Lecture Notes in Biomathematics

Managing Editor: S. Levin

46

Eric Walter

I dentifiabi I ity of State Space Models with application::! to transformation systems

Springer-Vertag Berlin Heidelberg New York 1982

Editorial Board

W. Bossert H. J. Bremermann J. D. Cowan W. Hirsch S. Karlin J. B. Keller M. Kimura S. Levin (Managing Editor) R. C. Lewontln R. May G F Ost.:>r A S P">rI~lson T Peggio L A Spg",,1

Author

Eric Walter L:lbor:ltoirQ dQS:: Sign:lux "t Sys::tomQS::, CNRS - ~colQ Sup4riQurQ d'~IQCtricitQ

Plateau du Moulon, 91190 Gif-sur-Yvette, France

AMS SUbjeCt ClassifiCations (1860): 92-02, 93 B 30, 34 A 00

ISBN 3-540-11590-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-367-11090-0 Sprill\:lt::I-Vt::"Cl9 Nt::w Yu", Ho::idelbcl9 Dedin

This work is subject to copyright. All rights are reserved, whether me whOle or psrt OT the material is concemed, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is psyable to 'VerwertungsgeseUschaft Wort", Munich.

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Acknowledgment

It is a pleasure to express my gratitude to those who helped me in the preparation of this monograph.

I owe special thanks to Dr. Pierre Bertrand, my thesis supervisor, Gilles Le Cardinal and Yves Lecourtier, whose friendly co-operation has made this book possible.

I also gratefully acknowledge many helpful discussions with Dr. Pierre Delattre, Professor Michel Fliess and all my friends of the "Laboratoire des Signaux et ~ystemes".

The translation from the original French would never have been under­taken WltnOut tne encouragement Of PrOfessor Slmon A. Levln and of tne revlewers, and tkc re~ult;ng [ngl;~n n~~ been muen improved by tne effo~tc of Guy WaltQ~ and

Dr. Kenneth Avery. (Of course I assume full responsibility for the errors that pro­bably remain).

Finallv I wish to thank Annick. Isabelle and Pierre for their patience.

Contents

INTRODUCTION .............•.••.•••..••.•••.••••.•••.•••••••••••••••••••••••• 1

CHAPTER 1. TRANSFORMATION SYSTEMS ••.••.•••••••••••.••.••••••••.••.•...••.. 5 1.1 I ntroducti on •••••••••••.•••••••••.•.••••••••.•••••.••.••••... 5 1.2 ro.-rnali3m ' •.... 1 ••••••••••••••••••••••••••••••••••••••••••••• ~ L3 An example: nonlinear chemical kinetics..................... 8 1.4 Spec1rlc problems or transrormat1on system mOdell1ng ••••••••• 16 1.5 Conclusion ••••••••••••••••••.••.••••.•••••••.••••••••••..•••• 20

CHAPTER 2. STRUCTURAL PROPERTIES AND MAIN.APPROACHES TO CJ.l£CKING TJ.l£M ••••••••••.•••••••••••••••••••••.••••••••••.••• 21

2.1 Introduction ............... I ••••••••••• II •••••••••••••••••••• 21 2.2 Definitions •••••••••••••••••.••••••.•.••••.•••••.•••.••...••. 21

2.2.1 Structural iproportioc and gonorieity ................... ?1 2.2.2 Connectabi 1 i ty •••••••••••••••.••••••••••.•..••••.•••.••• 22 e.e.J Stru(.turo1j observobi1ity ond

structural controllability I ... • .... • .. •• •• • ...... • ...... 24 Z.Z.4 ~tructural iaent1f1aDillty •••••••••••.•••••..•••..•.•.• l4 2.2.5 Relations between these notions ........................ 27

2.3 Practical methods for checking struct~ral observability ",nti dr"rt"r",l rnntrn1l.:1hi1ity ofjlino"r motioh ............ 11 2.3.1 All nonzero entries are free ........................... 31

2.3.1.1 Crolph thcorcti c opproolch ....................... 32 2.3.1.2 Algebraic approach ............................. 34 Z. 3. 1.3 eUII(;l u::.1 UII ••••••••••••••••••••••••••••••••••••• 37

2.3.2 Nonzero entries are dependent .......................... 37 2.4 Main approaches to structural identifiability .•.••.•.•....••. 41

2.4.1 Identifiable canonical representations .•..............• 41 2.4.2 Global optimizatien .................................... 41 2.4.3 Borman aAd Schoonf .. 1d'c approach. _ .......... _ ..... ___ .. 44 2.4.4 Transfer function approach ............................. 45 e.4.5 Minimal representation opproach •••••••••••••••.•••.•••• 49 2.4.6 Local approali:hes ....................................... 49 l.4.7 power ser1es approach .................................. ~o 2.4.8 Identifiability of large-scale linear models ......•.•.. 51

2.5 Cone 1 us i on •.•...•..•..•....•.•.••.•....•...••.........•.•.... 54

CJ.lAPHR 3. LOCAL IO£NTI~IABILITV ... _ .. __ . _ .. ____ _

3.1 Introduction ................................................. 56 3.2 Methods ..•.••..••••.•.•.••.•••.•..•....•..•.••.•.•..•••..•.•. 57

3.2.1 Use of the implicit function theorem ....••.•.••.•.•.••. 58 3.2.2 local stability of identification algorithms ..•.•.•.••. 58

VII

3.Z.Z.1 NewLun tlnd Gtlu~~-N~wtun dlyurltlllll:> ••••••••••••••• (jO 3.2.2.2 Gauss-Seidel a199rithm •.•••••••••••••••••.••••.•• 61 3.Z.Z.3 Quasi linearization algorithm .••••••.••••••••••••. 62

3.2.3 Observabilitv of the extended state ••••••••••.••••.••••.• 64 3.2.4 Information matrix....................................... 64

3_ 3 Ungar modG1s: ___________________ .. ____ . _____ . ___ . __ . ____ . __ . _. _ 65

3.4 Computer aided design of models ••••••••••••.••••••...••..•••••. 66 3.5 Implementation for linear t,-an:sformation 3'y3tem3 •••••• , •••••••• 69

3.5.1 Method A •••••••••••••••••••.••••••••.••••••.••••••••••••• 69 J.5.l 5~ruc~ural na~ure or the result ODtalnea ••••••••••••.•••. 71 3.5.3 Method B ................................................. 71 3.5.4 Examples.. ....... ........... ...... ................ ....... 74

3 _ 6 Cnnt:1u<; i nn . _____ . __ .. _ .. _ . __ .. _ ..... __ . _ . _ .. _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 77

CHAPTER 4. GLOBAL IDENTIFIABILITY OF LINEAR MODELS ••••.•••••.•••.•.••••.•••• 79 4.1 Introduction •••••••••••••..•...•....••.•••.••••..•..••••.••.•.• 79 ~.2 Proportioc of tho trancition matriK ____________________________ 70

4.3 Parametrization of the transition matrix •••••.••••...•••••.••.. 82 4.3.1 All th" ";9,,"Yo111":; or A 0'" ",,01 ........................ ae 4.3.2 Some eigenvalues of A are complex conjugates............. 83 4.3.3 \;Onnectlon wlth Lagrange-::;Ylvester pOlyngmlalS ........... lS3

4.4 Application to checkinQ S.Q. identifiability ..••••••.••••••••.. 84 4.4.1 The experimental data are entries of •••...•.•..•.•.••..• 84

4.4.1.1 No con~trainto~i~t~ on A _. __ . __ . __ . 95 4.4.1.2 General procedure................................ 87 4.4.1.3 [Knmp1e ; two-c1a33 trnn3formation 3'y3tem3 ••••••• SS

4.4.2 Method fer any B afld C ................................... 90 4.4.3 Pru~lem~ ral~ed by InequalIty ~on~tralnt~ ••.•••••••••••.• 9~

4.5 Conc 1 us ion .••.••.•••••.•.•.••..•...••..••..••....••............ 95

CHAPTER 5. EXHAUSTIVE MODELLING FOR L1NEARMQDELS ........................... 97 ~.1 Introauctlon •.•.•..••......•.•.•.•...•..••......•.••..•..•...• OJI 5.2 Class of the studied models................................... 97 5.3 The matrices B'and C are known •.••..••.••..•••••..••.•.•.•..•. 99

5 3 1 ThQ matricQ~ Band C arQ ~taRd3rd QQ 5.3.2 The matrices Band C are known, but non-standard .•.•.... 110

5.3.2.1 Standardization of CD ••••••••••••••••••••••••••• 110 5.3.2.2 Standardization of B ~nd.C .•.••.•.......•.••.... 112

~.4 The matrl~e~ B ana C are par~lally unknown •••••.•••••••••••••. 117 5.5 Connections with Kalman's canonical form ••.••...•...••••.••... 120 5.6 Applications of exhaustive modelling •..•.•••.•..•....••.•..... 122 5.7 Conclusion _ .. ______________________________________ . __________ 127

CHAPTER 6. EXAMPLES 6.1 I ntroduct ion •••..••.•••••.••.••...••••.••••••••.••...•.••..••. 128 Ii 2 ChQmothQrapQutic modQl ............... _. _ ......... ____________ . 129

6.2.1 First experimental set-up ••••.••••••••.••.•••••....•..•. 129 6.2.1.1 Connectabi li ty, 3tructura 1 ob:oervab.i li ty

and s tructura 1 contro 11 abi 1 i t;y .................. 130 6.Z.1.Z JLru~Lurtll lo~al Identlfla~11'ty •••••••••••••••• 130 6.2.1.3 Exhaustive I116de11ing .... ..... ................... 130

b.Z.Z ::;econa experimental set-up ••.••••••••••••••••.•••.•.••.• 134 6.2.2.1 Structural local identifiability ••.••••••••.•••• 135

128

6.3

6.4

VIII

6.l.l.l EJlhdU:sLIvt: IflUtlI!111119 .••••••••••••••••••••••••• Hepatobil i ary kineti cs , of B.S. P ........................... . 6.3.1 connectability~ structural observability and

s tructura 1 contro 11 abi 1 i ty .......................... . 6.3.2 Structural local identifiability ................... .. 6.3.3 !;:xhauttivQ modQlling ................................ . Metabo 1 i sm of i odi ne •••.•••••......•........•••••••••••••.• u.4.1 St,u,;,tu, 'ol lv,;,ol illenti rialli1 HI .; ................. .. 6.4.2 Structural global identifiability ••.•.••••••••••.••••

ti.4.Z.1 1nput-output tranSTonmatlon ••••••••••.••••••• 6.4.2.2 Standardization. '!" ••••.•••.••••••••.•••••.• 6.4.2.3 Determination of A ........................ .. 1\.4.1,4 Tntro..tllction of ~h" <trll .. tur"l

constraints on A .......................... .. 6.5 Sy~temie di~tribution of Vincamine •••••••••••••••••••••••••• 6.6 Conclusion .•...••••.•..•.•...••..•...••....•.........•.••...

13~

136

137 137 137 139 141 141 141 142 143

144 H9 151

CHAPTER 7. GLOBAL IDENTIFIABILITY OF NONLINliAR,MODEL~ •••.•................ 153 7.1 IIILr ·vllu~Llvll ................................................ 1:i3 7.2 Series expansion approach .•.......•.•..••......••..•........ 153

7.2.1 Time-power series.............. ........ .......... .... 1~j 7.2.2 GeneratinQ series ........... .................. ....... 155

7.3 Linearization approach ........ ............ .................. 162 7.3.1 Prineip10 .. ________ __ , __ 11\1 7.3.2 Application to nonlinear transformation systems ••••.. 163

7.3.2.1 ~hy3iea1 linearization by tracer inclu~ion , ••• 16~

7.3.2.2 Mathematical linearizatien ••••••••••.•••••••• 171 7.3.l.3 What 15 the best l1near1Latltlll ! ............. 173

7.3.3 Generalization ..•........•.•.•...........•.•......... 175 7.4 Conclusion.................................................. 176

CONCLUSION

REFERENCES .................................................................

17B

181

SUBJECT INDEX ..••••••.••••..•.•.•••..••..•.•.••.••••.••••••••....•••.•..... 198

Introduction

It is the objective of Science to formalize the relationships between observed quantities. The motivations of such a modelling procedure are varied, but can rougnly be collected around two pOles.

If one is concerned with process control, one wants to find a model which wl11 De aDle to predlct tne process Denavlor, taKlng lnto account tne applled lnputs. The model will then be evaluated on it5 ability to mimic the ob5e~ved input-output

behavior under c:onditione; ae; vari"d ae; po<;e;ibl". In th" limit. a mndel built on an

irrational basis might be considered satisfactory, provided it appears efficient in the design of a control law.

If one aims at a better understanding of the system under investigation, one requires more from the modelling rules, since one is trying to reach what P. Delattre calls "the underlying level of the 'real' behavior of objects". Besides mlmlcKlng tne lnput-output benaVlor ot tne system, one wants tne model to 1nclude all a pdori knowledge on the 3y3tem and to nove 30me internal conerence within the the

o~y whirh ha~ l~rl to it~ rl~v~loD~nt

The 5eQ~ch fo~ theo~et;eQl mode15 with thc help of method5 de5;gned fo~

huilding predic:tion models has produced a number of models whose only justification

lies in the similarity between the response curves of the system and that of the mo­del ; and this situation has to a great extent discredited "System Analysis" among experimentalists.

This monoqraph is aimed at those who look for a theoretical model. by pro­viding some tools for a critical reflection on the structural properties of such a model. Special attention has been devoted to the notion of structural identifiability, the study of which remains quite an open problem and to which an increasing number of 1'01'~1·:' Ol·~ u~u;'.oL~u. IL j:. :.LI·f~llIy Lu ,,~~ U,ol lI~ol·ly 011 l~I.Lul·~I·" oL Lh~ 5LI! IrAC

Symposium on Idontificat;on and Syctom Pa~amoto~ [ct;mation of Soptombo~ 1979 in

Darmstadt mentioned the test for identifiability as a neCeSSary step in the modellinQ process. Indeed this step is of obvious importance : if the structure chosen for the model is not identifiable, then, whatever the quality of measurements may be, several

2

"",deh with the ~ame ~t,u ... tu,,, uuL <Ii rr",,,,,l IlCl"ClIII"l",' vCllu,,:. ,'''11,''':''''IL LlI" lllJs"rv,,1l

r .. " 1 i ty '''1 "a lly w" 11 Thue th" valli". found for tho paramotor~ aftor any i donti fi ca­

tion procedure are highly questionable, if not meaninQless.

Th .. t .. rhniI]IlPc; to h .. Ilc; .. ti in i ti .. ntifhhil ity t".tin() diff"r appr"ci ab ly

depending on whether or not the model output is linear with respect to the input. If one has at one's disposal an analytical expression of the observed quantities as functions of the parameters for linear time-invariant models, this is not true for most nonlinear models.

All linear time-invariant models to be considered here are described by the following finite-dimensional state equation

{X = A(ij)X + B(ij)U,

'L = C(~)~ + D(~)u,

!at,,) ,

where ~, ~, L' ~ arc re5pectivcly thc state, input, output and parameter vector~.

Thp solution of surh an pl]uation is lin .. ",r with .... c;p .. rt to th .. initi;1l c;blt .. or to

the input, but remains nonlinear with respect to the parameters. Note that the tech­niques described here for continuous-time models are easily transposed to discrete­time ones.

In order to build a theoretical model, care must be taken not to intro­duce any hypothesis (either explicit or implicit) whicn cannot be justified. Thus one nas at one's alsposa1 for tne moaell1ng process:

The propel-tie3 of finite-dimen3ional 3tate-3pace repre3entation3

- The a priori knowledge on the system, which usually results in some constraints on the structure of the state matrix A ;

- The a priori knowledge about the nature of the interactions bet­ween tne system and tne experimenter, which results in some constraints on the struc-tU"e~ or the ('ollt.-ol matd" B Cllld the ouse, vatioll mat, i" C.

rosslOle unlaentltlaOl Ilty ot tne moael correSponalng to tne parametrl­",atioll defined above h Q 3eriou3 diffi.;;ulty (Chaptel-3 1 and £) , whi.;;h we mU3t be

",hl .. to tii",onosp (f.haptprs ]. 4. ~) hpforp finding ways to rpsolve it whpnever possi­

ble (Chapters 3, 4, 5) or to cope with it (Chapter 5).

Fur lIulIl ill~Clr' Uf' l.illlt:-vGf'yiny l1Io\.h:l!Jo, Lhe pr-oblem i~ even more c.omplex,

and hac ."ldom bo"n tack.l"d lip to now, W" .. hall conc;;ti .. r th .. 101',,1 ",nti th .. n th .. 010-

bal identifiability of a rather gereral class of models described by the state equation

3

While the initial problem is of a theoretical nature, its consequences are, to a high degree, of practical importance. This ambivalent situation raises a difficult problem in the presentation. In order to stress the concrete nature of the problem, an~ not to nloe any or the alfflcultles resultlng trom tne use ot tne pro­pOGed methods, an important place h03 been re3erved for exomple3. All co.-re3pond to

traMfnnnlltinn <:y<;tpm<;_

In thi~ formQli~m, the model is defined as a set of equivalence cla33es,

between which transformations may occur. Ea~h rlass rnntain~ all thp plpmpnt<; whirh have equivalent properties within the considered system. Transformations between clas­ses correspond to phenomena of exchanges (of material or of energy for example) or of reproduction. Depending on the nature of the laws which govern the transformations, tne moael may oe 11near or not.

Three main reasons justify the choice of transformation systems as ex­amples of appllcatlons.

Firstly this choice is by no means restrictive, for these models have all " .. ~r·aun.lillar·y var'i,,~y ur lIu~~iLJlt: ~~r·u(;~ur·t:~. TI.u~ ttlt:y form a partlcularly sulta­blQ fiQld for a ctudy on tho ctructural propertiec of modele. In particular, the con

crete meaninQ of the state comoonents makes it useless to resort to what is called canonical representation in the field of Automatic Control. Consequently the potential number of the parameters to be estimated is quite large. For example, when no a priori information on the system is available, one has to estimate all the elements of A in tne case Of a l1near mooel.

Secondly the formalism of transformation 3ystems (of which compartmen­

tal models. that have been used in Bin1ngy fnr m"ny YP""<;. ""p " <;lIh<;pt) wa<; wn,,~prl

out by P. De1attre for interdisciplinary purposes, and may be an efficient tool, at­tractive to specialists of Biology, Chemistry, Pharmacodynamics, Ecology, Economics, Population Dynamics, Automatic Control and so forth.

Last Iy to] S torma 11 sm 1 s deduced trom ax; oms, ; ntended to ; nsure the internal coherence of the model mentioned abovt:.

The f1rst Chapter of thls monograpn 1S deVoted to transformatlon sys­tom£. Their formalicm ;~ rcc~11cd, ~nd the notation to be used later ;3 introduced.

As an examole we show how one can. under suitable exoerimenta1 conditions. deduce a linear transformation system from any nonlinear chemical kinetics, without needing to linearize the model around some operating point. The structural difficulties raised by

4

the modellin9 of :>uch "y"tem" an: ""L ru,LI ••

rne secona cnapter glVes a survey Of the various available approaches for 5tudyi n9 the mai n structurn 1 properti e:> of 1 inenr mode 1:> (connectalJil H,y. uu:>","­v~b;l;t~. rnntrnll~hil;ty ~nrl irlpntifi~hility) anrl ~tatQ~ thQ rQlation~hip~ ox;~ting between these notions. The fact that the usual definitions of the model structure lack generality and do not make it possible to study all kinds of structures induced by the formalism of transformation systems is stressed. A more general definition is proposed, which leads us to develop a heuristic approach to the study of structural uu:>"nal./iliLy a",) "L,"ul;Lu,"al l;u"tr"ulldulllty when some cons"[ralnts on tne parame"[ers aro pro~ont"

Chapte.- three pre:>ent:> a :>Yllthe" i" of the ,,It:thu<l:> ""aul illy UII" Lu t"" t thp fl~r~mptpr~ of a rlynami c mod,;. 1 (whi ch may bo 1 i noar or not. timo-i nvar; ant or not) for local identifiability. The need for an active policy with respect to unidenti­fiable models is stressed. Some tools are developed, allowing determination of addi­tional measurements or structural constraints that may remove the indeterminacy. They result in fairly easy-to-use routines for computer-aided design.

Chapter four has the same objective, but this time from a global point of view, and for linear time-invariant models only. The proposed approach relies on the .JlufJ","Li,,:> ur t,a'I:'; L;uII lIIaL';l;"" all<l allu .. " u" Lu fJ'"UV" "UIII" y"",,,"al ,"""ul L" auuuL global idontifiability_

Unfortunately both policie:> which may make identifiable n model which i:>

not arp ofh.n '1"itp "n~ati~factory , th,;. fir~t on". which r"'1uir"" th" maldn!) of addi­tional measurements on the system, because such measurements are difficult, or even impossible, to carry out; the second one, Which relies on suplementary hypotheses about the model structure, because these hypotheses cannot be justified. For this reason Chapter five develops, for linear models, another approach to the problem, l;all,,<1 ,,~'.au"L;v" lIIu<l"l1 ;IIY. IL l;UII" ;"t:, ill !>"a,"U'illY rU'" LII" !>"L ur all III;II;lIIal '""­prooantationo havin9 tha oama input-output bohavior and compatibla with tho otruc­tural constraints on A. Band C which result from the a priori Knowledae about the system and its interactions with the experimenter. One then arrives at a model which preserves the ambiguity, if any, in the considered experiment. In this way, the iden­tifiable parameters are exhibited, and relations between unidentifiable parameters a'"" ,""v"al,,<I.

~napter SIX apPlIes tne tecnnlques presentea In tne prevIous cnapters to :>ome ,-eali3tic and ,-athe,- complex example3 of t.-an3fonnation 3Y3tcm3.

F1nally <:nap"[er seven aescrlbes several metnulfs wlilUI Cdll lJe u!>e<l tu !>tu<ly the global identifiability of the p3r3mctorc of a nonlinoar and/or timo-varying modol.

Chapter 1 Transformation Systems

1.1 lNIKOUUCIIUN

for two main reasons. Firstly they form an invaluable framework of experiments for testing methods to study structural properties of models, since they present a wide variety of structures. Secondly they were developed for interdisciplinary studies, and one may hope that algorithms designed for them will naturally find a wide range of QPpli< .. otiofl~ ill voriou~ ui~l.ifJ' illt:,~. Mureuver we re'L iL wuulu be u~eru' onu moti­

vating to do<cribo in <oma datail a domain with practical application<. on which tho

various proposed methods could be checked. Note however that the methods presented in the following pages can be used for many other kinds of models.

After recalling the formalism of transformation systems, we shall show, as an example, how it can be used to study nonlinear chemical kinetics. Finally we shall isolate some specific structural problems raised by the use of such models.

1.2 FORMALISM

The formalism of transformation systems has been introduced by Delattre [Ul - 03, 012, DU] • It 1S a general1zat1on Of compartmental mOdels [:>1, AI, Li, Ki,

J1, 913, S9] ,widcly uscd in Biology for thc study of mctabolisms, and which rccci

ved ~nnsiderable attentinn with the exnan~inn nf tra~er mPthAd~.

Transformation sy'stems are defined as sets of objects, some characte­

ristics of which are chanaina with time. because of the interactions taking place. The studied objects are distributed among equivalence classes. Each equivalence class, associated with the notion of compartment, contains all objects having identical func­tional characteristics within the considered system. It is represented graphically by d circle, (lnd the cardlnal or the class J ls noted x .• The posslble mlgratlon or

J

6

elelillmt:> 1"rulH une clilss to ilnother 1s called transfonnat1on. BetWeen any tWO Classes tho trancTonnationc may bo non-oxictont, unidir.ectional, or bidirection;)l.

Three kinds of laws are considered to define the possible rates of trans­fonnation between classes

(1) L1near transformat1on from class J 1nto Class 1

(1.1)

it describes radioactive disintegration. for example. or first order chemical kinetics [ B11] •

(ii) Bilinear transfonnation induced by a field with intensity ~

( 1.2)

Such eqU;)tion5 m;)y model the r;)di;)tion eTTect on chemic;)l or biologie;)l 5Y5tem5 [D10] .

(iii) Nonlinear transformation

r = ~ l:T x a ijk lJ lJ I< I< ( l.SJ

Thi<: Hnt! 01' hw. whi,..h rpl'rp<:pnt<: intpr:ll"tinn<: hptwppn c:pvpr:ll nhjPI"t.c:. is 'luitp

common in Chemical Kinetics. Coefficient aijk is the partial order of the reaction with respect to the kth constituent, and belongs to the set of the model parameters to be estimated. Bimolecular reactions are an important particular case : if two mo­lecules J and K have to meet for the reaction to take place, the reactlon rate Wlil depeml un lhe fr"e\juell\;,Y uf lhe \.ull b iUII" u\.\.uI"i1l9 !Jet",een the:;e mole~ule:;, a.od will thoroToro bQ proportional to tho product 01' thoir concontration~

such nonlinearities also arise in Population Dynamics.

Note that some nonl1nearltleS wnlcn are commonly used, partlcularly 1n Pha.,"ma.cokinctic3 [CIO, Wll, WI:!) , !luch ;)$ tho:)e modelled by the Michaelic-Monton

7

t:4UdLiuJI ;

are not directly taken into account by the formalism of transformation systems. Ex­Changes OT thlS klnd may reSUlt Trom slmpllTylng asSUmptl0ns, SUCh as some classes being in a ~teady ~tate, but the~e ~implified notation~ need, in the context of a

formillism whi~h Jlrptpnlic; to somp rigor. c;omp jllc;tifi~ilHons not illways pa~y to provi­

de.

Finally, the troll~ro,,"oLion rru", ,-lo~~ j illLu l.lo~~ " per unit time nroy

involvo a difforont numbor of olomont~ in tho initial cla~~ and in tho final ono.

This is the main difference between transformation systems and classical comoartmental models, besides the possible introductioh of nonlinearities. It enables one to take into account ·phenomena of multiplication or reproduction [018] . As a consequence, transformation systems are not necessarily stable, unlike compartmental models. Al­tnougn the stUdy oT non-conservatlve transformatlon systems wlth the methoos to be deccribed in the following chapterc doee not preecnt any particular difficulty, we

shall limit ourselves to conservative models. which make it oossible to use a simoli­fied graphical representation. The possibilities of transformations between classes will then be represented by oriented edges between the circles associated with the classes involved. Near each edge we shall indicate the value of the corresponding transtormatlon rate rij [I'lgure 1.1] .

~ig. 1.1 Trancformation from clacc j into cla~G i.

The outside of the system will be considered as a class, with the index O. A symbo­lic eye ( ~ ) will indicate the experimentally observed portions of the trans­formatlon system. Ihey correspond to What we shal I cal I outputs, tOllowlng the voca­bulary of automatic control theory, and to what Oelattre cal13 "categorie:> of experi­

mentill idpntifi~ilt.ion". t.o ilvoirl ;InY ~onfllc;ion wit.h t.hp t.rllnc;formllt.ionc; from t.hp

system to the outside. The state equations for any transformation system with N classes are

obtained by writing down the instantaneous balance equations for each class. For a

8

l;VII!>t:I"VClL;vt: IIIv<.I"l, Vllt: VULCl;II:' ;

N - L r ..

j=o Jl (i=l, ... ,N). (1.4)

j!i

1.3 AN EXAMPLE: NONLINEAR CHEMICAL KINETICS

A method to study the dynamics of nonlinear chemical systems through a compartmental model associated with a tracer has already been proposed [WI, L2, W2] . At about the same time Happel [HI, H2] showed how to use transient isotopic tracing to ldentlty the parameters ot some nonllnear Klnetlcs. ~o let us see, as an example, how nonlineClr kinetie3 CCln be de3cribed in term3 of tron3formotion 3Y3tem3.

cons1der a chem1cal system 1nY01Y1ng, 1n a cont1nuously st1rred open tonk reoetor of constont volume Y, N constituents, of which n orc lobclloblc with a

trilrpr [ Fi OUrI' 1.?]

Y, !

Fig. 1.2 Stirred open tank reactor.

Tht: :.LClLt: .=. vr Lht: n:Cll;Liulli!> the ve.:;tur "hu.se cumponent3 ore the coneentrotion3 of thQ£Q N rQactant~ in~idQ tho vo~~ol It can bo partitionprl into

e (1.5)

where Yi is the concentration in the ith labellable species, while zi is the concen­tration in the ith non-labellable one. If we suppose that all other factors which may influence the system are kept constant, the state equation can be written

9

( 1.0)

Now consider the subsystem constituted by the set of all labellable atoms. Two atoms will belong to the same equivalence class if they have the same functional behavior, that is to say if they belong to the same chemical species and are submitted to the same external constraints. We shall suppose that labelling does not modify the dyna­mic behavior ot the reactants, so tnat laDelled and unlaDelled elementS can De col­lected into Q 3Qme clQ33. With ni the number of labellable atoms in eaeh entity of

th" ith r"artino c:rpri,":. thp r~rtiin~l of thp ith rl~c;c; will hI'

( i 1. ...• n). (1. 7)

Similarly the flows of labe11able atoms resultinCl from exchanoes with the outside wi 11 be

The change of variables (1.7), (1.8) enables one to find, from the first n rows of (1.6), Lh~ IIvlIl ill~(H· ~l.{uQLivlI~ 5"Li~rieu l.Jy the labellable Qtom~

't ,. t .t xi - f i (~ ,!) T I i 0 t ( t ) - 'Oi ~ ,! (i-I, ..• ,n). (1.9)

The n clQ33c3 exchange atoms with one another and with the outside. These flows of

;!tnmc: ;!rp nonlinp;!r fllnrtionc: of thp c:t.;!tp x Thp rpc;1I1 tina tr;!nc;formlltion c;yc;t!'m ic; .

when no a priori structural constraint exists, extraordinarily complex [Figure 1.31.

10

t r 1O---1~

FiQ. 1. 3. Three-cl ass transformati on system without any a ori ori structural constraint.

We ch~ll now chow how cuit~blc cxpcrimcnt~l condition3 may make the

transformation system linear. thus allowing us to study its structlJrnl nronf'rtif'~ in a far more extensive way.

. t x; = -

Equation (1.9) can also be written

t tnt t t r J.; L~ • .f.) + 1 r ik L~ • 1.) + riO

k=l k;!i

(i 1, .... n) . (1.10)

the flow r~j of labellable atoms is the sum of the flow rij of labelled ones, and of the flow r .. of unlabelled ones. Similarly the number x~ of labellable atoms in the ith class ~~ the sum of Xi' the number of labelled ones, and Xi' the number of unla­u~l1~u VII~~. WILli ~llb IIV~C1llulI ~II~ UYIIClllliI .. t!yuCl~iulI~ vr lClut!11t!u C1I1U Ulilout!l1t!u

atomc aro rocpoctivoly

11

n n

I r .. + I r1K +rlO (i:l. .n). J=U J1 k=1 (1 11)

j!i kli

:. n n -I rji + I r ik + riO (i=1. ••.• n). J=o "wI (l.12)

jli k!i

If the probability for an atom leaving class k to be labelled is given by Xk/X~ • the flow of lallelle<1 atoms from class k lnto class 1 1S

Similarly tho flow of unlabollod atom, from cla,e k into claee ie

Taking (1.13) and (1.14) into account. the dynamic equations of labelled and unlabel­led atoms become

I. I. ~)

t (~t. ~) n r .. (~ . n r ik L Jl

" + L "I< + r lU • j=O x~ 1 k=l xt jfi 1 kfi k

(LIS)

-Xi

r~. t !:)

t t !:) n (~ . n r ik (~ .

I Jl Xi + I Xk + riO t t J=O l\i 1<.=1 l\k j!i k!i

( 1.16)

The similarity of these two equations should be noted. Moreover. if ~t and ~ are not functions of ~ and ~, these two equations are linear, possibly time-varying. This linearization is performed by introducing the labelled elements in an inclusive way. 1.11011. I:. lJy n:~iJldl;lll!l. ill I.h~ rluw:. ~1I1.~rlll!l I.h~ :.y:.I.~1I rn.1I11 I.h~ uul.:.hJ~. :'UIII~ ullldlJ~l­

lod olomonte by labollod ono, without changing tho total flowe, which aro euppo,od to

12

Loe rUIIl.Liu"" ur Lime ullly [Filjur'e 1.4] .

t 1"0;

u

Fig.1.4a : [racer inclusion resultinQ in a tracer state ~l(t).

t o t

Fig.1.4b : Tracer inclusion resultinq in a tracer state ~2(t).

FiQ.1.4. Linearization throuQh tracer inclusion

Fig.1.4c : Tracer inclusion resultina in a

t

tracer state ~1(t)+~2(t).

(Initial conditions in tracer are supposed to be zero).

IIntlpr thpc;p ~nntlitinnc; thp c;up!'rpnc;itinn prindpl!' lIppli!'':; tn th!' mndels describing the kinetic behavior of both labelled and unlabelled elements.

Thus, although Equation (1.10), which describes the total system, is nonlinear, labelled and unlabelled quantities satisfy identical linear differential equations. Unlabelled material tollows the same I<lnetlcs as laDelled one ana can alSO be con3idel-ed 0.3 0. tro.cer. The lineo.riz:o.tion l"e3ultin9 from trocer inclu~ion i3 of a phy~ical naturo . It i< valid whatpvpr tho proportion of l~hpllprl miltPriill ~y hp_ It should not be confused with a mathematical linearization around some average operating point. Chapter 7 will show that these two kinds of linearization lead to fairly dif­ferent mode Is.

It is essential, for the linearization to be effective, to choose as state varlaDle x~ Llle number' ur ldbelldble "Lum" ur " yivell chemic"l species j. neck,

J . for example, ha3 pre~ented in [Fl] a counter-example to. tho linoarity o.f the tracer equations. which yields linear equations when the formalism of transformation systems is used.

Oy replo.cin9 the consumed rcoctant3 and keepin9 tho reactor volume con<­hnt. nnp riln ~inbin thp olnbi!l prncess in a stationary state

.t ~ .Q. '

~ 0

13

Equat10ns (1.1') and (1.10) then descr10e stat10nary l1near systems

where

~ Ax + rO'

.:. A;' ~ - ... !.O'

- -rO I i = riO' rO I i riO'

a·· - -11

n L

j=O Hi

a·· -lJ

Linear transformation systems satisfy the constraints

(1.17)

( 1.16)

( 1.1!»

(1 20)

Conservative systems, such as those obtained when one studies chemical kinetics, ~atl~ry addItIonally

n L cI •• ~ 0

j=1 lJ (I 1, .•. ,n). (l.Zl)

TUl/t:U,t:, (1.1:0) ellII.l (1.<:1) illll'ly u,t: "~cluiliLy ur (1.17) clIO'" (1.16) [HJ] , ,,"iI .. " lot:­

sidos is physically ovidont.

The inclusion of tracer in the permanent inlet of a reactor in a sta­tionary state therefore results in two linear stationary kinetic processes evolving in opposite directions so that the total system remains in dynamic equilibrium. The parameters of the linear kinetics are related to those of the nonlinear one, and it may be possIble to est1mate the parameters or the nonlInear kInetIcs by study1ng how the entriec of the matrix A change with the operating point of thc rcactor.

