MATHEMATICAL ANALYSIS OF
* DYNAMIC PROCESS MODELS
• KIndex, inputs and interconnectivity
^ I , J i J : I ri • • I •-1 i'*:-l " p '^ CAi . f .' •-,'a; "7 r
Llfl-A.T.M. Judi Soetjahjo
Delft University of Technology
I) 2) 3) 4) 5) 6) 7) 8) 9) Stellingen
Procesmodellen worden gebruikt uni hel gedrag van een proces te voorspellen en om het ontwerp van regelaars en processen \c optimaliseren. Bij de uitkomst van een modelvourspelling betreffende hel regel-. en slüorgcdrag en de procesopiimahsatie dient in de praklijk meer rekening Ie worden gehouden met de onzekerheden en begrenzingen van het procesmodel. Dit zou kunnen gebeuren door uitbreiding van het gebruikte model met een onzekerheidsmodel waarin de onzekerheden modelmatig gekwuntillcccrd worden. üplimalLsering vindt dan plaats over de hele verzameling van modelJen die door hel on^ekerheidsmodel worden beschreven,
Ervaring leert dal een succesvolle afronding van een project sicchls mogelijk is indien het projeclmanagcment de risicofactoren van het project weet te minimaliseren. Dit is slechts mogelijk indien de structuur (wal en hoe) en de eigenschappen (invloed en tijdstip) van de risicofacloren op systematische
wijze in kuarl gebracht worden.
Het ontwerpen van processen en de bedrijfsvoering van processen is de afgelopen decade een stuk complexer geworden. Het op afsland waarnemen en registreren van een proces (remole monitoring) wordt door de proces operalor over hel algemeen ervaren als *big brolher is wachting y o u \ Gezien het feit dat remole moniloring wezenlijk bijdraagt aan een betere en veilige procesvoering zou hij dit behoren te ervaren als ^big brother is helping y o u \
De laatste jaren hebben Nederlandse onderzoeksinslellingen het onderzoek met betrekking tot procesengineering en procesvoering voor een grool deel afgestoten. Opvallend is het feit dal juist in Aziatische landen de laatste jaren veel is geïnvesteerd in onderzoek met belrekking tot deze vakgebieden. In Nederland lijkl men zich niet ie realiseren dat het juist voor een kenniseconomie van levensbelang is continu
te investeren in de ontwikkeling van nieuwe producten maar ook in industriële procesengineering.
Door de jaren heen hebben het Nederiandse bedrijfsleven en Nederlandse universiteiten moeite gehad aansluiting op elkaar te vinden en ervaring en kennis met elkaar uit te wisselen. Het is daarom belangrijk dat mensen uil hel bedrijfsleven en industrie gemotiveerd worden (in deeltijd) college te geven en dal de universiteiten worden gestimuleerd meer op hel bedrijfsleven gerichte 'business orienled' projecten en onderzoeken te gaan doen in nauwe samenwerkingsverbanden met dat
bedrijfsleven-De Nederlandse industrie heeft m het verleden haar 'business orienled* onderzoek voornamelijk in Nederland uitgevoerd. De laatste jaren is dergelijk onderzoek door het bedrijfsleven verplaatst naar zogenaamde lage lonen landen. Als gevolg hiervan dreigen universiteiten de aansluiting met het bedrijfleven te verliezen en daarmee hun kans een innovatieve rol te spelen binnen het kader van een kenniseconomie»
Tol nu toe overheersen in de media voornamelijk negatieve aspecten met betrekking tot een eventuele toetreding van Turkije (ot de Europese Unie. Zoals gezegd ^onbekend maakt onbemind\ Hel is om die reden belangrijk dat media, regeringen, industrie en universiteiten met open vizier op zoek gaan naar de positieve elementen en voordelen van een mogelijke toetreding. Turkije zou een geweldige economische en culturele brug kunnen vormen lussen de continenten Azië en Europa, [let niet toelaten van Turkije als Nd van de
Europese Unie is alleen al om die reden een gemiste kans voor ons allen.
De lerm 'deadline' in een projectplanning wekt de indruk dal de einddatum in een project een vast statisch gegeven is. Het dynamische gedrag van de 'geplande' einddatum binnen een project heel\ menigeen echter vaak onaangenaam \'errast.
Het feit dat de meeste mensen er niet graag oud uitzien valt niet te rijmen met het feit dat iedereen graag oud wil worden.
Deze slellingeii wonien opponeerbaar en veniedighaar geacht zijn als zodanig goedgekeurd door de promotoren:
Mathematical Analysis of Dynamic Process Models
Index, inputs and interconnectivity
Mathematical Analysis of Dynamic Process Models
IndeXy inputs and interconnectivity
A
^1
PROEFSCHRIFT
ter verkrijging van de graad doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof.dr.ir. J.T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 4 december 2006 om 12:30 uur
door
Aloysius Thomas More Judi SOETJAHJO
geboren te Malang, Indonesia
werktuigbouwkundige ingenieur
TR diss
4879
^
Dit proefschrift is goedgekeurd door de promotoren
Prof. ir. O.H. Bosgra
Prof. ir. J. Grievink
Toegevoegd promotor:
Dr.S. Dijkstra
Samenstelling promotiecommissie:
Rector Magnificus,
Prof. ir. O.H. Bosgra,
Prof. ir. J. Grievink,
Dr. S. Dijkstra,
Prof. dr. ir. P.M.J. Van den Hof,
Prof dr. ir. G. van Straten,
Prof. dr. ir. A.C.P.M. Backx,
Dr. ir. P.J.T. Verheijen,
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft, promotor
Technische Universiteit Delft, toegevoegd promotoi
Technische Universiteit Delft
Universiteit Wageningen
Technische Universiteit Eindhoven
Technische Universiteit Delft
Dedicated to my mother: Lia Indriawati
Teriniü kusih ufas doü-doaimi, Tithan hesevta kita
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Ya Bapakii daJam siirga. akii bersyukur padaMu
Pimpinlah akii selahi imtiik memhagi kasihMu,
Preface
This thesis is the result of my research expcriences during the period of 1992 to 1997 at the Delft Univcrsity of Technology, on dynamic modelling of a complex interconnected DAE {differential Algebraic Equations) of a Coal Gasification Combined Cycle Plant using Speed-Up (Aspcn Tech). This thesis describes specific mathematical problems, namely a high index problem and consistent (re-) initialisation problem that I have encountcred during modelling of a proccss and during
interconnecting of process sub models. The literature refcrences used are dated until year2003.
