. PROOF OF (a). See Lemma 5 for the proof of thispart.

PROOF OF (b). Let the closed subspace E � X be such thatImH� � E with A�(E \kerB�) � E . Then, under assumption (A4),we get for the orthogonal complement E?:

(3.3) AE? � E? + ImB:

Now, if P denotes the orthogonal projector on the subspace E?; thenthe inclusion (3:3) becomes

(3.4) AP (E?) � P (E?) + ImB:

At this point, the situation in (3:4) is exactly the same as (2:4) fromChapter 3, with E? instead of =?: Thus, the same demonstrationprocess leads to a relation similar to (2:7) from Chapter 3. Rewriting(2:7) in our present context, we get the existence of some operatorC2 2 L(X ;U) for which:

(3.5) (A BC2)x 2 E?; for all x 2 E?:

But (3:5) means that (A BC2) E? � E?; and thus, with noting

M0 = C2; we get:

(A+BM0) E? � E?;

whence (by taking the orthogonal complement in both sides):

(A+BM0)�E � E :

3. A SEMILATTICE OF KALMAN CONDITIONED INVARIANTS 45

Finally, since ImH� � E ; the obtained subspace E = EM0(H) is ac-

tually the Kalman observability subspace of some free system in theform (1:3) (1:4) : Hence, E = EM0

(H) 2 F : �

Lemma 7. For any index set �; and thus for any subfamilyfKM�

(H))g�2�

� F ; the following hold true for the intersection\

�2�

KM�(H):

(3.6) A�

" \

�2�

KM�(H)

!\ kerB�

#�\

�2�

KM�(H);

(3.7)\

�2�

KM�(H) 2 F for any index set �:

Proof. PROOF OF (3:6) : For any index set �, let us show that

� 2

\

�2�

KM�(H)

!\ kerB� =) A�� 2

\

�2�

KM�(H):

So, for such a �; and for all � 2 �; it follows as in (2:3) thatKM�(H) 3

(A+BM�)��: Besides, B�� = 0, whence

KM�(H) 3 (A+BM�)

�� = A�� +M��B

�� = A�� for all � 2 �;

and hence A�� 2\

�2�

KM�(H):

PROOF OF (3:7) : The proof comes on one hand from Lemma

7 by putting E =\

�2�

KM�(H); and by using (3:6) to obtain

(3.8) A�(E \ kerB�) � E with ImH� � E :

On the other hand, from (3:8) and part (b) of Lemma 6, we see thatE 2 F : �

Theorem 4. Under assumptions (A1)-(A4), and with respectto � and \; the collection F de�ned by (3:1) and (3:2) is a lower


semilattice10 of all the (A�; kerB�)-conditioned invariant subspacescontaining the given subspace ImH�:

Proof. (i) It is clear from Lemma 6 (and even from Lemma7) that F is the set of all the (A�; kerB�)-conditioned invariantsubspaces containing the given subspace ImH�:

(ii) It is also clear that F is partially ordered by � : Besides,from (3:7) ; F is closed11 for \; and thus F is at least a semilattice.

(iii) Finally, from (3:7) ; it comes that\

M2L(X ;U)

KM (H) 2 F ,

and hence the collection F is actually a lower semilattice admittingas in�mal element:

(3.9) infM2L(X ;U)

F =\

M2L(X ;U)

KM (H):

�

Remark 9. The inclusion (3:6) implies in particular that thein�mum of F ; described by the intersection (3:9) ; is an (A�; kerB�)-conditioned invariant subspace containing the given subspace ImH�;i.e. :

(3.10) A��

infM2L(X ;U)

F\kerB�

�� inf

M2L(X ;U)F :

Theorem 5 (The Main Theorem). The ideal observability sub-space = of the perturbed system (1:1) (1:2) is the in�mum of thelower semilattice F ; and as such, it is exactly the least subspace whichcontains the initially given subspace ImH� and which is conditionedinvariant under the pair (A�; kerB�):

10A lattice $ is a partially ordered set in which for any pair x; y in $ there

exists a least upper bound (l.u.b.), i.e., an � 2 $ such that � � x; � � y andz � � for all z 2 $ such that z � x; z � y; and a greatest lower bound (g.l.b.),

i.e., an � 2 $ such that � � x; � � y and z � � for all z 2 $ such that z � x;

z � y.11the intersection of any two elements of F remains in F .