Consider the reactional graph

14

kI A + R ... r

k2

It is associated with a reactional mechanism proposed in [NI] to describe a Friedel­Crafts reaction catalysed by iodine, with

CH - C l /

0

B 0 (in e,,<oe~~) ,

[;H \'(; U 3

CH - C 0, 1/ " , , C - 0 ,-12

\ , CH3 - C 0"

CH - C 0, 3 / , ,

, + U U )

-"'C /

CH3 0'

E

F I; The corre3ponding nonlinear kinetic model ic

15

[A) - 11.1 [A) T 11.2 [ C) - 1<.4 [E] [A]

[ C) k1 rAJ - k3 [ C] - kz r C 1 •

[D) k3 [C) ( 1.22)

[ t. J k3 [q 11.4 [t.) [A) ,

[F J "4 [F) [AJ

A, C, D, E and F are associated with five equivalence classes, containing respective-t t t t t

ly "I' )\2' )\3' )\4' amI )\5 atulII:> ur lalJ"llalJh, iudi"". Ttli~ l"ad~ lu lll" fullowi,,!!

changvc of variablvc in (1.22)

[AJ ----. xiJ2V ,

[C] --+ t x/2V ,

[D]- t X3/V ,

[E] --+ t X4/V ,

[FJ ............ X~/3V

If V is maintained constant, (1.22) can then be written

t t k4 t t 11.1 xl T 11.2 x2 - -V xl x4

t t k1 xl - (k? + k1 ) x?

k3 t 2 x2 •

kJ l k4 l l ~ x2 - ~ xl x4 JK4 t t 2V X4 xl •

( 1.23)

16

IL \..all lrt: :>I,u,," [W1] LllaL .. iLIl CI I,;UII:.LClIIL now r-lo ur ClLulII:' uf iUl.JiIlt: llILu 1,;1(1:>:> 1

ono can put tho ~y~tom in a .tationary .tato, if tho volumo V i. kopt conctant by

extracting a flow E of reactional fluid. The state eQuation of the transformation subsystem formed by classes 1, 2, and 4 is then.

(1. 24)

Constant flows r~. take place between classes. Any flow leaving a class (therefore with a minus sig~1 without entering another one (with a plus sign) is leaving the system. It 1s thus pOSSlble to spl1t the varlous flows taK1ng place 1n the SUbsystem formed by cla~~e~ 1, 2 and 4

t t [ k J t E t] t k j t 0, X2 = kl Xl - r x2 + Ii x2 -k2 x2 -r x2 ---------t r21 t r02

t. r 12 t r42

·t k3 t k4 t t E t o . Xq - T Xz - [ "IT xl Xq ... V "4] ~

-----~ t rA?

t rOA

Classes 3 and 5, which are not involved, have been put together with the outside, and indexed O. By using (1.19) one can now write the tracer state equation

17

xl k4

(Kl .,. V t )1.4 .,. I) V "'2 0 )1.1 1

x2 kl - (k2 + k) + ~) 0 x2 + 0 rlO" (1.25)

0 k j k4 t t.

0 X 2 ("2V xl + Y) x4

Figure 1.5 describes the corresponding transformation system. Note that the various flows are not independantly parametrized. For example both transformations from clas­:.t::. 1 dJl\! 4 Lu Lilt: uuL:.I\!t: iJlvulvt: Lilt: :'1I111t: fJdJ-III1It:Lt:J- 1<.4. Till:. \!t:pt:JI\!t:JI(;t: will imlu(;t:

~PQcific problQm~ during thQ ctudy of tho ctructural proPQrtio~ of modol~, ac WQ ~hall

see later on.

FiO_ l_li Trilnc;fnrmiltinn c;yc;tpm fnr iI Friptlpl-r.rilftc; reaction.

I\!t:JlLi fi(;dliUJl uf Lilt: pllrlllllt:lt:r:. uf tilt: liJlt:IIJ- lIIu\!t:l (1.Z5) frUIII IJlpuL­

output data QnablQ~ uc to Q~timatQ quito cimply tho paramQtgrc of tho nonlinoar ono

(1.22).

we must stress tne tact tnat ~nem1cal K1net1cs 1S only an example ot the potenti;)l ficld::; of application of t."on:>fOl"mation 3Y3tem3. In oddition to field:>

where compartmental analysis is extremely common. such as Biochemistry andPharmacn­dynamics, one can find applications for them in Population Dynamics [AI) , Ecology

18

[rl, A£) or [conomic:> [DZ) • Huwtm:~r · ~I,,: "uuvt: t:J!.CIlllfllt: of non11near cnem1cal )(1netlCS ha( allowod 1I~ to illll~trato tho notionc of clac£ and of .trancformation concretely, and to show that the study of linear transformation systems mav Drove to be useful for the modelling of even strongly nonlinear systems.

1.4 SPECI~IC PROBLEMS OF TRANSFORMATION SYSTEM MODELLING

The aim of the formalism of transformation ~y~tem~ ;~ tn ~rnvirlp a mP­

thod of constructing coherent models of observed phenomena, avoiding as far as possi­ble any implicit or unjustifiable hypothesis.

To get a coherent model, one must always keep in mind the following question. Suppose that the studied system has exactly the structure defined during the model characterization. Suppose moreover that there are available noise-free observat10ns of the system outputs for known 1nputs, opt1mally cnosen to senSltlze the model p~r~meterc. Wo~ld it then be p055ible to b~ild f~om thi3 info~<ltion <I mo del havino exactly the ~alTlP ~aralTlPter~ a~ thp real ~y~tpm ? Tn nthpr wnrrl~. i <: thp model identifiable?

Unfortunately the answer to th1s quest10n 1s frequently negatlVe for transformation cyctoms, and this for three reacons :

(i) When no <I pdod infOl-matiol1 exbt:> about the model structure, it db­play". ovon in tho linoar ca<:Q. a trQl1IQndOll~ <:trllctural val"iQty : dncQany cla~<: may be at the oriqin of transformations into any other one and to the outside. an n-c1ass model with one input and one output may lead to a vector of n(n+2) parameters to be estimated. For nonlinear models, where each transformation may be a function of seve­ral parameters, the problem is even worse.

(ii) The components of the model state vector all have a precise physical meaning, since they are aSSOCiated with the characteristics of classes. This forbids the U3e of what <lutom<ltic contl-ol en9ineer3 c<lll a <;;anonical state-3pace repre:>enta­tinn. pvcppt a<: a po<:<:ihlp intormortiato in thp con<:trllction of tho final ropro<:onta-tion.

(iii) Finally the possible measurements on the system under investigation

are often extremely limited, especially for living systems, since some classes may not be accessible.

Nnw. nooloctino thp inpntifiai'li1ity Ilrnhlpm whpn nnp i~ lnnking fnr thp parameters of a model may result in serious difficulties, of a numerical as well as

19

theoretlca1 or experlmenta1 nature.

Numerical difficulties : Some identification algorithms are unstable if the :.Lu<lit:<I,"u<l"l b nuL 1ul.olly i<l,,"Li riot.1". Mo,'"oye. the behoyior of ~ome identifi­

cation a1gorithmc cannot be oxp1ainod without rosorting to tho notion of idQntifia­

bilitv. We became aware of its imoortance when testinq an identification oroqram using the quasilinearization technique [52] . We had simulated the transformation system described by Figure 1.6a to check whether the identification algorithm was able to recover the parameters of the generating model from these simulated data. We were surprlsed to flnd that lt converged elther to the parameters of the generatlng model. or to tho~e of the model de~cribed by Figure 1.Gb. In both c~~e~ the value of

the quadratic criterion on the outout error was almost zero. so that we oot two fair­ly different models but with the same input-output relationship.

t 0.101 "2

Fig.1.6a. Fig.1.6b.

The.netil.o1 difril.u1tie:> ; A~ on example. L.C and C.R. lIu haye de~c.ribed in

[114] an idontification IlIQthod for tho p~ramotor~ of an n-ch~~ linoar modo1, of which

(n-l) classes are observed. This method is supposed to enable one to identify the n2

exchange parameters as functions of two of them. But one can show (see Chapter 5) that if inputs can be fed into m classes. the number of degrees of freedom of such a problem is (n-m). so that these degrees of freedom cannot always be represented by Lwu 1'0 1'11111" L"":' •

Experimental difficulties: Let us take the example ot blologlcal systems. Ihe pa.olloeL".,;; o"Loi""d roo Lh" ,"ud"l may 1.1" u:>ed roo a t.ette. l.o,"prehen~ion of internal

mochanisms, for tho diagnosis of disoasos, for thQ sQ1oction of a troatlllQnt ~lIitod to

the oatient's individual characteristics. or for an evaluation of the effect of drugs for homologation purposes. But if the model is unidentifiable, some of the parameters obtained do not have any meaning, which may have some striking consequences. Moreover the actual reconstruction of the system state, expressed in a basis detlned trom phy­:dc.a1 c.on~ide,'ation~, i~ po~~ible only i r the Illodel i~ identifiable, exc.ept fo,- 30me

20

A critical analysis of the possible model structures is thus necessary. But general global results are available only for linear models described by

- 11K Bu ~(o) - !!O' (1.26 )

Cx 1" Du

As is well known [Kl] • the only part of a linear model which is acce~­sible from the input-output data is the controllable and observable one. This will lead us to present in the next chapter some methods of practical interest to check the structural observability, the structural controllability. and the related proper­ty ot connectaOl Ilty Of models descrloed oy (l.~b)

In (1.26) the matrix A expresses the transformations taking place bet­ween tne model classes or from these classes to the outsIde. We shall call lt~ ~truc-

tu~e the model ~t~uetu~e. The m~t~icce Band C eKp~ecc the inte~actionc between the

~ystpm ~nd the experimenter. i.e. the ways in which the dynamic behavior of the clas­ses will be affected by the inputs and how the classes will be observed. They define what we shall call the experiment structure. The matrix D, which is introduced here for the sake of generality, expresses an instantaneous influence of the input on the output which seldom exists.

1.5. CONCLUSION

Transformation systems, because of the wide variety of their possible structures, form an attractive framework for the methods of studying connectabi lity, ooservaOll1ty, controllablllty dJld ItltmllflcliJil ily ur lIIull"l "l,ul.lu,,,:.. 6e:.ill,,:. it 1:.

not by pu~e accident that moet of the ~ecoa~ch on ~t~uctu~al id~ntifiability ha~ ro­

sulted from work on compartmental models. which are particular cases of transformation systems. We would like to stress again, however, that all the methods to be presented below apply just as well to other types of models. The notion of identifiability in particular plays a central part in the-elaboration of parametric models of all kinds, as In the synthes1s of upt.lmdl experlment~ fur ltJ(:~ .,,,LiIllClLiulI ur lhe f'C1'·ClIlJeter-s of

thece modele.

2-1 INTRODUCTION

Chapter 2 Structural Properties and Main Approaches to Checking Them

It i~ appropriate to define the structural propcrtics wc arc concerned

with mnrp prpri~ply Aftpr h;winO ~tiltpl'l whilt. wp mPiln hy t.hp st.rll~t.llrp of iI miltr;x.

and linked the notion of a structural property to that of genericity, we shall re­call the definitions of various structural properties: connectability, controllabi­lity, observability and identifiability. We shall see that close but complex links eXlst between these propertles, betore examlnlng the maln approaches avallable ln the literature fOI~ checldng the ... Thi3 ... ill enable u~ to rai~e ~ome open que3tion3,

anl'l to put in prop .. r p .. r<:p .. ,.tlv .. th .. m .. thorlc: to h .. propoc:prl nnt nnly in thi~ rhilptpr

for checking structural observability and structural controllability when constraints exist on the nonzero entries of A, Band C, but also in the following chapters for checking structural identifiability.

2-2 D~FINITIONS

2.2.1 Structural properties and genericity

Lin r L31 published the first results on structural properties of linear models. He defines the structure of a matrix by the position of the zero entries, the others being supposed free. If such a definition permits the constructi~n of very simple algorithms for checking structural observability and structural controllabi­llty or llnear models, such as those descr10ed 1n Sect10n t.J.l, IL lIt:vt:r·UIt:lt::.:. tid:'

the drawbaek of not taking into account modele where nonzero entriee in the matriceo

A. Band C are related. Yet this is a freQuently encountered situation when dealinQ with transformation systems. That is why we prefer to define the structure of a ma­trix by its parametrization. It will be supposed that the matrices A, Band C can be expressed as functions of a vector of free parameters ~€ ecl<v, with the usual

22

Eu\.l hJiclII LUl'ulu!:l)'. TilL> will 1o:a<1 u:> Lu <1t:yt:lul' a :.uaalllt: IIIt:Lhu<1 rur· Lfrt:l;kirry "Lr"u\..­

tural ob£orvability and £tructural controllability_

Remark : As one may expect, the structural properties depend on the structure charac­terlzatlon. An example of sucn a dependence wlll De presentea ln ~eCtl0n 2.2.~.

We will consider as structural the properties which hold for almost all the values of !, and possibly fail on a hypersurface (regarded as atypical) of the ddlllbslble pdrdllletric SPdce \:I. This corresponds to the notlon of d generlc property [WJ] • More preci3ely let A, B •••• be roatrice3 with entrie~ in DR, letTl(A, B •••• )

h" II prnp"rty th .. t th"c:" "'IItrir"c: roilY h .. v". All th" prnp"rti"c: nf int"r'"c:t tn riC: will

hold for all the admissible values of ~, except for those corresponding to the common zeros of a finite number k of properly defined polynomials ~i (Section 2.3.2 will gi­ve examples of such polynomials). Let VC~v be the algebraic variety constituted by the set of common zeros of these polynomials:

v = I!: ~; (!) = 0 ,i € 1 1,2, ••.• k! ! . v i" "aid to bo propor if' V ~IR" and nontrivial if V ~ PI A pnint ~t nf t.h" pilrilRlPtrir:

space is typical (relative to V) if at belongs to the complement of V in nRv. Proper­tynis said to be generic if a proper variety V exists such thatnis not satisfied uniquely for ~ € V. nwill then hold in an open subset which is dense in RRv. This subset corresponds to all the typical points of the parametriC space.

2.2.2 Conncct~b;l;ty

The nu L I orr of l;urrrrt:l; Loll i 1 i Ly wo" i" L. uduced by Day; .)on [ 04] • It U3e3

an oriented gr3ph accoeiated with tho coupling ox;£t;ng b~tw~~n th" c:tat" rnmpnn"ntc:.

the inputs and the outputs. The following example shows both how this graph is obtai­ned and how close it is to those used to describe transformation systems. Consider the model shown in Figure 2.1. One has first to create an oriented graph expressing in qualitative form the relations existing between the state components. For example tne relatlon

results in the subgraph descrlbea Dy ~lgure z.z, wnere tne lOOp arawn 1n d dotted 1 im: i~ shown 03 0 reminder, :t;nce one is only intcrcctcd in tho intoract;on~ bot­

wo"n ~lIh-c:yc:t"mc:.

Similarly one then expresses in a qualitative way how the state varia­bles are reachable, either through the control matrix B or through the observation matrix C. The resulting graph is presented in Figure 2.3.

z

Fig. 2.1.

4

3

Fig. 2.3 Graph corresponding to the transfor­m~tinn ~.Y~t<.m nf .io" .... ? 1

24

Th", :.LaL", va,",Clul", "j will 1.>", (;VJlfI",(;LCll.>l", Lv IlIpUL Ui If, dlld ulIly 1f, tho~o ox;,t, a do,conding path f~om u. to x., S;mila~ly x. will bo connoctablo to

1 J J output Yk if, and only if. there exists a descendinq oath from Xj to Yk. So on the proposed example only xl' X2 and x3 are connectable to input u1' but the whole state is connectable to the outputs. A structure in which the state is connectable to the inputs and the outputs is said to be connectable.

The prOfound s1mllarlty eXlstlng Oetween the dlagram of a transforma­tion ~y~tem and the 9~aph u~ed he~e enable~ one to ~tudy the eonnectability of a triln~fnl"ll1i1t. inn "y"t.pm "imply hy iMppdinO it" rpp~p~pnt.ilt.ivp tiiilorilm. pvpn fnr hr­ge-scale models. One may also easily check the connectability of any model structure on a computer by using the properties of Boolean matrices, which enable one to find all the paths of order 0, 1 •... between any two vertices of the graph successively. The resulting method is qUlte S1ml lar to tnat developed, tor compartmental mOdels, ln [C1, C2] . Note that what is lOolled ~tl' ulOturol ob~ervability (.tructul'ol contl'olla­bility) in tho~o ~oforonco~ i~ in fact tho output connoctability (input connpctani­lity). Another approach to check connectability is to look for arborescences contai­ned in the graph associated with the structure of the studied model [04]

2.2.3 Structural observability and structural control lability

The notions of observability and controllability are well known in the field of Automatic Control [K() . ~y definition we Shal I say that a mOdel 1S struc­Lur'ally vu""naul", (~tructurally cOlltrollC1ble) if the property of ob.ervC1bi lity (con­trollability) io gonoric in tho oon~o of ~oction ? ? 1

2.2.4 Structural identifiability

In a quite remarkable paper in 1950, Koopmans and Reiers6l [K3] had already stressed the importance of the notion of identifiability in such widely dif­fering fields as Biology, Psychology, Sociology and Economics. The first attempts on this subject concerned models described by algebraic equatl0ns WhlCn were 11near DOth tn the vl.>~erved vdr·iaul",,, alld ill Lit", UII""OWll parallleter~, and included linear con3 traintc on tho paramotoro [ 1(4) , Wald [WlJ] <Inri _ M _ic:hpr [_~. _Ii 1 nilvP pxtpntipd

these results to models which were nonlinear in the observed variables, ahd possibly included nonlinear constraints on the parameters.

For dynamic systems described by differential equations, it was only in 1963 that Kalman LKIJ mentioned, in a ratner allusIve way, the notl0n Of ldentl­riauiliLy. Shvr ·Lly arter, Y. C. 110 and Whalen in 1963 [115] , and thc:n LEo _i"hcY' in 1965 [_2). gavo riofinitionc: nf iripntifiilhilit.y in t.he special case of models whose state is completely observed.

25

o But. Ullt: IId~ Lu wdiL ullLil Lilt: f'af't:/ ~ ur 60::111110.11 o.nd A:;trOrn in 1970[01]

to. got a el~ar-cut dofinitio.n o.f ~tructural idontifiability fo.r dynamic cy<tgmC ~ It

is worth notinQ that the theory of compartmental models seems to have motivated them, since the paper ends on thi s cone I us ion : "the questi on of i denti fi abil ity of a com­partment structure when several but not all compartments are observed is an interes­ting and important problem that has not yet been resolved",

sInce then, the derln1tlon of structural ldtmtlfldlllllt.y IId~ 1Jt:t:1I ~ulJ­

jected to. vario.us slight mo.dificQtio.ns, Qnd there ic ac yet no. univorsally accoptod

c;pt nf !lpfi niti nn .. [n1 fi. I R] " Wp c;hi\ 11 IIC;P thp fo 11 owi no ones : Consider two state-space models M(~) and M(~) with the same structure,

that is to say with the same parametrization, receiving the same input ~ and having identical outputs ~ = 2 for any ~ belonging to the admissible input class U (Figure ,.4). Such models wi I I be said to be output-1nd1st1ngu1sna~ [57]

& Sarno paramgtrization ,

M(!!)

rig. '.4 Output-indistinguishable models.

The dimensions of the state ~, the input ~ and the output ~, which are part Of the defInItIon of tne model structure, are supposedly known:

dim (~) = n, dim (~) = m, dim (y) = p.

The vector e of the parameters to be identified is assumed to belong to some admissi-- v

ble parametric space 0 elR •

Definition 1 Ine parameter 8i 1S structurally locally 1dent1t1aOIe (Sit.1) tor tne

input c1os~ U if, ond only if, fOl" olmo~t ony 2 e 0 there exists Q neighborhood

V(fl.) c;lIr:h thi\t

26

(2.1)

Definition 2

The fJdrdllieter ~i Is structurally gloDally 1aentlt1aDIe (s.g.1.) tor the

input class U if, and only if, for almost nny 2 e 0 one ha3

(2.2)

Definition 3

The moael M(~) 1S S .... 1 (s.g.1.) It, ana only if, all its parameters

0i arc o.~.i (S·9. i .).

Remarks

(i) Structur<ll idcntifi<lbility mcnns identifiability in a gene.·i" 3e113e.

(ii) III whd~ rulluw~ we ~hall speaK lml1fferen"Lly of 5.1.1 (s.g.1.)

modQlc or of locally (globally) identifiable ctructurcc.

(i ii) Tu yiY" " <,ulll.,..,~e III"dllill!:! bu ~lle fJr ·ufJeny uf ldentlf1aDll1ty as

defined above, one may contidQr that M (2) is a model which has ~een obtained for ~

physical system. If M is not S.Il. i .• there exists at least one ~ ~ ~ such that M(~l

and M(~) are output-indistinguishable. This makes any further use of ~ questionable.

Linea ... case

Tn thp r:iI~P nf c:t.iltinnilry linPilr mn<ipl<: with finitp-<iimpn<:inn"l <:t"tp.

the model structure will be defined by the parametrization of the state equation

~ x(t) = A(~) ~(t) + B(~) ~(t) , ~(to) = ~o(fjJ,

(.l(t) = C(~) ~(t) + DW ~(t) . (2.3)

unless otnerw1se ment1oned, we Shal I suppose in what tol lOWS that the

initial 3tate ~(to) i3 ~ero and that the input cla33 U is the set of piecewise con­

tinllnlle: fllnr:tinne:. IIn<ipr thpe:p "<:<:lImptinn<:. nlltpllt in<iie:tinOllie:h"hility i<: P'llliv"lpnt

to the identity of impulse responses

D(~) = D(~) ,

(2.4)

27

D(~) + C(~) [sf - A{~)rl B{~) = D{~) + C(~) [sf - A(~)rl B(~J. (2.5)

3!1 1 ~ : !l'~€e, A(!l) = A(~), B(~l) = B(!2) , C(!l) = C(~), O(!l) = D(~) , (z.ti)

because it is clear that output indistinguishability would never imply parameter identity ;n .>ulOh 0 lOo.>e.

2.2.5 Relations between these notions

These nOtl0n~ are closely relatea Input conncctobility i3 0 ncccssory, but not 3uTficient, condition for

struc:turlll controllilbility [041 _ I ikl'wi<;1' Olltput connl'ctithility ic; it nl'cl'<;<;ilry. hilt

not sufficient, condition for structural observability. S.t. identifiability is a necessary, but not sufficient, condition for

s.g. identifiability [G1 J • The links that may exist between structural identifiability and other

.>trulOturol propertie.> have ut:t:n 5urri ... it:IILly ut:uaLt:u Lu ut:~t:nt: ~1't: ... iCll ClLLt:flLiufI.

The first papers [H5, F2, B1, E2, E3] which mentioned the links exis­ting between the structural properties of identifiability, observability and control­labIlIty dealt wIth modelS glven oy

p = A.! + B,!!

Ll: = x.

"or ~uch modoh. whon no conctraint oxictc on tho ctructuro of A, a nococcary and

sufficient condition for S.Q. identifiability is that the model be structurally con­trollable.

Note that there exist autonomous - thus non-controllable -models which are nevertheless identifiable (in celestial mechanics for example)[S3]. The same pro­blem appears when considering linear transformation systems in free motion

~ - A ~ • ~(o) - ~ •

One can return to the usual problem by simply considering the impulse response of

28

Lltt: IIIUIl I r; t:1l '"11Ilt: 1

~ - A x

The model will be "controllable" if the pair (A,xv ) is controllable.

S Ion I 1 Clr-ly, rur- d 1l1U1lt:l dt::'l:r IlJt:d lJy

A ~ + Q . ~(to) = Q

c x

a n .. c,,~~al'y and ~lIffici .. nt condition for L!J - idQntifiability, ",h"n no con~traint exists on A, is that the model be structurally observable.

A nasty generallzatlon lea some autnors to state tnat structural obser­vability and ~tl'uctural controllability wcrc nccc~~ary conditione for etructural idpntifhhility in ~11 I":)I<;P<;_ Rpl":pntly <;nlllP pXilmpll!<; hy Cnhl!l1i I!t al. [C3. C4] cle­arly showed that this was not true. Figures 2.5 a, b, c present three examples of transformation systems which prove that structural controllability and structural observability are neither necessary nor sufficient conditions for the model to be S.1.. 1. or s. g. i .

IIho.t i3 more, the po.ro.mete,-, of 0. po.rt of a model whi ch is nei ther in put-con"",ctahl .. nol' nlltrll,t-rnnnprhhl .. mily prnv .. tn h .. <;-O.i .. if thl!Y ilrl! rl!liltl!d by structural constraints to s.g.i. parameters, as can be seen in the example of Figure 2.5d, inspired from [D5] . Note that, in a recent paper (C4] , Cobelli et a1. showed that, with their definition of the model structure, connectability is a neces­sary condition for structural identifiability. However this result is not relevant nere, slnct: our :.trUl:turt: d'dr-dl:lt:r-ILdLiulI I-ClII IIII-lullt: .... ooo:>t.(lint:. betwecn nonlOc'-o cntricc of A, e, C and D.

29

x = [d11 "12] ! + [1] u

d21 022 0

Fly. l.Sd CUIlt.rollaOle ami observable, but non S.1(.1, moael.

JJ ).. aU 0 0 1

YD !.. 01 21 0 0 1\ .,. 0 U

--. a31 0 0 0

~ - [ g 1 OJ x X. 0 1 -

1=";9· 2.6b S'9';" -but uncontrollable, model [ CJ 1 •

,..-~

)81 ~ -(a?1+a31) 0 0 I I

~II u 1 I I ! a21 0 0 X + 0 u

-~! - a Jl 0 0 0

\ 3 : "\ I'

= l J! ' .... _ ...... y 0 0.5

Fig. 2.5c 5.g.i., but unobservable, model [ C4 J •

30

u

~ig. 2.6d S.g.i., but unconnectab1e, model.

we are tnus Doun~ to admit that Checking connectability, observability and controllabiHty of a given "tructul-e doe:> not enable U3 to IOom:.1ude "heU,e. V.­

not thp Jl;tr;tlllPtpr<; nf thi .. <;trllf"tllrp IIr,. itl,.nHfiab1,. In partkular if a ~tructurQ

is not observable, one should still study its identifiability. What is more, most of the methods checking structural identifiability are concerned with the controllable and observable part of the model only. Thus they elimi.nate by themselves any redun­dancy in the considered representation.

Call vile ,h::uul.e rnJlIl Lllb LliaL I.hel."'"\! :.L,-uI.Lu,-a1 vlJ:.e,valJi1' Ly allu

ctructura1 controllability of tho modal i& of no usa whon ono i& intoroctod in tho

structural identifiability of the parameters? We do not think so. Indeed. techniQues such as those presented in Chapter 5 which try to generate the set of all output­indistinguishable models having a given structure apply only to the observable and controllable part of the model. It is thus necessary to have at our disposal simple anO errectlve ways or checKlng the structural oDservaDl11ty anO structural control­lability of the ~tudied linear mode1~, in order to be ~ure that tho~e method~ apply.

Note moreover that the only parameters identifiable from input-output data are those which can be deduced from the knowledge of a minimal representation. For example the uncontrollable, unobservable or unconnectable models presented in Figures 2.5 b, c, dare s.g.i because their reduced transfer matrix, which is asso­clateO Wlth the control laDle ano oDservaOle part Of tne mooel, COnta1ns enough 1n­formation to enable one to compute all the parameter:> uniquely.

Nntif"P finally that. with th .. pXf"Pption nf "p;tthnlogical" parametri­

zations satisfying (2.6) [MI] , s.g. identifiability is a necessary, but not suffi­cient, condition for a model with unknown parameters to be actually controllable or observable. As a matter of fact, all methods which allow one to control the state Wlth the 1nputs or to reconstruct lt rrom tne outputs requ1re the Know1eOge or tne

31

moael parameters.

2.3 PRACTICAL METHODS FOR CHECKING STRUCTURAL OBSERVABILITY AND STRUCTURAL CONTROLLA­BILITY OF LINEAR MODELS

Several effective ana easy-to-operate methoas now exist to check struc­tu,-al ob:;ervability 0'- :;t,-u(;tu,-al (;ontrollability of a model the :;tru(;ture of "hi(;h

ic dofinod only by tho pocition of tho zoro ontrioc in tho matricoc (A,H,C)_ Two main

approaches are presented in the next section. We shall then see how to tackle the problem when nonzero entries of A. Band C are related.

2.3.1 All nonzero entries are free

Lin [L3] introduced the notion of structural controllability for scalar­input models in 1974. This notion has been extendea since then to the multivariable (;a:;e [az. 54, L4J • A IIe\..e:;:>a,y alld :>urrh .. ient \..ondi Lion ro, a II,ul tiva,iablc 1 inca,

timo-invariant ctato-cpaco modal to bo ctructurally controllablo ic that noithor of

the two following conditions is met

Type cOnaltlOn:

Max [Rank [A(~) I B(~)l] < n ~

where A is of dimension (nxn)

w_2~ondiJjon :

There exist permutation matrices P and 0 such that

(2.7)

(2.8)

~lnce. tor moaels aescrlbea by (Z.J), the observabiIity of a given mo­del i:; equivalent to the cont.-ollobility of it:; dual, :;tru,-tUlal ob:>e,-Yabil iLy \..all be

rhprkprl with thp hplp nf ~imil~r rnnrlitinn~

The existing methods can be arranged into two main categories the !,.-a(Jh-LheureLI\.. (I(J(Jr-u(l\..h (III'" Lhe (ll!jelw(I i\.. (I(JIWU(I\..h.

32

Ih1S approacn, due to Lin [L3, L4) , seems attractive when studying the :ltruetural propertie!l of tran!lfonnotion systems, given the simila,-Hies e ... isLiu!I

Ilptwppn tllp twn tyl'P<: nf rpl'rp~pnhtion.

Lin considers the structured matrix M = [A I Bland the associated oriented graph, which is constructed as follows: to any column of M corresponds a node of the graph, and to any nonzero entry mij of M corresponds an oriented edge trom node j to node i. The nodes associated with the columns of B are thus the sour­ce:; 0 r Lhe !I, ·o.pll.

Lin dofino. two particular typo. o.f graph. and cubgraphc, which aro

easilY shown to be associated with structurallY controllable models : the stems (Figure 2.6) and the buds (Figure 2.7).

Fig. 2.6 A stcm.

Fig. 2.7 A bud.

Tile g'o.p" e 0.5501 .. io.Lecl with 0. po.i' (A,.!:.l i:; a ~ if, and only if

(i) It can bo writt~n

(2.9)

where T iQ 0 stem origin~ting from the node accociatod with ~, and whor~ tho B,

(i .. 1.2 .... fl are buds. (ii) The origin of the jth bud is also the origin of an edge in the

subgraph S such that

33

(Z.lO)

Th" !J .. aph co .... "~pond;n!J to a pai .. (A, B) ;~ a "cacti" (I) if, anti only

if, it is constituted of one or several disjointed "cactus" (Figure 2.8). Lin has shown that a pair (A, B) is structurally controllable if, and only if, its graph is spanned by a "cacti", that is to say if it can be transformed into a "cacti" by de­leting existing edges whithout making any node inaccessible from the sources associa­L~<l wiLh Lh~ ~U1UIIIII" ur B.

Fig. 2.8 Example of "cacti" .

Con~ido .. again tho ~t .. uctu .. o dofinod by ~igu .. o 2.Gb, but cuppoco now

that the nonzero entry of Q is free. M is then Qiven bv

o u

M a?l o o o (2.11 )

"31 o o o

The associated graph is described by Figure 2.9

34

3

4

Fig. 2.9 Graph associated with Figure 2.5b.

On!! cannot extract a "cacti" from it without isolating node 2 or node 3. Thus this structure is not controllable.

The mo in odvontnge of thb method i:s to ffiQke it pO~3ib le to check 3;­

multanooucly that typo 1 and typo 2 condit;on~ ar" not m"t Th" _thnt! nf J:'r~nk~"n

et a 1. [F3] • whi ch tests the connectabil itv of the state to the inputs and outputs, is simpler to use. but less general since it assumes the model structure to be such that

M.:IX [ Rank [ M J] = n

a

Note ttlat l,;onulLlulI (t.lto) i~ ~ClLbrit:u ir

Max [ Rank [ AC~))] = n e

12.12a)

(2.12b)

Such is the case for all compartmental models which contain no closed subsystem. i.e. no subsystem receiving elements without being able to clear them to the outside (sink of [Zl) , true endotropic grapb of [D2] ).

Structural obcQrvability and dructural controllahility ~r" ~lgphr~;~

properties. and it seems reasonable to try to check them in a purely algebraic way. Shields and Pearson have formulated a method [S4] which enables one to

find simply, by block triangularization. the generic rank of a matrix, the structure of which is defined only by the position of the zero entries. (hiS methoa allows us to check the unul>:.e,·vCll>ill Ly (u,· Lht: UlIl;.ollt.ollobil ity) originoting from Q type 1

condition. Tho algorithm. for any matri~ M of dimon~ion (nvm). with n ~ m is as fol­lows :

Step 2

:,tep j

StI,p 4

Step 5

Step 6

Step 7

35

SeL M(O) = M , J(O) = 0 , v = 1 .

Find itO) the number of rows of M(O) entirely composed of ze,-o:;. Form M(l) by ,-emoving the:;e i(O) ze'-o rows f,-om M(O).

:,et J(V) equal to tne m1n1mum numoer OT nonzero entr1es 1n any row of H(v). Set ;(v) equal to the maximum numbe,- of

structurally "'l,,;vall>nt rowe: nf M(v) havin!] j(v) nnn71>ro

entries.

lJc;p row and rnlumn PlIrhanopc:: tn Pllt M(v) intn thp fnrm

iCY)

j(v) -t [-~--~-~-l

M1 I M" I ~

Set v = v + 1.

If M2 has more than one row and one column, set M(v) = M2 and go to Step 3. Otherwise go to Step 7.

Successive permutations of rows and of columns have now put M 1nto tne Torm :

0-------------------- v

(lJ------ ---- ------ V

where Rk i3 of dimen:;ion (i(~) x j(k».

The generic rank p of M is given by q

p [M] = n-max [~ [ i (k) - j (k)] • 0] • g bO

O,q<v

(2.13)

36

L~L U~ 0",,1.1 thh ,"ethoq to o;;heo;;k the generio;; rank of the matril< [AID) of tho oxampl0 do~cribQd by ~iourQ 2 _1

The method yields the following matrix, after exchanglng rows ana COlumns

j(l) j(2) j(3) j(4)

:"'--'141 ·1.....--.' ......

a44 I 0 0 1 0 ,0 I I, ----,----------'----r---t o I a32 a33 I 0 '0 I I I

-------------------~----

i (1)

i(2)

1(3) 1 I I' o 1 <122 0 I 0 21 I 0 , I I ----,---------'----r---

~ a34 lOaD I an I b1 ;(4)

The generic rank of this matrix is thus

4 - max [0. -1. - 1. n. n) = 4_

If the corresponding model is not structurally controllable, it can only result from a type 2 condition. Shields and Pearson [ S4] have proposed an alge­braic method which also enables one to check conditions of this type. However the result 1S more aHf1cu1t to obtaIn tll<ln wi Lh Lh~ ""'Lhuo.l o.Ieveloped by Glover and

Silverman [C2) , who con~ider the controllability matrix

(2.14)

and find the position of the zero ~ntrie~ which define its structure. They use for this purpose the Boolean matrices A and B such that

a;j if aij is free,

aij 0 H d ij = 0 ,

and the Boolean scalar product

37

AT A ~ ~ l. ~ ("1 ClIIoJ YI)or("2 OlloJ .12) or· ••• or· ("n am! Yn> ,

from which the Boolean product of matrices is easily deduced. Structural uncontrol­laOl1lty caused oy a type l condltlon WIll result In at least one zero row In

A fl- A - - -

Mr: = [B I A * B 1 ···1 ~ ~ . .:2 ~ * B] . n-I

(2.IS)

The amount of computation can be reduced by using a recursive procedure operatinQ on a matrix of reduced dimension.