At this time, I wish to express my gratitude to all the pcople who have contributcd to this thesis, although unfortunately I am able to mention just a few! Firstly I would like to thank my promoters Prof O.H. Bosgra and Prof. J. Gricvink, and my co-promoter, Dr. S. Dijkstra for providing a valuable research project. Thcir support, advice & encouragement even after my resignation from university, ensured that I did not give up on completing this book. Thank you for opening the door to an incredible educational joumey for which I am thankfiil! I am grateful to Dr. P.J.T.Verheijen for our discussions and his critical remarks regarding the presentation of my work. Thank you also to Prof P.M.J. van den Hof, Prof G. van Straten, and Prof. A.C.P.M. Backx for scrving on the thesis committce and for thcir valuable comments.
Secondly, I would like to thank SBM (Single Buoy Mooring) Chief Operating Officer, Dick van der Zee and GustoMSC Hcad of the System Engineering Department, Bertus Bemhard, both of whom have always supported me and activeiy encouraged me to fmish this thesis in conjunction with my daily work and responsibilities at GustoMSC. Thank you to Andries Mastenbroek and Paul Spoeltman for their understanding and support.
Thirdly, 1 would like to thank for all my colleagucs and students at the Delft University of Technology, all of whom supported me during my research period; Peter Valk, Arie Kowing, Gideon Go, Agung Hcmiawan, Habicb, Trias Hermanu, and Folmer de Haan. My special thanks to Angela Hemandez from GustoMSC for her assistance with linguistic and layout corrections.
Finally, thank you to my dearest friend Ton Peters. Thank you for your unconditional support and your cverlasting care.
Delft, December 2006 A.T.M. Jiidl Soetjahjo
Summary
Process modcUing plays an impotlant role in process engineering and opcration. The process models are to be used for: designing of process cquipment and process control, prediction of process behaviour and optimisation of product propcrties during the
process operation.
Usually, a complex process model consists of interconnected sub models, i.e. as a composite network of sub models in different laycrs with incrcasing detail of model components. Modclling of cach sub system as a lumped system described with flrst principlcs cquations, i.c. balancc and constitutive equations, gives a set of first order non-linear DAE (Differential Algebraic Equations) model.
A consequence of modelling with DAE is the possibility to get a high index and/or a consistent (re-) initialisation problcm. An (differential) index of DAE model can be casily deflned as the number of differentiations required to transform the origina! DAE model to an explicit ODE model. A high index occurs if this required diffcrentiation has to be donc more than oncc.
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I iA high index probleni can bc caused by: the wrong choice of input variables, intemal state constraint cquations (c.g. rcaction equilibrium, phase cquilibrium), or sub models intcrconnection.
A consistent (re-) initialisation problcm means Ihat it is not possible to set arbitrary values of the accumulation/storage variables at any time. A high index model and a low index model with non minimum differential variables always pose a consistent
(re-) initialisation problem.
Basically, solving a high index problem needs diffcrentiation, instead of integration only. Bcsides that most numerical algorithms do not cover a diffcrentiation algorithm,
it is not practical to solvc an interconnected high index DAE models, because a high index DAE poses also a consistent (re-) initialisation problem. A modelier can get unexpccted behaviour on the defmed/chosen accumulation/storage variables; i.e. these storage variables can 'jump' in their time response. Whcn modclling a large scale interconnected system one should avoid a high index DAE model.
Therc have been many rcscarchcs carried out to solve a high index DAE model. The distinction of the work in this thesis compared with others is that the detection of a high index problcm and an index reduction tcchniques is developed from the original theory of Kronecker and Rosenbrock [107], i.e. a strict system equivaicnce opcration. A strict system equivaicnce operation pcrforms the required differentiations while maintaining the input-output behaviour of the system. The theory of Kronecker and
Rosenbrock [107] is extended for a first order nonlincar DAE model by using the
structural information of the cquations, rathcr than the exact numerical value.
This thesis shows, for a first order DAE process model, that it is possible to fmd a procedure (based on a strict system equivaicnce operation) for a high index reduction,
which does not apply complicated non-lincar algebraic manipulations. With this procedure the original model equations and variables will not be destroyed.
Moreover for a givcn DAE model it is possible to detect which input variables set will not raise a high index problcm, provided that therc are no state constraints cquations. These state constraints cquations can he detected by any choice of an input variables
set which give the maximum numbcr of state variables of the model.
Interconnection of DAE sub models requircs Information on possible input variables of each sub model. This input variables set does not have to be unique. Interconnection of sub models will not raise a high index problcm, when there is no state constraint equation arising due to the interconnection, i.e. the interconnectcd low index models resulting to a low index composite model.
Despitc of advantagc on applicability of the structural method, for a rare case it is possible to have an analytical cancellation problcm on a high index dctcction; i.c. the
stiTjctural algorithm shows that the DAE has a high index; but analytically the DAE does not have a high index. A combination of symbolic and structural approaches can solve this problcm and the theory in this thesis can bc cxpandcd to such application.
Moreover the extension of the high index theory for a distributed system, PDAE (Partial Diffcrcntial Algebraic Equations) is neccssary, since a state constraint problcnis can arise either in a time domain, or in a spatial domain or a combination of tliem.
