System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den
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1 System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1996 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Hof, J. M. V. D. (1996). System theory and system identification of compartmental systems s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:
2 System Theory and System Identication of Compartmental Systems Jacqueline van den Hof
3 RIJKSUNIVERSITEIT GRONINGEN System Theory and System Identication of Compartmental Systems Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnicus, Dr. F. van der Woude, in het openbaar te verdedigen op vrijdag 8 november 1996 des namiddags te 2.45 uur door Jacoba Marchiena van den Hof geboren op 14 december 1967 te Emmen
4 Promotor: Prof.dr.ir. J.H. van Schuppen
5 Voor Wiljan
6 Voorwoord Als laatste voor dit proefschrift (dat ik in mijn dromen al een tiental keer verdedigd heb) schrijf ik nu dit \eerste" hoofdstukje. Hierin wil ik iedereen bedanken die bijgedragen heeft aan het tot stand komen van dit proefschrift. Een aantal mensen wil ik in het bijzonder noemen. De afgelopen vier jaar heb ik heel veel geleerd. Hierbij speelde Jan van Schuppen een onmisbare rol. Zijn uitstekende begeleiding en constante interesse waren een enorme stimulans om dit onderzoek tot een goed einde te brengen. De duidelijke systematiek van Jan heeft mij geholpen om niet in de vele onderzoekrichtingen te verdwalen. Regelmatig heeft er overleg plaats gevonden met Peter Janssen, Wout Slob en Peter Heuberger van het RIVM in Bilthoven. Peter (Janssen), bedankt voor de goede gesprekken, ook de meer persoonlijke. De simulaties in Hoofdstuk 7 zijn tot stand gekomen dankzij de Matlab berekeningen van Peter (Heuberger). Wout wil ik bedanken voor het schrijven van het gezamenlijk artikel voor een toxicologisch tijdschrift. Twee keer per jaar vonden er bijeenkomsten plaats van de begeleidingscommissie. Deze bestond uit de bovengenoemde mensen van het RIVM, en verder Odo Diekmann, Arnold Heemink, Hans Metz, Gerrit van Straten, Jan Verwer en Jan van Schuppen. Deze bijeenkomsten waren heel nuttig, bedankt. De laatste veranderingen zijn aangebracht dankzij de grondige lezing door de beoordelingscommissie, bestaande uit Malo Hautus, Rien Kaashoek en Gerrit van Straten. Ik wil hen bedanken voor het plaats nemen in deze commissie. Heel belangrijk voor de theoretische ontwikkeling zijn de cursussen van het Netwerk Systeem- en Regeltheorie geweest. De internationale contacten zijn grotendeels te danken aan de workshops van het SCIENCE Project System Identication. Hen allen wil ik bedanken. Voor de prettige werksfeer wil ik de collega's van het CWI bedanken, in het
7 bijzonder mijn kamergenoten Hein en Coelho, waarmee ik ruim 3 jaar kamer M138 heb gedeeld. Verder bedank ik Kai, Jeroen, Wayan en Ben, die het afgelopen jaar mijn kamergenoten waren. En de koe om 16.00u? Jammer dat dit niet meer een echt afdelings-evenement is. Verder wil ik mijn familie en vrienden bedanken, vooral mijn ouders Ineke en Wim. Jullie altijd aanwezige steun en interesse, zij het op afstand, zijn heel belangrijk voor mij. Aan mijn broer Wiljan draag ik dit proefschrift op. Ik weet dat je enorm trots op me bent, maar ik ben ook heel erg trots op jou, omdat jij je, ondanks je problemen, staande weet te houden in deze maatschappij. Tenslotte, zonder Ard was dit proefschrift er niet geweest. Jouw liefde en steun hebben me geholpen om door te zetten. Zelfs in de laatste periode bleef je een rustpunt, ondanks dat je zelf ook met de afronding van je proefschrift bezig was. Amsterdam, september 1996