Applyin9 tllh metllod to tile example de:)cribed in rigul'e 2.1 lead:) to

1

o

o o

1

o o

1

1

1

o

1

1

o

Thus the model is not structurally controllable, and this originates from a type Z <.olloJi Lioll.

2.3.1.3 ~Q~£l~~!Q~

The point to stress is that easily operated algorithms exist which ena­ble structural observability and structural controllability of linear models to be checked in so far as the nonzero entries of the matrices A, Band C are free parame­ters. unfortunately tnls IS not always tne case. ~ome nonzero parameters may Oe known a priori, other~ may be dependent: for example when one ~tudie~ nonlinear chemical

kinetics with the help of the formalism of transformation systems (see Figure 1.5) or when a class does not have any flow to the outside, forcing the sum of the entries of the corresponding column of A to be zero. It is thus necessary to design "ad hoc" methods which account for the possible unobservability or uncontrollability resulting trom tne relatlonsnlps eXlstlng oetween parameters.

l.J.l Nonzero entrIes are dependent

The structure of the model is no longer defined by the position of the zero entries. It is now defined by the parametrization of the triplet.

ACe), B(~'>. CW (2.16)

3ul-'l-'0::'~' ror' ::.lolll-'ll<.ILy. ~CI<.II ~IItry of A, C amI C to Oe a ~olynomlal functIon of We

38

PQrQmeter~.

ThQ modo1 will bo ~truetura11y eontro11ab1o if, and only if, the con

trollability matrix

MC [ B 1 A B I... 1 An- 1 Il] (Z.ll)

is generically of full rank

(2.IS)

Similarly the model will be "'~ru,,~ur~a11'y ulJ::.t:nalJ1t: Ir, dll<.J ull1y If, tilt: ull:oervalll11-

ty matril<

M ~ o

c C A

(2.19)

is generically of full rank

(2.20)

Let E be the set of all the admissible values of a such that the con­trollability (or observability) matrix is singular

(2~21)

where

(l.ll)

the followinQ procedure can be used to check in a practical way the structural con­trollability (or structural observability) of the triplet (A, B, C) :

Step 1 , Define on admi:.:5ib1e parameter~ :5ub:5et e ' :5u<;;h th(l~

(2.23)

StPIl? I:hnnsp rllnclnm1y 1I V;r111P Al for PlIrh Il;rrilmptpr AI. SlJllllnsino Al

uniformly distributed in [aim' aiM].

Step J , Te:5~ the ~ont'~o! lobil ity (01" ob:.e.-vabil ity) of the t,-ip1 et

(A(~). R(~). r.(~».

In order to understand the reasoning behind this algorithm, assume that

39

the fi r ~ t (v-1) 0 i have been chu5en. The equa L iUIi

det [ M(~) ] = 0

thQn rQ~ult~ in a polynomial oquation of finito dogroo in Qv' which corrocpondc to

one of the polynomials ~i(~) introduced in Section 2.2.1. when Qenericitv has been defined:

n ~j tv (~J - E f j (ol'" ,ov_l) - o •

j=O v (2.24)

Two cases must be considered

1) 3j f j (OI ' ... , °v-l) t- o (Z.Z5)

Equation (2.24) then has ~ finite number of roots, and the probability of hitting on one of then when picking 6v is zero. Thus one will almost surely detect the structu­rdl controllablll:y (or.struc:ural observaDl11ty) Of tne model. Note tnat if tne nu­merical model (A(~), B(~), c(~» i5 fo.und co.ntro.llable (o.r ob3ervoble) , the 0330eio­

ted structural property ic; then r:ert.~in.

2) f. (6 l , .•. ,6 1) = 0 Vj. J v-

(2.26)

This may occur for two reasons :

2a) As the result of.a coincidence in the partic~lar values taken by 61 ..• 6 l' One can then apply to 6 1 the reasoning used for a . Going back step by v- • v- v step to 61 one can tnus snow tnat tnis case almost surely Wll I not nappen.

2b) Because the model is structurally uncontrollable (or unobservable).

Note that checking the rank of a given matrix is a numerically diffi­cult task, which deserves special attention [LIO, B2] . Quite often, a matrix which is theoretically degenerate will be, strictly speaking, of full rank, because of the numerlcal errors lntroduced during the computation of its entries. Inversely, becau-3e the eomputoti en i3 cor"; ed out with 0. fi nite numbel 0 r d i 9 i L5, Llle IJI"uiJaiJ i 1 iLy U r

~ nlll1lPrir~l1y dnOllhr matriy addn!) accidontally i~ not :zoro.

The 5ingular value deeo.mpo.:dtio.n metho.d [GJ 1 i3 to. be ,'ecOJII1Iended. It

enables one to write any mat.rix M in t.hp form

M u y (2.27)

40

with

(2.2B)

U and V are unitary matrices. The 0i are the nonzero singular values, the number of which is equal to the rank of M. In practice some of the singular values which theo­retically would be zero may happen to be small but nonzero. The problem is thus to u\::l.iu\:: wIJIl;1J ~Inyular values are taken as zero. Tecnnlques ot numerlcal analysiS are now available to estimate fairly precisely the number of 3ignificant digits of a nu­

merical result. and to know if it r.~n hp rli~tinOlli~hprl from 2pro. takina into account

the precision of the computer used [VI 1 . One of the interesting properties of the singular-value decomposition,

in addition to its great numerical stability, comes from the very concrete meaning of tne nonzero slngular values. Let M be the matrix obtained by setting to zero the r-r 3malle3t 3;ngular valuc3 ;

0 1 , , \ ,

M U 0 v . '" r

ttl

M is the matrix of rank ~ which is the best approximation of M in th~ ~~nse of thp

criterion j, defined by

haec [ ~ Tel ,

M - M , ('.3UJ

and it is easy to show that j is equal to the sum of the squares of the discarded ::.lllyular valut!s.

r .i I (2.31)

k:~+1

One can thus relate the approximate rank estimated in this wav to the entries of the matrix M. In case of ambiguity, one may be led to pick several 6 in order to be sure of the structural nature of the result obtained.

Another. and seeminolv more straiohtforward. approach would be to use an algebraic programming facility such as REDUCE [H11] to compute an analytical ex­pression of det [M(~) 1 as a function of ! . However this can be extremely expensive in terms of required CPU time and memory, even for rather simple examples. This is

41

wily d (;dn:rul lIulllt:";o.al al'l.".uao.h b !>Lill or intere!>t.

Z.4 MAIN APPROACI~S TO STRUCTURAL IDCNTlrIAOILITY

Before examining the various methods available in the literature for checking structural identifiability, we first present two modelling approaches that avoHI the problem. We ~huw theIr cUIlt.rlbuLlulI~ clfnl 1I111ILdllulI::'.

2.4.1 Identifiable canonical representations

Two models described by (2.3) are said to be equivalent if, and only if, there exists a state-space similarity transformation which changes one into the other. A set of construction rules makes it possible to associate a unique (canonical) re­presentat10n w1th any equIvalence class (or orbIt). CharacterIstIc parameters (orbI­tal inyo~ionto) eo~reopond to thic cononico1 ~ep~ecentotion. Any complete cot of or­

bit~l invariants completely characterizes the associated equivalence class. The idea of considering a model as identified when one has estimated a complete set of orbital invariants follows from this. A model will then be completely identifiable [S3] from an input-output trajectory if, and only if, one can associate with this trajectory only one equivalence Class, characterizea by ltS orbltal lnvarlants. My uSlng sUltable c(monical !>tl' uctul'e" [DC, S3, OZ, Tl, T£], model:> can be obtained, the identifiabili­

ty of which dop"nd~ only on th" natll~" of appli"rl inrI/to ~nrl r .. <:I/1tinO nl/trlllto_

Such a black-box-type modellinq, althouqh attractive for control pur­poses, is not relevant here. Indeed the physical meaning we attach to the state im­plies that we choose a particular representation (which may be called a-canonical) incorporating our hypotheses and knowledge about the nature of the interactions bet­wt:t:1/ Lht: :.LaLt: o.lJI"I'Ullt:IIL", Lht: itll'uL:. a 1111 Lht: uuLl'ub. NoLt: that, he ... : too, the ~

po£toriori idontifiability of tho modol (i.o. whon actual data aro takon into account)

will iaraelv depend on the experimental procedure (shapes of the inputs, measurement scheduling, choice of the sensors ... ). This is why research on experiment design ma­ximizing a posteriori identifiability under realistic hypotheses is so important [M2, L9 J •

2.4.2 Global optimi zati on

The aim of olobal~optimization methods is to find the values of the pa­rameters corresponding to the minimal value of the criterion j, by exploring the who­le admissible parameter space 9. Their use as identification methods should thus ena­ble one to find all the possible models and to solve the identifiability problem sImultaneously.

42

In all co~e~. one lou~~ ru,' 0 ~uch ~llc~

j (~J ~ j (~) V! € 0cIRV

Existing algorithms riln hI" r:liI<:<:ifipti into four .. atp!Jor;Q~ (13 1 _

(;) Sp<>ce- eovcr; n9 teehn; qUC3 [ E1] •

(2.32)

(i i 1 Procedures searchi ng for thE' rF'gi nno; nf ilt.t_rilrti on of t.hp vari ou~

minima [C5) • (i i i) Trajectory methods [B3] • (iv) Random-search algorithms [G4]

We shall limit ourselves here to presenting the principle of one trajec­tory method. Indeed this approach is not restricted to problems with a very small nu­mber of parameters as are approacnes (1) ana (11). MOreover. un11Ke approacn (lV). It pcrmit~ U~ to draw <>n intcrc~tin9 parallel with condition3 for local identifiability

to be pr@s@nt@d in thp npxt r:hil~tpr_

The procedure consists in defining. by means of a differential equation, some trajectories passing through some (if possible all) stationary points of the criterion. Among the various existing methods, the first to appear, and which is also the more commonly used, is due to Branin lB3 1 .

Let .[(~) be the gradient ur the c,ite, ' iun with 't::~I't::\..L Lu Llrt:: I'C'-Clllt::­

tore, which ie euppocod calculablo_

(2.33)

The method consists in tracing, in the parameter space, trajectories along which the yrdtlltmL ~ hd~ cI ylven IIlre(;LtulI ~(~o) ;

(2.34)

A stationary point of the criterion will be reached whenever f(t) tend to zero. A trajectory satisfying (2.34) can be obtained by solving

d + +t -.9. = - .Q. ~ Q(tl = e - .2.(0) -dt

Let G be the Hessian of the criterion

(?-~~)

(2.36)

43

one has

.!..,a= G .!..e. dt dt -

(2.37)

Taking (2.36) into account, the equation of the trajectory. at each point wh"r" th"

Hessian is invertible, is then:

d +-1 -8= -G n dt - 4

(2.38)

Th;c: "l1ullt;nn ;c: v .. ry c:;m;hr tn th .. nn .. ct .. c;rrih;no thl> r,~IJ<;c;-Nl>wton illoorithm for

the local minimization of a criterion j by using a second-order Taylor's series ex­pansion. The sign changes, in the right-hand side of (2.38), enable one to leave the obtained extrema in order to continue the exploration.

If there exist some pOints in the parameter space at wnlcn tne Hesslan bcc;omc:s :sin9u1,,,", onc c;an ,"cplac;c (C.:JO) by

-h- 8 = ! adj [ G J .lIo (2.39)

but an extraneous s1ngulilr1ty may then appear 1n ! suet, thdL

~ .ll(~);. Q, ~ adj [ G(~) ] .ll(~) = Q. (2.40)

Br·o.nin I,a~ "'ade Lht: rollowin9 l.onjt:l.tun, ; in the ab5enc;e of any cxtrancou:s :sin9ula­

rity, all th" traj .. ctori ... d .. fin .. d by (2 . 3B) Dr (2.3Q) pace throlloh all th .. c:htin­

nary points and hence the method finds all the global extrema of the criterion. This conjecture is still open [T3] . The absence of any extraneous singularity results in

Rilnk [G(E,)] - " v E, c 0 c /TP • (2.41 )

this condition is formally analogous to the one expressing the 10C~1 identifiability of a model, to be presented in Chapter 3.

The implementation of Branin's method raises critical problems, and, snOUld ~ranln's conjecture be proved, one WOUld neVertneless Tlnd praCtlcal SltUa­tion:> whcre the method would fail to locate the !lobal minimum [116] •

Whatever the global opt1m1zat1on method used, lt requlres a great nu­mber of computationc of the criterion. The re&ult i. obtained for particular valuet

of the data. and it is difficult to infer from it anv conclusion of a strutural na­ture. In particular it is impossible to know the influence of a given supplementary constraint on the model structure (or of a given additional experiment) on the number

44

of 910bal minima without havin9 to ,-cpcaL all Lilt: 1..(lIIII'uL"UUII:>, whil..l, an: "IIL,.."""ly

timo-coMumina_ Thp pYhal.l~tivp mod"l lina approach. to bel pr,,£ontod in Chaptor ~. hat

the same ambition of finding all the global minima, but from a structural viewpoint, so that it becomes far easier to study the consequences of any modification of the parametrization.

As far as we know, researchers in compartmental analysis seem to have been the first to ask themselves to what extent it was possible to estimate the para­meters of a given state-space model from experlmental data [HI] • It has been known for a long time tho.t if one did not ob3ene the evolution of the o.ctivity in eoch

compart"mont (i _0_ tho comp1oto £yctom .tato). on" ha~ to introttucp ~omo ~trlll-tlll"a1

constraints to avoid difficulties in determining the model. But the credit for having clearly stated the problem while presenting some elements of a solution probably be­longs to Berman and Schoenfeld [84] .

Consider an n-compartment model, with distinct eigenvalues Ai (i= I, l, ... ,n), descrllJel.l lJy

(2.42)

Thp activity of tho ith compal"tmont .ati.fio~

n A.t I;" I" ~. J ~ lJ ~ j=l

(7.43)

Tf th" init;~l ronrl;t;on~ !u ~r" known. thp matrix R of thp DrpPXDonpntial factors rij have only n(n-1) independent parameters since one has

II L r iJ· = xi(O) . J=1

(2.44)

the eigenvalues Ai of A and the independent entries of R thus form a set of n2 para­meters, i.e. as many as there are entries in A. If the observation of one compartment permits the estimation of (2n-1) parameters (i.e. the eigenvalues Aj and the associa­ted preexponent 1 d1 fdctur-:» Lire UlJ~"I"VC1 L iUII (I r u'"'' 1II,)I'e compo,-tmcnt pCl"mit3 the e3-

timati on of on 1y (n-l) now paramotol" •• d nco tho pi a"nva 1 LI"~ a"l> ("olTlMn _ Tt i ~ thll~

possible to calculate the number of deqrees of freedom. as the difference between n2

and the number of the independent parameters which are present in the observed out-puts. For p observed compartments, • is given by

45

(2n-1) (n 1) (p 1) (n p) (n 1). (2.45)

It is easily shown that the state matrix A is related to the preexponential factors by

A1 Q) \ \

A R \ -1 \ R , \

\

\ (() 1n

and that any similarity transformation T such that

A' _ T A T- I

modifies the preexponential factors as follows

R' = T R

(2.46)

(2.47)

(2.48)

Berman and Schoenfeld call a generat1ng model any model A wh1ch ls cons1stent wHn the d~t~. Onee ~ gener~ting model i~ obtained, for example by fixing arbitrarily the

entries of R associated with the comoartments unobserved. the outout-indistinOlli~h~hl others can be deduced by similarity transformations, the matrix T being restricted by any constraint on the initial conditions or on the values of some entries of R and R' that may exist, through (2.48). If some entries of A are known a priori ,(2.46) Wlll De used to reduce the numDer Of degrees Of freedom Of I.

Among 011 the gene,-oted model", "OIne - and in po,-ti cul or the gene,oti n9

motipl-may hp nnn-rnml'~rtmpnt;ol Thpy h;ovp thpn tn hp tiic:r;ortipti c:n th~t nnly t.hp mo­

dels which are consistent with the data and which comply with the compartmental cons­traints are retained.

The main idea of this approach, which we shall reintroduce in Chapter 5 in a multi variable perspective, is to generate, in case of indeterminacy, the set of all Li,,, vuLl'uL-;mJbUII!lubloa"l" IIIv\l"l:;, ;II:;t"all or ;II'I'V:;;II!I lllU'" vr 1,,:;:; or"il,a,-y

constraints in ordQr to insurQ thQ uniquQnQss of thQ modQl_

2.4.4 Transfer function approach

It io ~ccumed th~t the oboerv~tion of inputc ~nd outputo h~ve permitted

unioue determination of the transfer matrix H(s) of the real system. The identifiable parameters will then be those which can be uniquely deduced from the parameters of the transfer matrix.

46

Ii ~tJe model ~tructure 1s dei1ned by the parametr1zat10n oi

~: = A(a) ~ + B(l) y. ~(O) Q .

= C(Q) x ... D(~) .!:!.. (2.49)

the transfer matrix is related to the model parameters by

M(~,.E.) - C(~) [ ~ 11 - A(~») -1 B(~)' D(~). (2.50)

the model will be s.g.i. if, and only if,

M(l·s) = M(l.s) ~.fr. = .!l.. (2.51) o

Bellman and Astrom lin] were the first to use this technique tor stu-dying the identifiability of linear modeh. The many outhol-" who followed include

H£jak [HB] • Barnt~an 9t a1. [B5] • Glovar at aL[G5) • Mi1anll~1l at a1. [M3] •

Cobelli et a1. [Cl, c21 and DiStefano [051. Whenever there are few exchanges between classes, it is easy to test a

for identifiability with the help of condition (2.51). The method becomes more awk­ward if the structure of the model is more complex (in such a case the method to be presented 1 n Chapter 5 orten leads to s1mpler compuLCItlurl~).

It is useful, at this point, to describe further the method proposed by Cobell i et a1. , and the discussion to which it gave rise [Z2, 07. C4] • for. while the method is Simple, easy to implement on a computer. ana frequently usea, It 1I1CI1 Uti rur·~utla ~t:l, I ead to er roneou~ re~Ul t~.

MethOd prInCIple If we C\33ume thot 0, always identifiable, has been diceard9d, the tranc­

fllr matriy r~n hp writtpn

M(s) 1 n 1 n-Z

---- [C B (s - + a S + ••• + an-I) de ~ ( ~11 - A) 1

... CAB «n-2 + ~1 ~n-3 + .+ ~n-z)

+ ...

+ C An- 1 B 1 • (2.52)

det [511 - A 1 (Z.5J)

47

Suppose that the model has m 1nputs and p outputs. M(s) then has mp entr1es mij w1th the ~ame denominator. Identifying the tran~fer function~ mij(c) will thu~ enable one

to determine the n parameters common to the denominators plus the parameters of the numerators (i.e. n parameters for the m .. of which the degree of the numerator is

lJ (n-l), (n-l) parameters for the transfer functions of which the numerator is of de-gree (n-2) (which corresponds to CABl ij ~ 0 with CBl ij =0) and so forth). It is thus posslDle to evaluate the numDer ot parameters that can De aeaucea trom the Knowleage of H(3). Cobelli prop03ed checking the 3tructural identifiability by comparing the

nllmhpr of ll:lr"mPtpr~ of M(~) with thp nllmhpr " of IIn!cnown Il"r"mptpr~ "Illlp"ring in thp

compartmental model.

Main objections Zazworsky et a 1.[ Z2] and then Del forge [07] have poi nted out the dan­

ger of such reasoning : the parameters appearing in the expression of M(s) are non­linear functions of the compartmental parameters, and having at least as many equa­tions as there are unknowns by no means guarantees that the soJutlon wllJ De unlque (and thU3 the (;ompa,-tmenta 1 pa,-ameter3 ; dent; fi abl e). It i 3 only 0 ne(;e330ry (;ondi­

tion for local idontifhbility_ It may bo not"d- that it ic: ",ac:y to built! " non" !J i

model which still satisfies Cobelli's condition, as is the case, for example, for the model described by Figure 2.10.

FiO_ 1'_10 Non ~-O-L mndpl c:;"tic:;fying r.ohplli'" f'ont!ition_

Its transfer function is

M(s) (2.54)

It depend3 on three pa,-amcter3, a3 many a3 thc,-c a'-c (;ompa,-tmcntol pa,-ametc'-3. Out it

ic; PIIc;ily c;hown (c;pp r.hlllltpr 4) thllt <:Uf'h 11 mndpl ic: only 10f'"lly it!pntifbhlp :lnt!

that there are two models having exactly the same input-output behavior. Later Cobel­li et al. [C4] have defined structural identifiability, in a way which we think not restrictive enough, as a property of a model which is almost everywhere locally iden-

48

tirio"'h:. Ev.,1l "ith Lhi:> lI.,rirriLiuJI. Lht: I...-U",U:.t:lI IIIt:LhulI ",,·uvidt::. unly CI nt:c:t::':'Clry condition ~or ~tructural idonti~iability_ Sinco como con~ucion ~till ~oom~ to ramain on this subject r C6, G61 , it is probably best to Qive a very simple example of a non s.t.i. model which satisfies Cobelli's condition. Consider a two-compartment model on which two distinct experiments can be performed: first one observes the response of compartment 1 to an injection in compartment 1 (Figure 2.11). and then the response uf c:ompClrtment l to Cln 1nJec:t10n 1n c:ompClrtment l (F1gure l.ll).

Fig. 2.11.

rig. 2.12.

One shows easily (see Chapter 4) that this second experiment does not bring any tur­ther 1nforlllCltlon wlth rt::.",t:c:L Lu Lht: riDL Ullt: a 1111 LilaL Lilt: lIIuolt:l i:> nut. !I.ot. i. N.,­vcrthclc~~. when applying the method propocod by Cobolli et al. ono con~idor~ tran~­fl"r functions

mu (s) 1

[s + a1 + aU]' s2 + al S + a2

(2.55)

mll(~) 1

[ s + "1 + all]· s2 + a1S + a2

(2.56)

which actually depend on four different parameters, as does the initial compartmental mooel. From tnat consl0eratl0n one may wrongly deduc:t: thdL Lht: lItullt:l J::, 11ItmLiria"'1.,.

To conclude this discussion. the method proposed by ~ODel 11 et al. ulily ",enIlIL:. Ullt: Lu klluw if Q n., .. .,:.:)(U', o.ondition for ~ identifiQbility i3 met.

49

It con only be u::>ed to prove thot 0 given model i::> not ~.l..i., while the methode to

h~ rr~t~nt~rl in ChaptQr J allow ono to know wh~thpr thp ~tl~;prl marlp' i~ locally iden­tifiable.

Tn arrlpr tn ~ tudy <; _ g _ ; dpnti fi ~bi lity with thp tr~nsfer functi on appro­ach, one cannot merely consider the number of the parameters appearing in the trans­fer matrix. One also needs to consider the analytical expression of the transfer ma­trix as a function of the parameters to be identified. This expression is rather easy to obtain when the matrix A is sparse. When A is not sparse, calculations are much 1fIU'"~ '.1Irrfc.;ul~, whlc.;h IIIdl<.~~ OilY "vu~llI~ o~~"dl-LiY~ ~hdL I-VllIput.es t.he ollolyt.i~ol e><-

procci on of tho tranl:for matri x ucoci atod with a 9i von ctato oquati on [ (6) Not<!-

however that a systematic study of the nonlinear equations relating the identifiable parameters of M(s) to those of the compartmental model remains to be carried out.

2.4.5 Minimal representation approach

It is well known [Kl] that all minimal state-space representations equivalent from an input-output point of view can be deduced one from another by a state-space similarity transformation T. Glo~er and Willems [G5, GI] have proposed studyi ng the number of the so I uti ons for (I ,~) of

A(~) T

B (.§.)

r (!:) C (!:) T

D(~) - D(.£.)

(2.57)

Indeed, if this set of equations has a locally (globally) unique solution (t ,~), the corresponding model will be locally (globally) identifiable. Tse and Weinert l T4, T5] IodV~ oJ~v~lup~oJ d ~ illlild'" dPIJl"VOc.;ll Lv ~LuoJy ~Io~ ioJ~IIL i ridlJil i Ly vr LIo~ pd"OIIl~L~'"~ ur d

stochastic linear model and of the associated stationary Kalman filter. Glover and

Willems have obtained very eleoant - but difficult to implement - sufficient condi­tions for local or global identifiability. Pohjanpalo and Wahlstrom [P2] have applied this idea by taking advantage of the particular structure of a matrix C commonly en­countered in compartmental modelling, so as to reduce the number of parameters in the prOblem. A slm1lar 1dea wll1 be used 1n a more general framework 1n Chapter ~ on ex­houstive modelling.

2.4.6 Local approaches

Checking ~.g. identifiobility presents some intrinsic difficulties sum-

50

RlCl.-iotcd in (07) • whil .. I, IIC1Ve lIIuLlvdLeu IIUlllerous studles on local ldentlf1aOl11ty [ts~, T6, G7, W4, R2, V2, M6) _ Indood local idontifiability ic far cimplcr to check and i3

a necessary condition for Qlobal identifiability. Among the approaches employed. one may mention: the implicit function theorem [GIl. the local observability of an ex­tended state [A3. B5 1 • the stability of a second order identification algorithm (W4. L5] , and the properties of Fischer's information matrix [F4. T6, R3 ]. We shall de­ta1l these approaChes and thelr relatl0nsnlps 1n tne next cnapter.

~_4_7 Powpr ~pr;p~ ~rrrn~rh

Pohjanpalo [P3] has considered models described by

~ x(t) - !J!!.(t) .~(t) .t •. £), ,!!.(O) - ~(.£.) • t c [ O.T) •

~ l.(~. t) - ~(~( L) .~) • (? IlR)

He has proposed check!ng their global identifiability by studying the possibility of determining uniquely! from the experimental knowledge of each term of the Taylor se­r1es expanSlon of l(t) around t = 0 , 1.e. from

1 im t + D k=D, ••• ,oo • (2.59)

A sufficient condition for global identifiability is that the set of equations

1 im t + U k=O, ___ .= • (2_60)

hac:: a IIniC'JIIP c::nllltinn fl = A

We 5hall rC5trict our5clvc5 hcre to thc lincar ca5C. thc nonlinear onc

being deferred to Chapter 7, It is easy to demonstrate that for a linear model des­cribed by

A(~J ~ + B(~) u , ~(D_) = Q •

C(~J ~

of which the impulse response matrix

Y(t) ~ C(O) eA(!) t 8(0) - -

(2.61)

(2.62)

51

i:. :>uI'I'v:>t:d Lu lit: vll"ened, thb identiriollil it.Y I..unditiun b ohu ne(.e""or'y • Indeed

for ovory pocitivo finito T, ono hac

V (t) I r A" R t" '"

\I t,,[n.T) (1_61)

AllY IlIrUf"lIIOLlulI UII r(L) I:. Lhu:. ~UIILilll1~d III tts successIve derlvattves, evaluated at time t - 0, and a necessary and sufficient condition for a model defined by (2.61)

to be s.o.i. is that. for almost any ~ belonoino to the admissible parameter set A

one has

~ 0 ~ g-I . k -= C(~) A (~) B(~) , "=0,1 ... ,m (2.64)

ThIs condItIon has already been stated by Grewal and Glover [G7 ).

Note that it is out of the question to use the successive derivatives vr Lht: "y"Lt:II' vuLI'uL" wIU, ,-t:"I't:~L Lv Llmt: If! unl.,,- Lv ~"Lh"oLt: .!. Tht: I'~-v~t:du,-t:

followod 1'101"0 concictc in docign!ng an imaginary mothod for idontifying!_ If thic

method enables one to determine ~ = ~ uniauelv from the data aenerated bv a model with parameters !. global identifiability is proved. Other sufficient conditions for global identifiability might be developed along the same lines, particularly for non­linear models, as will be seen in Chapter 7.

~.4.a Identifiability of large 3cale linear mode13

The complexity of chec"ing s.g. identifiability increases extremely quic"ly with the order of the model considered. That is why specific methods have 1It:t:1I dt:yt:lvI't:d ru,- la'-!;It:-"I..olt: IllUdt:b. A fi,-"L Ol'I',-uCll..h [61, C7) b Lv I"-t::>t:IIL CI

cyctomatic ctvdy of como commonly uced model ctructurec, the global identifiability

of which is easv to test with the help of the associated transfer functions. So for mammillary or catenary compartmental models (or for particular combinations of these structures) one can "now if the model is s.g.i. or s.t.i., and, in the latter case, the number of local models.

A second approach consists in dividing a large-scale model into smaller parts, the identifiability of which is easier to check. Of course this is only possi­ble for particulol- model :>tl-uctul-e". ro,- e"omple We hoye "hQwn [W5)thot if there

pxic:tc: II ornllll r, nf cnmllll\"tm.;.ntc: ... 11 outflow .. of which ar .. (tirpct .. t'I towart'l .. tn .. ~amo

compartment (Figure 2.13) then one can build a partial model which only involves the

52

compQl"tment3 cxtcl-nal to G. Thc ~U"''''I · t:~'t:U !jJ·uuIJ I, a(;LIJI~ unly through the t1me de­

riv~tivp nf it~ tnt~l activity 0_ tno paramatarc of tna partial model are eacily de­

duced from those of the complete model (Fiqure 2.14). As far as 0 is observed. it can be considered as an input (known but not chosen) of the partial model, the identifia­bility of which is much easier to check than that of the original model.

L u. iCC 1

Fig. 2.13.

>;9- 2_14_

- b

r u_ ieG'

Pohjanpalo and Wahlstrom [P2j have studied models havino the structure

described by Figure 2.15, where aij and aji are assumed S.9.;.

53

Fig. 2.15.

(2.65 )

It can bo calculatod from experimentally mcacurcd quantiticc, and thuc concidcrcd oc

an input which is known but not chosen. This enables one to "break" the existino links between i and j, resulting in the scheme of Figure 2.16.

Fig. 2.16.

When the structure of the model is adequate, one can in this way divide it into par­tial models, the identifiability of which is then easier to study.

Note that if one cannot prove by a preliminary study that aij and aji

are S.9.i. , one can nevertheless consider xi and Xj as known inputs and get the sche­me of Figure 2.17.

54

"j----..., Ie---'::--- xi

Fig. 2.17.

In this case entries of A will app!!lIr in thp cnntrnl miltrix R nf thp moripl Slirh an example will be treated in Section 4.4.2.

One must however be somewhat cautibUs when usino methods of this type. Indeed it is well known that the identification of an artificially isolated subsystem may prove to be impossible if the whole system forms a closed loop [GS]. A systematic study of the effect of the feedback loops of the whole system on the identifiability Of the parameters Of a transformat10n SUbSYStem rema1ns to be carr1eo OUt. Moreover. although thc5c tcchniquc5 cnable one to reduce the volume of computations considera

hly. thpy nnly ilpply tn pilrticuhir ~trllctllrp~_ Rp<;irip<; nnp milY hp <;1I~pirinll<; lIhnllt the usefulness of methods for checking the structural identifiability of very-large­scale transformation systems. Indeed the actual identification of the parameters of such models remains problematic. even if the corresponding structure can be proved to be theoretically identifiable. It is well known. for example. that it is unreasonable to e:)timate the pa,-amete,-:) of a multiexponential model with mo,-e than three exponen­

tial to .. m~ from tho data corro~pondin9 to only on" impuho ro~pon~o (L6. M4. DR. B14.

J3 ]. and this althouqh all these parameters are s.q.i ..

2.5 CONCLUSION

Thi~ chaptor ha~ onablod u~ to ""plain tho notion of' " ~trllctllral pro­

perty. to define those we shall be concerned with. and to underline the relationships they may have with one another. It has also attempted to give a comprehensive view of the results of the literature. which makes it possible to set apart the open problems

55

We now have a set of easy-to-use methods at our disposal for checking, in any case of practical interest, whether d 1 illeor" IlIU"'el i:. l.ulII.t:I.Ldble, :.lruI.Lurdl­

ly oh~" .. vahl" 0" ~t .. uctl ... ally cont .. ollabl ... W .. had fo .. that pu .. poc .. to p .... cont a

heuristic approach for studyinq structural observabilitv and structural controllabi­lity which is able to take into account possible structural constraints on the non­zero entries of the matrices A, Band C. Indeed such constraints appear frequently when one studies transformation systems, and this makes useless any method which sup­IN:.e:. L1.e lIulILeru ellLrfe:. uf A, 6 and c to oe free.

The situation is far from being so advanced for structural identifiabi­lity. Numerous methods are available, all'" eol." uf t"e.1I Llwuw:. CI <.llffer"enl 1191.1. un

thi~ notion. but non .. of th .. m ean p .... t .. nd to colvo all tho p .. oblem~. The main diffi­

culty results from the nonlinear character of the deoendence between the model out­puts and the parameters. In the case of models which are nonlinear with respect to the inputs, the problem is even more arduous.

The unsatisfactory features of the existing methods lead us to center our attentlon on ldentlflaOl11ty teStlng, wnetner tne model 1S 11near wltn respect to the input~ (Chapter~ 3, 4, 6 and 6) 0" not (Chapters 3 and 7).

Chapter 3 Local Identifiability

3.1 INTRODUCTION

The local approach to identifiability testing is both restricted and ge­neral. It is restricted because of its unknown domain of validity, and general becau­se It allOws us to SWdy dynamlc models whlch mayor may noL 1Jt: llnt:,H" <1m! Lilllt:-illv<I­riont.

Thi, ~h~ptAr prAsAnts. in a comprAhAnsivA way. various methods for chec­king the local identifiability of such models by providing sufficient (and formally equivalent) conditions for local identifiability. All these approaches rely on a li­mited expansion of the model output or of a criterion around some nominal pOint! in the parametnc space. ::>trlctlY speaklng the reSUltlng cona1tlons are not J1t:ct:~~dry

Lu ill~ul'" local identifiability [D14) , and when thcy arc not met, one hac to tako highllr ordgr t9rmo in thQ QVl"andnn t.n "AA whAthAr thAY can rAmove the indeterminacY. Calculations may however become so complex that the appeal of a local approach is less evident. Moreover effective parameter identification by local methods will al­ways lead to serious numerical difficulties, because of the necessity of taking into account high order infinitesimals. In that sense one can say that the condltl0ns to De presented here are, prdcUc<llly, lJuLIr 1It:'-"'~5al-y and 3ufficient fOI- local identi fiobility. Note th"t a global approach of thll idllntifiability prohl .. m ran rA,'llt in ,tri~tly nAcAssary and sufficient conditions for local identifiability (see [017]

and Chapter 5). Fairly often, especially when the number of parameters is large and the

measurement possibilities are restricted, the result of the test is negative. IhUS we have tned to De more SPeC11'1c oy extractlng from ttlt: Lt:~L~ all availabl", infol-mation characterizing the local degeneracy of the problem. By 00 doing it i~ poooibl .. to oturly th .. inflllAn~A nf~ny additional experiment or sUDPlementary constraint on the s.~. identifiability of the model very simply.

Finally two of the methods presented, considered as representative, are implemented for Computer Aided Design.

57

J.l MtIHOU:'

<;lll1l1n~ .. th~t. th .. ~t.rlldllr .. nf th .. mnd .. l M(~) is defined by the pilrilme-

trization of

.f(! • .!!. !. t)

.. 9,(! • .!!. !. t) (J.l)

There it no need to attume here that the initial ~tate ~ it known. ~o that th .. para­

meter vector can be written. in the most Qeneral case

~ [T TJ T ~ - .!o' .~ . (3.2)

In order to present the various existing methods synthetically. it is useful to in­troduce the notion of an exhaustive summary of an experiment.