4. CONCLUSIONS AND FINAL REMARKS 47

Proof. According to (1:8) from Chapter 3-Corollary 7, andwith (3:9) into account, we are led to the equalities

= = infM2L(X ;U)

F =\

M2L(X ;U)

KM (H);

and hence = is actually the in�mal element of the semi lattice F : So,it is evident from (3:10) that12:

A� (= \ kerB�) � =:

But, according to Theorem 4 above, F is made up with all the(A�; kerB�)-conditioned invariants containing ImH�: Thus, the in-�mum = of F ; as the intersection of all these invariants, must benaturally (and obviously) the least one among them, i.e. the least(A�; kerB�)-conditioned invariant containing ImH�: �

4. Conclusions and �nal remarks

Following Theorem 4 and Theorem 5, among the results workedout in this chapter under assumptions (A1)-(A4) for the perturbedsystem (1:1) (1:2), it is worth to raise especially the above es-tablished optimal properties of the ideal observability subspace =,namely:

� = = \KM (H)M2L(X ;U)

= infM2L(X ;U)

F ; where

F := fKM (H) j M 2 L(X ;U) g

with

KM (H) = SpnIm [(A+BM)�]

iH�; i 2 N

o;

� = is the least closed subspace containing ImH� with the(A�; kerB�)-conditioned invariant property

A� (= \ kerB�) � =:

12This conditioned invariance property of = was already established (in

Chapter 3-Lemma 2) independently, just by means of the ideally observable di-rection concept.


By duality, and still under the same assumptions (A1)-(A4),these two properties lead naturally to similar optimal formulations13

for the subspace =?; the orthogonal complement of =; namely:

=? =[

M2L(X ;U)

K?M (H) =

�inf

M2L(X ;U)F

�?:

� =? = maxM2L(X ;U)

F?; where F? is the upper semilattice with

respect to � and + de�ned by:

F? :=n[KM (H)]

?j M 2 L(X ;U)

o

with

[KM (H)]?=\

i2N

kerH(A+BM)i;

� =? is the greatest closed subspace contained in kerH withthe (A; ImB)-controlled invariant property

A=? � =? + ImB:

Now, as a conclusion, we recall that this �nal chapter wasintended to be a geometrical synthesis establishing the linkbetween the two approaches developed up to now. Sev-eral lemmas have been used to introduce the two principalresults of this synthesis. These lemmas translate the ex-istence of certain properties of invariance and conditionalinvariance; properties shared between the subspace of idealobservability = and the speci�c family F resulting fromthe directional approach of Chapter 3, with F only madeup of Kalman-observable subspaces. These preparatory de-scriptions lead to Theorem 4 where it is proved that thiscollection F is in fact a semi lower lattice formed of sub-spaces conditionally invariants, all having the property to

13The expression of F? is skipped to avoid non needed developments in thepresent context.

4. CONCLUSIONS AND FINAL REMARKS 49

contain the initial observable subspace while being inacces-sible to the in�uence of the perturbation u(:). These lastconsiderations lead in return to the fundamental Theorem5 where it appears that lower bound of this semi latticecoincides exactly with =; and represents in this respect thegeometrical print for the ideal observability subspace of thedisturbed system (1:1) (1:2). In other words, and apartfrom the intrinsic expression (well too theoretical) reservedfor inf

M2L(X ;U)F , one can thus conclude that the algorithm

of construction of such an in�mum is well that of Theorem1 resulting from the ortho-projectional approach developedin Chapter 2.