Table of Contents
Pre/ace i Siimmary ii Chapter 1: Introducfion 1.1. Background 1 1.2. Moti vation 2 1.3. Related works 71.4. Problcm definition and scope 9 1.5. Approach and outline of the thesis 10
Chapter 2: Mathematical Properties of a First Order DAE model
2.1. Introduction 13 2.2. Kronccker Canonical Form 14
2.3. Low index criterion of a DAE model 18 2.4. Calculation of the index of a DAE model 19
2.5. Solving a consistent initialisation and a re-initialisation problcm 21
2.6. Tndcx rcduction of a DAE model 27
2.7. Conclusions 38
Chapter 3: Structural Properties of a First Order DAE model
3.1. Introduction 39 3.2. Stmcturc of system matrices £ , T{s), T{s) 40
3.3. Stnictural properties of a first order non linearDAE model .... 44
3.4. Gencricity of stnictural properties 49
3.5. Examplcs 54 3.6. Conclusions 61
Chapter 4: Application of Low Index Detection and High Index Reduction of Non-linear DAE Process models
in
I.
4.1. Introduction 63 4.2. High index reduction through variables differentiation 65
4.3. High index reduction through cquations differentiation 77
4.4. Consistent re-initialisation problem 84 4.5. High index problem due to interconnection 89
4.6. Conclusions 94
Chapter 5: Input Variables Assignments of a DAE Model
5.1. Introduction 95 5.2. Model entitics and variables classifïcation 98
5.3. Low index input variables assignmcnt 105
Table of Contents
5.4. Maximum dynamic degrcc input variables assignment 118
5.5. Conclusions 126 Chapter 6: Interconnection of DAE Models
6.1. Introduction 127 6.2. Process model ports 128
6.3. Process model connectors 133 6.4. Interconnection of sub models 138
6.5. Conclusions 152 Chapter 7: Conclusions and Recommendations
7.1. Conclusions 153 7.2. Recommendations 155
Appendix A: Structural Frame Work 157
Appendix B: Graph Algorithm 161 Appendix C: iModeJIing of an ICGCC Plant 171
List of symbols and abbreviations 183
References 185
Samenvatting vi Ciirriculiini vitae viü
Chapter 1: Introduction
Design engineering praciices always reqnire process mode/s as a predicting and calcululiün looi. The model niodii/arisafion. interconnectivily and Iransparency ave essential in model building and model re-iise. The malhemalical models of process resnlt mnsllv info o set of interconuected Differenlial Algehraic Eijiialions (DAE). The research motivation ofthis thesis is analysis of the mathematical process model stnictiire and its intercunneciivily.
"A system is oiir 'projection ' of a striicüire or an ordering of 'mechanisms'"
1.1. Background
Mathematical models are uscd in many areas of sciencc or engineering to undcrstand the system's behaviour and/or design of specific system behaviour. This model represenis intemal behaviour of a system subject to the purpose, assumptions, and
simplifieations of the modellcr. The system intemal behaviour is generally described with baiances and othcr constitutive equations.
A system can be anything of a single part, organ, or mcchanism or a set of interconnected parts, organs, and/or any physical process. that may perfomi a particular flmction or composite functions. A dynamic system is a system where its intemal propcrties or quantities undcrgo a progress of changc in time. A mathematical description of a dynamic system is ealled a mathematical (dynamic) model.
A mathematical (dynamic) model {Willems [131]) is a mathematical description of a dynamic system (for a given boundary) that consists of aspects; time and behaviour equations. Behaviour B of a dynamic system manifests to a set of signal trajectories
Z ( / ) G R " on some time interval te (a h)\ a.,hE'R , satisflcd behaviour equations,
B = { r ( 0 : M —> M" |Behaviour equations are satisfïed] (1.1)
Modelling of a complex process consists of interconnection of sub-models each represented by a set of behaviour equations (1.1).
Prior to 1960 - 1980"s, it was common practice for modelhng of a system behaviour by a set of Ordinary Differcntial Equations (ODE) with fixed chosen input variables.
x = f{^.iiJ),
(1.2)n-^k-i-l
where £:U"^'-'' -^R", x^xeWare the model solutions, z/e variables, and / e 7" denotes the time at some interval 7" cz M .
A-are the input
-Chaptcr 1 : Introdiiction Chaptcr ! : Tntroduction
Simulation of an ODE model was donc by transiating it into a Computer Language as interconneetion of mathematieal libraiy operations, from which it was diffieult to recognise the original set of equations. diffieult to modify the model, and diffieult to re-use the model for other pui-poses.
Nowadays mathematical process modelling and simulation are bascd on balanee equations and constitiitive equations resulting into a set of Differential Algebraic Equations (DAE), given by :
f : ( r ( / X r ( 0 , / ) = 0 ,
o r £ ( f ( / ) , z ( O , i / ( / ) , O = 0 ,
(1.3a)
(1.3b)
simulation tools on the market give different ways for model reprcsentations, model aggregation and model intereonnectivity of proeesses from different physical domains. The dynamic modelling process and representation should be standardised in universal-ways. The standardisation in this context also brings a 'bridge' between
Conceptual Modelling Process and its translation into Computer Language for solving the problems (see Figure I.I.). Dynamic process model beeomes transparent when the 'gap"" between conceptual and simulation language is not too large, which makes life easier for a lot of cngineers to communicate with eaeh other, re-using models from different sources/authors, and doing maintenancc or modification of the model when required. On the eomputation level the computer model should be translated to be able
to solve with a numerieal engine.
where F_:V —^ W , V is open in M ""^' is a vector defming the model; _r : T —> R" denotes its solution, witli m < n and / 6 T denotinti the time at some interval T <Z M ,
is a common approach, where the modeller always has the transparent original DAEs model in the simulation engine. This has many advantages for re-usability or model modification.