Definition An exhaustive summary of an experiment is a vector which contains only.

but all. the information about! that can be extracted from the knowledge of the in­putS ana OUtPUtS for any .!! € u ana any t ~ U.

Example For linear time-invariant state-space models. the set of all the coef­

ficients of the associated transfer matrix in reduced form is an exhaustive summary of all the Infonnatlun tllilL call ut! cullt!cLt!U i rub LIlt! ~t!L ur fJit!l.t!wi~t! l.ulILinuuu~

functione. So.ctionc 3.3 and 7.2.2 will p,"ovide oth~r e)(ample~ of e)(hauttive ~urmiir;e~.

for linear and nonlinear models.

Let ~(!e.l) (i=1.Z ..... 1) oe

- either the output I(ti ) of a model with parameters !e'

- or some ith vector corresponding to a partition of an exhaus-tive summary of the planned experiment.

output lndtstingubhatJl1lty w1ll rt!sult In

(i=1.2 •...• I). (3.3)

Ipt IIC c:.hnw that, with thic nnhtinn, th .. var;nnc "l'l'rn:lrh .. c: I\rnl'nc: .. tt in t .h .. lHprl't­

ture yield formally identical conditions for local identifiability. We shall use Vetter's convention for noting the matrix derivatives [V3] • and suppose the exis­tence of all the required derivatives.

58

3.~.1 U"e uf t.he lmpl h. i L fum.LiulI Lh"un::m [61)

Let g be the mapping

. . Identifying ~ from { s(e .i) • i = 1.2 •...• I } • or equivalently from { s(~,i) ,

--.. --"'e -1 ---.. i = 1.t ••••• n • \;(111 11" vi"""u (I:' luuklng fur ilil IlIv"r:." 1Ildlllling II • ThIs Is po:.:.I-ble (locally) if the Jacobian of i is nonzero, that is if the matrix H defined by

. . 1 s(e .1) 1 s(~.I) 1 T!(~.I) ~g -~ :>g -el en+v a~

H ~ (3.4) .

a ~(~.I) l ~(~.I) l T!.(~.I) ae - aee -e, n+\J a.!!..,

has a rank equal to the number of unknown parameters. This amounts to writing

H o~ = Q =:;> o~ = Q. (3.5)

Remor-k:)

(i) If ~omg componont~ of ~. for oyamplo ~Omg initial condition~. aro

known. it implies that the correspondinq components of o~ are zero. and the associa­ted columns of H have to be discarded.

(ii) The entries of H are the sensitivity functions of s with respect to the parameters for all the values of i.

3.2.2 Local stability of identification algorithms

The principle of an approach using the local stability of identification algorithms is very simple [W4, L5] • It consists in computing

s(e) ~ { s(e .i). i= 1.2 ..... 1 } _---Q -...;..0

(3.6)

ror Cl model M(~) of the eho3en 3trueture. whieh is supposed to represent the real

pro~~ss. Th~n on~ initializ~s at i., an id~ntification aloorithm which reauires local identifiability in order to converge. and checks its stability at the nominal point e • Since the "process output" s(e ). is identical to the model output s(~) (due to -"'e --"'e .---.. our idealized experimental conditions) the "output error" ~(~) is identically zero. and the tradItIonal scneme of the model method (flgur" J.1) \;dll 11" '""UU\;"U Lu UrdL

shown in ~i9urQ 3.2.

59

"Process" ~(~)

M(~)

/ "Model" -

- ~(~) ~ M(~) _

initializod at ~ - ~

~ t Identification alqorithm

requiring local identifiability

Fig. 3.1 Model method.

Identification algorithm requiring local identifiability

Fig. 3.2 Reduced form.

+ -~l~) = u -

-

.=.(~) - 0

This reduced scheme illustrates the fact that any identifiability study is, by necessity, a reflexion on the model and not on the real system, which is essen­tially out of reach.

Now we shall present four second-order identification algorithms, and show that they y1elo equ1yalent con01t1ons for local 10ent1f1a01l1ty. In eacn case wo chall U&O a quadraHc critorion in tho "output orror" ,

60

where the Q(i) (i=1.2 ••••• n) are symmetric positive definite weighting matrices.

~uppose tnat tne crl~erlon J{~} can De expanaea as a laylor serles ana con3ider a di3placement ~~ about~. 3mall enough 30 that the term3 of higher order

th~n twn ~rp npgligihlp

j(~ He 1 = i(; l+ [+ i(; l] T~A +.!. liA T [~ j(; 1j -e -e -e d~ -e -e L -e T-"e a!!.oa~

(J.ts)

Ne~ton's method [87] looks for Ii~ which minimizes the second order expansion of J{~ + o~}. It saTlsTles

o. p.9)

Newton's algorithm will be lecally stable at ~ if, and only if, (3.9) implies

Q. (3.10)

One has

~[~ ~T(; .i)] I)(i) [ <:(A .i)- <:(~ .i l ]= 0 - L - - -'-e --e--e i=l a~

(3.11)

anrl

f l2 =.T (~.i) ] Q(i) l-=- =-(~>i) 1 i=l ~~ ~~

(3.12)

wln:~n~ 'lSI :.loll\b rv,· Un, K,v"" ... "",. ",,-v.Ju ... l.

Tho idontity of ~(~.;) anrl =-(~.;) ;mplip~

61

~2 - I [a --J(tI)= l -~o ~9T -'-e i=l ~9 -'-e -e -e

(3.13)

so that the exact Hessian involved in the Newton method is strictly equal to the ap­proximate one of the Gauss-Newton method [B7] • This makes the two methods equivalent for local identifiability studies. In both cases the condition of local stab'ility can be wrltten

Define Q as

Q(l)

Q ~ Q(2) <D

Q(I}

011 =0 ~OI1 = O. -'-e - -e-

This symmetric Dositive definite matrix can be Dut in the foHowinQ form

where R is a square matrix of full rank. Condition (3.14) becomes then

(nH)T (nH) o~ -O"""'i>o~ -0. -e -'-e-

(3.14)

(3.15)

lj·lb}

(3.17)

It will be met if, and only if, condition (3.5) is satisfied. Checking local identi­fiability by the Newton method or by the Gauss-Newton method is thus equivalent to <.heddng it t.,r the approa<.h of the impli<.i t fun<.tion theorem. Note that the introdu<.­

tion of tho woighting matrico~ Q(i) ha~ (at loa~t thoorotically) no influonco on tho

result of the test.

3.Z.Z.Z ~~~~~:~~!~~1_~19Qr!~~~

While the two preceding algorithms use a Taylor series expansion of the I-I'i t~I'iUII, tll~ Gdu:.:.-S~iu~l lII~tlluu [H9] u:.~:. elli ~"pelll:' iUIi ur ~

1C!l.e + 6~, i) = 1(~, i) + [1. T 2(i, 0] 6~ + 0 1I1i~ II . a~

(3.18)

62

Suu)LauLIII~ rur ·.~(~e'" O~,i) iL~ ""J.lQII~iun in O~ tnml.ated at the or-del une in the

oxproccion of j(~ + &~) givon by (3.7), ono 90t~ an approximation ~ of tho valuo of

the criterion :

~(2.e I o~) -i it I'!(~'i) .!(~,i) - [:!! :>(2.e'i)] s~l T • Q{i)

. [ i~.' ) -iii.. j ) - [ ;.;; ili..' ) 1 '!e j. The oie minimizina 1 satisfies

p.l!1)

which, taking into account the identity ot ~(~,l) ana ~(!e,i) can also De wr1tten

(J.lI)

Thus the Gauss-Seidel algorithm will also be locally stable if, and only if, Condi­tion (3.5) is met.

3.2.2.3 QY~~iliD~~ri~~!iQD_~lgQri!~~

This method r 52. xl1 aDDl ies only when

Denote by ! the following extended state vector

which, taking (3.1) into account, verifies

p.Z.:!)

(3.24)

63

Ltn U) lIJlt:d,-I £~ p.l4) di.>uuL LIr~ t'dje~to,y ~(t) a~~ociated with the value 2. of the

model parameter,

O!(t) ,,[ ;!T io(~,.!!,t) ] o!(t)

===->O!(t) " t(t,tO) O!(tO) ,

(3.25)

(3.26)

where t(t,tO) is the transition matrix associated with the t1me-varying Ilnear equa­Liu" p.l:;).

(3.27)

The fir~t order appraximation of the model output at time ti for the trajectory

(! .. ~!!.) i ~

(3.28)

which, taking (3.26) into account, can also be written

(.3.l9)

SUb!St1tut1n91(~:t"O!,ti) for l(!,ti ) 1n t l1~ cr1t~rlun, ,wd IIIlnlllli£lny wILlI r -~)fJ~J,;L Lu

~~(tO)' ono gete

itf[;!1 ~(!.~.tl) ] "'(t1.tu)1 T Q(i)r[-;!T~(!.~.tl)] d>(t1 ·tU) ].s!(to)

it [[ !,!:,T ~(~,.!!,ti) ] t(ti ,to)] T Q(i) ll(ti ) - ~(~'.!!'ti)] (3.30)

Since!e =!e' the right hand side of (3.30) is zero, and the condition for local stab; 1 ity is

64

It r~lIIalnl> tu U~ l>lluwn ~lia~ Cuml1t1un (3.31) h ~4u1val~nt tu Cundltlun (3.14). Th~

proof follow£ tho linoc of [Xl]. From (3.2) and (3.23). ono hac

O!(to) = cS~,

besides, (3.22) and (3.27) imply

With the help of (3.26), (3.33) can also be written

.a. T ~(!e,i) ; [.a. T !Ie(!'~'Li)] 'I'(LpLO)· a~ a~

(3.32)

(3.33)

(3.34 )

CUlII.li Lioll:> (3.31) Clnd p.14) Clre thu:> equivCllent, Clnd the qUCl3i1ineClriz:ation algorithm

wi" bo locally ~tablo und",. tho ~ .. m" "ondition~ a<: th .. pr"violl~ on .. ~

3.2.3 Observability of the extended state o A3trom [A3] , and then Berntcen and Balehen [B6J • havo propocod ctudy-

ing thp lo~al idpntifiability of models described by (3.1) bv checkina the local obser­vability of an extended state vector defined as in (3.23) around a nominal trajectory !(t) [L7] • This approach is similar to the preceding one. Indeed solving (3.30) in cS!(tO) corresponds to locally reconstructing the initial state from the observation ot the outputs.

1 . ? 4 Tnfnrm~tinn matrix

Consider now a stochastic framework. If ~ 1s an unOlasea estlmator for !e' iL" l,;uvddclIIl,;" :)dLbrie5 the Crame.·-Rao inequality [G9 J •

(3.35)

where Me is Fisher's information matrix. For the covariance of the estimation error c

65

tv "':OlldiJl nJlitt:, Fbllt:,"'", iJlrvnlldtivJl IlIdtJ"i1l JIIU",t "'t: iJlvt:,"ti"'lt: V'", t:yuivdlt:Jltly "'t:­

cau~e of it~ ctructure. pocitive definite (T6 ] • Let uc chow that. in the ca~e of

Gaussian additive white noise ni€ N(Q.r(i)) on ~(~.i). this condition is similar to the previous ones, thus extending a result of [T7] •

The likelihood function is

=.n r[(21T)p det(1:(i ))r~ exp[-H~(~' i )-~(~. i )]T 1:(i) [~(~, iJ-~(~, iJ]]] '-l~

(3.36)

and Fi~her'~ information matri~ follow~

y. [l ~T(e ,i)] I:-(~} [l T ~(e .i)] i=1 3e -'-G 1 ae -0

-'-E! -'-E!

(3.37)

It ic thuc equivalent to cay that Me (3.14) is satisfied. with -'-E!

ic of full rank or that Condition

. -1 Q( 1) = 1: (i) . (3.38)

In this p~rti~ul~r ~~~p. ~to~h~~ti~ ~nrl rlptprmini~tir prohlpm~ of lnr~l irlpntifi~hi­

lity are thus equivalent. Note, however, that some authors [S5, S6] are more restrictive in their

definition of stochastic identifiability: for them the model is stochastically iden­tlTlaDle IT every asymptotIcally effICIent estImator converges to! 1n quadrat1c mean. If one uses this definition, Me needs not only to be of full rank, but 0130 to have

-'-E! all 1ts e1genvalues tend1ng to 1nf1n1ty as I ~ m.

3.3 LINEAR MODELS

In the special case of linear models, several exhaustive summaries may be cons1dered. For example F1sner [ F4], and tnen GreWal and GlOver [G7] , nave propo­ced ~tudying the ~.~. identifiability ef linear state spaee models by supposing D(~)

66

and the 2n fir3t coefficient3 of the matrical Taylor :>erie:> eJlpon:>ion or C(!)exp [A(!)t] B(!) tn np known. Thnw~pn [T7J ~imply ~hnw~ why thg~o torm~ on~blo ono ,to reconstruct the impulse response of the corresponding model entirely. Thus they form an exhaustive summary of the available information On! and one has just to set

V(~.l) ~ VQct[D(~)J •

? ~-'!, i+2) = vect [C(!} Ai (!) B(!} 1 (i=O, ... ,2n-l) , (3.39) .

in order to be able to test the S.t. identifiabilitv of the correspondinq model with the help of Condition (3.5). Note that the initial conditions are supposed to be zero here, and thus do not belong to the vector of the parameters to be identified.

Similarly Berntsen and Balchen L B5] have considered linear models dri­v~" liy I "IIU L~ n~~ulL i rly rr·urn c1 1 i rr~c1r· (;uHlli i IIC! L i urr u r C! rill i l~ rlurnli~r' I u r ~ 1 nu~u'ruc11

tignalc. It corrQcpondt to thQ catQ U . (tinw1t.i.l.2 •...• IJ. and Qnablet one to ttu­dy the effect on identifiability of periodical input siqnals, which freQuently occur in biological systems. The transfer matrix M(!,jw) associated with the model satisfies

(3.40)

and an exhaustive summary of the information contained in the component of pulsation Wi is given by M(!,jwi ). So one can set

(i=I, ... ,I), (3.41)

emu llr~JI Lt::~l lhe 101;.01 identifiability of 0 for the con3ide,-ed input:> with the help of Condition (3.5).

3.4 COMPUTER AIDED DESIGN OF MODELS

If Coml1tluJl (3.5) b Jlul rll~l, lhoL i:> Lu :>oy i r Lhe :>tudied model is

not pr~ctic~lly locally identifiable, thrQe different policiec. cummarized in ~i9urg 3_3. can be considered.

ihe tlrst one, ana eVlaentiy tne nest, amounts to cOllectIng complemen­tary information on thc studicd systcm by incrcasing the number of inputc and outputt. that is to say the number of interactions between the svstem state and the experimen­ter. The techniques presented in Section 3.2 enable one to see easily whether it is possible to make the model locally identifiable in this way. Unfortunately these com­plementary experiments are often unrealistic, especially when dealing with biological systems.

67

Tilt: ~e: .. umJ "ulh;y <;.un:;i:;t5 in tryin9 tu find thc $ct of all modch ha­

ving tho ct~UctU~4 dofinod by tho chocon pa~amotri7ation and which cannot h~ rli~tin­

Quished from an input-output point of view. This approach, that we named exhaustive modelling, will be developed in Chapter 5 for linear models.

rinally thc third policy i3 to modify the model poromctri%ation (i.e.

th~ mod~l strur.tur~) with th~ purpos~ of suppr~ssino the existino deories of freedom. It can be achieved by introducing a reduced-order made] with aggregated state varia­bles [B8, S7, MS]. One can also keep the dimension of the initial model, but intro­duce some constraints between the parameters. It is what Milanese and Belforte call parameter aggregat10n 1n [M; J ,wnere tney glve examples snow1ng tnat, 1n some Ca$C3, Q model obtained by 3tate-variable aggregation may be le33 identifiable (in

th~ C"M~ of th~ nLlmh~r of irl .. ntifi~h'" l1ar~m .. t .. r~) th~n th .. initi~1 mnd"l.

If, in the past, most authors interested in testing models for local identifiability have limited themselves to examining the rank of H, it is possible to go further [M6, W4, LS, Lll] , and to exploit the information contained in H to characterize the linear relationships locally existing between unidentifiable parame­Lt:n,. Till:. b ht:l",rul wlot:1I runlluloLill!l 0 IIt:W ",o.ollle:L.iLOLioll or the: modcl to makc it

s.~.i •• Indeed one cannot arbitrarily chooee the conetraintc which will relate tho

parameters. If. for example. one introduces a relation between some already identi­fiable parameters, one gains no additional information capable of removing the inde­termination.

A pn~~ihl~ prnr.~dlJr~ [W4. Ui 1 is as fnllows :

(i) Compute r = Rank [H) . The number of degrees of freedom is then (ve-r), where ve is the number of unknown parameters.

(ii) Choose r linearly independent columns of H to form a basis.

(iii) Express the (ve-r) remaininQ columns as linear combinations of the basis columns.

So one gets (ve-r) relations between columns of H, which express local linear dependencies between the associated unidentifiable parameters. In order to ma­ke them locally identifiable, one must state at least (ve-r) additional relations in­volving the (ve-r) groups of locally unidentifiable parameters. Note that the method ",.-UflU::.t:d uy MilclIIt:::.e: ill [M6] pruvidt::. It::.:. Ilirunllc1Liuli UII Lilt: ullidt:IIU rioule: ",or-oll""­

tore, cinco it only dividoc tho paramotorc into a locally idontifiablo and a locally

unidentifiable subset. As already mentionned in Chapter 2, determining the rank of a matrix is

a numerically difficult problem. It is even more complex when the columns of Hare obtained from SenSit1Vity functions computed by numerical simulatlon Of a set Of non-

68

- D~rllll Llull ur ~h~ lIIul.lo::1 ~L, ·u(.t.un=

- Experiment design

New eXDeriment desiQn (Policy#l)

no

Arc other experimcDt~

Dossible ?

no

New d~f;n;t;on of thp mOrlp1 (Policy #3)

Identifiable?

no

Are additional structural constraints or model reduction reasonable?

yes

Fig. 3.3. Three possible ways of dealing with unidentifiability.

69

lineo, difre,-entiol equotion:>. One DlU:>t therefore be e:>pedally careful when :>elec-

t; ng drnul ati on a 19ori thm~ _ Onc" a!lai n. a ranI< t .. ~t ha~ .. 1'i on th .. cd n!llll ar-va 1 III' l'iprnm­

position of H is recommended, because of the great numerical stability of this tech­nique. Moreover such a decomposition enables one to substitute for H the matrix of rank r which is the closest in the sense of the Euclidian norm, when studying possi­ble local linear dependencies with a standard algorithm [II] .To do that, one simply :.et:. Lu "eru Lhe (ve-,-) :.llIyuldr" value:. l.ulI:.ilJe,"ed a:' IlIdbLlllyubhalJ1", r'"UlII L"""U

whon tho numorical proci~ion of tho computation~ i~ takon into account.

3.5 IMPLEMENTATION FOR LINEAR TRANSFORMATION SYSTEMS

We have selected two algorlthms. The flrst one allows the :.tudy of lo­cal identifiability in a purely ~tructural way. and doee not require any difficult

simulation. The second onp. on the contrary. enables one to study the influence of non structural factors such as the measurement schedule, but at the cost of more cri­tical computations.

3.t>.1 Method A

This method uses the exhaustive summary defined by (3.39). The computa­tion of the jth column of H is fairly simple. It contains

- thp mp pntripe; nf -}~. [ n ] • J

- the 2 nmp entries of

.!. k ae .l CAB] =

J

(k=U .... ,zn-1J. (J.4Z)

At any nominal point! one can thus compute H from A(!). B(!). C(!). D(!) and their first derivatives with respect to each parameter. Nevertheless, calculation of deriva­tives is rather tedious. It can be avoided for a broad class of problems, whenever the jJo,"olllt:L";LOLiulI ur A, B, C olld 0 1.011 IJ", :>Londo,diLed. Thi:; i:> 0 highly ,Je:;i.oble

foaturo for a C.A.D program, co in tho routino wo havo wr;tton thoro i~ an option

corresponding to the parametrization.

A(.!!) ! Til.!!.,

ex . (3.43)

70

Th~ IIldtrlt;~~ 6 Clml C, el\IJre~:;1n9 tile 1nterdct1uns between the system ana the exper1-

montor, ar~ cuppocod Booloan and known, co that tho paramotor~ ! correcpond to the

matrix A alone. Moreover these parameters are assumed to satisfy

n ekE: {a;jla;j F 0 and i;: j} LJ {aJ.J. I 1: a;J';: O} .

i=1 (3.44)

Two kinds of structural constraints, which besides are the most frequent, can then be considered

- no transformation exists from j to

- no transformation exists from j to the outside

n

I aiJ· i-1 o .

(3.45)

(3.46)

This particular structure for the parametrization makes it easy to com­pute the derivative of A with respect to the parameters without requiring the use of Clny symbultc cumIJutdLlurl fdclllLy. Wh~1I IIV ~Lr·ucLur·dl CVrI~Lr·dilil ~"t:,l:> VII A, VII~ 100:>

a A I - = o .• 0., , H mJ vd· •

lJ R.m

i,j=l ..... n (3.47)

Anv constraint of type (3.45) or (3.46) decreases by one the number of the parameters to be estimated. In addition a constraint of type (3.46) on the jth column implies

LA I - (0. - 0 ).s • a H R.mIDJ aij vm

{i-l, ... ,n

i#j .

(3.49)

For models having this parametrization, one has just to provide the program with the structure uf A, ll~rill~1l lJy LIo~ :>~L vf 011 .,,,i:>Lin!:l l.OIl:>t,·oillts of type (3.4~) ond

(3.46) and tho ctructuro.of tho Booloan matrico£ B and C. The rOlltine then (l"n"r"t .. ~

a model with oarameters ! satisfvina these constraints and checks its local identi­fiability. If the model is not found locally identifiable the routine supplies the linear relationships existing locally between unidentifiable parameters. One can then use this information to modify the model structure in oroer to remove tne lnaeterml­IIdLiuII. or l.uur·~~ lh., :>ludy of oth." tyP.,s of parolllt:triz;otions is aho possible. With

tho procont routino it i£ noce~~"ry to ~p .. cify th .. loral rl .. p .. nrl .. nr .. nf A. R. r. ~nd D

with respect to ~ by supplying the algorithm with the matrices :!, ~~, ~~ and ~~

71

evaluilted at tt • Thh re::iul L~ In an Input. rluw uf llaLa LliaL l,;an be ,oaLhe,- la'-!le, cJlIll

dotractc from the convercational features of the routino. Symbolic computation faci­

lities. when available. may be useful here to liahten the user's task.

3.5.2 Structural nature of the result obtained

Fx~ept when ~he~kino r.nnditinn (~_5) by mean~ nf a ~ymbnli~ ~nmputation program. which may be extremely time-consuming, the local identifiability of the structure defined by t~e model parametrization is studied in the neighborhood of a given numerical value! of !. One may therefore wonder whether it is possible to infer any conclusion of a structural nature trom the result Obtained. Method A Wli I

find the model unidentifiable if

Rank [ H(!) ] < " (3.49)

A<; lnng a<; ea~h entry nf A. R. r. and n i<; a pnlynnmiill fundinn nf the parameters, a reasoning similar to that of Section 2.3.2 leads to two possible cases

(i) The studied model is locally identifiable at~. so that its S.t. iden­tifiability ;s certain;

(ii) The studied model is not locallv identifiable at~. It is then almost surely non s.t.i.

The exhau3tivc modelling method, to be pre3ented in Chapter 5, will

enilhle ue; to give e;nme examples Af input-nutput rehtinn<; thilt re<;ult in iI model which is locally identifiable at a particular point of the parameter spac~, and lo­cally unidentifiable along a curve. However such examples are not structural ones, for they are associated with some very particular values of the parameters.

J.:;.J Method D

We wanted to have a method at our disposal for studying the influence on local identifiability of unknown initial conditions or of non-structural factors ::iu~h a::i the measurement s~hedule. The approa~h presented In Se~tlon J.l.l seemed at­

tractivc, for it allowe caey integration of a local identifiability teet in an iden­

tification aloorithm. In such a case. if the condition of local stabilit~ proves not to be satisfied, it means that the model is not locally identifiable at!, taking in­to account the characteristics of the algorithm used, the model structure, the shape of the inputs and the measurement schedule. The result is thus more realistic than t.he olle obtained wIth method A, whIch ~orresponds to purely structural propertIes of tho matrieoc A. B. C and D. Among all the algorithms presented in Section 3.2.2, we

72

have reta1ned the qUcI~lllrn~clr ' l£clLlulI L",c:;tllllyu"" wlllc:;11 fJJ"UY"',j Lu 1.1", w",l1 ~ui L",,j Lu Lilt::

modolling of tran~fo""ation 'y~tom~ [Wl).

Many feature3 of the implementation of thi3 method are identical to

tho~p of m~thnrl A. ~nrl will nnt hp rppp~tprl . Only ~pprifir prnhlPm~ will hp nntpd.

Consider, for simplicity, a system with one output freely evolving from an initial condition ~ (which may, for example, result from an impulse input).

~ ~ - J\~!)~. ~(to) ~ ~ •

lY=£.~. (3.50)

Suppose moreover that the parametrization of A is defined by (3.44). The extended state vector then satisfies

-- [~ :j ~(t) \V \V f(t) .

£quation (3.25) rgoulto in

~~(t)-

IXT(t) : V

~~-------t----------------------------A ! I xT(t)

i V ! - - ----~-_~ xT(tl I' -___ , ________ -L ___________________________ _

ID! Q) ,

which can al~o bo writtgn

o.

The Jth CU1UIIIII ur LII", Lnll15it i on matrix -I>(t,tO) a330ciated with (3.52) i"

of thi~ oquation with thg ini t ial cnnrlitinn

(:l.5~)

(3.53)

the solution

to applY t;onalt10n (3.31) Tor local 1dent1T1abl1l Ly, UIII:: II,,:> Lv \.UlIlpute

73

(3.54)

or, taking (3.61) into account

(3.55)

k Thus we are only interested in the first n lines of ~. Let 1 (t) be the vector of the f1rst n entr1es or tne Ktn COlumn. It can De computea from

t ~k(t) = A .l(t)

1k(t ) .s o Ii ik it k " n (3.56)

~ iK(t) = A 1k(t) + .sA !(t) •

Ie _ 1 (to) = £ ' 1f K > n (.3.::i7 )

Thp ~nllJmn .l(t) ~~~n~ht.prl wHh thp rar.:.mptpr ':'lJ i~ nhtainpn hy dmlll.:.tina (3.1)7)

with

.sA (3.58)

The matrix H can be written

H (3.59)

The specific difficulty of method B is that the +k(ti) must be very ca­refully computed. for one will then compute the rank of H and study the linear depen­d~ncte~ that poss1bly exlst between lts columns. Slnce (3.57) 1s an lnnomogeneous aquation, it i. preferablo to roplaco it by tho following homogonoouc ono, to avoid

any problem of interpolation.

74

i"(t) . A a A

~(t) o A ~(t)

(3.60)

(bn) .

The e1genvalues of the state matr1x assoc1ated w1th th1s homogeneous d1fferent1al equation aro thoco o~ A, with twico thoir multiplicity. Thoro~ore they are indepen­

dent of Ie. This malees Harris's method [HlO 1 particularly suitable for the simulation of (3.60), and actual results obtained with it are quite satisfactory. The extension of the method to multivariable models with inputs is straightforward [WI] and only complicates the notation.

3.5.4 [xomplc3

Example 1. Consider a two-class model, where one observes the result in class I of an unit injection in class 1, the system being in zero initial condi­t1on. ~uppose moreover that there 1s no a pr1or1 restr1ct1on on the model ~Lru~Lure

(Figurc 3.~).

II

Fig. 3.4 Two-class model with no a priori ctructural roctriction.

75

If one assumes the number of measurement 1nstants to be suff1c1ently large, both me­

thode givo the came reeult. The model ie not e.~.i. There ie one degree of freedom;

and if all and azz are s.t.i .• alZ and aZI are locally linearly dependent. We shall see in Chapter 4 that all and a22 are actually 5.g.i., while only the product a12 a21 is s.g.i., which effectively corresponds locally to a linear dependency between al2 and a21 •

One may try to remove such an indetermination by setting a constraint of type (3.45)

but then the programs conclude that the model is not s.t.i., with two degrees of free­dom. Only all is s.t.i .• So one gets the apparently paradoXlcal situation where a mo­del or 3implel" ~tl"uctul"e h le~~ identifiable than the fir~t one. It can eo.:>il)' be

undor$tood why by looking at ~iguro 3.5. Class 2 doo~ not rocoiv~ any mat~rial from

the outside. and is not input-connectable. The model cannot be identifiable since the observed output

does not contain any information about a12 and a21 •

u

Fig. 3.5 Non s.t.i. model with three parameters.

On the other hand. if one suppresses the flow from class I to the out­side (Figure 3.6), whiGh imposes a linear constraint of type (3.46) between a s.l.i. parameter an and a previously unidentifiable one a21 , the resulting model is found to be s.t.1..

76

~ a2l Xl

u ~l. 'y,2 ~""'Clrl-2--V-"2----'

a02 x2

Fig. 3.6 S.i.i, model with three parameters.

E"C1lllplo:: Z. Co,,~iuo::r· nuw Llro:: rivo::-\'lo~~ model, withz:ero initial condi­

tion. do~cribod by Figuro 3.7. Cla~~o~ 1. 2. 3 and 4 aro ~uppo~od to bo ~oparatoly

injected and observed.

Fig. 3.7 Example 2.

Without any ~tl"uctul"al condl"aint on A, thi~ IIV)rl"l n"'~ ?Ii unknolllfl param!!t!!rs. Beth DrOQrams find it non s.i.i., with one degree of freedom. Figure 3.8a shows the trans­formations associated with s.i. i. parameters, while Figure 3.8b corresponds to local­ly 1inear1y.dependent parameters. Note the S.1. identifiability of aSS' which will be explained in Section 4.3.1.1.

77

Fig. 3.8a Transformations associa­ted with S.Li. parameters.

Fig. 3.8b Transformations associa­ted with locally related parameters.

Tho programs hav .. boon appliod to about f'orty t..st ('a~o~ of' tr"n~f'or­

mation systems ranging from 2 to 5 classes. and the results have been checked with the help of the methods described in Chapter 4 and 5. They have always proven to be correct. Care has to be taken, however. not to restrict the choice of ei to too par­ticular values (such as small integers). since the probability of getting atypical r·dll"-" fur" H(~) wuull.1 1Jt! 1I1!jIlt!r".

3.6 CONCLUSION

A suitable formalism has allowed us to present in a unified way the va­rlous met nods enaollng one to cneck tne local ldentlTlaOlllty of models wnlcn mayor moy not be linear and time invariant. Moreover. when the model i5 not 5.~.i •• we have

seen how it is possible to get some information on the n~ture of the local degenera­cy of the problem. This information can be used in a conversational program to design S.Li. models.

We have shown that all approaches available in the literature result in formally equivalent local identifiability conditions. and that the possible presence

78

of CClu~~i C1n oddi the whi te noise on the output:; \Iu~~ IIU L I..hClrl!:l~ Lh~ fie! tur~ uf the

nrnhl",m .

~or linear time·· invori C1nt mode 13. the exi 3tence of fi ni te-dimen:;ionCll

exhaustive sUJll1lllri@c; nf th", infnrfMtinn IIhnut th", param"'t9r~ containod in tho outpuh

enables one to avoid any simulation in the study of purely structural properties of

the matrices A, B, C and D. It is however of interest to have at one's disposal me­

thods to test the influence on identifiability of initial conditions. measurement

SChedule and Shape ot lnputs. ~uch methods can be included in identification algori­

thm3. in order to check 0 p03tedod 10~ol identHiClloilHy, wh~1I L"~ ~Lu\ly ur C1I..LuCll

I!xpl!rirnPnbl ebh. linn nnt d"'l'ly of th", modo' ~tructuro. h contidorod .

Now we have ~ornc eQ3j' to U3e tooh C1't our di3P03C11 for de~i9nin9 3.t.i.

models. The next chapt@r Jlrl!c;l!ntc; II mPthnn fnr c;tunyino thp 01nhal inpntifillhilityof

the selected structures, in the special case of linear time-invariant models. The ca­

se of nonlinear and/or time-varying models is deferred to Chapter 7.

4.1 INTRODUCTION

Chapter 4 G loballdentifiability of Linear Models

This chapter describes a method [WI, W6 - W9] for testing linear time­invariant models for s.g. identifiability, as a result of a study on compartmental IIIvllel~. Wile II Ileal ill!:! wi~II ~vI.II· IIIvlleh, vile I.all r.eyuell~ly I.oll:.ide. the e"pe.imellt to

bo compo~od of a ~oquonco of olomontary oxporimont~. oach of which ontail~ ob~orvina

in compartment i the result of a unit impulse injection of tracer in compartment j, which mayor may not be different from compartment i. The resulting output then cor­responds to one entry of the transition matrix ~ associated with the considered mo­del. This has led us to study the properties of these matrices in order to see to wildt extent the knowledge of ~ome entr1e~ of d given trdn)I~luli IIId~rlx errdLJle~ n:­

conctruction of tho othorc, thuc making tho modol ~.9.i ..

III ~Ire rir'~~ :.el.UvII :'Vllle fJf·Vl'er·~ie:. ur ~"e t.afl~ition matr-i" a'"c c3ta­

bli~hod. Tho ro~ult~ obtai nod aro thon appliod to tho ~tudy of tho idontifiability of

the entries of A. with the help of a parametrization of ~ which exhibits the eigen­values of A, generally identifiable, as explicit parameters. Finally the method is extended to the case where the matrices A, B, C and D are functions of the parameters to be identified.

4.2 PROPERTIES OF THE TRANSITION MATRIX

Let ~(t) be the transition matrix associated with the model

(4.1)

For simplicity, assume the eigenvalues Ai of A to be distinct. The transition matrix

80

ecn then be written c, Cl 'urn of .:;on,tcnt rnctrice~ Xk, lIIul LiiJl it:\J uy ~(;dl(H' t:l\iJumm­

tials

~(t) (4.:»

EQuation (4.2) expresses ~ as a function of n(n2 + 1) parameters. necessarily inter­dependent. Indeed. since A has only n2 entries. ~(t) = exp [At] can always be expres­sed as a function of at most nZ parameters. For that purpose we shall now study the properties of the matrices Xk (k = 1,2 ••.. ,n).

Let ~i be a (nonzero) eigenvector of A associated with Ai ; and define T and A as

(4.3) .

(4.4)

v

Wi Lh Lin:: d~~umllt I un ur db tl nctness of the el genval ues, T 15 1 nvertl bl e, and one can d .. finQ w~ (i - 1,2, ... ,n) by

-1

T ~1

T ~2

T !:!n

The similarity state-space transformation T diagonalizes A, so that

-1 n T A = TAT ~ A = L !it ~ Ak

k=l

and ~can be written

<I>(t) = ""I'[At)= T "VI' [At) T-1

(4.6)

(47)

81

(4.8)

The tran3ition mo.trix t i3 now cxprc33cd a3 a function of n(2n + 1) parametcr3. "c­

liItf.d hy n2 Pqullt; nne; c;; nl"P

and

r1 T = 11. ~ w: v. -1 -J

From (4.2) and (4.8) one has

(k=1,2, ... ,n) ,

-> Rank [ Xk 1 - (1<-1,£ ••••• 11) •

X· X. = v. w: v. w: . 1 J -1 -1 -J -J

Tllking (4.Q) intn IIrrnllnt. (4 . 1?) im"li.,~

(i ,j=1, ... ,n) .