APPENDIX A

Some extensions of the directional

approach to Banach space

The reader may notice that this appendix is presented as a self-contained chapter. Following [6], it is given to recall how a nonneg-ligible part of this thesis matter can be extended to Banach space,with some suitable adaptations, and despite some known comple-mentation problems and other closure di¢culties, due to the in�nitedimension and, in a more signi�cant way, to the absence of a an innerproduct [16].

As explained in the introductory chapter for the case of Banachspace X , it seems natural to describe the observability properties inthe dual space X �. To do so, the concept of ideally observable direc-tion introduced for X = Rn in [20] will be readapted here througha somewhat di¤erent formulation.

1. Introduction

The concept of ideal observability will be considered in this ap-pendix for linear systems in Banach space with bounded operatorsunder the action of a disturbance generating a given output on some�nite interval of time. Some properties and descriptions are estab-lished for the ideal observability subspace within the meaning of theweak-(*) topology, including the lack of state reconstructibility.

Let us consider a dynamical system described by the equations:

(1.1) x0(t) = Ax(t) +Bu(t)

51

52A. SOME EXTENSIONS OF THE DIRECTIONAL APPROACH TO BANACH SPACE

(1.2) y(t) = Hx(t)

where x(t) is the state of the system, and y(t) the output observedon a �nite interval of time [0; T ]. The unknown function u(:) is adisturbance supposed locally integrable in the sense of Bochner, i.e.u(:) 2 L1loc([0;+1[;U). The spaces of phase, output and distur-bance, noted X , Y and U are real and separable Banach spaces. Theoperators A, B and H are linear and continuous, i.e. A 2 L(X ;X ) �L(X ), B 2 L(U ;X ) and H 2 L(X ;Y), with H non invertible.

With the knowledge of the coe¢cients A , B andH , the problemconsists in obtaining from the output y(:) the maximum of informa-tion on the trajectory x(:) on [0; T ], specifying under which condi-tions the given observation makes it possible to rebuild the state ofthe system.

2. The ideal observability subspace

The functional f 2 X � will be called an ideally observable direc-tion for the system (1:1) (1:2) i¤ the following implication is trueon [0; T ] for every perturbation u(:) :

(2.1) y(t; x0; u(:)) � 0 =)< x(t; x0; u(:)); f >� 0:

In this de�nition, x(t; x0; u(:)) stands for the solution of equation(1:1) generated by the perturbation u(:) with the initial conditionx0 = x(0), whereas y(t; x0; u(:)) is the corresponding output.

Definition 7. The set = of all the ideally observable directionsin X � will be called the ideal observability subspace of the system(1:1) (1:2).

Proposition 1. The set = � X � is a closed subspace in theweak-(*) topology �(X �;X ).

Proof. It�s clear that = is in fact a subspace in X �.Moreover, let us denote by Fz the functional de�ned on X

�

by Fz(f) :=< z; f >, where the expression < z; f > means thevalue of the element f 2 X � at the point z 2 X . So, we can write

2. THE IDEAL OBSERVABILITY SUBSPACE 53

= = \z2M

KerFz with M = fx(t) 2 X : Hx(t) = 0; t 2 [0; T ]g. Thus,

= is the intersection of the kernels (the null subspaces) of contin-uous functionals in �(X �;X ) , and it�s consequently closed in thistopology. �

According to this proposition, the next de�nition follows natu-rally.

Definition 8. The system (1:1) (1:2) is said to be ideallyobservable i¤ the subspace = coincides with the whole space X �.

In other words, De�nition 9 relates the very situation where theobserver can determine the whole trajectory x(:) on [0; T ] solely fromthe knowledge of y(:) .

Proposition 2. If KQ(H) is the Kalman-observability subspaceof the system:

(2.2) x0(t) = (A+BQ)x(t)

(2.3) y(t) = Hx(t);

then = � \KQ(H)Q2L(X ;U)

(i.e. = � KQ(H) holds true for every Q 2

L(X ;U)).