8 1 Table of Contents Table of Contents 1 1 Introduction Motivation :::::::::::::::::::::::::::::::::::::::::: Approach :::::::::::::::::::::::::::::::::::::::::::: Overview of the thesis :::::::::::::::::::::::::::::::::: 11 2 Compartmental Systems Introduction ::::::::::::::::::::::::::::::::::::::::: Continuous-time compartmental systems :::::::::::::::::::: Discrete-time compartmental systems :::::::::::::::::::::: Graphical representations of compartmental systems ::::::::::: 23 3 Structural Identiability Introduction ::::::::::::::::::::::::::::::::::::::::: Structured linear systems :::::::::::::::::::::::::::::::: Structural identiability from Markov parameters and initial parameters ::::::::::::::::::::::::::::::::::::::::::::::: Structural identiability from input-output observations ::::::::: Discrete-time ::::::::::::::::::::::::::::::::::: Continuous-time ::::::::::::::::::::::::::::::::: Conclusions :::::::::::::::::::::::::::::::::::::::::: 51 3.A Proof of Proposition ::::::::::::::::::::::::::::::: 51 3.B Proof of Lemma ::::::::::::::::::::::::::::::::: 54 4 Positive Linear Algebra and Polyhedral Cones Introduction ::::::::::::::::::::::::::::::::::::::::: 57
9 2 4.2 Presentation of primes :::::::::::::::::::::::::::::::::: General denitions ::::::::::::::::::::::::::::::: Primes :::::::::::::::::::::::::::::::::::::::: Equivalences ::::::::::::::::::::::::::::::::::: Apartial classication of primes ::::::::::::::::::::::::::: Positive matrices and doubly stochastic matrices :::::::: Polynomial representation of doubly stochastic circulants :: Primes in the doubly stochastic circulants ::::::::::::: Polyhedral cones :::::::::::::::::::::::::::::::::::::: Positive matrix factorization ::::::::::::::::::::::::::::: Positive rank ::::::::::::::::::::::::::::::::::: Extremal cones ::::::::::::::::::::::::::::::::: Positive linear dependence ::::::::::::::::::::::::: Determination of positive rank :::::::::::::::::::::: Conclusions :::::::::::::::::::::::::::::::::::::::::: 94 4.A Linear equations over a doubly stochastic circulant ::::::::::::: 94 5 Realization of Positive Linear Systems Introduction ::::::::::::::::::::::::::::::::::::::::: Problem formulation ::::::::::::::::::::::::::::::::::: Existence of a positive realization :::::::::::::::::::::::::: Case of a system without input ::::::::::::::::::::: General case :::::::::::::::::::::::::::::::::::: Characterization of minimality :::::::::::::::::::::::::::: Positive rank ::::::::::::::::::::::::::::::::::: Positive system rank ::::::::::::::::::::::::::::: Equivalence and classication :::::::::::::::::::::::::::: A relation between minimal positive realizations ::::::::: Classication ::::::::::::::::::::::::::::::::::: Realization of continuous-time positive linear systems :::::::::: Problem formulation ::::::::::::::::::::::::::::: Existence of a positive realization :::::::::::::::::::: Characterization of minimality :::::::::::::::::::::: Conclusions :::::::::::::::::::::::::::::::::::::::::: A Proof of pos-rank(h(4; 4))=4in Example :::::::::::: Positive Linear Observers Introduction ::::::::::::::::::::::::::::::::::::::::: Problem formulation ::::::::::::::::::::::::::::::::::: Continuous-time case ::::::::::::::::::::::::::::::::::: Discrete-time case ::::::::::::::::::::::::::::::::::::: Conclusions :::::::::::::::::::::::::::::::::::::::::: System Identication Introduction ::::::::::::::::::::::::::::::::::::::::: Stochastic approach of the approximation problem ::::::::::::: 160
10 3 7.3 Deterministic approach of the approximation problem :::::::::: Example :::::::::::::::::::::::::::::::::::::::::::: Conclusions :::::::::::::::::::::::::::::::::::::::::: Case Studies Introduction ::::::::::::::::::::::::::::::::::::::::: Case study 1: Nitrate model ::::::::::::::::::::::::::::: Case study 2: PBPK modelling of dioxin :::::::::::::::::::: Case study 3: Benzo(a)pyrene model ::::::::::::::::::::::: A general mammillary system ::::::::::::::::::::::::::::: Conclusions :::::::::::::::::::::::::::::::::::::::::: A Maple V procedures and sessions :::::::::::::::::::::::::: A.1 Maple V procedures :::::::::::::::::::::::::::::: A.2 Maple V sessions :::::::::::::::::::::::::::::::: Concluding Remarks 197 Bibliography 201 Notation 211 Samenvatting 213
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