(i ,j=1.2 ••.•• n)

~rom (~.6) and (~.10) A can be written

(4.9)

( 4.10)

(4.11)

(II .1()

(4.13)

(4.14)

Multiplying both sides of (4.14) on the right by Xj • and taking (4.13) into account. one OOtalns

(j=1.2 ..... n) •

so that one can choose any nonzero column of X. as v .. Moreover one has J -J

T T Trace [ Xk 1 - Tracc [ ~ ~ 1 - Tracc [~ ~ J •

and (4.9) and (4.16) imply

Tracc [ \ 1 - 1

Finally t(O) =11 implies n k~l X" = 11

(k-1.2 ..... 11) •

(4.15)

(4.16)

(4.17)

(4.1R)

82

4.3 PARAMETRIZATIO~ OF THE TRA~3ITIO~ MATRIX

In this section we show how to write the riQht hand side of (4.2) as a function of n2 parameters only, including the n eigenvalues of A, which are s.g.i. if the model is connectable [P2) • For that purpose we have to take into account the real or complex nature of the eigenvalues of A.

4.3.1 All the ei~enva1ue3 of A are reo1

Since all the columns of X. are eigenvectors of A with the eigenvalue Aj , one can build T by choosing for the jth column of T any of the nonzero columns of Xj (J=l,Z, ... ,T1). WheTi IIU l;ull~Ln1illL e"bL~ V" A, all the model modes appear in 011

the co1umnc of the trancition matrix and ono can thuc rotain columnc in tho ~amo po­

sition in all the matrices Xj • which makes it Dossible to use (4.18) in order to ex­press the nth column of T as a function of the (n-l) first ones. T is then a function of n(n-l) parameters, and (4.7) can be used to express ~(t) as a function of nZ para­meters :

~(t) n Alo;t I Xk e

"=1 ---- --­n(n-l)' n

(4.19)

When constraints are present on the structure of A, one must assume (and this assump­tion is easily checked by inspection of the model) that at least one state component \ say the 1tn) , s such that there exl s t de~t;emJl II!! fla Lh~ r,"V'" it to all the other3.

One will then U3C the ith column of each X. a£ tho jth column of T. J

Example : Parametrization of a second-order transition matrix.

If one retains the first columns of Xl and X2, T can be written as a function of two parameters a and b

T = [a I-a ]. T-I = [' 1,:] b -b 1 - 0

(4.20)

Taking (4.7) into account, the transition matrix can now be expressed as a function of four parameters a, b, Al and A2

83

a Q(g-o) I-a ~ t(t) e

Alt +

A2t e • (4.21)

b 1-a -b a

With the help of (4.14). the relation between the parameters of t and those of A is easily deduced from (4.21).

A (4.22)

4.3.2 Some eigenvalues of A are complex conjugates.

w~ 1 illli~ uur · ~~1v~::. h~r~ ~u ~rCln::.i~lun matrlc~::. all ~ntrle::. u1' which are

roal. Lot Ai and Aj bo two complex-conjugate eigenvalues

(4.23)

X. = 1. 1 .1

(4.24)

so that the lth ana Jtn COlumn Of I can De aeflnea as Tunctlons Of ~n parameters only, provided tno.t tne collMllns retQined nQve tne 3Qme posit.ion in X. Qnd X .. Under this

1 J condition. (4.7) c;till yipldc; l'I Ill'lrl'lRlPtri7l'1tion of cI>(t.) wit.n n2 paramotort, which can

be written in the form of (4.19).

4.3.3 Connection with Lagrange-Sylvester polynomials

From Sylvester's theorem [A4. K5] . any analytic function f can be con­sidered as a function of a square matrix A with distinct eigenvalues Ai through

n f(A) L Xi f(Ai)

1=1 (4.25)

with

84

n

j~1 (A -A-.1

t)

1L Ii (4.26) 1 n

II (Ai - A) j=1 ,1

provided the power-series expansion of f converges for all Ai' Equations (4.2) and (4.14) correspond to the Lagrange-Sylvester formula (4.25). respectively with f(x) = eX and f(x) = x.

The parametrization presented here has proven that it was possible to express the Xi wlthOut USing the eigenvalues Ai' which was not obV10US from l4.Zb).

4.4 APPLICATION TO CHECKING S.G. IDENTIFIABILITY

4.4.1 The experimental data are entries of +

To hl>!)in with. <:lIl'flO<:I> that <:0_ pntrip<: "'lJ of thp tr~no;ition m~trix can be experimentally measured

n Ak L

~i j (t) y. x" k e k';l 1J

(4.27)

where

)<.ijk ~ xkl ij (4.20)

Al U,vu\jh the (lI;.tual e~timat;on of the "ijk and Ilk from the numerical knowlcdgc of +lJ(t) at var;ou~ ti"",. t rlli.",c: w",ll Irnown rliffi,.,,,lt;p<: [I fi. M4. 1lR. R14 •• n] . th!'s!' parameters are s.q.i ..

For linPilr tr~no;formiltion syst!'ms. 4>IJ(t) is the result at time t in class i of a unit injection (Dirac impulse) in class j at time 0, since the observed output will then be

y(t) = lO ... OlO ... OJ +(t) l'

ith

o

o 1 ... jth o o

l4.Z!I)

85

nn: t:xllt:I-inIt:IIL ~LI-u\;LUl-t: will Ilt: ~ulI.lldl-iLt:d Il'y d !>'1uan: dlTd'y wiLt. 112

divicionc, aach of tham baing accociatad with ona antry of ~. Any divicion corracpon­

ding to a measured entry of t will be shaded. This concise notation makes it possible to describe experimental situations which cannot be characterized by merely stating the structure of Band C. For example, the experiment structures described by Figures 4.1a and 4.1b are both associated with

B = [~ C [ ~

FiO 4 h

o 1 ~] .

FiO 4 lh

Fig. 4.1 Two experiment structures leading to the same Band C.

4.4.1.1 ~Q_~Q~~!r~i~!_~~i~!~_QD_~

(4.30)

The previously stated properties of the transition matrix imply some very general results.

Result 1 A sufficient condition for aij to be s.g.i. is that +ij be measured.

Proof: the parameter a .. can be deduced through (4.14) from x-' k and Ak lJ lJ

(k=1,2, ••• ,n), which are s.g.i .• One thus retrieves a classical result in compartmen-tal analysis [ R4] : if one knows the result in compartment i of a unit injection in compartment j, then the entry aij of the state matrix A is s.g.i ..

Result 2 It all out one Of the entries of the main diagonal of ~ are measured,

the unmea3u.-ed ent.-y ." jj can be I t:\;UJI:> ~ i ~u Lt:d from the n.ea:>ured one:>, (and .:.on",,­

QIIPntly it<: mPll<:llr .. mPnt would not provido;o any ~tructural information)_

Proof; the mi33ing entry Xjji of the moin diagonal of each Xi con be computed

by means of (4.17) to oht;dn thp ;In~lyti(" PXf\I"Pc:c:ion of oi>J/t) _ Notp thllt II JJ ;c: thpn

s.g.i. from Result 1. This is a global confirmation of the local result obtained in Example 2 of Section 3.5.4.

86

R",:>ul L 3

A necessary condition for structural (lo~al or glnhal) irlpntifiahility of an unconstrained A is that at least n entries of ~ be ebserved.

Proof: When no constraint exists on A. the dimension of the parametric space is n2, but the maximum number of independent parameters that can be deduced from the observation of m entries of the transition matrix is n + m(n-l). i.e. n parameters ~k

plus mn parameters xijk ,related by m equations since (4.18) implies

II

k~1 xijk = 6ij • (4.31)

For all the entries of A to be identifiable, m must be such that

2 n + m( n-l) ~ n ~ m ~ n . (4.32)

Result 4 A suft1cient condition tor A to be s.g.i. is that a row or a column of

the tron~ition matril< be ob:>e.-ved.

PrOOf: From a cOlumn Of ~ one knows the n eigenvalues Ai Of A. and n corres­pondin9 independent eigenveetors, which con be eho3en 03 the eolumn3 of 0 3imilority trilnc;fnnniltinn T rliilonnaH7ino A. Thllc; thp matrill Arlin hp rnmrlltprl thrnlioh thp in­verse transformation

A = T 1\ T-1 . (4.33)

Similarly. the analytical knowledge of a row of ~ makes it possible to compute AT and then A.

Result 5 A sufficient condition for A to be s.g.i. is that one observes n+j-l

entr1es ot ~, located on J rows 1n SUCh a way that (i) No ob3erved entry eon be deduced from the other3 throu9h (4.11),

(ii) At lpac;t nnp pntry h nhc;I>rvprl in parh rnlllmn nf '"

Proof: Whenever this condition is satisfied. a column of t can be reeonsti­tuded with the help of (4.11). so that the sufficient condition of Result 4 holds.

Another sufficient condition is obtained by exchanging the words rows and columns in Result 5. These conditions generalize Result 4.

Figure 4.2a presents some examples of experiment structures which make A s.g.i •• The arrows suggest a possible progression for the reconstitution of a column of ~ using (4.11).

n = 4 j = 1

n = 4 J = Z

87

n = 4 j = 3

n = 4 j = 3

Figure 4.2b gives some examples of e~periment structures for which the sufficient condition of Result 5 does not hold, in spite of the great number of ~ij measured.

fill. 4.ZIJ f}\l'~rlm~IIL ~tru(;tur~~ fur whl(;11 tile (;ond1tlon of Result 5 does not hold.

4.4.1.2 ~~~r~!_~r9£~~~r~

If none of the preceding necessary or sufficient conditions is conclu­Slve, 1t 1S adV1saOle to cneCk tne model Tor local ldentlTlaOlllty, Tor example Wltn one of the mcthod5 prc5ented in Chapter 3, before any global 3tudy. If the model pro­

ves to bp lnr.~lly idpntifiahlp. onp r.an thpn tp~t it for alohal idpntifiahility hy

working on the expression of ~ as a function of n2 parameters. ~ will be s.g.i. if and only if these nZ parameters can be uniquely determined from the analytical expres­sions of the observed +ij(t) and from the constraints on the structure of A, which

88

con be token into oc.c.ount th,ou9h (4.14). E'lucaLiulI (4.14) also proves Wat wnenever ~ i~ ~-D_i • A i~ <.D.i. too. An important part of tho computation con~i~t3 of cxpre3 sing the transition matrix as a function of n2 parameters. but this can be done once and for all for any given order n.

Anothor po~~iblo approach ic to koop tho ontriac of ~i and ~i (i=I.2 ••••• n) as 2n2 parameters involved in the followinQ three tvpes of bilinear equations :

(i) equations (4.9) ; (ii) equations resulting. through (4.10). from the s.g. identifiability of

Un~ ellLrle:s lI" k ur xk (~=l.l •.••• n) for any 1 ami J sucn tnat cpo .(t) ls oOservell ; lJ lJ

(iii) equations resulting from structural constraints such as

tnrougn l4.b).

n , or.L aij = U

1=1

One then h03 twice the initiol number of porometer3, but thi3 ollow3 utilh",tinn nf vpry intprp"tin(J tprhnif'l'IP~ in nrtipr to rhprlc (Jloh;ol itipntifi;ohility by a succession of linear stages. See [N2, N3, 815] for further details.

Oelattre [09J has shown that, for any linear transformation system such

a .. <: 0 , a k' ~ 0 (j~k;j,k=l, .•. ,n) , 1.1 .1

(4.34)

the eigenvalue w1th the greatest real part 1S real anll ot mUlt1pl1c1ty one. In1S 1m­plie3 thot, fo.· ony two-c1033 tron3fonnation 3y3tem 30thfying (4.34), the eigenvo lUQ~ of A arQ alway. di.tinct and roal Untior tho<o rontiition~. thp l";oramptri7;otinn (4.21) can always be used.

Consider, for example, the experiment structure described by Figure 4.3.

Fig. 4.3.

89

The oOserved entry or the trans1tlon matrlx b ip12(L), wllil;h \;urn::. ... umb Lu Un: n:­

~ult in cla~s 1 of a unit initial condition in class 2.

With no restriction on the structure of A the necessary condition of Result 3 does not hold. Consequently we introduce the following structural constraint

(4.35)

From (4.21), ~21(t) can be written

(4.36)

and the parameters whlch can Oe estlmated rrom ip21 dre

Al • Az. a(!-a) _ (4.37)

Constraint (4.35) is accounted for with the help of (4.22)

(4.36)

~, and thererore A, wlll Oe 5.9.1. 11', and only lf, tne set Of equatl0ns

(4.30)

where kl and k2 can be estimated from the data, has a unique solution for a and b.

The set of eQu~tions (4.39) is eQuivalent to

~ al _

l h = (4.40)

which has two distinct solutions (a1.b t ) and (a2,b2). associated with two output-in­distinguishable models: This is why we have found two different values of ! when Hlent1tY1ng the mollel wHh th1s structure presented 1n {;hapter 1.

The most common two-class models are easily studied in the same way. f19ure 4.4 summar1zes the results oDta1ned. 1n the rorm or a two-d1mens10nal array. rach row of this array is associated with an experiment structure, and eaoh column

corresoonds to a model structure. In all cases. the s.Q.i. and s.l.i. oarameters are

90

i"oJi~aLt:oJ. NuLt: LliaL Lilt:! uLllt:!I' ~LnH;Lurt:!~ ubta1ned by excnang1ng Classes 1 ana z nave not boon includod.

- '- 1 Z 1 Z 1 Z 1 Z

Exper

~ ;ments

~ll·~Zl'~ · !l i S. g.; S. g.; S.q. ; S. q.; 5 .q.;

.. of sol. 1--

a11 ,a12 s.g. ; s.g. i s. g.; S.g.; S. g. i S.g. i

.. uf ~o1.

an,aU ~.g. ,

°U,022 3 • .t.i 3.g.; "·9·; ".9. i :)'!I' i :>'!I' i

two <;01 .

all ,a22 S.g.; all ,a22 S.q. ; all ,a22 S.q.; S. g. i s. g.; S. g.; .. of sol. .. of sol. .. of sol.

all ,an s. g.; all s.g. ; all s.g. i s.g.; s. g. i s. g. i

- of :;01. - of :;01. - of :;01.

a2l ~'!I' i °21 :..y. i a2l ~.g.1 .,.g. ; £ .g.; an ,aZ2 c.I,.; £.g.;

.. of sol. .. of sol. two sol.

4.4.2 Method for ;lny B ;lnd C

We shall assume that the model ;5 cOnnectable, and that ~ can be part;­tioned into a vector ~ of the parameters of A and a vector !r of the parameters of 6, c ana D, so tnat tne equatlOn Of tne moael can De wrltten

~ ~ = A(!a) ~ + B(!r) ~ , ~(O_) = Q '

t 1. = C(!..) ~ + D(!..) ~ • (4.41)

rrom the definition 3toted in Chapter 2, thi3 model will be 3.g.i. if, and only if, fo!" ~lmn~t ~ny ~ ~ A nn~ h~~

91

. . . q!r)exp [A(.!!a)'t] B(!r) := q!r)exp [A(!!11)'t] B(!r) (4.4Z)

Following th~ sam~ 5t~P5 a5 b~for~. w~ shall r~param~triz~ th~ transi­

tion matrix so as to exhibit the eigenvalues of A. The parameters to be considered will be

(i) the n eigenvalues Ai(~) of A • (11) the parameters £(~, ot all Xi (1=1 ••••• n).

The last equation of (4.42) can also be written

A.( a )t C(~r)Xj(~(~))B(!r)e J --0

(4.43)

Sinl':p thp mndpl is ~onnp~hhlp. thp pigpnvalllPc; of A arp 5.g.1.. and. with a c;llihhlp

ordering of the eigenvalues. one has

(i~l ..... 11> • (4.44)

Condition (4.42) then becomes

! € 8 ,

. Ai(~) = Ai(~) • (i=I •... ,n) ,

For Condition (4.45) to be satisfied, it is sufficient that the corresponding set of equations has a unique solution !r' £(.!!a)' Indeed B, C and D are then uniquely defi­ned, whilst A can be uniquely computed through

(4.46)

Two oxamploc. corrocponding to two very common cituationc where the

matrices Band C contain some unknown parameters are now presented.

92

~onsider the two-class system described by Figure 4.5.

ng. 4.6.

After having injected a unit amount of tracer into class I, one is supposed to mea­sure the proportion of tracer in the total quantities in each class, i.e., with the notatlon ot ~hapter 1 , Xl/X~ and X2/X~. Ihe control and observatlon ~atrices are then

c (4.47)

From (4.2l) and (4.45). the parameters AI' Az •

(4.48)

and

(4.49)

are s.g.i •• They make it possible to compute a, X~ and b/X; • Thus, from (4.ZZ), all dlllJ d 22 eln: :.. \/. i.. "II ilt:: el12 elml el2l elf-t:: IIUII :. • .t. i ••

~rom a purely structural pOlnt ot vlew, measurlng tne speclTlc actlVlty of clQ3~ 2 add3 no information. In practice, however, 3uch Q mco3urcmcnt h~5 obviouo-

93

If a £tudiod trancformation system is too complex to be handled 03 Q

whole. one may wish to break it UP into several subsystems. Consider. for example. the transformation subsystem represented by Figure 4.6.

f19. 4.0.

It is a part of a model associated with the iodine metabolism in the rat [LI, WI] . After having injected labelled iodine into class 1, one observes xl' considered here as the input of the subsystem, and (x2 + x3 + x4), corresponding to the labelled io­dine in the whole thyroid gland. The dynamical behavior of the subsystem is described Loy

x2 -a32 aZ3 0 x2 a21

x3 a3Z -(aZ3+a43 ) u x3 + u xl'

x4 n "43 n x4 n

y = [ 1 11· [::J ' (4.50)

To"ing intu o<;.<;.uunt U .. , :>t,"u<;.t".-.. ur A, th", o:>:>u<;.iot",d L,"on:>iLiuII 111(1-

trix can bo immodiatoly doducod from that of tho two-cla£~ ~y~tom ,

94

" t "t Ot I ;II' 1 +(1-;1)1' 2 +np I ;1(1-;1) "1 t "?t Ot : _

I b (e -e J+oe : u --------------------------~-------------------------------------1----A tAt Ot: "t "t Ot I

~ = b(e 1 -I' 2 )+01' I (I-a)e 1 +ill' 2 +np ! 0 --------------------------r-------------------------------------1----

"It A2t Ot i " tAt Ot ! Cl43b[ ~ - ~ T(l - l)e ] l Q [..l.:A cIT A. e 2 _(I-a T A.)" ]! 1

"I A2 "2 Al J 43 "1 "2 "1 A2 : (4.51)

thp p"r"mPtpr "4.)" whirh <:til1 "ppp"r<: in (4.1;1).i<: nhhinprl ;1<; ;I fllnrtinn nf ;I. h.

"1 and "2 by writing, with the help of (4.14). that the sum of the entries of the se­cond column of A is zero.

Thus, with the given experimGnt £tructurg , ~1' AZ'

ana

are s.g.i •• From (4.53) - (4.55) one deduces

a+h 1 [ C("IXl+A2X2)B - lZ] = k1 Al - },2 qXftX2TX3)B

a43b ~l AZ C XJ B

= "1 - "2 C{X1 + X2 + Xl) B

Taking (4.57) and (4.58) into account, (4.52) implies

k2 + "1 kl a - (1-k1)("2 --AJTIT

= k2

(4.5Z)

(4.54)

(4.55)

(4.56)

(4.57)

(4.58)

(4.59)

95

Liun5 (4.5(j) - (4.59) ","kt: iL I"u:.:.iLJlt: Lu ut:Lt:nlllllt: II, LJ, "43 IIIIU "'21 ullllju\::ly. tho paramotorc of tho matricoc • and Q ar~ thuc e.g.i •• and tho paramotorc of ~

S.Q. i. tOO. since they can now be uniquely comDuted with the helD of (4.46).

4.4.3 Problems raised by inequality constraints

Tn ~ny ca~p~. tho ~rlmi~~ihlo ~paco Q ann ~v aro not i~omorphic. bocau­

is restrained by some inequality constraints induced by the physical nature of studied phenomena. For example, linear compartmental models satisfy

(;,Ij ; i ,j - 1, ... ,n) , (4.60)

{J 1, ... ,n}. (4.61)

J now, lnequallty constralnts nave not been accounted for, bUt one can lmaglne : where the ctudied modcl would be only lOCQlly identifiable in ~v , but glo­

~ identifiable in A. fnr t.hp inprpJ~lity rnn~tl"~int~ wnlllrl pliminato al1 tho 10-

solutions but one. To our knowledge, no systematic approach exists to check a model for

tifiability when inequality constraints are present. As the computations invol-tnese constraints rapidly grow intractable, the Simplest approach seems to be to

y <2 pust",-io,-i (i .". on(;" th" ""'CI:>U,·t:II't:llb IrClV\:: LJeel1 I"e"rUIlII\::U .1IIU CI lV\;dl IIIvuel 1) whothol" othor pocciblo loeal modole (if any) caticfy tho eonctraintc defining

f not. one will be able to conclude a oosteriori. and thus in a non-structural that the definition of the admissible parameter space 0 confers global identi­

ility on the model. To generate the set of all possible local models to be con­red, one may use the technique of exhaustive modelling to be presented in the (;lrdl" Ler·.

CONCLUSION

The approach presented in this chapter has enabled us to prove some rdl re~ults on the structural global ldentlflabll1ty Of l1near tlme-lnvarlant a-cpaee modele. When thece generQl recultQ do not leQd to <2 eonolu3ion, it i3

sab 1 e to reserve thi s aooroach for models whose structlJr~ 1 1 nr~ 1 ; npntifhhi 1i ty

Jeen previously checked with the help of the techniques presented in Chapter 3, e local identifiability is a necessary condition for global identifiability.

96

Ttlt~ computCIt.lona1 I.lurd~n IlIIp11~LI I.ly ~II~ ~~)L ur CI ITlUd~l rur ~Lr'uI,;Lur'C11

gl oba 1 i dentifi abi 1i ty growe very rapi d1y witn tn .. mod .. 1 ord .. r, wni en confi mi. tn .. interest in breakina uo the whole model into submodels. the identifiability of which can then be studied separately.

No" let u~ eXClmine to "hClt extent thh ClpproClch preve, to. be ,uited to tn .. imp1 .. montation of tn .. thr .... policio~ concornin(J unirlpntifiah1p mnrlph pnll_r~tprl

in Chaoter 3. Anv additional constraint on the model structure is easily introduced with the help of (4.14). Similarly any complementary measurement results in new in­formation on entries of the transition matrix. However it may be very difficult to see the influence of a given constraint or of a given measurement on the model iden­tlrlabl11ty w1thout ~tartlny ttJ~ wllo1~ prol,;~uun:~ C1\jClill. A n:C1~on rur' thi5 i5 that the cnoocn paramctcro p(!a) do not correcPo.nd to. tno dogr .. e. of fr .. odom of tn .. prob1om. but are associated with a oarticular oarametrization of the transition matrix. This gives us a first motivation to look for another parametrization of the problem, ex­hibiting as many parameters as there are degrees of freedom in the problem.

As to the third policy, which consists in looking for the set of all the output-lnalstlngulsnable moaels, 1t wou1a reqUIre us to expres~ tile :.up~rCll:lumJcluL w­

efficient3 0.3 functions of so.me judiciously cno.ocn paramctorc. But such computatio.n. arp pradirally rliHirll1t. Thi.; is ~ <;econd motivation to modify the oarametrization of the problem, with the help of a method which forms the subject of the next chapter.

5.1 INTRODUCTION

Chapter 5 Exhaustive Modelling for Linear Models

The aim of all the methods presented in the previous chapters was to enable us to design s.g.i. models, i.e. model structures such that the set of output­InllbLlnyubhill)lt! mollt!h rt!duct!. to ont! t!lemt!nt only. UllfortunClLt!ly .uch ClII Clllllroac;h

of ton compolc uc to impoco on tho ct~uctu~o of tho modol como ~oct~;ctionc which havo

no justification and result from our inability to make other measurements. In this chapter a method, which we have named exhaustive modelling, is

proposed to obtain the set of all output-indistinguishable models, without trying to reduce it to one element by imposing arbitrary structural constraints. This set of models may De cons1dered as a un1que one, wh1cn preserves tne amD1gu1ty Of tne or1g1-nal experiment.

C\lIrh ... flnint nf vipw WAC; fi~c;t. fl~pc;pntpn hy Rpl"IIIAn Ann <;rhnpnfplti [R4 l. and later by Rubinow and Winzer [R4] ,for autonomous models with partially observed states. It is extended here to the multi variable case by taking advantage of proper­ties of minimal representations. For that purpose, a particular structure for the ma­trices Band C will be of some help. As it is a rather common one, we shall name it Lilt: :.LC2I11JC2nJ :.L,·uI..Lun: (VI· 6 121111 C. TIl!:: I..C2:.t:: vr I<.IIVWII 121111 :.LC2l1lldl-1l 6 dl,1l C will LJt::

concido~od fi~ct. bofo~o 90no~ali%in9 tho p~ocodu~o to any known o~ pa~t;ally unknown

Band C. The connection of this standard structure with Kalman's canonical form will then be developed. Finally some examples of applications will be presented.

5.2 CLASS or THE STUDIED MODELS

We shall consider systems which can be modelled bv linear time-invariant state-space representations of finite dimension. For the time being, these models are supposed structurally controllable and structurally observable.

98

Th~ lIIuu~l :.Lru(;Lur~ Is dei1ned by the parametrlzatlon Of the TOllowlng oquation. accumod nonpathological in the sense of Section 2.2.~ :

~ i(t) = A(~) !(t) + B(~) ~(t), !(O_) = Q '

~ tJt) = C(~) ~(t) + D(~) ~(t). (5.la)

It _ilnc: thilt thp pilrilmptpr vprtnr ~ ("an hI> Ilni'llll>ly dl>dllcod from tho knowlodgo of A, B, C and D.

Nonzero initial conditions ~(!) can be dealt with by considering the equivalent model

! i(t) = A(!) ~(t) + [B(!)I~(!)][;;:~] ,~(O_) = Q '

L(t) - c(~) ~(t) T O(~) ~(t). (~.I~)

Another approach is proposed in [ T9] •

(Ao, BO, Co, DO) @ (A(!o), B(!o), C(!o) , D(!o)), !u€0. (5.l)

This model may be non-numerical or numerical, depending on whether the study is struc­tural or not. We are interested in buildino the set (AS. BS • CS • DS)1 of all control­lable and observable models having the structure defined by the parametrization of (5.la) and output-indistinguishable from the generating model (Ao, BO, Co, DO). From this set, the associated set {!~} of the possible parameter values will then be de­aucea easlly.

Each of the3e minimal repre3entation3 (AS, DS, CS, Os) can be deduced frnm (AO. RO. rOo nO) hy ~ (nvn) <:tatp-<:parp c:imilarity tr~nc:fnrm~tinn T [KI). Tho mo­del thus has n2 degrees of freedom when the structure for A, Band C is free. Note that D is always s.g.i., since it is invariant under any state-space similarity transformation. The existing constraints on the structure of the model and of the ex­periment will reduce the number of degrees of freedom, and we shall use the tree en­L,i~;) vr T Lv ~uihJ 0 ,,~ .. )Jo,o"'~L,iLoLiv" vr LIr~ lIIuu~l lIIoLrh.t::>.

Three situations may occur, depending on the number of possible T.

(i) The solution T ~ t , which always exists, is unique. The model in then globally identifiable, and the final set of models reduces to the generating model.

(ii) The set of the solutions T. for T is finite, or at least denumerable. , ---The model is then locally identifiable, and all possible models are generated by

99

{AS} .. {T i AO Til I i~l.Z •••• } •

{Bs} = {Ti BO I i=1.2 •••• } •

{Cs } = {Co T~l I i=1,2, ... } ,

{O"} = DO.

(5.3)

(5.4)

(5.6)

Globally identifiable parameters. if any. are easily detected. since they take the same value in all elements of the model set.

(iii) The set of the solutions for T is not denumerable. The model is then unidentifiable. and the model set obtained expresses the ambiguity of the experiment considered. Insight is given into the relationships existing between unidentifiable parameters. Illooally or locally l<1entlflaOle parameters are agaln easlly foun<l.

Thus the method will make it possible to detect which parameters are s.g.i. or s.t.i •• and to find how the unidentifiable ones are related. In what fol­lows two cases will be distinguished. depending upon the degree of our knowledge dbout 6 dnd C.

5.3 THE MATRICES B AND C ARE KNOWN

The only parameters which must be tested for identifiability are the unKnown entrles o.t A, slnce

To begin with. we 3hall con3ider a particular - and rather frequent 3tructure for

R lind r..

5.3.1 The matrices Band C are standard

By dofinition. Band C aro standard [l5) if thoy sati~fy

, t i 4) q : q.m-q -i>------------.------<i)-----------

gn.m - q;I-_<!!L _____ i------.Pi<!!..~~--m-o.o I m-o --------------,------------------o , 4) n-m-p+~.~ ! n-m-~q.m-q

(5.0)

100

c - [ lop p,n «>p,n-p] , (5.3)

For a transformation system. it means that the experiment structure is as follows:

- Tho fir~t q ctato componontc aro ac~ociat~d with thp cla~~p~ which ~rp both injected and observed ;

- The p-q following components are associated with the classes which are obseryed but not injected;

- The m-q following components are associated with the classes whlCh are 1nJe~ted but not ob~erved ;

- Tho n-m-p+q romaining componontc aro ac~ociatod with tho cla~~p~ which are neither observed nor in.iected.

Tho ~tato variab1o~ aro thuc groupod into four mlltlllllly pyrludvp fl)lrtc:. which we shall respectively name observed-controlled, observed-uncontrolled, unobser­ved-controlled and unobserved-uncontrolled. Note that, with our assumptions, unobser­ved and uncontrolled state variables are nevertheless observable and controllable.

Now 1pt lie: PY)lminp thp c:trllrturf' of thf' state-space similarity trans­formations T preserving the standard form of Band C. let T be partitioned into six­teen blocks Tij , the dimensions of which are compatible with those of the blocks of Band C, as defined by (5.8) and (5.9).

• Tll T12 T 13 T14 Q (observed-controlled)

-t-T21 T22 T23 T24 p-q (observed-uncontrolled)

T -t-131 T32 T33 T34 m-q (unob~ervell-(;urrLr·u11 ell)

1-T41 T42 T43 T44 n-m-p+o (unobserved-uncontrolled) ,

(Ci 10) In order to preserve the standard structure of Band C, T must satisfy

II Q) i(I

~l fl T12 T13 '14] q C = CT# (5.Il)

«> .. «> «> T21 T22 T23 T24 p-q

B = TB~

t q

I[)

()

I[)

t m-q

I[)

101

Tn T13

T21 T23 (5.12 )

TJ1 TJJ

141 T43

The only blocks which are not determined by (&.11) and (&.12) are T32 , T34 , T42 and

T44 • The associated entries of Tare arbitrarv as far as T remains invertible. 50

that the transformation T can be written

t () q () ()

I[) t I[) I[) p-q T

() T;,z t T;'4 In-q

0 T42 0 T44

where the dimen~ion~ of the block~ of arbitrary parameter~ are

dim [T 32] (m-q) x (p-q) ,

dim (T 34] (m-q) x (n-m-p+q)

dim (T4?] (n-m-p+q) x (p-q)

dim [T 44] (n-m-p+q) x (n-m-p+q)

(5.13)

(5.14)

(5.15)

(5.16)

(5.17)

Lt:!L a bt:! LIlt:! vt:!(;Lur uf LIlt:! t:!lItrlt:!!> uf Lht:!!>t:! blu(;b. Tht:! ()llnell!>lulI uf a b equal to

the number + of degrees of freedom present in the problem when the structure of A is

free.

~ = Olm [~j = (n-m).(n-p). (!l.llS)

This relation generalizes (2.45). As pointed out by Delforge [DIS] , this result im­plies that there must be at least (n-m).(n-p) constraints on A for the model to be (locally or globally) identifiable. In the case where all classes are observed (ncp) or o;;ontl-olled (n~"'), A b :>.!:j. i. wiLhuuL (111.1 I.UII"L'-dillL. A !>imild'- It:!:>ul L (;dll bt:!

found in[Bl] • and in Chapter 4.

It is easy to show that the inverse 5 of T is qiven by

102

4q c c 4)

S ~ T- 1 = II> ~ () ()

p q (5.19) 4) -1 T42 ) 11 -1

(- T32 T T34 T44 m-q (- T34 T44)

() -1 (- T44 T42 ) ()

-1 T44

NOte tnat T IS invert1Dle 1t, and only it, T44 is invertible, which is true for any T44 3uc:h that

l T~~ € IR (n-m-p+q) - {T44 I Det[ T44 ] = O} • (5.20)

Thus T is generically invertible. The set of all values of T44 such that it is not invertible is atypical, and has not to be taken into account for structural studies. Tin" t:lltr-it:!> ur T 44 I..dll thu!> I.>t: I..um.ldered a:; free.

When there is no constraint on the structure of A, the set of all ma­trices A~ compatible with the standard structure of Band C can be written as a func­tion ot tne arD1trary parameters ~

S 0 -1 0 A (.z) = T (.z) A T (.z) = T (.z) A S (.z) (5.21)

where

a € A ~ (a I Det(T44]1 Ole /R(n-m).(n-p) . (5.22)

Ir AO dllll S d't: I'd,tiLiullt:1l d!> T illLu !>iALt:ell 1J1ul..k!> Aij ami Sij' Lhell (5.Z1) re!>ult!>

;n (5.23).

Equation (5.23) deserves some remarks :

(i) Four out of the sixteen blocks in AS, namely A~l' A~3' A~l and ~3 do not depend upon T or S, and are thus s.g.i .• These blocks are associated with the L'-dll!>rUl"llIdtiulI!> r'-UIII d l..uIILrulled I..la!>!> Intu dn ulJ:>erved une. One retrIeve!> a cla!>:>I­

c:al result (already ment;oned ;n Sec:t;on ~.4.1.1.) : ;f one observec the result ;n

class i of an injection in class.i. then aij is s.o.i.. (ii) All parameters a would appear in linear equations if structural cons-

o s s AS -s tra1nts on A31 , A33 , 41' and A43 were properly chosen. (iii) It stands to reason that the choice of T32 , T34 , T42 and T44 as the

:.eL uf free pdrdmeter!> !! I::. arbItrary. Another possIble choice would be to select

(n-m}.(n-p) froo paramotorc among tho 2(n-m).(n-p) ontr;oc of T32 • T34 • T42 • T44 •

a AS

o 1\11

1 1 1 I 1 1

o 0

0 A

12 +

A13

$3

2 ..

A14

5~2

~3

o 0

All

5 34

+ A

14 5

44

I I

1 _

__

__

__

__

__

_ ..J

. __

__

__

__

__

__

__

__

__

L _

__

__

__

__

__

__

~--------

__

__

__

_ _

I I

I

o A21

i 0

0 0

I ~

1 I

A22

+ A

23

S32

~ A

t4 5~

2 i

A 23

: I

I 1

o 0

A23

5 3

4 +

A24

544

1 1

1

----

----

----

-t--

----

----

----

----

-L--

----

----

----

i---

----

----

----

---

I 1

1 0 0

)

1 0

0 _

o.