Proof. Recall that Kalman observability is a particular case ofour present problem, since it supposes that the function u(:) is nota disturbance, but a control known to the observer. In this context,the subspace KQ(H) is known to be:(2.4)

KQ(H) := Sp�(ImH�; Im(A+BQ)�H�; Im((A+BQ)�)2H�; :::)

where Sp�(:) is the closure in the topology �(X �;X ) of the linear

span of the subspaces between the brackets (with ImA� standing forthe range of A� , the adjoint of A).

Now let f 2 = , so that (2.1) holds true for every u(:) 2L1loc([0;+1[;U), and in particular for u(t) = Qx(t). Thus, f is aKalman observable direction for the system (2:2) (2:3), meaningthat f 2 KQ(H). �


Remark 10. Proposition 2 means (naturally and) in particularthat:

(2.5) = � K0(H) = Sp�(ImH�; ImA�H�; ImA�2H�; :::)

where K0(H) indicates the Kalman observable subspace of the sys-tem (2:2) (2:3) with Q = 0 . So, a necessary (but not su¢cient)condition for the ideal observability of (1:1) (1:2) could be taken asK0(H) = X � (and even more as \KQ(H)

Q2L(X ;U)

= X �). Besides, under

some additional assumptions for the case of Banach spaces, otherproperties and descriptions of the subspace = can be given (on onehand) similarly to the projectional approach of chapter 2, but withannihilators instead of orthogonal complements. On the other hand,similarly to the situation in Chapter 3, we state the following funda-mental invariance property.

Proposition 3. The subspace of ideal observability = containsthe subspace ImH� and veri�es the invariance property:

A�(= \ kerB�) � =:

Proof. Let us show that (f 2 = \ kerB�) =) (A�f 2 =) .By de�nition of = , we have the successive implications:

fy(:) � 0g =) f< x(t); f >� 0; for all f 2 =g

=) f< x0(t); f >= 0; for all f 2 =; a.e. in [0; T ]g ;

whence:

< x(t); A�f > + < u(t); B�f >= 0; for all f 2 =; a.e. in [0; T ] :

Thus

< x(t); A�f >� 0; for all f 2 = \ kerB�;

and hence

A�f 2 =; for all f 2 = \ kerB�:

Now, to end the proof, notice that, among the preceding im-plications, the �rst one states indeed and obviously the inclusionImH� � = . �

3. ON THE LACK OF IDEAL OBSERVABILITY 55

3. On the lack of ideal observability

Among the Banach spaces X ;Y;U ; we consider in this sectionthe situation where Y = R , with the intention to focus on the lackof ideal observability in the case of a scalar output y(t) = Hx(t); t 2[0; T ] .

So, let us take the element h 2 X � such that < z; h >:= Hz

for z 2 X . Then the Kalman observable subspace speci�ed in theformula (2.5) becomes:

K0(h) = Sp�(h;A�h;A�2h; :::);

and can be seen as the annihilator of the closed subspace1\i=0KerHAi

.

Lemma 8. If ImB �1\i=0KerHAi, then the ideal observability

subspace = of the system (1:1) (1:2) with a scalar output is givenby:

(3.1) = = K0(h) = Sp�(h;A�h;A�2h; :::):

Proof. = � K0(h) is clear according to Proposition 2 (see alsoRemark 10).

To establish the opposite inclusion, let us suppose that y(t) =<x(t); h >� 0 and show that < x(t); f >� 0 for all f 2 K0(h) . Bydi¤erentiation (as in the proof of Proposition 3), we obtain y0(t) =<x0(t); h >= 0 a.e. in [0; T ], from which < Ax(t) + Bu(t); h >= 0a.e. in [0; T ] , and thus < x(t); A�h >� 0 since ImB � KerH . Byrepeating the process of di¤erentiation, we get < x(t); A�ih >� 0for all i 2 N, and by continuity we arrive to < x(t); f >� 0 for allf 2 K0(h) . �

Remark 11. The above assumption ImB �1\i=0KerHAi is too

strong. It means that the action of the disturbance is inside theKalman non-observable subspace. All occurs as if the disturbancewere null. The following result avoids such situation by supposingthat ImB is not contained in all the subspaces KerHAi (i 2 N).