1.2. Motivation
1.2.1. Industrial needs
The deveiopment of numerieal solving methods and computer hardware in the last decennium make it possible to do large-seale mathematical model
simulation/calculation even with a personal computer.
Process model simulation on the process industry consists of two main items, namely: the process model building and the process model re-use. On the proeess model building the modeller needs to make the mode! equations. For the process model re-use the modeller uses the existing made models and intereonnect them with other models when it is neeessary.
Commercial software such as Aspen [5] and Hysis [61] built a lot of Standard model libraries, which can be re-used and interconneetcd for designing complex processes. Also there is a tendency to migrate the software programmes to open concepts, where different model libraries built in different programmes can be re-uscd and intereonnected.
Nowadays engineering design activities include integratïon of processes from different physical domains, e.g.: electrical, chemical, physical, and mechanical processes. Engineers necd to be able to build, re-use and intereonnect mathematical models from different physical domains into one common simulation platform.
Despitc the fast developments in different fields of software programming and hardware availability. industrial practice shows that there are still several problems to be solved on dynamic model building and model interconneetion. The available
Conceptual Mathematical Dynamic
Model Mode! biiiUling
Computer/Simulation Language
Translation
Numerieal Language Translation And Computation
-• Simulation
Figure 1.1 Modelling and Simiilalion stages.
In eonclusion, the industries application fields need: an open modelling platfonn,
universai modelling standardisation,
flexibility on model modularisation, and intereonncetivity,
flexibility on model re-use and modification (model transparency), and easy use and fast model building.
1.2.2. Academie research
The mathematieal modelling of physical systems consists of the decomposition of complex systems and composition/intcrconneetion of sub-models. This topological decomposition (SCQ figure 1.2) is neeessary for several reasons:
1. The real physical processes exist of intereonnected sub-systems.
2. The decomposition and composition of sub systems are hunian tools to 'create' a new system.
\
Chapter I : Introdiiction Chapter I ; Introdiiction
Topological structure assumptions Condensor Topology Decomposition
£
Sub-model1
Interconnectioni
Sub-sub-model1
Interconnection±
Elementary model±
InterconnectionE i e men tar>' behaviour Behaviour
assumptions Figiire 1.2. Model decomposiliou
Preis'ig [104] defmes that an elemcntary model/system is a finite volume body or a
finitc volume single-phase system. If the elementary model/system bas spatial uniform physical properties theii this system is called a lumped model/system. If the physical proporties are not uniform thcn it is called a distributed model/system.
Figure 1.3. shows an exampic of a topological decomposition of a distillation system. Part A of this figure gives schematically that this distillation system is a composite of interconnectcd models fa distillation column, a condenser, a rcboiler, scveral control valves, and controllers). Part B gives schematically that the distillation column consists of interconnectcd distillation trays model and finally part C shows that a tray sub-model consists of interconnectcd two elemcntary phases (liquid and vapour phasc), where mass and energy are transferred through the interface.
CSslilaüort Cdunn
Intefsce rress S energy [ranker
A. Inlenxmneded rmdds
of a dislillslion sysiem B InlerconneOed Iray submadels d a distllalion cdiJmr
C. Inlercomeded elenenlav liqud & vapojr phases in a distllalen tray
Figure J.3. A lopologica! ciecompnsifinn ofa üisliHution model
Basically the modelier will make two kinds of assumptions during the modelling o f a dynamic system, namely:
the topological assumptions the behavioural assumptions
The topological assumption on the process model decomposition or composition of lumped models dcscribe:
the interconnection structure of the elementary models, and the geometry of the elementary models.
The set up of the behavioural equations within an elementary model depends on the behavioural assumptions made by the modeller. Basically there are two types of behavioural assumptions {Marquard! [85]), namely:
1. The primilive behavioural assumptions:
The primitive behavioural assumptions are the first assumptions that should be made by the modeller to decide the relevant physical balance equations or balance mechanisms within the system, e.g. component mass balances and energy balance. These must fonn independent balance equations; therefore the accumulation or storage terms in these balance equations should be also independent. The primitive balanee/storage variables are principally the state variables of the model.
Z The constiiutive behavioural assumptions:
Chapter 1 : Tntroduction
>
Ciiapter I : Introduction
the transport behaviour assumptions the reaction behaviour assumptions
the physical propertics behaviour assumptions other behavioural simplifïcation assumptions
The constitutive assumptions must also include the cquations.
validity range of the used
A sub set of behaviour equations (I. I) for a mathematical model representation based on the balanee equations and constitutive equations are given on a set of flrst order differential algebraic equations (DAE (1.3a), (1.3b)).
In the last decade it is has been realiscd that DAEs have two essential different propertics than sets of Ordinaiy Differential Equations (ODEs), namely the term 'high index' and 'consistent (re-) initialisation problenV of DAE model.
A high index DAE model
What is a high index DAE model ? A high index DAE model is, roughly speaking, if in the model equations there are (hidden) state constraint equations or better to say, constrained differential variables equations. A high index DAE process model may cause conflict with the primitive assumptions of the modcller about the choice of storage variables and related model cquations. These storage variables are no longer independent.
In some cases the reaction rate. or the mass transfer rate, or the energy transfer rate are not known or to be considered as the phases or the components are in cquilibrium, examples; applying a thermodynamic equilibrium or reaction equilibrium assumptions. These assumptions will result to non-independency of the phases orthc components.
Basically there are three causes of a high index DAE, namely: a) intemal states constraints, e.g.:
introduction of behavioural constraint equation, e.g.: reaction or phasc equilibrium
introduction of geomctrical constraint equation, e.g.: volume constraint on multiphase fluid system
choice of coordinate system, e.g. pendulum with cartcsian coordinate
{Mcillsson [86])
b) non causal rclation of the chosen 'input' variables of the model, c) interconnection of DAE sub models
i
Finding an input variables set for a given DAE model (1.3a), which result to a low index DAE model is sometimes not easy. The wrong choice of input variables set can resuh in a high index model {A.Lefkopüidos [74.75]).