I T3

2 A

22 +

A32

~

T34

A~2

: (T

32

A Z3

+ A

32

+

34 A

43. 53

~ 1

1 I

1 o

0 0

1

0 0

0 0

0 0

1

T 32

A21

+

A31

+

T34

A411

+

(132

A 2

3 ~

A33

+ T

3~ A4~)5~2

T 32

A23

+

~33 +

T3

4 ~4

31

1 1

I I

I +(1 3

2 A~4

~ A~4

+ T3

~ Ag

~)5~

2 :*

(T32

A~4

+ A~

~ +

-34

A~4)

54

t I

I -

---

----

----

-1--

----

----

---

____

___

.l. _

__

__

-----

------t-

-----------

-------

o 0

42

A22

+ T

~4 ~

42

o 0

(T 4

2 A

23 +

144

A43

) S3~

o ~

T42

A21

~ TL

4 A

U

(D

C

+ T4

2 A2

3 of

T4~

A~3)

$32

o 0

T42

A 23

~ T

~4 A

~3

+ (

T42

~4 of

T

4~ A

~4)

$42

o 0

+ (T

42 A

24 +

144

A

44)

54~

4 q

~

• p-

q -----

4--

m-~

..

• n-

n-p+

. •

(5 .Z

3)

104

332 , S34' S42 1111\1 s44· Tilt: IJ~!> L I,;hv h;~ IIl<Iy \I~I'~II\I VII Lh~ I'v::.::.llJl ~ ::. Lr"ul,; Lural I,;UII::'­

traintc accignod to AS.

(iv) If one considers the blocks T3£, T34 , T4£, T44 , S3£' S34' S4£' S44 as a set of 2(n-m).(n-p) related parameters, then constraints such as

~ T J!

s aij ()

~ !> () aij , ,

will result in at most quadratic relations. (V) Any strUctural conStralnt asslgnea to A~ must alSO oe asslgnea to AO,

3;nce

~ Consider the four-class model described by Figure 5.1.

Claccoc 1 and 2 aro ob~orvod. whil~ cla~~9~ 1 ~nrl 1 ~rp rontrollpd. The generating

model can be written

o

1

o o

o

o

o 0

o

o

105

.!!'

(5.24)

Tne matrICes 6n anO cn are stanOarO. wnen no constra1nt on A ls taken 1nto account, th;3 model h~3 16 p3r3meter3 but only 4 de9rcc~ of freedom from (6.1S) . Any ctato­

space similarity T preserving the structure of BO and CO can be written

1 0 0 0

0 1 0 0 T(~} (5.25)

0 a1 a2

0 a3 0 a4

So that (5.21) yields (5.26).

If one wishes to make the model s.g.i., one must assign at least four structural constraints to AO and AS. As will be seen in the next three cases, exhaus­tive modelling makes it easy to study the influenc"e of these structural constraints on the identifiability of the model.

AS (2

.:

°

all

i I I I I o

0 a3

( 0

0)

a 12-

al

al3

+ a4

a2

a13

-a11

. o a 1

3

o 0

aH

-a2

all

a4

I --

----

----

-t--

----

----

----

----

----

--+

----

----

---t

----

----

----

----

----

----

o I

° 0

a3

0 0

a 21

! a 2

2-a

l a 2

3+ a

4(a 2

a~3-a

~I.)

a~3

a24 -

a2

a23

I I

I a4

----

----

---r

' ---

----

----

----

--I

L --

----

--,-

----

----

-, -

----

----

----

----

----

--:

I I

I I

I I

'I

I o

0 0

I 0

0 a3

I

I J

+

'J Q

0

0 °

a 31+

a 2 a4

1~a1 a

21 i a

32+a

2 a4~

+al a

~z-a4(

a~4+a2

a~l.+

al a~~

) l a~3

+~ a

: 3+

al a~3

:a34

02

a~4

+01 a

Z4-~(a33+a2a43+al~23)

I I

I I

I I

I a"

a3-1

1a,

I I

+

' '(

Q

0 0

I I

a33+

~ a4

3+~

a'3

) I

I I

a~

'I

I

------

---!--

----

----

----

----

----

---+

----

----

--J-

----

----

----

----

----

--I

I o

C

I 0

0'3

0

I 04

a n

+a3

a 21

: :1

4 '4

2+1 3

a22

-¢'4

(04 a4

4+~

a~4)

: I

I I

I I

I I

a2~3

-ola

I

° °

I ~

4 (

0 0

0 0

I a

a -a

(:1

,

+x

aO)

I a4

a43

+a3

a2)

&4

a 4

3+a

a I

a (

+ 3

24

2 4

43

3 23

I

a 4

3 23

I

~4

I I

I I I

04 :14

(i .2

6)

8

107

C4:.e 1

Assign to A tho following ctructural roctrictionc

(5.27)

which result in the model described by Figure 5.2.a. Constraints (5.27) imply

(5.28)

ami the 5et uf' all pus51ble T redU(;e5 tu the 1dellt1ty matr111. Thus the mudel 1s s.9.1..

Case 2 Suppose now that the constraints defining the model structure are

o s 0 s 0 s 0 s °14 - °14 - °24 - °24 - °31 - °31 - °41 - 041 - 0, (5.f9)

which corresponds to Figure 5.2.b. Constraints (5.29) imply

'" 1 = "z = '" j '" 0 • ('> 10)

but a4 is free, so that the model still has one degree of freedom. Indeed, once 05 .. . . 0 5 414 = 414 = 0 J:, L4kell IIILv 4\;WUIIL, Llle f4\;L LII4L d24 = 424 = 0 \live:. IIV ddd'LlvlI4l

structural information.

The set of all possible state matrices is given by

0 an

0 a12 0 a13 0

n a2l n a22

n a23 0

A!: (a4) 0 (5.31) 0 0

a34 0 ilJZ ilJJ (14

0 0 a4 a42

0 a4 a43 0 a44

'he model 1S un1dent1tlable, but al I lts parameters whlch do not depend on (14 are ".y.i ••

Case 3 Finally assume the structural constraints to be

4 c =.I ai4 = u. 1=1

108

Ttn:~ curre~poruJtng lIIullel I~ de~l,;rlbed by ftyure ~.Z.c. ConHralnt~ (~.JZ) lmply that

( 0 0 0 0 tho~o are t wo dictinct colutionc for Q. The firct one Ql - QZ - Q 3 - 0 , Q4 - 1)

corresDonds to T = t . The second one ;s

ao a404 aO 1 1 1 44 1 _ -..4.4.

~1 - 0 • ~2 - 0 • ~3 - -0- (1 - --0--) • u4 0 a24 a22 a22

(~.33)

The set of all possible state matrices thus contains two distinct elements

0 °11

0 "'12

0 "'13 0

0 a21 0 a22

0 a23 0 a24

AS(!!.o) = AO = 0 u a32

0 a33 0 a34

0 °41 0 0

°43 00) (°24 I °34

and

0 all

u a12 u a13 0

0 a21

As((h

0 0 0 o a22 a44 a23 a24 -0-

a44

0 00 _ 00 0

o + 0 (44 22) aO 0 °22 a12 a14 n :n a34 0 a24 a44

o

where

109

Fig. 5.2.a 5.g.i. model.

r;9. ~.Lb Un;dt:IILi r;"LJlt: lIIudt:l, Fiy. 5.l.l: 3.!:'.1. model, with one degree of freedom. with two distinct solutions.

110

The model .:;on"ide,-ed i:; :> • .t.i., amI thb n."ult lIa" IJ~~I ulJtdiu~d from

alnh~l rnn~;rl~rat;nn~ Contrary to tho ro~ult~ pro<ontod ;n Chaptor 3. ;t do~e not

rely upon a limited expansion around some nominal point in the parametric space. Now we know the exact number of the possible models, and the values of the parameters as­sociated with each of them. The parameters all' a21 , al2 , a13 , a23 and a33 , which are identical in A3(~0) and A5(~1), are s.g.i .•

Conclusion

By using the standard structure of the matrices Band C, the set of all minimal and output-indistinguishable realizations (A", B", C", D") has been determi­ned. When no constra1nt on A 1s taken into account, there are (n-m).(n-p) independent paramete,-:; in AS, whi.:;" an: a5:>o\"idt~d wiLh th~ ",,,;,,tluy d~yr-~"''' ur rr-~~dulII.

Any a prior; algobraic conctraint on tho etructuro of A yiold< an al­

aebraic relation between the p~rameters characterizina the transformation T. This ma­kes it possible to study the effect of these constraints on the identifiability of A.

It has to be noted that any possible a priori knowledge on the sign of some entries in A has been ignored up to this PQint ; it may allow one to restrict the final set of output-indistinguishable models further.

The complexity of the procedure mainly depends on the dimension of ~ ; if the number m of the state variables controlled and/or the number p of the state var1ables observed are close to the model order n, then the number ot degrees ot tree­dom i:; :;mall, even if the model order i:; large. Thu3 the method i3 parti,:;ularly well

<uitod to ouch ca<o<_

E"ten,,;on of the above ,-e3ult3 to the ':;03e of a.-bitrory matrice3 D and

C [W10. W15] ic di<cu~<od in tho novt ~oction_

5.3.2 The matrices Band C are known, but non-standard

Assume now that Bond C arc known, arbitrary but of full rank. Thie

la~t hYnnthPsi~ amounts to assuming that the redundant components of X and Q have been discarded. In order to take advantage of the previous results, a number of li­near and biunique transformations will be performed on the state, the inputs and the outputs of the model.

5.3.2.1 ~t~o~~rQi~~tiQO_Qf_~~_

First of all note that the standard Band C satisfy

C B = [~ :J, (5.34)

111

where q 15 the number or ob5erved-~ontrolled 5tdte vdr1dble5.

Since the product CB is invariant under any ~t~tc-~p~cc cimilarity

transformation. such a transformation cannot be used to put Band C in standard form. unless CB already satisfies (5.34). Hence we look for transformations to be applied to the inputs and outputs of the models (Ao, BO, Co, D°) and (As, BS, CSt Os), and such that the resulting models (Ao" BO" Co" D°') and (A3 " B3 " C3 " 03 ,) satisfy

(5.35)

The singular-value decomposition [G3] of CB yields

° ° s S [H~] C B = C B = C B = UCR ~ ~ VCR ' (5.36)

where

and where UCB (pxp) and VCB (mxm) are unitary matrices

(5.38)

(5.39)

Ai is the ith nonzero eigenvalue of (CB)T CB, and q is now defined, in a more general woy, 0:0 Un:: '-0111<. u r CB.

LQt L bQ thQ diagonal (pxp) matrix defined by

L = [WI : j, o p_q

(tI.4U)

and let us perform the following transformations

112

u' (:) .41)

1,.' (5.42)

It 1s then easy to see that 100k1ng for the set of models

~: AS 1 + RS .!! • 1(°_) = Q

CS x + OS u (5.43)

is equivalent to looking for the set of models

~~, = AS, x + BS' u' , !(O_) = Q

= cs , ~ + Os, ~. (5.44)

where

AS' AS (5.45)

BS ' BC VT CB (!>.4b)

CS, L UT CS CB (5.47)

OS. T s T L UCB 0 vCB (:).40)

From (5.36). (5.37). (5.46) and (5.47) it is then clear that CSt BS ' does satisfy (5.35).

The same transformations can be applied to (Ao, BU. CU, OU) to obtain (Ao" BO" Co,, D°'). Note that the state matrices AO and AS are not affected by tnese 1nput-output transformat1ons

(5.49)

5.3.2.2 ~!~~g~rgi~~!iQ~_Qf_~_~~g_~

Now we are looking for a state-space similarity transformation Tl which changes the minimal representation (AS., B"", C'>·, 0'>') into a representation (A 3 ",

B"", CCII, OCII) where Be" and CCn are standard. In order to 00 that we oer1Ve two con­secutl VI! trarl'> fUl'llid L I Uri'> ; Lht:: r i r · ~ L om, T C tro.nsforms the Ob3erVo.ti on mo.tri II CS I

into a standard CSII : and tho ~"cond ono TIS rhllngp<: thp rontml matrix into a stan­dard BSII • while leavinq CSo invariant. The resulting transformation Tl will then be

113

(5.50)

and the set of standard representations will be deduced from the set of representa­tions (A)', B~', e~', 05 ,) through

AS" T AS, -1 1 T1 (5.51)

IS"" I IS'" 1 (5.5l)

eS" eS' -1 T1 (5.53)

Os" _ os, (6.6~)

a) ~:~:~~~~~~~~_~~_~~. The simil arity transformation T e must sati sfy

Since eSIl is standard, (5.55) implies that the first p rows of Te are those of eS '.

The (n-p) remaining rows have, to be chosen so that Te be of full rank. A possible POllCY 1S 'to cnoose l1ne vectors Whlch are orthogonal to each others and to the :;ub-

3paee generated by the rOW$ of CS '. By doing $0 we guarantee that the tran$formation

\ is inv~rtihl~. sinc!! eS ' is of full rank with our assumptions. Howev!!r. in prac­tical situations, Te may be completed in other ways to simplify calculation.

b) ~~~~~~~~~~~~_~~_~~. Let Bc be the control matrix resulting from the

trancformation TC

B T B5, e = e (5.56)

We now look for a transformation To chanqinq Be into a standard BS". while leaving CS" unchanged. Such a transformation must satisfy

(5. 57)

(5.58)

Since CSIo is standard. (5.57) imDlies that the first D rows of To are those of eS ".

Thus (5.58) can be written, using an obvious notation:

114

\ CO B C~II

Cll B C,?

G) G) B C2l B C22

-----------

() f...-q T T T T Bjl B3Z B33 B34

B B ' (5.:'9)

c31 c3Z

() () T B41

T B42

T B41

T B44

B C41 B C42

which implies

[ :'11 B 1 r :] el2 q

Bczz = () Cn

From (5.60). the four upper blocks of Be must already be standard. This is the case, because one has

(5.61)

Taking (5.59) and (5.60) into account. the last (n-o) rows of To satisfy

(5.62)

(5.63)

Gauss's pivotal method allows determination of some TR which satisfy (5.62), and 'J

insure the invertibility of TB• Equation (5.63) then gives the corresponding values of TB and TB • The blocks TB and TB ,which are not present in (5.62) and (5.63),

31 41 3Z 4Z can be chosen arbitrarily.

115

!:uch that

(5.64)

and

(:>.o:»

where BO", Co". BS", CS" are standard.

w~ "C1V~ ~"UWJI IIUW Lu \.."C1"!l~ L"~ III"ul.l1elll or rindinS the )et of all pO)5i­

b10 (AS, 9S , CS, OS). with any known gS and CS , into tho prob1om of finding tho cot

of all oossible lAs". BS". CS". Os"). with standard BS" and CS". The results of Sec­tion 3.1 can then be applied to obtain

(5.66)

Taking (5.49), (5.51) and (5.65) into account, the set of all possible models, when all entries of A are supposed free, can be written

s -1 0 -1 -1 A (a) = T 1 T(a) T 1 A T 1 T Cal T 1 • a € A (5.67)

(5.68 )

The comolete orocedure is summarized in Fioure 5.3. As in Section 5.3.1. any a priori algebraic constraint on the structure of A yields an algebraic relation between the components of ~. and the model identifiability depends on the number of solutions for ~ of these equations.

Example Consider a fourth order model. with no constraint on A. and such that

o

o o [20 B~. C

o o (5.o~) B

o o

116

Elements Sets

Element , Set of models I (Ao,Bo,Co.Oo) output-

ln01stlngulsnaDIe {(As .Bs .Cs .Os)} of the set of models I

I Bijective Bijcctive , applications applicati (5.45) (5.40)

ons 48) (5.45)-(!).

~ • Standardizable 5et of ;standa,-diLable

, '-"I""""ULC1Liuu repreSentatlons

(Ao .• ~o. ,Co .• 0°,) {(A'" .B'" .C 3 , ,03 ,)}

I

1-- ---- II -

Tl

Standard I Set of standard representation

olltrllft-representations

indistinguishable (Ao".Bo",Co",Oo,,)

I {(A"",B"",C"",O"")}

, J I 1 I'rODlem ot :>ectlon ::>.j.l r

Fig. 5.3 Procedure for exhaustive modelling, with any known Band C.

-

117

Applylng the proposed procedure yleld~

CB (5.70)

'l = ROllt,. [ CD) = 1. (5.71)

The matrix T(~) is thus identical to that in the example of Section 5.3.1, since n,m, p and q have the same values. Performing a singular-value decomposition of CB (here tr1v1al) and then uS1ng (~.4b) and (~.4/), one oDtalns

c" • [: 0 0

:] B~' = B~ • 1 1

(5.72)

A corresponding standardization matrix is

0 0

~l u 1

T1 -1 0 0

0 0 1

(5.73)

;lnli A\~) ~;ln hp mmrllltpd with thp hplp of ('i_67)_

5.4 THE MATRICES BAND C ARE PARTIALLY UNKNOWN

rrequently 0 and C al-e not completely known. ror example the 3tate of

a cla<:<: may bl> m<>a<:url>d with <:oml> unknown mllltiplicativ.:. factor. or a tr.,r"r inj"rt"rl

may have some unknown distribution in a few compartments. This is why Band C are now assumed to be functions of a parameter vector ~

B - B(~) > C - c(~) • (6.711 )

We suppose moreover that Band C are generically of full rank. Ihe extens10n ot the procedure summar1zed 1n ~lgure ~.j to the present

ca3e i3 ea3Y. rir3t notice that -the pl-oduct CO i3 3.g.i., 3ince it i3 invo.-iont under

;I"y ~ht,,-c:(,\;lr" dmibtrity tr""~form,,tin" Thll~ R ~"ti~fi,,~

118

The tran3formation:> (5.45) - (5.48) \;dll :.Llll be dppl1etl to (AC;, B", c.", nS) ~nd (Ao. RO. rO, nO). for thoy only do pond on CO gO, whieh is supposod known. But one must now distinguish between the standardization transformation Tll~) for the set of representations (AS" BS', CS', Os,). and the standardization transformation Tl(~o) for the particular representation (AD', BO" Co,, D°').

Let B be the subspace of IH n{mlp) such that (5.75) holds for any ~ be­lUII!!ill!! Lu 15 • EtjudLlun:. (:l.07) dntl (5.68) become

BS(~) = T-1(S) T (so) BO 1 - 1- B(~) (5.77)

c.\~) = (.0 li\!t) 11 (!!) C(~) (!> • III )

DS = nO (1;.79)

where

(5.80)

Any algebraic 3tructural con3t,-aint on AS and AO ... ,,:>ul t:>, Lh. UU!!" (5.76), in an algobraic rolation botwoon tho compononts of ~ and ~. and tho modol idontifia­bility depends on the number of solutions for ~ and ~ of these eQuations.

we have therefore assembled some tOOlS to generate tne set Of all POSS1-blc AS when Band C are totally or partially known. Now let us present some results on the S.D. ;d~At;f;~b;l;t~ of thp p~r~mPtprc; for R ~nd r..

Property Wh~fI 11\1 \;UII:.Lr·diIiL ~A(:,L:. UII Lh~ :.Lr-u\;Lun~ ur A, Lh~ ullly tmlr·i~:. ur 6

and C which aro c.g.i. are thoce which can be deduced from the knowlodgo of the pro-duct CB

~ • I='or "'ny ! € B • th" I"roc"lft,r" of "yh",,,c:tiv,, molf"l1ing can h" """If to find a matrix AS(~,~) such that the model

(5.01)

is output-indistinguishable from (AD, BU, Co, OU). Thus there is no way to impose fur­ther restrictions on B •

119

Ol.lvluu:.ly Lh~ \1I"~(lL~" Lh~ numb"," ur ulilJ,,"Li riaule \..umpunenL" in~.

the more difficult the ~olution of the problem. In the limit, when none of the en­

tries of Band C is known. the problem loses its meaninQ. and there are n2 degrees of freedom. since any similarity transformation T is acceptable.

Some simple examples are now presented, to illustrate common cases for Band C.

Example 1 If Band C satisfy

BT = [1 0 0] • C = l6 0 0),

B is s.g. i. ; but with C = [ 0 sol, 6 is unidentifiable and there is one degree of freedom for Band C.

~)(ample 2

With

o

1

both 61 and 62 are s.g.i. but if the observation matrix is chosen as

C [: o

only 61 is s.g.i. and there is one degree of freedom for Band C.

(5.83)

(5.84)

However the previou~ eonelu~ion~ may happen to be modifiod if the ma­

trix A is constrained. as shown in the followino examole.

Example 3

CUII~ioJ"r" Lh" Lwu-... la~" lIIuoJ"l oJ"" ... ,.;ueoJ uy figure :;.4.

Fig. 5.4.

120

A33ume the output to be "2 time .... oo.e unknvwlI "o.olllt:Lt:.· ~. Tht: lIIul.lt:l t:\juilLiun b Lhen

x [-a21 0 j X + [1] u. = 021 -002 0

v = [ 0 B ] 2S. (5.85)

tlparly a i~ not ~.9.i. from CB. ~owovor ono hac

(5.66)

:ilnCe CAll lS S.g.l.(fOr 1t 1S lnvarlant unaer any state-space Slmllarlty transfor­mation) Dnd 5inec ( a21) corresponds to on ob3ervoble - thu3 3.g.i. - mode, then ~ i3

<;·O·i

5.5 CONNECTIONS WITH KALMAN'S CANONICAL FORM

U" Lu IIUW, wt: hovt: o,;ulI:.II.lt:.·t:1.I o,;uIIL"ulloiJle 0111.1 uiJ:.e,·voiJle ,"ul.lel:. ulIly.

Wo havo chown that cuitablo biuniquo trancformationc on tho ctato, tho inputc and tho

outputs always allow a dec6lllPosition of the model into four parts .• namely observed­controlled, observed-uncontrolled, unobserved-controlled, and unobserved-uncontrolled. The dimensions of these four parts only depend on n, m, p and q, where q " Rank [C8].

Successive application to a linear model (which is not necessarily obser· voiJle 0111.1 o,;uIIL"ulloiJle) uf 1\01111<111':' o,;OIlUUio,;ill l.Ieo,;ullljJu:.ILiulI ClIII.I ur Lhe l.Ieo,;um"u:.iLiulI

resulting from the ctandardization procedure yioldc a ctructuro ac procontod in Figu-

re 5.5.

To study the s.g. identifiability of a non-minimal model with the techni­que of exhaustive modelling, one has first to apply Kalman's decomposition. Then the structural identifiability of the parameters which appear in the observable-controlla­ble part of the moael can be stualed. Any parameter whlcn cannot be (Unlquely) deduced from the knowledge of 3 minimal realization i~ not (globally) idcntifiablc •

. txample Consider the model de3cribed by

y [ 1 0] ! {~.IH}

u == ~

121

Observable-controllable part

F1 UDserveo-controlleo part t VCB -~ r ODserveo-uncontrolleO part t ~UCBL-l~

~ Unobserved-controlled IJdrOL 1 I UlluLJ~t;:.-vt::U-UII\.uIIL, ·ullt:u po,otl

Observable-uncontrollable part

Unobservable-controllable part

unobservable-unCOntrOllaDle part

"i9' b.b Structural decomposition of any linGar time-invariant state-space model.

F=

.....

r

:;"me inte.oc:>ting """",ent:> hQve Q],°eQdy been mQde on mode13 of thb kind [D5,C3,J2 J Thi< on.;o 1< not conno:>ctakl". not cnntrnllakl" ~nrl nnt nk~"rv~kl". hilt it~ fl~r~m"t"r

~ is s.g.i. from the controllable and observable part, which can be described by

(5.88)

122

~.O APPLICATIONS OF EXHAUSTIVE MODELLING

A first application of exhaustive modellinQ. illustrated by the pre­vious examples, is the test of linear time-invariant state-space models for structu­ral identifiability. Figure 5.6 s~rizes the steps to be followed.

A second application is the search for the set of all numerical minimal representatIons whIch are compatIble wIth the structural constraInts and OUL!IuL-lndb­tingui~h~ble from a given reprecentation, obtained from real data. ~igure 6.7. illus­tr~tp~ tnp pro~edure_ Notp. howpver. that true model set so obtained expresses the structural indeterminations only. To these models have to be added th6se resulting from measurement noise, from numerical errors and from model inadequacies. The in­fluence of these factors can be studied with the help of usual techniques [R5, M7, VI, St!, Ill,] .

A more unexpected application is the design of models with quite spe­cific identifiability properties [W15] • Let us, for example, build a model such that the set of all output-indistinguishable models is represented in the parameter space by an lSOlatea pOInt l plUS a curve r. ThIs nontr1vlal l:!l\dlll)J11:! )Jr'uvl:!~ LhaL a lIIucl,,1 locally idcntifi~ble ~t an i~olatcd point £ is not necessarily locally idQntifiable ~t p~~h point of the p~rameter space correspondina to the same inout-outout behavior.

Consider the three-class model described by Figure 5.8. The experiment structure is

o - [ 0 o

o 1

From (5.13) am! (5.19). Lhl:! L.on:.formotion T con be writtcn ~s

T =[~-~~-l T32 I 1

with

(5.69)

(5.90)

(5.91)

(6.92)

123

Initial model structure

Eliminate 3tructu rally uncontrollable or 3tructurally s, if any. unobservable part

Non-numerical minim~l 0 0 0 generdt1ng model (A ,~ ,e ,D )

Transform the inp uts and outputs with (5.45)-(5.48).

Non-numerical standardizable representation (Ao, ,Bo, ,Co, ,D°')

Transform the sta te with Tl(~o) (non-numerical).

Non-numerical standard representation (Ao",Bo",Co",Oo")

Tran~form tho ~ta •

Set of standard representations (A~·,~~·,~~·,u~·J, parametrlzed oy ~

Transform thp st~

Set of standardizable representations (As',Bs , ,Cs',Ds ,), pdrdmetrlzed by ~,~

Return to the ori (::;.4::;)-(::;.48),

~et of all minimal mode13, when no constraint on A is accounted for, parametrIzed oy ~,~

Solve the system the l;on:>trd1nts 0

I"ocponding {(As ,B

Final set of models

to with T (~) .

ginal inputs and outputs through

of algebraic equations associated with n A for a and ~, and generate the cor-s.Cs.Ds)~ ~nd (~s)

tlg. ~.b tXhaust1ve modelling for structural identifiability studies.

124

Input-output data

P~r~mptpr irlpntif ication with an alaorithm which doo< not require local identifiability [R5,E2j.

NumericRl minim~l

generating model (Ao • BO. Co, D°)

Transform the ino uts and outouts with (5.45)-(5.48).

NUmerlCa I stanllardlZable o-epO"e3ento ti on (AO, ,00' ,co, ,D0')

IranSfOrm tne sta

Numerical standard representation (Ao·,Bo·,Co·,oo.)

Transform the sta te with T (~) .

Set of standard representations (AS",Bs",Cs",DslI) parametrized by a

Transform the sta -1 te with T 1 (~).

Set of 5tandardiLable n:po"e3entoti on5 (451 .BS I .cs I .os I ) paramptri zpd by ~.~ .

Return to the ori 9inal input" and output" through (5.45)-(5.48).

Set of all minimal modole, when no constraint on A is accounted for, Pdrdmetrlze~ by ~,! .

. -

Solve the System of alQebraic equations associated with the constraints on A for ~ and ~, and generate the cor-respondlng t(A~ ,~~.c< ,U~)j ana {!~ } . __ -------------L----~--~~__,

Set of all minimal models, when all algobraic con~traint< on 4 are accounted for.

Discard all models which result in non-admissible values of!: !¢ e.

Set of all admissible models which are output-indistinguishable.

Fig. 5.7 Exhaustive modelling for identification.

125

Fig. S.S.

Equation (5.21) gives the set of possible AS as a function of a, and a? The structu­ral constraints on AS lead to a set of nonlinear algebraic equations. We would like this set to have both an isolated solution and an infinity of solutions depending on on~ Pdrdmet~r. To LII b ~1I\.1 IL b :,ufn l; I ~rl L to U,oo:,~ for u,~ y~n~r'd tiny 1II0\.l~'

1.5

- (j 1 •

0.5 o 2 1

- 6.5

Any possible matrix A is then given by

l-a1

0.5+5al+a2-ala2-2a~

l-ZU2

-b-a2

al+0.5a2-2ala2-a~

I : £ I

I 1 I

! -6.5+2a1+a2 I

There ore two 5trueturol eon5troint5 on AS which hove to be accounted ~or

Equ~tion~ (~.9~) and (~.96) have two ~ets of ~olutions

(5.93)

(5.94)

(5.95)

(i) An isolated one (at = 1. a2 = 0.5). which corresponds to the locally identifiable model of Figure 5.9.

126

Fig. 5.9 Locally identifiable model.

Note that the ze~o coefficients alZ and aZl a~e pa~amete~c of the model. and locally

identifiable as well as the nonzero ones. (ii) A set of solutions such that

£ "l + ..,z - 0 , (5.97)

which results in an infinity of matrices AS, each of them being associated with a particular value of "l

-1. 5 - Z al

I - "1 -6 ... 2 "1 1 (!\.!lB)

0.5 + 3 at o -6.5

A~(al) will be compartmental for any al € [-!,~].

These results can be confirmed by testing models belonging to these two sets of solutions for local identifiability, with the help of techniques presented in Chapter 3. If the model described by Figure 5.9 proves to be locally identifiable, on the other hand the mOdel deSCrllled oy (5.98) W1th al z aZ = 0 h founu uniul:flLi fiClU11:,

with one degree of frecdom. Moreover the locally linearly depondont paramotorc aro

aZI' 3JI • A IZ ~n~ ~ll' while a13 • al3 aRd a3l are lecallv identifiable. This result is coherent with (5.98). since all was not considered as a parameter. because of the constraint aOl = 0 .

127

It :.hould be noted thot :.u,;,h ,e:.ult3 ore not 3truo;turol, fo,- they ore

a~~ociatl.d with atypical valu,,~ of th" param"t"r~ af th" a"n",rat.in!) mod"'l AO Thu~

such examples are not contradictory with the structural nature of the results which can be obtained when testing local identifiability with the techniques of Chapter 3.

5.7 CONCLUSION

Exhaustive modelling is a systematic procedure to generate the set of all models which are output-indistinguishable and compatible with the assumptions on the model structure. This method enables the model set to be built in a rather simple way, even for relatively large-scale models.

Three situations may occur. 'In the first one this model set reduces to one element, and the model is globally identifiable. In the second one the model set i:> dt:llulllt:I'dlJlt: ~u llidl lilt: IlIUlJel b lOl,;dlly Idtmtlfldble. FlnCllly In ttle thIrd sltuCl­

tion thQ modol cot ic not denumerable. The model is then unidentifiable, but one

knows which parameters are globally or locally identifiable. and how the unidenti­fiable parameters are related.

Wh"n"v"r th" mAtI"l ic: nnt !Jlnh~lly itl"nt-ifhhlp. ilntl nnp rpfuc;pc: tn ;n­

troduce any unjustifiable structural constraint, the procedure of exhaustive modelling generates a set of models expressing the structural ambiguity of the considered expe­riment.

The next chapter will illustrate the power of this approach, which ge­nerally leads to far simpler calculations than the one presented in Chapter 4.

6.1 INTnODUCTION

ChFlntp.r 6

Examples

The examples considered in the previous chapters have been designed for the purpose of illustration. Now, in order to prove the ability of the described me­thods to solve realistic problems, we present some examples extracted from the lite­rature concerning compartmental modelling. Ihese examples are tairly typical. They will ~how in pcnti~ulCl" the: ~illlpl i~ity of the: lO~Cll ~tudy, the: puwe:r of e:"lIClu~tive:

modQllin9 and thQ cohQ~QncQ of thQ ~Q~ult~ obtainQd with thQ~Q two app~oachQ~.

6.2 CHEMOTHERAPEUTIC MODEL

Ifr i ~ rlluc.l~l, wh i <.;h hd~ 1J~~rr u~t:c.I t.u c.lt:~<.;r· i IJ~ lll~ uyrrdllr i" lJ~hCl vi vr' u f tilt:

antieaneQ~ou~ d~u9 NSC-130106 ;n~idQ thQ body [Tsl • eomp~icQc th~QQ eompa~tmQntc. ThQ

first one is associated with the liver. the second one with the plasma. and the third one with the tissues. The structure of the model is described by Figure 6.1.

tli Ie Urine

f1g. 0.1 A c;hemUUrerdfjeuLI<.; muc.lel.

A known amount of the drug is injected into the plasma at time 0, the system being in zero initial condition. If the state variable Xi is the total amount

129

uf UIt: lIruy ill I,;Ullllld r'llll~lIl i. lh~ d!)!)ULidl~lI ~lCll~ ~4uClliuJl b

au a12 0 0

~ a21 a22 a23 ~ + 1 u. ~(O_) = Q • (6.1)

0 a32 -a23 0

where

(b.Z)

drug concentration in compartments 1 and 2. If vi ;s the (unknown) volume of compart­ment i. the corresponding observation equation is

o

(0.3)

The second experimental set-up consists in measuring the biliary and urinary secretion rates. The observation equation then becomes

o

:] ~(t) (6.4)

Consider the model structure defined by (6.1) to (6.3). Let the parame-ter vector Oe

(6.5)

and noto that a01 and a02 are oacily deduced from the knowledge of ~ by

(6.6)

130

G. £ .1.1 ~!:.'!:!!:!!:!:!~~i!i!.ll_~!!:!!!::!!!!:~L!!!!:!!:!.!~!!.!!i!L!!!!~_:!!!:!!!:!!!!.!!!

£~!:!!!:~ll~!:!ili!t

By merc inspcction of thc compartmental schcme, onc can easily make sure that thl' mndl'l statl' is mnnl'r:tahlp tn thp inputs and nutputs

Thc parameter5 in A are con5trained, for the 5um of the entrie5 of the third column has to be zero. Therefore the usual methods for testing structural obser­vability and structural controllability do not apply. On the other hand the procedure described in Section 2.3.2 leads to a conclusion. Note, however, that it is sufficient to find a value of the model parameters such that the observability and controllability matr1ces are OT TUll ranK to De sure tnat tne oDta1nea reSUlt 1S strUctUral. one can thus mcrely check that a triplet (AO' BO' CO) Q"ociated with some simple values of thc p~r~mptprc; ic; rnntrnl1~hlp ~nli nhc;prv~hlp. Tt ic; trivi~l tn vprify th~t c;llrh ic; thp case for

o I

(6.7)

However. should the resultina observabilitv or controllabilitv matrix have proven to be singular, one would have had to pick at random the parameter values, for those which allow easy calculation are quite specific, and often lead to nonstructural degeneracies.

Using method A of Chapter J, one finds that the modol ic not ~.~.i .. Thoro is nnp dl'grl'l' nf freedom. and all' azz' azs ' asz and Vz are s.t.i •• while al£, a£l and vl are locally linearly dependent.