Lemma 9. Suppose that the inclusion ImB �1\i=0KerHAi is

not true. Then the subspace = of ideal observability of the system(1:1) (1:2) with a scalar output is reduced to the �nite dimensionalsubspace de�ned by:

(3.2) = = Sp(h;A�h;A�2h; :::; A�kh);

where k is the minimal integer verifying HAkB 6= 0 .

Proof. We shall prove in the following that = = KQ(h) , where

KQ(h) = Sp(h;A�h;A�2h; :::; A�kh)

for some linear and continuous operator Q 2 L(X ;U) .By assumption of Lemma 2, there exists k 2 N such that

HAkB 6= 0 with HAiB = 0 for 0 � i < k.Take u0 2 U such that HAkBu0 6= 0 (here 0 := 0R) , and

consider the operator Q de�ned by Qz = HAk+1zHAkBu0

u0 for z 2 X . It�s

obvious that Q 2 L(X ;U) .Now, let us calculate the subspace KQ(h) , taking into account

the condition HAiB = 0; 0 � i < k .We then get successively:

H(A+BQ) = HA+HBQ = HA;

H(A+BQ)2 = HA(A+BQ) = HA2 +HABQ = HA2; :::;

H(A+BQ)k = HAk 1(A+BQ) = HAk:

Besides, we have H(A+BQ)k+1 = HAk+1 +HAkBQ = 0 since (byde�nition of Q) the relation HAk+1z +HAkBQz = 0 is true for allz 2 X . Thus, H(A+BQ)k+p = 0 for all p � 1 , leading consequentlyto the �nished (and thus closed) subspace in (X �; �(X �;X )):

KQ(h) = Sp�(h; (A+BQ)

�h; (A+BQ)

�2h; :::)

= Sp(h;A�h;A�2h; :; :::; A�kh):

Now, = � KQ(h) derives from Proposition 2. The converse= � KQ(h) can be obtained by successive di¤erentiations in thesame way as in the proof of Lemma 9 (knowing that HAiB = 0 for0 � i < k) . �

4. CONCLUSION 57

Remark 12. Under assumption of Lemma 10, which impliesB 6= 0 , the state function x(:) can only be recognized along a �nitenumber K of directions (see also Remark 13 below).

Proposition 4. The system (1:1) (1:2) with a scalar outputis ideally observable if and only if the two following conditions aresimultaneously checked:

(CC1) Sp�(h;A�h;A�2h; :; :::) = X �

(CC2) B = 0:

Proof. Notice that condition (CC1) means1\i=0KerHAi = f0g :

Now, if the two conditions are checked, the system (1:1) (1:2)with a single output is obviously observable within the meaning ofKalman observability (i.e. condition (CC1)), with the additionalabsence of any disturbance (i.e. condition (CC2)). It is thus ideallyobservable.

Reciprocally, ideal observability implies condition (CC1), and,according to Lemma 10, condition (CC1) itself shows the lack of any

integer K such that HAkB 6= 0 . So, ImB �1\i=0KerHAi = f0g ,

whence B = 0 . �

Remark 13. Proposition 4 means that the system (1:1) (1:2)with Y = R cannot be ideally observable whenever the disturbance isnot null. However, Example 1 of Chapter 2 proves that two outputs(say Y=R2) may well su¢ce for the ideal observability of the system(1:1) (1:2). Proposition 4 could be seen as an extension of a �nisheddimensional approach proposed in [17].

4. Conclusion

We have presented here some properties relating to the subspaceof ideal observability for a class of abstract linear systems in Banachspace. Some results remain partial and likely to be improved foran extension to the more general and more practical case where theoperator A is the in�nitesimal generator of a C0 semigroup. That


would make it interesting to investigate systems generated by partialdi¤erential equations or systems with delay.

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Spécialité: Mathématiques Option: Théorie Mathématique du ... · Ahmed BARIGOU Jury de soutenance TERBECHE Mekki Président Professeur Université d’ORAN MESSIRDI Bekkai Directeur