It is known that a high index DAE model is numerically difficult to solve. The problems on numerical solving of a high index model are:
break down of the numerical solver error control, e.g. the error control of implicit numerical solver backward formula (Brenau, Petzold, Campbell [17],
Bujakiewicz [21]) increases for a high index DAE model.
I
II
f
finding consistent initial conditions and solving a consistent re-initialisation problcm{S, Leiwkiihler et.al [76], C.C. Panleiides [98])
DAE models inlerconjiection
A complex mathematical model consists of interconnection of sub-models. The interconnection is the way of 'transferring goods' belween sub models, 'the goods' herein are e.g.: infomiation, material, energy. and momentum. These are generally given in the interconnected sub models variables, e.g.: mass flow, pressure, mass
fraction. specitlc enthalpy, heat flow, tempcramre, entropy, position, speed, force, electric cuirent, and clectric voltage.
The interconnection equations can act as constraint equations for the state variables such that the interconnected DAE sub-models gives a high index model.
The interconncctions between DAE sub-models are important during model building and a model replacement. The questions here are how to interconnect DAE sub-models and to find the extcmal input signals, such that the interconnected DAE model does not have a high index structure.
Thus it is important to rccognize whether a DAE model has a high index and to detect what is the cause of this high index problcm, i.c:
to detect if the state variables constraints are due to intemal constraint equations within the model, or due to the choice of input variables, or due to the interconnection of sub models,
to detect which equations result in state variables con.straints, to know if the pre-assumed storage variables are independent,
to look for the solving method for the high index model, either by symbolic 'index' reduction or special numerical handling
Moreover for interconnccting for DAE sub models, it is important to know: which input variables sets will not result in a high index model, and
how to interconnect DAE sub models that it will not result in a high index composite model.
1.3. Related works
A lot of research has been done to explore propertics of a DAE model (1.3b). The related works are briefly described in this section.
1.3.1. The index of a DAE model
The index property of a lincar DAE model was analysed by Rosenbrock [107], G.C.
Verghesse. B.C. Levy and T. Kailalh [127], Van Dooren [32], J. Demmel and B. Kagstrom [31]. The theory of Rosenbrock [107] is very nice, since it is based on an
Chapter I : Introdiiclion Chaptcr 1 : Introduction
to calculatc the index and reduce the index of a linear DAE model. The disadvantagcs of these opcrations are practically difficult to perform and destroy the original variables and equations of the model.
Luen berger [81] did an attcmpt to pcrfomi an index rcduction of a linear DAE model
which includcs only row manipulations, i.e. model equations differcntiation and variables elimination and substitution.
The works of C.W. Gear. LR. PetzoUi [44], and Unger. el.al [125, 126] on index detection of non linear DAE model have the root from the work of Luenhcrger [81]. These algebraic manipulations usually difficult to apply for non linear DAE model and destroy the original charactcr of the equations. The Gear's algorithm and LJngcr's algorithm are practically suitablc to calculatc the index of a DAE model
The work of Pantelides [98] is based on the solving of a consistent initialisation problem of a DAE model. This algorithm does not involve the structural eliminations and substitutions of variables on the equations. Tt dctcrmincs only the minimal subset of the model equations that must be differcntiated of which impose constraints on the initial conditions. The disadvantage of this algorithm is possible introducing unnecessary equation differcntiation. Matt.sson and St'kler/ind [86] developed a symbolic index rcduction technique base on the algorithm of Pantelides for finding the
necessary equations to be differcntiated.
Biijükiewics [21] proposcd a numerical solution of a high index model by numerical
scaling of the error control using the Information from the DAE model structure, without perfomiing any change on the original model equations. The index of a DAE model can be calculated using the structural Information of the model.
Since a high index and a consistent initialisation problems are different, we want to have an algorithm that can detect a high index and a consistent initialisation problem, calculates the index of the model, and gives proposals to perform an index rcduction. The index rcduction procedure shall result in an equivalent model. This algorithm should not perform complicated non linear algebraic manipulations, which can destroy the original forni of the model equations and model variables.
1.3.2. The interconnection of DAE sub models
The work on the linear models interconnection was done by H.H. Rosenbrock, A.C.
Pitgh [109]. This work is Hmited to linear state space models with fixed input and
output variables.
BreedveJd [15, 16] gives the concept of bilaterally coupled models interconnection,
i.e. that sub models are interconnected tlirough pair of 'effort' and 'flow' variables.
Marquardt [89] uses almost the same concept with bilaterally coupled connection, i.e.
the interconnection between sub models is given by the fluxing relations. This fluxing rclation calculates the 'flow' variable as result of the differences of the '•flux' or
'effort' variables values. The extension of this work is reccntly done by B.Maschke.A.
van der Schaft [90] for the interconnection of mechanical systems. The new concept
of the work of Maschke and A. van der Schaft is formulation of interconnection
equations without pre knowlcdge of the directionality of the input and outputs variables on the interconnected sub models.
In the process modelling, the interconnected sub models can consists of model equations and/or cncapsulatcd sub-routines (e.g. physical properties routines). Moreover the interconnected variables can be bilaterally coupled variables and/or non bilaterally coupled variables. Due to the complexity of interconnected DAE sub
models in process modelling, wc want to avoid a high index problem due to interconnection. Therefore we need to examine what is the necessary condition to get interconnected low index sub models.