Ine exnaustlve moaell1ng approacn Wl1l allow a more preclse characterl­zation of the identifiable parameters and of the relation5 existing between unidenti­fiahlQ onQ<;. ThQ prnrQlillrQ tn hQ fnl1nwpli h~c; hppn dl'sr:ribed in Figure 5.6. Since the model is structurally controllable and structurally observable, the minimal generating model can be written

0 0

:!'] [J all au

AO 0 0 BO a21 a22

0 0 an -a2J

131

o :] . (6.8)

!~2~!~~~!2~!_!~2~~f2~2!i2~ Thp initi1l1 c:nnditinn nf t.hp imll"l~p rp~llnn~p~ nf t .hp mntt",lc; ic;

(0.9)

Thus the set of all possible v2 reduces to v~. and the volume of the second compart­ment is s.g.i •• Since CO BO does not satisfy Condition (5.34). a singular-value decom­position has to be performed

CO BO . r: 1] [ 1/':] o· 0 . [11 • (6.10)

which yields

uCB -[: J (G.ll)

VCB [ 1 1. (6.12)

L t :1· (6.13)

Then. taking (5.45) to (5.49) into account. one gets the standardizable models resul-ting from the input-output transformation

AD. AD AS' AS (e>.14)

BO , BO BC ' Be (6.15 )

f,,: ,

: 1 co, (6.16)

0

132

ce , _[0 - l/v! o : 1

§t!!Dg!!rg1~!!t1QD A similarity transformation standardizing Co, is given by

1

a a

The o.~~oci 0. ted cont,-o 1 mo.td x i ~

l b, 11)

(6.18)

(6.19)

It is already in standard form, which makes the similarity transformation TB unnecessa­ry, and one can choose T1(v1) equal to TC'

~~!_Qf_~~~~~~r~_~~~!~ The model considered is such that

n = 3. m = 1. p = 2. q = 1 (6.20)

From (5.lJ). all slmllar11:y transformatIons whIch preserve the St.cHlddrd furlll uf B dud C o.re given by

a 1

~~~_9f~~~~~r~i~~~!~_mQ~~1~

(6.21)

A S1mllarlty transformatIon whIch allows a return t.u tile s~drlddrdiLclLJle

S 0 model~ i3 obto.ined by 3ub3tituting vI for vI in the expreccion of Tl

o o

(6.22)

Di3regording, for the time being. tho roctr;ct;enc en tho. .tructuro. of A. tho. ~ot of

133

IIIdtrlces A~ 1S g1Ven oy (~.70)

(€i.n)

Vs i 0 -k "12

1 <111 I 0

I VI I 1 0

s s 1 [0 0 0.1 0] 0 I a23 ~ A (~.vl) - Vs vI 421 - 0.2 423 422 I (O.N)

I I 0.2 1 I I I

ul [ 0 0 ] ul o 0 I 0 "( a23 + aU 0 a12 + 0. 2 an I -a23 VI vI I

I

Thi~ m ... t.rix i~ nnt ~ff"rtprl hy th" r"vpr~" inpllt-nlltrlllt t.r ... Mfnrm ... t.inn whirh hrin(J~ II~

back to the initial problem.

!D1rQQ~~1igD_gf_51r~(1~ral_(gD5traiDts_9D_~= The models must satisfy the following structural constraints

3 $ 0 ~ 1 I a' 1 0.2

1 =1 ' -(6.25)

3 0 ~ = 0 a31 0. 1 (6.26)

$ "13 - 0 (yicldc no inform"tion).

Thus the set of a 11 possible models i$ given by $ v

0 -.l ,,0 0 "11 VO 12 1

o

0

AS(v~) vI 0 0 0 BS 1 "( a21 a22 a23 • vI

n 0 ~.3l

0 -'l.3 n

[ S

0 0

1

l/v 1

$ $ (6.27) C (v 1)

0 llvo 0 2

134

Th" l'a,-allot:L",.,. an' a22' a23' a32 ami v2' .. hj~h t.Iu lIuL a I'I'"a,- a:. rUII~LiulI:' ur vi ill (6.27) aro ~.g.i .. [quation (6.27) alco g;voc tho oxact nonlinoar rolationchipc oxic-

ting between the unidentifiable parameters a12 , a21 and vI

(6.28)

rquation (6.29) confirms and complotos tho local rosult of Soction 6.2.1.2. No to that

aOI and aoz ' which were not retained as components of ~. are unidentifiable. for one has, through (6.2) and (6.28)

(I'> ?Il)

<;

o vI + al ,,(1 - -)

~ VO 1

(6.30)

If AO is compartmental, v~ must satisfy

V~ 1

< VO 1

(I'> ':11)

for A<;' to De compartmental too. Any roodel gefleratetlOy (O.Z7) lind :.u~h t.hdL (6.31) i:. verified will be compartmental, ond output-indistinguishablc from the gcnerating modal.

RemarK. InequlIl1ty const.ralllts :.uch a!> (6.31) 1i"l<l jllrunllaLiuII UII Un: volut::~ .. hh.h

ca~aken by unidentifiable parametors. Supposo for oxamplo that AO ic oqual to AU

in (6.7). (6.31) becomes then

c

0.5 " vI

" 2 (6.32) 0

vI

~ 0.5 s 2 (6.33) .. 012 ..

~ 0.5 s 2 (6.34) " a2l "

6.2.2 Second experimental set-up

Now considor tho modol structure dofinod by (6.1), (6_2) and (6_4) Jt

135

6.2.2.1 ~!r~f!~r~l_lQf~l_i~~~!ifi~~ili!~

All parameters are found s.t.i ..

The method de:.c:r1IJed In Chdpter 4, whlc:h dllow:. te:.tlng the model rl;)r

c.g. identifiability, dace not apply here, for some parameter~ are cemmon to the ma­

trices A and C. Therefore we shall use the procedure of exhaustive modellino. It is sufficient to SUbstitute a01 for 1/v1 and a02 for 1/v2 in the previous calculations. Constraints (6.25) and (6.26) still hold, and imply

So that the set of all possible models satisfies

- (a~l + a~l) o

o

Be .[!] .

. [ s 0

: 1

<101

(;~(a01) (6.36)

° ° <102

From (6.36) s a02 ° a02 ; thus a02 is S.g. i •• This makes the model S.Q. i .• for one has

J S a O (...Q.!. - 1) ° S 0 r a' 2 a12 0 ~a01 a01 (6.37)

i-O 1 S <101

~ (As, BS , CS) (Ao, BO, Co) • (6.38)

136

!:o the 3econd experimental 3et-up i3 bette,. thClIl lh" ri"l u"" ct, n~!:Ictnh ,lructural

irlpntifi~h;lity .

6.3 HEPATOBILIARY KINETICS OF B. S. P.

The model p,.e3ellt",", i" F i !:I'"'' O. Z IICI' Uti,," U~' I ~rl~u ( M3. M:l ) lU Utt,­

crih~ th~ dynamic behavior of bromo~ulphthalein. widely u~ed in clinical invectigation

on liver functions.

Drug

---r-----------------

Motabo 1 i tee

Rlood Liver

Fig. 6.2 Compartmental model for B. S. P.

The correspondinq structure is characterized by the triplet

- a31 0 a13 0

0 - a4? 0 a?4 0

A • B

"31 0 3 33 0 0

0 a42 a43 a44 0

. [ 0 0 0

1 c (0.39)

0 1 0 0

137

"i Lh

(6.40)

6.3.1 Connectabi 1 ity, structural observabi 1 ity and structural contro 11 abil i ty

A!jClill Cl III'"'' ill"p,,\;LivlI vr Lh" \;UlllpCl,·LIII,,"LCll "\;h"n,,, '" :.urrl\;I"JlL Lv IIICl­

ko ~Urg that tho modal ~tato i~ connectable to tho input and output~.

The PCl'CllIlCte,:. ill A 01·" \;vII:.Lla ineu. 0"" loa:. Liru:. Lv u:." Lh" p,o\;edu,,,

of ~pction 2 3.2. which onablo~ ono to provo that tho triplot (A.B.C) ;~ ~tructurally

controllable and structurally observable.

6.3.2 Structural local identifiability

Thi <; modpl i <; not <; _ P. _ i _. it hi! <; onp dpgrpp of frppdom_ Mpthod A of I:hi!Jl­

ter 3 allows one to find that a21 , a12 , a1J and a1~ are locally linearly dependent.

6.3.3 Exhaustive modelling

Now we look for the set of all outDut-indistinQuishable models. Band C are already standard, and the parameters characterizing T(~) are

n - 4 • m - 1, P - 2 ,q - 1. (6.41)

Thus the number of degrees of freedom would be six if there were no struc­tural constraint on AC , and T(~} can be written

1 0 0

0 1 0 T(~)

0 "'1 "'(

0 "4 "5

It will be nonsingular if, and only if,

fi ~ a2 a6 - a3 as ~ 0

Any possible AS satisfies

0

0 (6.42)

"'j

"6

(6.43)

(6.44)

138

The 3tl-uc.turol c.on"troint:. UII A \..011 IIOW LJ~ Lol<.~11 into (I!;!;ount

s aU

0 - an s a31

n a31 112 ~

s s au + a31 = 0

1112=1 11

===bA = "0 ~ n

S 314

s "u

So that T(!!) hl!r:nml!s

T('"4''"6) -

and (6.44) yields

1

0

0

0

1115 = 01

(6.45)

0 0 0

1 0 0 (6.~6)

0 I 0

114 0 116

o I I 0 I

:-~~~-~----------------~---------------~--~~~--~-------~------I 11 I I aO

o i_au - aU ~ i 0: Z4 : 42 24 116 I : 116 I I I

------~--------------------------------+-------~--------------I I I

a31 ! 0 : a33 i 0 I I I

------~--------------------------------t-------1--------------: 0 1l4( 0 0: 0 I 0 114 0

o ! 042(u6 u4) - 116 u6°44 T u4°24) : u6°43 I °44 I 116 °24 I I I

(6.47)

A last :.trul..turdl l..un:.LrctinL n:lllctin:. Lu LJ~ l..un:.hJ~,·~<l

(6.49)

gec~uce of the c;9n eonct~a;ntc on tho pa~amoto~c, (6.~Q) ;0 tho oquation of a hypor-

139

lJula. Nu~t: that tht: I'uillb :>U"", that (16 - 0 IIIU:>t lJt: ,eje .. ted, ro,- they "o'Te:>pond Lo a singular T(~).

The re3ult obtained i~ quite ~on~i~tent with that Qf the lQ~al 3tudy, cinco a~l' which ic a function of "4 ann "'b' wae: not rpta;npn <Ie: " I'"r,,_tpr. hpr,,"e:p of the structural constraint a22 = - a42 .

C;hollln nnp hp intl>rpc;tI>li in mllicing thp mnlipl ".a.i. loy ilddina iI rnns­traint on the parameters. this constraint must involve one or several of the parameters which are functions of (14 and a6. It is of no use, for example, to impose a13 = 0, since this parameter is already s.g.i .. On the contrary a constraint such as

(6.49)

Wl I I lmplY a6 = 1. tquatlon lb.4HJ nas tnen two S?lutlons 1n (14' eacn or wn1cn 1S as-3Qciated with a 3.~.i. mQdel. The 3ecQnd model (A ,D,C) con be deduced frQm the gene rilting nnp (Ao.R.r.) hy pxrh;onging IIg4 lind iI~Z. Bnth mndplc; lIrp rnmplIrtm@ntlll.

HQweyer one may refuse to impose such a restriction on the model ctrue­ture without any biological justification. and prefer to keep the model ambiouity. For any numerical value of AO, it is now easy to generate the set of all output-indis­tinguishable models and to eliminate the noncompartmental ones, if any.

G.4 METABOLISM OF IODINE

Consider now the model described bv Fiqure 6.3 •• which have been deve­loped for the study of the metabolism of iodine in the rat [L1] •

lowe

1 plo3mati~ iQdide , 2 fact hormonal placmatic ionino 3 fast orqanic thyroid iodine, 4 slow organic thyroid iodine , 5 peripheral hormonal plasmatic iodine.

140

Fig.6.3 Model for the metabolism of iodine in the rat.

Aft~r lin intrll-('I~ritnn~lIl inj~rtinn nf II knnwn lImllunt nf "1; intfitf~. th~

following outputs are observed: (il the specific activities of compartments 1 and 2

t Y1(t) - xI(t) I xl

(ii) the total activity of the thyroid

(111) tne urInary ana faecal excret10n rates

(6.50)

(6.51)

(6.52)

(6.53)

(0.:>4)

The outputs Y4 and Yli have not been used in the identification process, for the corres­ponding measurements were too corrupted by noise. Therefore they have not to be taken into account in the identifiability study, so that the model to be conslderea 1S cna­rat;tt!r I LeI! lJy

141

an a12 a13 0 a15 1

0 a22 a23 0 a25 o

A a31 0 -(a13Ta23Ta43) a34 0 , B o

0 0 a'3 -a34 0 o

0 a52 0 0 -(a15+a25) o

t 1/X1 U U U 0

C 0 t 1/x2 0 0 0 (6.55 )

0 0 1 1 0

The triplet (A,B,C) is easily proven connectable, structurally controllable and"struc­turally observable.

6.~.1 Structural local identifiability

All thirteen parameters of this model are s.t.i ..

6.4.2 Structural global identifiability

The method of Chapter 4 would lead to extremely tedious calculations. It is far better to use exhaustive modelling.

The control and observation matrices satisfy

[ l/x~Ol [l/X~C 1 CO BO = CS BS = 0 = 0

o 0

(6.56)

A slngular value decomposltlon Of CO BO yields

(6.57)

(6.158)

142

['r 0

:] L = 0 1

0 0

(6.59)

Tilt: \;UJ.,.t::.pum.llny :.LclJlddrd1zdble models verlfy

BO' = BS' = [ 1 0 0 a 0] T • (6.60)

[:

u u u

:1 co, l/xto 0 a 2

0 1 1

(6.61)

C" • [ :

u u u

:] l/x~s 0 0

0 1 1

(6.62)

A cimilarity trancformation which ctandardizQc CO, ic givan by

1 0 0 0 0

0 l/xto 2 0 0 0

T (xto )= r. ? 0 0 1 0 (6.63)

0 0 0 1 0

0 0 0 0 1

ThiS transformation preserves the control matrix, which is already standard. There is no n .... d th .. ,-.. for .. for TB, and on .. o;.an "dL ..

with

Similarly the transformation which standardizes Cs , is

ts T1(0) - TC{x2 ) ,

with

(6.64)

(6.65)

(6.66)

143

s = X;3 (6.67)

6.4.2.3 Q~~~rmi~~~iQ~_Qf_~~

Since n = 5. m = 1. p = 3 and Q = 1. T(~) is oiven by

1 U U U U

0 1 0 0 0

T(.!!) 0 0 1 0 0 (6.6S)

0 at a2 a3 a4

0 a6 a6 a7 ag

It will be non-singular if, and only if,

• 6 - f 0 u - a3 Us u4 u7 . (6.69)

From (5.76), any AS satisfies

(6.70)

6.4.2.4 !D!rQ~Y~!iQD_Qf_!be_s!ry~!ural_~QDS!raiD!S_QD_~~

Now consider the relations that ~ and B must satisfy for AS to have the

same structure as AO.

(y1elos no relat10n) , (b.ll)

144

s s o ==9 "4"5 - "1"8 (a13 + a~3) = 0 a32 + a42 h

'> a4a~ = a1aO • (6.75)

5 a13 0 a13 (6.76 )

s ~ 0 a23 ~o a23 (6.77)

0

s 0 0 a23 0 ~33 a 13 a!3 - a! ~ - n3 ~4J

S (6.78)

0 c a23

+ n a43 a 1 --0 a3 a43 8

(6.7!)}

4 s o (S ==;) 18 tJ 2: aiJ = 0 ==> a"3 -0 - 1) 0 ;=1 ~ 8 (6.80)

(G.01)

(6.82)

(G.OJ)

(6.86)

145

which implies, taking (6.75) and (6.86) into account

Taking (6.86) into account, (6.82) implies

Equation (6.81) then yields

(b.!!/)

(6.88)

(6.89)

u

_ "la25 ] ~ 0

80

(6.90)

(6.91)

(6.92)

(6.93)

(6.94)

(6.95)

o . (6.96)

(6.97)

(6.90)

146

30 that, fn)fn (G. 9G) one ha:)

(6.99)

Thus the solut1on for a and ~ of the cOnStra1nt equatl0ns 1s un1que, and can De wrltten

(6.100)

The answer would have been different. had we shdied the model described by Figure 6.4., which was also considered in [Ll] , while keeping the same matrices B and C.

Fig. 6.4 Alternative model for the metabolism Of 1001ne 1n tne rat.

Indeed, if such a madel remains connectable, structurally controllable and structurally observable, calculations similar to the previous ones show ~hat it is no longer s.g.i., and that the set of all output-indistinguishable models can be written as follows

o (6.102)

147

a31 (a31 + a41 )(1 - a2) - a3 a41 (6.103)

I: a41 ( 0 0 ) 0 a2 a3l + a4l + Q3 a41 (6.104)

~ u aSl • (6.105)

s n a12 a12 , (6.106)

s C122

0 C1 22 (b. WI)

s C132 - 0 (b.lW)

s - 0 °42 , (15.109)

s - 0 °52 °52 • (0.110)

S ~13 - 0 • (6.111)

S aZj o ( -1 aZ3 '"2 .. '"3) "'3

o -1 - 324 "2 "3 (6.112)

S 3 jj (aZ .. OJ) 0;1 [-(1 - Q2){3~3 .. 3~3) .. (1 - "'2 - "3) a~3]

- a? a; 1 [ (1 - a2) a~4 - (1 - a2 - Q3)( a~4 + a~4) 1 • (6.113)

(6.114)

s a:;J O. (Ii . "")

s 3 14 - 0 • «j.1l6)

(6.117)

148

a~'1 (aZ ... "j - 1) ,,;1 [- (1 - "'Z)(a~3 ... a~3) + (1 - "2 - "'3) "~3]

+ (1 - a?) a;l [(1 - ( 2 ) a~4 - (1 - (12 - (3)(a~4 + a~4) 1,(6.118)

S ;115

s aC;C;

u

o ;!15

u

= 0

B ,

The parameters a2 and a3 can take any value, provided that a3 ! 0 and that

, Y i f j

(6.119)

(6.120)

(6 _12l)

(6.122)

(6.123)

(6_12~ )

(6.125)

(6.126)

(6.127)

(G.1t6)

(6_129)

Note that, when substituting (1 - (2) for a2 and - a3 for a3 in (6.101) to (6.125), UJlt: iflLt:n;hdflyt:::. Lht:: Lilli'll dml fuur-LII ruw:. dflll Lht: Llllr"lI dflll (uurUI !;ulullm:. u( AS. Till:.

QxprQ~~Q~ thQ fact (a priori obviou~) that compartmQnt~ 3 and 4 cannot bQ dictingui­

shed with the experiment beinq considered. so that amonq all possible models there are those associated with the permutation of compartments 3 and 4 .

Figure 6.5 presents the s.g.i. parameters. Note, as a curiosity, that, should an injection in compartment 5 be possible, it would not improve the struc­LUI al hlCIILi fial,;' i Ly vf Llrc ",vuc'.

149

Fig. 6.5 Transformations associated with s.g.i. parameters for tho modol of ~iguro 6.~.

6.5 SYSTEMIC DISTRIBUTION OF VINCAMINE

TIle mudel presented In Flgtlr"e (j.(j lIt!s(;rllJe~ the lIl:.lr"llJuliulI ur

Vincamine in~ide the human body [ R6 J • Th;~ drug is a v~sodilating agent conmonly used

for handlina various circulatory disorders of the brain.

4

fig. 6.6 Model for the distribution of Vincamine.

The physiological interpretation of the compartments is as follows

150

1 blood,

? kinnpy .

3 gastro-intestinal lumen , 4 tissues .

Amounts of Vincamine in the blood and in the urine are measured, after introduction of the drug either orally or intravenously; so that the state equation of the model is

-(a21+a41 ) a12 a1J a14 0

il21 -(il02"'ilI2 ) 0 0 0 0

~ - ~ I ~, ~(o_) -2,.. 0 0 a33 0 0 1

a41 0 0 "44 0 0

[ : 0 0 :j 1.. ~ (6.130)

il02 0

It is easily confirmed that this model is connectahle. <;trllrtllrally

controllable and structurally observable. Moreover its eight parameters are all s.t.i ..

trom (~./b). al I output-lndlstlngulshable modelS

o o s A (!!.B)

151

. [: 0 0

:] CsOS) • B 0

(0.131)

where

SO 0 B aOf!

s = aOf! (6.132)

The structural constraints on AS must now be considered

CIS 34 = 0

s - 0 °42 > "j = 0

II ~ 0

s 0 a32

(6.133)

(6.134)

Tnu~ tho modol i~ ~.9.i .•

6.6 CONCLUSION

These rew examples. extracted (ram some real prOblems. prove that it ;~ actually possible and useful to apply the method~ pre~ented in the prev;ou~ ch4p­

ters.

Tn" h~t of tho paramotor~ fer ~_Jl_ idont-ifhbility i. betn tU"O.

as the results obtained have never beefl contradicted by the global analysis which fol­lowed. and fast. as the total CPU time required for all the examples of this chapter was less than one second on a 470/V7 AMDAHL.-

152

[xhau:otive modelling p,-ove:o to be quite erril .. it:ll~ ~uu. It t:llaLJlt:~

one to ch@ck S.o. idpntifiilnility pvpn whpn thp mpthorl of rh~l'tpr 4 rlEl"~ not apply.

for example when the parameters in A are related to those in Band C, or when the di­mension of the state-space is too large. When the model is not s.g. i., it gives useful information about the identifiability of each parameter, and about the relationships existing globally between the unidentifiable parameters, and it makes it easy to ge­m"a~t: ~ilt: ~t:~ ur all uU~iJu~-imJi~~I,,!!ul~lldLJlt: lIIuut:b.

In the most complex cases, the calculations needed for the study of one mode 1 haye regu ired hal r a uay'" "u, " ~. Till ~ IUdY ~t:t:IU <';UJI~ I Ut:r"dlJl t:, lJu ~ lid~ ~u lJt:

rnml'~rprl with th" timp ann PnPray that may b .. wa<t .. d in carrying out an inappropriat ..

experiment. As an example the study described in Section 6.4 lasted fifty @ight days. Mor@over the use of algebraic programming facilities [HII] now greatly reduces the calculation time [LIZ] .

We have met few examples of trulY multi variable transformation SYS­

tems. If the observation of several class@s is fairly frequent, one generally contents oneself with using a single input. We think it ~orthwhile to stress the structural in­formation revealed by the use of several inputs, which is evidenced by (5.18), giving the dimens10n Of a as a funCtiOn Of the dimenSions Of the state, inpUt and OUtpUt vec­tor",

7.1 INTRODUCTION

Chapter 7 Globalldcntifiability of Nonlinear Models

Of course a global approach to identifiability is as important for non-11near models as 1t 1S tor 11near ones. However, Wh1 Ie many papers have oeen devoted to the 3. y. ;del1t; r; abn ity 0 r H Ilear lIIodeb, very rew [ DlD, 011, P3, WI, W14] have

hppn concprnprl with nonlinp~r modpl~ Amona othpr thinac:_ thi~ i~ ~ m~nifp,tation nf

the still embryonic character of nonlinear theory. Testing nonlinear dynamic models for global identifiability is especial­

ly difficult, for one cannot, except on some rare occasions, calculate the analytic expression for the observed outputs as functions of the parameters to be identified. COIl"eyuell~ly i ~ b <Ii rrh.ul t ~o IovI'e ev"r ~V arrive a~ a yeller·al llIe~Iou<lolo9Y vaH<I rvr

any Hnd of modol. On thp othpr hand onp can work out npcp .... ary and/or <ufficipnt con-

ditions for the Qlobal identifiability of some classes of nonlinear models which allow a conclusion in many cases. For example local identifiability, which can be checked with the techniques described in Chapter 3, is a necessary condition for global iden­tifi abi 1 ity.

In tnls cnapter tWO approacnes, wnlcn -lead to varlous suiilclent con­ditione for the global identifiability of nonlinear modele, are con3idcrcd. The firet

one relies on series expansions of the model outnuts. The second one uses linearizations of the model equations around one or several operating points. The methods resulting from these two approaches are shown to yield different - and complementary - results.

7.2 !;[RI[!; [XrAN!;ION ArrnOACIi

7.2.1 Time-power series

The u~e or t;,ne-powe, ~e, ;e3, p,vP03C<l by Puhjallpalu [P3 ] , ha3 al, "a<ly

hppn nrp~pntpti in r.hantpl"? Thp motiol ic; a<~\lmPd to ho d""cl"ib"d by

154

~(o) ~ ~(~). L e [0. T 1 •

(7.1)

The values of the outDuts and all their derivatives with respe~t to timP at thp initial time t = 0+ are assumed known. and noted

(7.2)

A sufficient condition for the model to be s.g.i. is that. for almost any! used to generate the ~. there exists ~ € U such that

(k=O.l ..... "') l>s=s.

(7.3)

Note that if f is Lipschitzian on rOo T 1 • this does not make Condition (7.3) necessary and sufficient. contrary to what is said in [P3] . Consider. for exam­ple. the model described by

x(O) - 0 ,

(7.4)

It is Lipschitzian with respect to the state x and even to the extended state e = [x. s] T. At time t = 0 the function exp [- l/t~] has an essential singularity, aM all ltS OerlYatlves are zero. though It IS not 10entlcally zero on [0, T 1 . ThUS Condition (7.3) ;s not satisfied. However should one meQsure y at some nonzero t, one

wOlllti hp "hlp to itipntify A.

in the form

Application to nonlinear transformation systems

The equations of many nonlinear transformation systems can be written

j ~(t) I y(t)

A(~(t) .!) ~(t) + B(!) ,!!(t).

C(8) X(t) • (7.5)

A sufficient condition for 91obo.1 id~ntifio.bility i:s tho.t the knowledge

of ,,11 !.t«~) mllkp~ it I"0c:c:ihlp to pc:timlltp IInil')lIply ~ =~. Takino (7.Ii) into ",~~ollnt.

155

tin:: ~(~) O!\::: \li."," l.Iy

~(~) C~) ~(O)

!1(!) C(!) ~(O)

C(Q.) r AlxlO) .i) xlO) + BI~) .Y,IO) 1 •

~(~.> r.(~.> !(O) • . . - C(~) [I\(,!!.(O) .~) ,!!.(O) I A(.!!.(O) .~) .!!.(O) I B(~) ~(O) 1 •

C(~) [A(~(O) .2.) ~(O) ... A2(~(O) .2.) ~(O) ... A(~(O) .!!.) D(!!.) .!!.(O)

+ B{!) ~(O)J . . !3(~J C(a) [2A(~(0l..~) A(~(O) .!) ~(O) + A(~(O) &) A(~(Ol..~) ~(O)

+ A(~(O).~) ~(O) + A3(~IO).~) ~IO) + ? A(x(O) .A) R(A) 11(0) + A2 (x(0) .9) R(~.l ,!;!.(O) - - - - --~ 1\ (.!!.(o) .~) B(~) ~(o) I B(~) ~(o) ] •

~(~J = (7.6)

As one can see. the complexity of the computations involved increases very rapidly with k and the dimension of the state space. 50 that this method is app1i­cab 1e on 1y when the model order is small and when one is able to prove the gl oba 1 iden­tifiability of ! from the very first~. The examples of applications presented in [PJ, Dl0J Sdst1fy these r~qu1rement5. It shoula De notea tnat [I'J J ana L BlO J give Q)(amp 1 QC of non 1i noa .. modo 10 wni en ,,,-e 3. 9';' whil e the cOl-re3pondi n9 1i neod z;ed one5

are not.

7.2.2 Generating series

Thp """ .. o",.h IIdno oono .. ating ~o .. io~ [lt6. 10116] applioc to modele

described by

'" i l(~.!) \~l u;(t) f (x,a). x(O) = xo(a) •

= .[(~.!) (7.7)

In some neighborhood of any attainable state X. the fi(~.~) (i=O.l •••.• m) are assumed to be analytic vector fields in ~ [A5) • parametrized by! ; and .9.(~.!) is supposed to be an analytic function of ~. parametrized by!. (To be more precise the state ~ be-

156

lon9~ to 0 reol analytic manifold on which the vel-to, ri.,hb ri """I (.110: uuL~uL rUII(;(.10Tl

.2. are defined everywhprp) . Timp ri\n h" inrl",l".-i in tn" thto variablot, in ordor to deal with time-varying models.

Each vector field !i can be written as a differential operator

n

L (i=O,1, •.. ,m) , (7.8) k-l

where f~ is the kth component of !1.

Fl i ess [ F7, F8] has shown recently tha t for any ~ € U, where U is the set of piecewise continuous functions, the model output is a causal analytic functional of the inputs ~, the generating series ~ of which is given by

s m jo jk 1(X)1 + l I f o ..• 0 f ol(x)1 v .... v.

- kO· . 0 - J J o :. Ju •...• Jk = 0 Ie 0

(7.9)

The v. are non-colllTIutative indeterminates Iletters) allOt/ina a svmbolic calculus which Ji

gpnpri\li7P, Hpi\vi,irip', onp . Thp ,ymhal I mpan~ that pvorything to itt lQft in a givon o

monomial hac to be evaluated at the initial ctote~. The model output Z(t) for any 9i ven.!! € U is obtained by replacing Pilch word vJ .,.vJ in (7.q) hy t.h" rnrrp"llnnriino iterated integral: k 0

(7.10)

where

~o (T) = T (7.11)

'l(T) f: Ill(rr) rirr (i=l ..... m). (7.l?)

if t is small enough for the series to converge. The computation of l(t) for larger t can be carried out through ana lytic continuation methods [ K5] •

ProbablY for historical reasons. ~lterra series are a much more com­monly used tool for functional expansion than causal analytic functionals ; but there is a strong relationship between them, and (7.9) can be used to express the kernels of the VO I terra seri es associ ated wi th (7.7). Furthermore the express ions of the Volterra kerne13 relo.ted to 0. differentio.l equo.tion o.re much mOI·e complico.ted tho.n (7.9) [R7] •

157

TII~ !l~II~r·C1UII!l "~r·i~" ~ 1'1C1Y", ru,· lIull1 ill~C1r IIIU<l~'" <l~"1.,-j1J~<l lJy (7.7),

a ro1o ~;mi1ar to that o~ tran~for function~ for 1inoar modo1~. It charactor;%o~ tho

inDut-outDut behavior of the model ; and the vector ~(~) of all the coefficients of ~ forms an exhaustive summary, in the seAse of Section 3.2, of the information that can be collected on the parameters, because:

(i) It contains no information on 8 which cannot be extracted from the knowledge of the input-output behavior. Indeed if such was the case, one would be able to find two distinct values ~ and ~'SUCh that !t~) ~ !t~·) and that Mt~) and Ml~') be outl'uL- ill<l;"Lill!lu;"II<11J1~ IL would mean tllere would eJd~t a nom:ero generating :>ede:>

s: - SIr ~uch that ~or any piocQWi<o continuou< u th .. a«ociat .. d output i< irlpntica11y

zero, whi ch is fa I se r F8 1.

(ii) It contain<: ~11 information nn ~ whirh r~n hp pytr~rt.)rl from thp inpllt­

output behavior, since the generating series allows computation of the output from the input.

Thu!: tho output-indictin9uiehability o~ M(~) and AI(~) ic Gquiva1c>ot to

the eQuality of the oeneratino series

Therefore, for models described by (7.7), a necessary and sufficient condition for s.g. identifiability, when the input .class is the set of piecewise continuous functions, is that, for almost any ~ € 0, one has

8 € 0

8 • (7.14)

Remarks (i) In fact, in most cases of practical interest, one will have to extract

from !(~) a finite dimensional summary, which will generally no longer be an exhaustive one, so that the resulting condition will then be sufficient only.

(ii) For a model with zero input, the generating-series approach is equiva-lent to the time-power-series approach. as will be seen on Example 7.1.

In order to use Condition (7.14), we need to be able to compute terms of the generating series. Two methods can be used.

The first one, which always applies, is to evaluate the coefficient associated with the word v .... v. by computing

Jk Jo

158

(7.15)

It will be used in Example 7.2.

However, especially when !(O) z Q, many words are associated with zero coefficients so that this method may lead to many unnecessary computations.

The second method is an extension of symb0lic calculus to nonlinear differential equations [Ll3 - Ll5J which uses transforms as given in Table 7.1.

"i(O} - "i (O)

Xi (t) ----. Si

f: xi(-c) d-c ----. Vo si

Table 7.1 Transforms used for symbolic calculus.

The shuffle-product operator UU appears each ti~e a product nonlineari­

ty is met. The shuffle oroduct of two words is obtained by mixino their letters in all the possible ways without permuting the letters of each word. For example one has

Vo VI uu Vo - Vo VI Vo + Vo Vo VI I Vo Vo VI l

Vo VI Vo + 2 Vo vI

S = L (s,w)w , w

(7.16)

(7.17)

where w Is a wunl amI (s,w) Un:! c1l>l>u(;ic1Lo:lI wo:rril.iO:IIL, Lht: $hurrlt: pnJduct of two

ccalar cories Si and Sj ic given by

159

Si W sJ' - L (s. ,w.}(s .,w.) w. LU W. W. ,W. 1 1 J J 1 J

, J

(7.19)

10 take advantage of the results of fab Ie 1,1, one lntegrates (7.7) Ullt: Lilllt: ;

.!(t) (7.19)

and writes the corresponding transformed equation

(7.W)

where ~Ii is the transform of !Ii •

The series ~ can then be computed with the recursive scheme

(7.21 )

which must be initialized with an element of the series. for example

(7.22)

Tn ~k th~ coefficients of the words of lenoth k are then exact. Each iteration only brings new terms to be included in the exhaustive summary of the information contained in the state, from which one then deduces the exhaustive summary of the information contained in the input-output behavior.

Th1S methOd has the advantage Of generatlng onlY wordS WIth nonzero coeff;c;enb.

Example 7.1 Cons;der the nonl;ne~r model descr;bed by

!(O) __ [01]

y - "1 . (7.23)

Its s.g. identifiability has been studied by Pohjanpalo with the time-power-series approach l P3 J • The vector fields involved are analytic, so that the generating-series 4ppn;J41Oh 4ppli",~. Th", tnll1~rvru""d "''1u4tivn rvr" Lh", ~tClt", h

160

(7.24)

am! ~ can be computed recur51vely oy

I 5~+1 = - (A l+AZ)Vo s~ + AZ AJ Vo -<:~ Lli <:~ +

1<+1 k k k k 52 = 82 Vo S1 - 82 83 Vo 51 UU 52 - 84 Vo S2 (7.25)

wnn

1

- 0 (7.26)

One gets

(7.ll)

(7.28)

r,-om Table 7.1,(7.20) implie3

so that the successlve entrles of the exhaustlve summary are equal to the !J< (~) of the

timc-powc~ccricc approach

SOC~) '" aO(~) '" 1 (7.30)

51 C~) a1 (~) -(81 + 82 ) (7.31)

s2(!!) a2 (.!!) [ (tl1 T tl2)? T tl~ tl3 ] (7.JZ)

lL b e<1"Y Lu "huw U'<1L CumJiLiuli (7.14) i" "<1Lbrie\!. The lIIu\!el (7.Z3) b Lbu:> :>.9. i ••

t.xampte I.t.

Con3ider now the "ame model, but with I:ero initial condition and on

input u on thp fir~t r.11l~c; :

161

y = l<t (7.33)

Frolll (7.7). (7.0) and (7.33) one has

fO = [ - (Al + 8Z ) Xl + A l A 3 X 1 Xl ] ~ x 1

+ [ 82 Xl - 62 83 XI x2 - 84 x2 ] tx · l

(7.34)

(7.35)

(7.36)

,·s fjo jk I From (7.15) the coefficient associated with vJ ... vJ 0 ... 0 f 0 c(x) . As ~n k 0 - - 0

ox~plo. tho coofficiont a~~ociatod with tho word Vo VI is

f1 0 fO 0 .2.(~) 10 = 1 ~Xl [[ - (81 + 82 ) Xl + 82 e'3 Xl X2 HXl l<t

+ [82 Xl - 82 83 Xl X2 - 84 x2JtX2 ~11 ~ 10

'

(7.37)

Similarly, one has

f1 0 fO 0 fl 0 fO 0 .a(~) I - 1 ~2 ~3 ' 0

(7.38)

fl 0 fO 0 f1 0 fO 0 f1 0 fO 0 .l(~) I - 3 2 °2 °3 ' 0 (7.39)

,t 0 fO 0 fO 0 fl 0 fO 0 .l(~) 10 - - 2 °2 °3( °1 + °2 + °4)

Equations (7.37) to (7.40) yield

162

01 I 02 - 01 I 02

e~ e 3 9~ 9 3

: 3 :2 A 3 02 V2 V3 - v2 3 ~ " ~

92 93(91 + 92 + 94) = 92 93(91 + 92 + 64)

=;> tI = tI (7.41)

Thus the model is s.g.i ..

A~ can bg ~ggn from thg prgviout gxamplg, thg ca l culation of thg cogf­

ficient associated with a oiven word is fairly simple. and can be eased bv the use of algebraic manipulation routines. The problem is to select the words of interest, i.e. those which actually yield some information on the parameters. For the time being, this is done 'on a case-b~case basis, but more systematic methods are under investigation. ,

As for Pohjanpalo's method. computational complexity is a highly in­creasing function of the model order, so that the models tractable for the time being are far simpler than those which can be considered in the linear case. The linearization approach, to be presented in the next section, does not have this drawback .

7.3 LINEARIZATION APPROACH

This approach has been considered for the study of physiological sys­tems (01) and of nonlinear chemical kinetics with the help of compartmental models [Wl,LZ) • It leads to sufficIent condItIons for global ldent1flalJlllty.

7 _ ~ _ l Principle

A sufficient condition t'or M(!) to be glObally lClentl!laDle is tnat 1.I1t: fwu\;t:uur 't: ut:~ .. r'ilJt:u ill fillu ... : 7.1 C1l1ow~ unique dete,-minCltion of 2. - 2.'

Such an approach makes it possible to use, for the study of nonlinear models, the techniques presented in the previous chapters.

163

Nonl;ncar I-systemll Linc .. riz .. tion Linc .. rizcd

M(~) "system"

~QtQrmination of identifiable parameters or exhau3tive modelling.

-Parameters e of the Synthesis of Parameters of the -

-nonlinear model M(!) studies at linear model various operating points.

Fig. 7.1 Linearization approach.

7.3.2 Application to nonlinear transformation systems

Consider a model described by (7.5). The notation of Chapter 1 will be used to distinguish the labelled. unlabelled and labellable quantities. Suppose that a constant total input ut can be used to set the system in a stationary state xt. If Acl.!J 1S 1nVert1Dle:lw111 tnen Sat1sfy -

_ A-1(~\.!!) B(!!) 1!t • (7.42)

- C(!) A- 1{l.!) B{!) l (7.43)

Any conservative transformation system which contains no true endotropic graph [D2J will yield an invertible A(xt.e). provided that no component of xt is zero. The equi­librium state l will be as-;um;d locally stable (i.e. the real p-;rts of the eigenvalues of tne l1near1zed model w1ll be supposed str1ctly negat1ve).

T .. king into .. ccount the hypothecec m .. de on the nature of the trancfor­mations between classes (see (1.1) to (1.3)). the entries a;j in A(Kt .!) are all linear combinations of terms of the following kinds:

164

1 i IIt'OI "ij (7.44)

bilinear (7.45)

nonllnear (7.'16 )

It is worth noting that the linearized model obtained may be different dependlng on wnetner tne Ilnearization is of a matnematical kind - around some average operatin9-p(lint -, 01 of a f'hy";,,al lIa~Ult: -lJy ;1I-.1U::';OIl or ~r~a"t:r ;11 ~lrt: "y::.~t:'" ;11-

p"t~ - [BIO.WI]

Consider, for example, the transformation system described by

(7.47)

Linearizino (7.47) mathematically around xt. one oets

(7.48)

On tho othor hand, thQ bQhavior of a tracor includoq in thQ inflow ut of thQ ~y~tQm in

a stationarY state xt will be described by

(7.49)

thp pqll~tion for Ilnl~hpllpd m;ttpri;tl hpino simi1;trly

(7.50)

unce tne equatlons ot tne Ilnearlzed model nave oeen OOtalned, tne identifiability of it~ parameter3 con bc 3tudied, toking into occount thc rclotion3

hptwppn thp<:p par;omptpr<:. if ;ony Thp tprnni'lIlP tn np Il<:pci for thp <:ynthpsis of thp

nonlinear model from the linear ones will depend on the physical or mathematical nature of the linearization considered.

The linea.-izeed model I~e~ulting from 0 tracer inclu3ion in the ~tatio

nari7Pci <:y<:tpm i~ immpciiatply nhtainpci frnm (7 ~)

A(l ,~) !(t) + B(~) .!:!.(t) •

C(~) !(t) . (7.51)

165

ArL"'f tlo", :>tudy or th", identifiability of the entrie:> of the mat'-;~e:> A, D, and C .. ith

tho holp of tho tochn;quo~ pro~ontod ;n tho prov;ou~ chaptor~. two ca~o~ may occur_

7.3.2.1.1 All entr;e" ;n A(~t,~) and D(~) are ".g.; •• If 011 unknown

entrIes of A(xt,~) ana 6(6) are S.g.l. from experIments on tne lInear system assoclatea w,th thc trac;r,-thc 3tat~onary total 3tatc xt ;" 3.g.i. too, from (7.42). Thus onc

knl1ol~. po<;~ibly for ~pvprill VillIJP~ of thp to;ill ron~tilnt infll10l IJt. thp VillIJP~ of thp

entries in A(xt,e) a~d xt. A sufficient condition for M{a) to be-s.g.i. is that this information implies! = !.

Example 7.3 Consider the model described by Figure 7.2.

I"g. 7.2.

All exchange flows between classes are supposed linear, r~3 excepted which is assumed given by

(7.62)

The parameters of the nonlinear model are

~ - [oOl,012,k 13 ,u131,u133,a21,a02,a23,o24,o32,o03,a34,o42,o43]T.(7.~3)

The state equation associated with the kinetics of the tracer included in the inflows

166

of the 3'y3tem when in d,yllomh. "'luil iL,..iulII b \;lJdrd(;Lerlz;ed by the rOllowlng trlplet

I

( QI33- l ) I

k [xt] 11 131 [ t] - ( am +a21 ) i a12 0 I 13 1 x3 I I

321 : - (302+3 12+332'~42) I

023 024 A - I

I I [ tf13l[ t] (a133- l ) 0 I 3 JZ !-( aOll-k 13 xI x3 +323+343)1 ~34 I

I I I 0

I I "4l I a4J !-(a24+a34)

I

(7.54)

o o o o (7.55)

o 1 I o

It has been shown in Chapter 5. with the help of exhaustive modellino. that all free entries in A(!t,!) are S.g.1 .• It remains to be proven that k1l , a1l1 and al33 are s.g.i. too. Since A(!t,!) is invertible and s.g.i., (7.42) can be used to compute!t as a function of the total inflow.~t. FOr any ~~, one has

. t t log ~13 - log k13 I "131 log xl I ("133- 1) log x3 ' (7.56)

where all' x~ and x~ are s.g.i .• One can thus build a system of linear equations in a131' a133 and log k13 which makes it possible to estimate these three parameters pro­vided one proceeds to measurements in at least three different stationary states !l.

A particular case of importance arises when the four following condi­tions are satisfied :

(1) eacn non-olagonal entry ln A(!t,!) cOntalns only one term or tne KlnOs (7.44) - (7.46) ;

(ii) the nartial orders "tJK of the terms of the kind (7.46) are supposed known (bimolecular reactions for example) ;

(iii) A(xt,e) is generically invertible; (iv) no-co;ponent of !L is zero.

In tnat case tne s.g. 10entlTlaOll1ty Of tne moOel l1nearlZeO oy tracer InclusIon 1s a 3ufficient condition for the PQrOmeter3 0 of thc nonlincor modcl to bc 3.g.i •• Indccd

the entries in A(~t.~). Btfr) and ~t bein; s.g.i .• one can estimate ~ with the help of (7.44) to (7.46) without needing to consider several stationary states !t.

167

ElIdmple 7.4

Consider again the model described by ~igure 7.2, but assume now that

the bilinear and nonlinear exchange flows are

t r l3 kl3 t

xl t

x3

t r23 K23 xt 2

t x3

t r'lj k'lj "t

4 t

"j

t , xt rn on 4

The paramoter~ of the nonlinear model are then

(7.58)

The tracer 3tate equation can be written

-(a01+a21 ) a12 t

k1JX1 0 0

a21 -(a02+a12+a32+042') t k23X2 °24' 0 0

.! x+ II

0 a32 t t t

-(a03+k13xl+k23x2+k43x4) a34 0 I

0 °42' t k43x4 - (°24 ,+a 34) 0 0

(7.5g)

.[: 0 0

:] y X • (7.60) 1 0

This linear model has the same structure as the previous one, and the entries in A{!t,!) are S.9.i. here too, so that!L is S.g.1. with the help of (7.42). This makes ! s.9.1., Slnce one nas

168

°1 - ("11 I "21) ,

62 a12

63 t

an/Xl

64 a21

65 -(a I2 + a22 + a32 + a42 )

- .t 06 ' 023/"'2

67 a24/tjr

Aa "'JZ

69 -(a13 + a23 + a 33 + a43 )

6 10 a34

6 11 ~ 042/'1'

6 12 = t a43/x4 (7.61 )

7. 12.1.2 ~omo ontriQ~ in A(~t.~) and B(~) arQ not ~.9.i .. In that

t -----------------------------------------------cose x moy not be calculable by (7.42). One h03 then to 3tudy whether the knowledge

nf th~ '.a.i. pntrip, in A(Xt.A). R(A) ",nn xt. ",nn nf thp rpl",tinn, pvi,tina hptwppn

unidentifiable entries (pos~iblY for-several values of ut ) allows a unique determina­tion of e = e .

Example 7.5 Consider the two-class model described by Figure 7.3.

Fig. 7.3.

169

rne state equatIon Of tne lncluOeO tracer Is

y = [1 o J x •

t k lZ xl

+

o

(7.62)

From Figure 4.4 this linear model is not s.g.i .. Exhaustive modelling can be used to obtain the set of all matrices AS corresponding to output-indistingui­shable models

(7.63)

trom (/.4(). one nas tnen

0 a2?

0 Q21

0 Q12

0 QU 0 Q22

~\a) 0 a a21 ut (7.64)

0 0 o 0 il ll azz - an a lZ

Equation (7.63) impl ies that the following quantities are s. g. i.

< ° - (a01 + a2l ) all all ,

s il ZZ

0 ilZl. -(ilOl + k II

t x 1)

s s a?l a 12

0 an

0 all'

t kI? xl a?l (7.65)

from (/.b4) x11S S.9.i .. It is thus sufficient to stationarize the system at two dif­fe,-ent ope'-Qt;ng p"dnb ~t Lu Lot: aLolt: Lu dt:Lt:nuillt: ulliyut:ly Lilt: l'''nllllt:Lt:r-~ ur Lilt: nun­

lin"al" mnti"l. nam"ly aUl' ~U. "n "nd "Ul Tho intl"odllction of a nonlinc>al"ity hat mad"

S.9.i. a model which, when linearized at ClI1_e operating point. was unidentifiable.

170

[ ... dllI!'l t: 7 •. 6

Consider the model described by FiDure 7.4.

Fig. 7.4.

The correspondino state equation i~

~t

(7.66)

The stat10nary state assoc1ated w1th a g1ven constant 1nput ut 1s oOtalned oy Settlng ~t _ 0 in (7.66). It yicld~

(7.67)

(7.68)

Therefore x~ depend" linearly on ut , .. hile xl doe" not depend on it. Thi" i" a moni­

fo~tation of tho nonlinoarity of (7 66)

The state equation of a tracer introduced by inclusion of a labelled inflow u in ut is

[ ~l xt a 12 2

~

!ell yt

-(lIOl. + l

Y 1 0 x • (7.69)

171

E:xhaust he model1! ny \;(111 In: u:.ed Lv yelltwa Le Lhe L .. o ~ La Le 11112 kh.e:) 0:):)0(, i oted .. ith tho output-indictin9uichablo modolt. Ono it obtai nod from tho othor by oxchanoino a~Z

o and aZ1

0

, 1 - a21 a12

v (a02 + a~2) a21

(7.70 )

u

" j - a02 a12

0 - (a12 + a~l) a02

(7.71)

Therefo,-e 012 i3 3.9· i ., while 002 and a21 ore only 3 • .!..i.

Now l!>t II~ ~turly thp id!lntifiability of ~Zl. Thp ~bt;nn~ry ~btp~ fnr

the labellable quantities associated with AO and Al are respectively

(7.72)

Taking (7.67) into account, (7.72) implies

(7.73)

(7.74)

so that both models yield the same value of k21' which is s.g.i. although a21 is not.

7.3.2.2 ~~!b~~~!l~~l_ll~~~rj~~!jQ~

The main difference between this and the previous section is that the state matrix appearing in the linearized state equation will not be A(!t,!) as defined by (7.5). Should all unknown entries of the matrices of the state equation of the mo­oel 11nearlzeO arounO an operating pOint xt prove to De S.g.i., th1S Will not 1mply that A(~t,~) i3 3.9.; •• The determination-of ~t from (7.42) may therefore be impo~~ible.

172

One will then have to determine Lin" i"ruJ"llloliuti UII lit pruvl11ed by tne Imowleoge or tne ;~pntif;anlp pntripc of tho linoari7od modol. and to oxamino itc conccqucnccc on the

identifiability of the parameters of the nonlinear model.

FXllmplp 7_7

Let the model be described by Fi Qure 7. 5a.

Th~ corr~:,polld1ng :;tat~ equat10n ls

. t x t t fix ,e,u )

Fig. 7.5a.

k(l-sx~)xi + ut 1 k(l-cx~)xI -~02 x~

Llnearlzlng (7.75) around ~t, one ODtalns

dy = [

t -k(l-sx?)

o 1 dxt.

(7.75)

01 1 du t ,

(7.76)

Tn; ~ 1i n"ar moil"l ; 0;: r"pr"o;:"nt"iI ; n ~i !Jur" 7 <;b A ngw "d!Jo ha~ app""r"d. wh; eh H 1 U~­

trates the information on class 2 being fed back into class 1 by the nonlinearity of the exchange r~l'

173

tlg. I.~O. txample ot tlgure I.~a 1 j'Jt::CI,·I£~cJ ClI·UUllcJ "t.

From Chapter 4, all entries of the matrices of this linear model are s.g.i., so that k{l-sx~) k s x~ and a02 are s.g.i •• From (7.75), the operating point XL satisfies

xl - ut I k(l-~"~) (7.77)

(7.78)

XL is thus s.g.i., since it can be expressed as a function of s.g.i. quantities. Linearizing the system around two different operating points, one obtains

~IIUU\j1r jllruI"I'ICILiul' Lu ~~LlllldLe unlquely the parameters k, sand a02 of the nonl1near

kineticc. Thuc they are £.9.i ••

7.3.2.3 ~~~!_j~_!~~_Q~~!_]jD~~rj~~!jQD ?

The two kinds of linearization correspond to quite different experiments, and the information which can be collected from them may also be fairly different. When ootn KlndS or experlmentS are teaslole, a natural questlon is : which one is better, from the identifiabnity 3tandpoint ? No 3Y3temati,; an3we,· ,;all be 9iven, and, depend;II9

nn thp rnn,irlprprl mnrlpl. onp or thp othpr may bp thp bgttpr .

[xampla 7.S

Consider again the model Df Ex~mpll' 7.7. hilt rl'phrl' thp IMthplMtirlll linearization by a physical one, by tracer inclusion. The resulting linear model is described by Fi gure 7. 5c, and sati sfi es

y 1

u

174

o ] x.

Fig. 7.5c. Example of Figure 7.5a 1 inearized oy tracer lnCIUSlon.

The only s.g.i. entry of this linear model is

lI.l9)

(7.80)

and one can only estimate k and s/a02 ' so that mathematical linearization has to be pretere~ tor tnls example.

Remark: As seen from Examples 7.7 and 7.8, not only the values of the exchange rates, OUt alSO tne number an~ loCatlon aT tne eX1Stlng excnanges ~epen~ on tne kln~ aT the linearization performed.

Example 7.9

Con!:ider the model de!:cribed by I'"igure 7.6.

Fig. 7.6.

175

The assoc1ated state equat10n for the labellable quant1tles Is

X = • t [

t t (k12 - k21 ) Xl x2 +

ll<21 - 1<12} x~ x~ -

(7.6l)

Note that x;(O) must be nonzero for an exchange from class to class 2 ever to take place.

Mathematical linearization will never make it possible to estimate 11.12 <11111 11.21 separCitely, s1n(;e only the1r 1I1fferell(;e C1ppeCirs 111 (7.61).

On the other hand, tracer inclucion recults in a dynamic behavior described bv

[ - '21 / 1<12 xt

1 [:j 2 1 x - ~ u,

kll yt t

l -(kllxl+ilOl)

y = [ 1 o ] x • (7.82)

The recultc of ~igure 4.4 then imply that k21 xi, k12 xl and a02 are c.g.i •• equation (7.42) allows uniQue determination of xi and xi. so that k1Z • kZl and aoz are 5.g.i .. Thus, for this example, a physical linearization gives more information than a mathe­matical one.

I.j.j uenerallzatlon

The method is immediately generalized to models described by

(7.83)

d",. -1

I u u X. ,u. ,6 -1 -1 -

a = -r g(x,u,e) I

ax - 0 0 x· ,u. ,tt

-1 -1 -

176

<lA • .,. -1

x~,u~.e -1 -1 -

o 0 x. ,U. ,tl -1 -1 -

duo -1

d'y";

around several operatino points K~ associated with several constant input~ Q~ (i=I ••••• 1).

Equation (7.84) can also be written

oj. B du 1 -1

(7.64)

(7.85)

If the knowledge of the 5.g.i. entrie5 of (Ai' Bi' Ci , Di ) (i-1 ••••• I) allows a uni Que determination of~. then ~ i~ ~.o.i ..

With the exception of some particular cases, such as the study of non­linear chemical kinetics, it is quite inappropriate to use this experimental procedure to estimate the parameters actually, just as it was out of the question in Pohjanpalo's metnoa to estlmate ! actually rrom tne successlve aerlvatlves or tne output at tlme zero. In both ca3e3 the aim of the method i3 to 3how that the 3tructure cho3en for the mnrlpl m~kp~ it~ irlpntifir~tinn thpnrptir~lly fp~~ihlp.

7.4 CONCLUSION

Fuur' "rt:Lllu<l~ rur' ~Lu<l'yill1:l Lin;: :>.9. idt:ntiriabiliL'y ur nUIIU",;:a, <I'yIIO­mic models have been precented. They can be cla~~ified into two approache~_

Tilt: :>erie:> (J.ppr'oo~h lead:> to ~omputation3 the "omple",it.Y of whi"h g'-ow"

extremely rapidly with the model order, However it pre~ont. the advantago of enabling the study of models which cannot be put into a stationary state. such as time-varyinq or unstable models.

The linearization approach yields simpler calculations, and the limi­tation to the complexity of the tractable problems only comes from the limitations of the linear methods.

The main point to remember is that, in almost any case of practical

177

In~t:rt::i~. dll ~1It::it: tnt:Lln:llb r·t::.ul~ in I;t"I"'i~iu"" fur· ylulJdl idt:,,~ifidlJili~y wllidl dr·t:

only oufficiont and which aro difforont. Thio i. why nono of thorn can protond to bo

the best in any case. On the contrarY one JOOst be thankful that there exist several tools amongst which one may hope to find the one which will allow a conclusion in the special case of interest ; and it can be anticipated that methods leading to necessary or sufficient conditions for global identifiability of nonlinear models will JOOltiply. In th1s respect two 1deas seem l1aOle to iuture development. The i1rst one Is UraL vi

building more or leee imaginary identification methode, which one trice to prove would

allow a uniQue determination of the model parameters. The second one is to look for new exhaustive summaries of the information available on the parameters.

Conclusion

In the lntroauctory Chapter of his book "System Identification", P. tykhoff stres3ed tne need for 30me limiting theon'lU~ VII lilt: 1IIC1111nlulII r~!)ul t obta1nable in the solution nf II giw.n Jlrohl,;.m. In !:om~ S:OMO, tho s:tudy of tho structural iden­tifiability of state-space models belonqs to that rationale. The point is to test. be­fore attempting any parameter identification, whether the structure chosen for the model does not render the attempt void of any meaning. From this point of view, one may say that the most significant property in practice is not identifiabi1ity, but unhlellli r; all il i ly. IL b lIul ll~(;aus~ a parameter 1 S s.9. 1. 'that one Wl11 aCtUally De ablo to o~t;mato it. On tho other hand, if 3 p3rometer i5 unidentifiable, one know3 one wi 11 not be able to estimate it. except by ca 11 i no upon ~om .. IIclcliti nnlll ~ Jlri ori knowledge.

[o.ch time one aim~ at e~timotill9 flo,allot:lt:':' willi Q ~ulI~n,Lt: 1IIt:,miIlY, t .. ~ting thpm for c:frlldur"l id,;.nt;Hability is: an indis:ponnblo ~top in tho modolling process, so that most of the areas of the applied sciences should be concerned with the notion. However, if its importance is now well acknowledged in Economics, Biology and Automatic Control, it remains strangely ignored in Physics, Chemistry or Ecology, all aoma1ns where 1ts importance seems manifest.

The methods which have been presented in this monograph make it possible to test state-space models for identifiability, either with a local approach or with a global one. The techniques differ depending on whether the model is linear or not, but, 1n both cases, a reSOlUtely active attltuae towaras unlaentltlable moaels has been proposed. Thus one e3n 3nolysc, at least locally. the nature of any recorded deoenerilcy. so ile; to clptprminp whirh 1It1t1it"ln"l drllctllral ronc:traint~ or compl""'~nta­rY experiments would be able to remove it. Such an aid in the choice of the nature of the interactions between the experimenter and the system is akin to experiment deSign, to which we shall revert later on.

Unfortunately the policy of makinQ the model identifiable at all costs is not always reasonable, especially if it results in unjustifiable structural cons­traints on the model. This is why we have developed, in the particular case of linear tlme-invariant moaels, the technique of exhaustive modelling, which enables one to

179

repla~e the :;ean;h for a unique lIIudel by tile :.edn.h rur the :.eL uf dll uuLpuL-illdi:.­tinguichablo oneco Thie technique aleo permitc one to know which parametore aro ~.~.i.

or s.g.i .• and how the unidentifiable parameters are related.

Une Of tne bottleneCkS ot a global approacn to lOentltlabl Ilty stuOles, e3pecially for large-3cale mode13, i3 the nece33ity of 301ving 3et3 of nonlinear 01-

ophrllir O:>lllllltionc:_ Wo:> fo:>o:>l thllt tho:> 1lC:0:> of tho:> proro:><1llro:> of o:>xhllllc:tivo:> mn<1o:>llino. or

of suitable exhaustive summaries (i.e. summaries leading to the simplest sets of e­quations), now allows one to arrive at a conclusion for most linear models of practi­cal interest. Programs designed for algebraic computations offer a promising way of furtner reouclng the user's burden, and sucn programs, currently under development, wi" a""" the 3 tudy of more and more comp 1 eJ< mode 13 •

For nonlinear and/or time-varying models, the situation is far from be­ing so advanced. There is no tractable necessary and sufficient condition for global IdentIfIabIlIty, ex~ept In some very specIal cases. At best one has at one's dIsposal necessary conditions and sufficient conditions for global identifiability which may

allow a conclusion in the particular case of interest. Local identifiability. for ex­ample, is a necessary condition for global identifiability. The methods which yield sufficient conditions for global identifiability of nonlinear state-space models are not numerous. We have presented four of them, and we feel that there is much room for otners to extend tne set ot eXlstlng tOOlS.

Whether the model be linear or not, these structural studies are per­formed independently of any ~easurement on the real system. That is why one often speaks in that case of a priori identifiability~ As we have seen, the study of struc­tural (or a prIorI) IdentIfIabIlIty enaOles one to know whIch parameters wIll Oe Im­possible to estimate. One must then proceed further, and try to estimate the parameters

(or the combinations of parameters) of which the a priori identifiability bas been es­tablished. Because of the uncertainties in the data, and of the ever approximate cha­racter of the model, it is illusory to hope to determine these parameters uniquely. On the contrary each s.g,i. parameter will be estimated with some uncertainty, which has tu be evaluated and depend:; on the experIment ~hosen. ThIs leads one to lntrodu~e the notion of a posteriori identifiability, i.e. taldng the collected data into account.

There is no longer any dichotomY between the identifiable and unidentifiable parameters. but the less uncertain a parameter, the more it will be a posteriori identifiable. If one is interested in an accurate estimation of some parameters, optimal experiment de­sign may be the determination of an experimental policy which maximises their a poste­rlor1 Ident1f1ab111ty. Tn1s raIses three prOblems.

180

Fir:>tly vile "eell" a <..-iLe.-ivJI Lv lfu'IIILiry a fJV!>L~rlurl ld~nt1flabll1ty,

in orrlpr to hp ablp to comparo tho idontifiability of tho paramQtor~ of tho £amo mo-

del (which are numerically known) under different experimental conditions. This pro­blem is related to the estimation of the uncertainties in the parameters from the un­certainties in the data, and, in spite of a number of papers devoted to this problem of tremendous practical importance, the results obtained are still very fragmentary. Mc:tny I,;holl,;~!> csre fJu:.:.lble, and these must be evaluated and compared. One of tnem, for oxample, ic to uce ac 3 criterion of 3 po~teriori identifiability the determinant of

Fisher's information matrix. With such iI choke it mod!'l which ;c; not c;.LL would hilv!' a zero a posteriori identifiability.

The :>eeond problem b to en:>ur-e LhaL the ofJLimi;<aLion al!:jv.-iLlulI;> will

yiold roali~tic o~PQrimont~ which could actually bo performed, taking into account

the practical constraints (often very exactinQ. especially in'Biolo<lv) on measurement schedules and input forms.

~inally there ic a need for come cuboptimal experiment decign, that i&

the desion of experiments which would not be optimal just for one very specific value ~ of the parameters, but simply acceptable for any value of the parameters belonging to the set of all the possible models, taking into account the evaluated uncertainties in e. In a sense this can be viewed as some "generic" or "structural" a posteriori ldentltlalJ1IHy.

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Subject Index

posteriori identifiability, 41, 179 = priori identifi~bility, 179

'H .. Livi Ly specific ,92, 140 tota 1 _, 140

aggregation I"~r~mptpr "fi7 state variable ,67

algebraic approach, 34

- programming, 40, 71, 152, 162 _ varl ety, 22

analytic continuation, 166 = vector field, 155

applications of exhaustive modelling, 1"

attraction r90i nn !'If _" 4?

~typicQl nypcr~urfQcc, 22

Berman and Schoenfeld's method, 44

bilinear equations, 88 = trancformation, 6

bimoleculo..- reo.ction, G, 1GG

Boolean product of matrices, 37 scalar product, 36

Branin " conjocturQ. 43 =' s method, 42

bromosulphtalein, 136

bud, 3Z

cacti, 33

cactus. 32

canonical roprocontation, 19, 41

cate9orie3 or ellperimental identi ri­cation, 7

catenary models, 51

~hpmi~~l kinpti~~. R. lfi?

cnemothcr~peutic model, 129

c;ld:':'

equivalence _. 5 two-_ models, 90

Cobelli 's method. 46

coherence intcrno.l • 1. 10

compartmental models. 7

complex-conjugate eigenvalues. 83

computer-aided design. 66

concentration d. U9 ,1Z9

condltlons necessary ,27~ 56) 86, 153, 157 sufficient- , 2/, 50, 85. 86, 154 157. 162. 1b6. 177

connoctability, 22 input ,24, 27 output- , 24, ::!7

\;tm5l!rVCI L I VI! mul.ll!b, 7

constraints ineQuality • 95 structural :' 55

continuous-time models, 2

control process _, 1

contro 11 abil; ty matrix. 36

structural , 24, 27, 31

controlled part, 100

convention for matrix derivatives, 57

Cramer-Rao inequality, 64

cr; t/:'r; nn. 'iQ

decomposition uf lCIrgt:-:'\;dlt: lIIul.lt:h, 51

singular-value _, 39, 69, 110

deaeneracies non-structural , 77, 127, 130

degrees of freedom, 44, 101

denumerable set, 98

dependent entries, 37

: flows. 17

dorivativo matrix ,57

design experlment ,41, 178

Dirac impulse, 84

discrete-time models, 2

dual, 31

dynamic equilibrium, 12, 166

eigenvalues compex-conjugate • 83 real , 82 -

equ il i bri urn dynamic ,12

state ,-163

199

equivalence ela~~c3, S

error output , 58

exhaustive modelling, 97, 116 = cummary, S7, lS7

t:'AJ.lII:Tilllt:'lIt

design, 41, 178 structure, ~U, B~

experimental set-up, 129

extended state, 62 ob3ervobility of thc _, 64

extraneous slngularlty, 4~

eye symbolic _. 7

feedback loops, 54

Fisher's information matrix, 64

Freedom degrees of , 44, 101

Friedel-Crafts reaction, 14

GClU:':'

-Newton, 60 --'s pivotal method, 114 :-Seidel. 61

gonorating modol, 19,4;, OS

gene .... ic. property, 22 = rank, j4

generi city, 21

global approach to local idcntifiobili

ty, 110 il.lt:IILifldUll i ty, Z6, 84, 98, 118,

153 optlmlzatlon, 41

gradient, 42

graph rcocti ona 1 , 13 _-theoretic-approach, 32

Harrls's metnoa, 14

hepatulliliaJy Idneti,,~ uf D.S.P., 13G

HeSS1an, tH

id~ntifiability. 18 a posteriori • 41, 179 a pri ori ,1iO global ,-26, 84, 98, 118. 153 lu"al ~ Z~, ~G, 90, 110

of 8-and C, 118 stocnastlc _, 65 structural ,24

idgntification algorithms, 58

imaginary _ method3, 51, 177

lmpl1Clt runctlon tneorem, ~6

impulse injection, 79, 84

inclusion tracer _, 11

lndhL I"gubll<llli 1 i Ly output ,25, 97

ineQuality constraints. 95

information matriy. h4

initial state, 57, as

lnput connectabl11Ly, Z4

internal coherence, 1

invariant orbital • 41 time-_ models, 79

iodine, 139

Jacobian, 58

Kalman's canonical form, 120

Kronecker product, 60

labellable, 8

labelled, 8

Lagrange-Sylvester polynomials, 83

large-scale models, 51

200

letters, 150

1 i ke 1 i hood, 65

linear modgl. 65. 69. 79. 97

- transformation, 6

linearization, 162, 173, 176 matnematlCal ,ll, 104, 171 physical ,1~. 164

I ip~chit7 condition. 1~4

local approaches, 49, 56

ylulJ<ll appr-u<l"I. Lu _ hJt::rourialJili­ty, 110

idenfifiabi Itty, Z5, 56, 98, 110 = stability. 58

manifold, 156

mathematical linearization, 12, 164

metabolism of iodine, 139

Mi chile li s-Menten eQuilti on. 6

minimal roprocontation. ~O

mudel ~tr-u(.tu.e, ZO

mOde 1 I,ng exhaustive .97.116

mod,;>l~

catenary ,51 continuous time ,2 discrete-time ,-2 generat1ng ,T9, 45, 96 large-scale- , 51 linear ,65, 69, 79, 97 manmillary • 51 nonlinear : 56, 153 output-indi~tingui.hablg _' 25, 97

theoretical ,2 time-invariant ,79, 97 time-varyino .-156. 176 unstable ,176

multiplication, 7

necessary condItIons. ll, 50, 60, 153, 157

Newton aloorithm. 60

non-cnnmoobti vt> i ndt>tt>rmin;ott>~. 11;h

nonl1near _ models, 56, 153

transformation, 6, 154, 163

non-standard matrices, 110

nontrivial variety, 22

number of degrees of freedom, 101

observabi 1 i ty m"tri". lA

- of the extended state, 64 5tructur~1 _, 24, 27, 31

ubs~rv~d ~drt, 100

operating point, 163, 176

optimization global 41

output, 7 _ connectabil ity, 24

error, 58 = indic;tinguishilbilit.y. ?5. 97. 157

parameter a!l!ln'!latiufI, 67

=: space, 25

parametrization of the problem, 96 = of thp tr~n~ition m~tri". A?

pathological ,27, 30

physical linearization, 12, 164

Pohjanpalo's method, 50, 153

pocteriori a _ identifiability, 41, 179

power series, 50, 153

preexponential factors, 44

priori a _ identifiability. 179

procedure for exhaustive modelling, 116, 1Z:3. 1Z4

process contrOl, 1

201

product Boolean of matrices, 37 II.rOneCker ,00 shuffle ,-158

prnper variety. 22

prop~rty

generic ,22 ~lrul.tu,Ql _, £1

Quasilinearization, 19, 62, 72

random search. 42

rank checking, 39, 67

reactional graph, 13

real eigenvalues, 82 = ~y&tom. 50

, "l.un~t,ulOtion of the "tote, 19

relatlons _ between structural notions. 27

between unidentifiable parameters, h7

rcprc5cnt~tion

minimal ,49, 98

reproduction. 7

scalar product Booloan • 36

~prip~. 176 generating ,155, 157 power • 50-Taylor- , 60 t1me-power ,153. 157 Volterra ,-156

shuffle product. 158

dn!Jular-va luo docompocition. 30. 60. 110

singularity ,,:.:.tmlldl • 1:14 extraneous-_. 43

solutlon denumerable set of _so 98 unique _. 98

space-covering techniques, 42

species. 8

specific activities. 92. 140

s tabi 1 i ty, 163, 176

~tanrlard matric~~. 99

sbndardi "ati on

state

of Band C. 112 of ca. 110

aggregation. 67 extended • 62. 64 recon,truction of the stationary ,12 -

stem. 32

19

stochastic identifiability. 65

"trllrtllrlll controllability, 24, 27. 31

- identifiability. 21, 26. 26, 27 - nature of numerical results, 71 non degenerd~le~. 77. lZ7. 130

observability. 24. 27. 31 properties. 21

structure experiment , 20, 96 model 20-

subsystems, 54, 93

SUTTlclent conaltl0ns, Z7, 5~, 65, 6~ 154, 157, 162, 166, 177

«;ummllry. 1 ~7 exhaustive ,57, 157

symbolic colculu~, 156, 156, 16e

- computation, 40, 71, 152, 179 - eye, 7

system real • 59 transTormation ,5 _ under~tandin97 1

~y~tcmic di~tribution of Vincominc, 14~

202

Toylul ~elh:~, GO

tneoretlcal moael, z

thyroid, 140

time continuous , 2 discrete .-2

-lnvdrld'ilL, 79 --power seri es , 50, 153 =-varying. l!lb, lib

tracer includon. 11

- injection, 79 -trajectory methods, 42

transfer function approach, 45

tranc;fnrmation. 6 bilinear ,6 linear ,-6 nonlinear ,6 ~fJ"d ri<o fJ"ulJl"lII~ ur _ ~y:)te", lIIudel-1 ing, 18

system, !)

transition matrix, 72 parametrization of thg . 92 properties of the _, 79-

typical points, 22

understanding c;yc;tf!m _. 1

unlaentlflaDle model,99

relations between parameters, 67

unstable models, 176

YO"; t: Ly, ze algebraic ,22 nontrivial- , 22 proper • 21

vprtnr fipldc;. 1~~

Vetter's convention, 67

Vincomine, 149

Volterra serles, 15~

word. 156. 158. 162