1.3.3. Other related works of a DAE model
a. Numerical solving methoda o[a DAE model
Research on the numerical solving methods of DAE are done for example by:
K.E. Brenan, S.L. Campbell and L.R. Pelzold [17]. E. Hairer. C. Lubich, and M. Roche [54], Wijckmans [J33].
b. Process modelling and simiilalion using DAE models
Research on the process modelling and simulation using DAE are done for example
by: P.L Barton [6] and C.C. Pantelides [99], Matts.son and
Söderlind[86]. Nilsson [93[. Marquardt [85].
1.4. Problem definition and scope
Despitc the work that has been done around properties of a DAE model, in my opinion the foUowing academie items are still ongoing discussion:
1. How to detect a high index, calculatc the index and perform index rcduction of a DAE model based only on the DAE model structure ?
2. How to assign input variables to a DAE model and what is the necessary conditions to avoid a high index model as a result of the interconneetions of low index DAE sub-models ?
These questions give motivation for my research of:
Mathemaücal anaïysïs of dynamic process models hiterconnectivity
Index, inpitts and
with main contributions being:
to clarify the index detection, index ealculation, and proposing index rcduction based on the mathematical structure of DAE model,
to fnid input variables for a DAE model, and
to avoid a high index problem due to interconnection of DAE sub-models.
M.R. Westerweele [135] works emphasize on the process modelling process, i.e. how
to avoid a high index in modelling process. Additional equations (rcstrictivc type)
Chapter 1 : Introdiiction Chapter 1 : Iniroduction
might be necessary to bc added, when during modelling a high index slructurc ariscs. The work presented in this thesis comcs from different view. namely how to analyze a given mathematica! model structurc of a process model and to rcduce a high index
from a genene system theoritical approach.
This thesis forms an extension of the works that have been done by
H.H. Roseuhrock [H)7] and Biijakiewics [21J. i.e. derived a method to detect high
index, perfoim index reduction, fmding input variables based on Kj-onecker fomi for non linear first order DAE model.
The scope will be limited to analyse the strueturai properties of a first order DAE model for a lumped system dcscnption. The DAE model here is the result of modelling using the balancc and constitutive equations. The issue of a first order DAE solvability is also outsidc the scope of this thesis; sincc the structiiral approach ean not detect indcpendcncy of DAE equations. The existence of a local consistent initial conditions, as it will be dcscribed in Chapter 2, gives only a local necessary condition
for an existence of a solution.
1.5. Approach and outline of the thesis
The first question in my thesis will be answered in Chapter 2 and Chapter 3.
Chapter 4 will describe several examplcs of high index model detection and reduction, based on the theory deveioped in Chapter 2 and Chapter 3. The second question in my
thesis will bc answered in Chapter 5 and Chapter 6, namely it describes a method for finding input variables of a DAE model and gives a sub-models interconncct ability condition.
A model diagnosis tooi is deveioped by use of a structiiral approach to help the modellcr during the model building phase to avoid a high index problcm.
The outline of this thesis is as foilows:
Chapter 2: Mathematica!properties of a first order DAE model
This chapter gives as a short description of equivalence system operations to calculate the index of a DAE model and to reduce the index of a DAE model. Further easier methods are derived for a low index detection, index calculation and detection of a consistent (re-) initialisation problem. Moreover this chapter also presents the conjunctures of high index reduction techniques on a non linear first order DAE model
and strict system equivalence operation on a linear DAE model.
Chapter 3: Strueturai properties of a first order DAE model
This chapter describes the strueturai properties of a first order non linear DAE model based on the theory deveioped in Chapter 2. Some definitions of mathcmatical
structure are given, where properties such as strueturai determinant, strueturai rank, strueturai low index criterion, and strueturai index calculation are derived. The strueturai theory is used to analyse the properties of a first order non linear DAE model.
Chapter 4: Application of low index detection and high index reduction of nonlinear DAE process models
This chapter describes several examplcs of high index dynamic process models. The high index detection and the high index reduction techniques derived in Chapter 2 are applied in this chapter. Moreover this chapter also gives the differcncc between a DAE model with a consistent (re-) initialisation problem and the high index DAE.
Chapter 5: Input variables assignment of a first order DAE model
Modelling with a DAE allows different input sets assignment, which result in low index systems. This chapter describes a stnictural approach of modelling diagnose tooi / algorithm for finding a low index inputs assignments or for detection of state
constraint equations.
Chapter 6: Interconnection of DAE models
This chapter includcs definition of model ports, model conncctor, interconnection, and interconneet ability structurc between sub-models. Based on the previous ehapters, it will be derived a necessary condition for interconnection of low index sub models that result to a low index interconnectcd model.
Chapter 7: Conclusions and recommendations
This chapter includes the main conclusions of my research and recommendations for fiiture work.
Appendix A : Strueturai frame work
This appendix describes the stiTictural frame work for development of the theories in Chapter 3.
Appendix B : Graph algorithm
This appendix describes the prineipal of a graph algorithm for implementations on the theories in Chapter 3 and Chapter 5.
Appendix C: Modelling ofan ICGCCplant
Chapter 1 : Tnlroduction
Chapter 2: Mathematical Properties of a First Order DAE
model
\
The system matrices E, {Es— A) . and [Es — A) ofa DAE model,
dF dF
^ ( £ ( 0 n 2 ( 0 ^ i ^ ( 0 . 0 - ü ' ^^''^^ -£" = ^r— üiid A=—— contam all svsïem
oz_ dz_
properlies of ihe DAE. i.e.: index, cousislent re-inilialisalion. state and non state variables.
%
\
A change of a phvsical system slriiclure may residt in 'an impiilsive behaviour'"
2.1. Introduction
In Chapter 1 wc have secn that mathematical modelling ofa physical system leads into intcrconnccted independent first order differential algebraic equations with the general forTn(1.3b):
f:(2(o,z(/),»{/),/)=o
(1.3b)Tn the last decade much research has been carried out into the properties of differential algebraic equations. Historically the properties of linear time invariant DAE models have been analysed (e.g.: Rosenbrock [107], Verghese el.al [127], Van den Weiden
[128]). Tn the 1980's anaiysis of a non-lincar DAE began as a research topic in
modelling and numcrical solving arcas (e.g.: Brenan. Petzold. Campbell [17], Gear
[44], Marqitardt[87],[88], Panlelides [97],[9S]).
Rosenbrock [107] shows the important properties o f a linear DAE modei (i.e. index of
DAE, consistent initialisation, index rcduction) via equivalence system operations to transfomi the DAE into a Kronecker Canonicai Fonn. Practically this transformation is difficuh to can7 out.
» The objectives of this chapter are to derive easier methods for detecting a high index
problem, calcuiating the index, and detecting a consistent (re-) initialisation problem bascd on the Information of the system matrices ofa DAE model (1.3b),
F(_r,_r,£,/') = ^ , Those system matrices are E,{Es — A) , and (Es — A)' , where
E = ^ = and A = -=•.
dz
d=
Further this chapter describes proposals of index reduction techniques ofa DAE model without perfomiing non linear algebraic manipulations. This is to retain the original
ferm of the model equations and model variables. The index reduction techniques will be derived in conjunetion with an equivalence system operation (Rosenbrock [107]).
' ^
Chaptcr 2 : Mathematicai prüpcrlics of first order DAC mode
This chaptcr consists of scven scctions. Section 2.2 briefly describes the Kroneckcr Canonical From (KCF) transfomiation, to kiiow the index of a DAE model and to pcrform an index reduction. Scction 2.3 describes mcthods to detect ïf a DAE model bas a high index problem. Section 2.4 givcs a method to calculatc the index ofa DAE model. Scction 2.5 describes a consistent (re-) initialisation problem ofa DAE model.
Section 2.6. gives procedures to reduce the index ofa DAE model and flnally, section 2.7 gives some conclusions.
2.2. Kronecker Canonical Form
Let a non-linear input-output DAE model be given by the equations:
y^gi^^n
(2.1a, 2.1b)where / e l denotes the time on some interval II c R ; the function F :V ~> W, V is open in M 2m+k + \ • is a vector space defining the model; the solution variables and the
input v a r i a b l e s z : I - ^ M " ' , ; / : I I — > R . Moreover the output functions g'.lV^Y,
in+\
where W and ï e R'"^' with the output variables y . I ^ M". The index of DAE model (2.1) is defmed as follovvs:
Definilion 2. IA. Index K of a DAE mode! (Brenan, Campbell. Petzold [17], Biijakiewicz [21])
Given a DAE (2. Ja): f ; ( r ( / ) , r ( / ) , i ^ ( / ) , 0 = O
77?^ index K ofa DAE is the miiiinniin niimber of linies ihal ihe complete or a part of
equalion (2.1a) must be differentiated willi respect lo l. such that the svstem equations:
dt B.L^t\z{t\i±{t\t)^^
d[''
can be transformed into an explicil ODE. i.e. i as a continuoiis function of z_ , li_^it_ ,...,11 and t, by algebraic manipulations. (note: 1</C).
A DAE model with a (differcntial) index greater than onc is called a high index DAE otherwise it is a low index DAE.
r
Chapter 2 : Mathematicai proporties of first order DAE mode
De/in i f ion 2JB. Perlurbüliun Index p ofaDAEnïodel(Hairerel.al[54])
Equation (2. la) has a perWrhation index p if p is the smallest integer such that, for all functions z_ (t) being solu/ions of the DAE perturhed by Sil{t)
Btpit\z^XtUjitlt):=Su{t) (2.Ic)
There exists a bound on the dijference between z_{t) and z_ At):
~U)-z^{t) < c [ r { 0 ) - r (0) +max|<^z/(^^)| + --- + max öu''-'\^{\ (2.ld) w'henever the expression on the right-hand side is sufficiënt smalL
Gear [46] has shown that fC< p< K+\, Moreover Gear derived that K= pfox DAE having an integral form of:
dfiz)
f(l.zas)=—^-g(z)-hiii) = 0
dt ~
which includes also DAE of the fonn (2.2J.
(2.1e)
Since in this thesis, wc will analyse the properties of a DAE based on the fonn of (2.2), then we will not niake difference between differential index and perturbation index. We call only the 'index' ofa DAE (2.2).
The definition (2.1) is closely relatcd with a strict system equivalence operations ofa iinear implicit DAE {Rosenbrock [107]). To understand, suppose
^n ~ C.^0'-^0'^0' ^0) ^'^^ ^^"^ ^ solution manifold of (2.1a) at time /„ G I and suppose (2. la) is locally time independent. Without losing generahty for analysis purpose we choose (lo>^0^ ~ (^"2) • Linearisation of (2.1a) at ^ gives:
Ezit)^Az{t) + Bu{f) (2.2)
where:
E = dF
aFl^^i'
^^JKi.
öTliJ'
B = -^^(^ dul^
uAfterLaplace transfomiation we get the solution of (2.2)
1{S) = [T{S))~\EZ,^BIS{S)) (2.3)
where: T{s)= KS)J: =[ES-A] is called the system matrix of (2.2) {Rosenbrock
[107]). Matrix (^E/l-A)\s called pencil of (2.2) (Brenan, Campbell and Petzold [l 77).
Let T(s) be non-singular, then there is a constant pre-multiplying matrix M and a constant post-multiplying matrix N (i.e. algebraic manipulations on the DAE model equations and variables) that transfomi (2.2) to a Kronecker Canonical Form (KCF).
14 15
-^
Chiipter 2 : Mathcmatical proporties offïrst order DAE model
This equivalence system opcration is also called a resiricted sv.siem cquivalence
operaiiou, {Rosenbrock [107]). This is given as follows:
MT{s)NN ^zis)^MENN ^ ZQ + A4BII_{S) (2.4a) MT{s)N={s) = I^ÏENSQ +ii{s) whcre
MT{s)N =
Sir - A, O O ƒ „ .. + sJ m—r , MEN -l-
O
O J tJi-r -I , z(^) = A^-'z(^),ü{s) = MBuis)
with J - block diag( J,,..., J ) ; where:
Chaptcr 2 : Mathematica! proporties of flrst order DAE model
- I
[i+sjr =
1 -s
i-[y'-'s''-'
o 1
o
-s
1
in the time domain this can be written as:
|,(/) = X(-l)'7'^'Vf,„ + X(-l)V'i?:"(/)
andi ( 0 = X(-l)^^''^"^^:u + É(-')'^'iiy'(0
;=ü ;=0 (2.7b) (2.8a) (2.8b) l Ü ' ) J: = O 1 O O • O 1o
, / = 1,2,...,9 (2.4b)sl,_ - A,.
O
O/.-. + ^J
A\iAs)
\ JI o
O J V-:üy+
'\LM)^ ii,{s) V (2.5) jEach of J, has a size /,. and the largest m a x ( / ) — ] is callcd the dcgree or index of nilpotency (J). For j > 1 the DAE (2.4) is called a high index DAE, otherwise it has a low index. Fory = O the DAE (2.2) is called an index one DAE.
In the time domain the first row of (2.5), the ordinary state space part, can be written as:
lU) = ALO)+iMit) lM = i
in (2.6)The solution of the sccond row can bc done by pre- or post-multiplication with unimodulair polynomial matrix M(s) or N(s) (i.e. algebraic manipulations and diffcrentiations on model equations and variables). This equivalence system operation is called as a slrict system equivalence operation {Rosenbrock [107]):
Mis)[l„^_,.+sJ%_{s) = M{s)Jt,, + M{s)il,{s)
or (2.7a)
[ƒ,,_,. +..j]A^(..)z,(^) = N{s)Ji,, + N{s)iiAs)
X
/
whcrc M{s) = N{s) = Diüg]^[ + sJX'\
and every block cntry of M{s), N{s) has a form of:
16
Where : S = i-th time dcrivate of impulse ftinction.
The foimulas (2.7a,b) show that for y > l , the solution of the second row of (2.5) requires row or column diffcrentiations, where this solution (2.8a,b) may introducé
impulsive behaviuiir in case of z^^^^Oand may rcquire Jifferenliation of 'input' variables if //.(O ^ 2 - Thus for j">l the DAE (2.2) requiresy times diffcrentiations
i w
to be able to transform (2.2) into an ODE with z_^{t),z_M)^?' functions of Il ( 0 . £ i ( 0 and 2 ^ (-1)'7''//'-,'•'(/) (seedefmition2.1).
Fory=0, i.e. an index one DAE, the solution (2.2) bccomes:
z,(t) = ih{t)
KI
Lennna 2.2
Consider (2.4a,b); (hen j - degree of nilpotency of J eqiials to the (dijferential) index K.
Proof: to soJve (2.5) it needs {j-\) diffcrentiations (see 2.7b). and one extra different iat ion to calciilate zft)(.see: (2.8a,b)). It follows ihat the (dijferential) index
K of (2.2) equals toj-degree of nilpotency of J.
With this KCF transformation, we can calculate the index of a DAE model as given on (2.5) and perform an index reduction through a strict system equivalence operation as given on (2.7). The problems on these equivalence system operations are:
difficult to get the transformation matrices M and N to bring to Kronecker Canonical Fomi
the mode! equations and variables are changed
17
\
Chapter 2 ; Mathcmatical properties of fïrst order DAE mode
2.3. Low index criterion of a DAB model
In case if we only want to know if a DAE model bas a low index, the foUowing Lemmas 2.3 or 2.4 give an easy computation procedure:
Lemma 2.3 Low Index Criterion - I Consider a DAE model (2.2):
T{s)z{s) = Bii(s) (2.3)
where T(s) = [Es — /l] and lel def T(s) i^ O ,i.e. non singiilar, then the DAE (2.3) has a low index ifand only if
det^detn^s-) = raiik^" (2.9)
Proof'.
Given a DAE (2.3), T(s) and E by a restricled systeni equivalence operalion transfoniied to (2.5), T{s)^ E': whcre:
T \s) = Si y, /l.-O
0
/ +S-/ ^ in~r "-^-^ _, £" =
j'A-
f0
0"
J
J w defïned as in (2.7b). Since deg det[/„^,. + sJ^ = O, we have
degdet T{s) = degdet T'(s) - degdet[.?/,. - 4 ] =
f-Fiirther rank E = rank £"'= rank
O J = r + rank J". but since DAE model has a
low index ifand only if rank / = O, we have rank E — r ifand only ifihe DAE model has a low index.
When the system matrix T(s) is not in KCF, then r can be calculated from deg det T(s), as follows : ;
Given a square matrix T{s) =
i^'Mj
e R""'"[s] and S is the set of allpermutation <T of numbers 1,2,...,m, then the determinant of T{s), d e t r ( ^ ) i s defmed (J. G. Broida [18], F. Lancaster [72]) as;
det T{s) := X ^'gf<o-)t,,^ (s^ja, (^)-La^ (^)
aeS
m
= Ya^ign{a)]\t.^{s)
{2.I0a)
tres ,=l
where 5/g-;ï((T)equaIs-l for a odd and 1 for G even.
18
'•
\
Chapter 2 ; Mathcmatical properties of flrst order DAE model
Then
r = degdetr(.s')
m
= deg(X^v/g/7(ö-)n^../^))
(2,10b)fTE5 / = l
Lemma 2.4. Low Index Crilerion - 2
Consider a DAE model with the foUowing equations
^(/) = 4,^(/) + 4,z,f/) + S,^(7)
O = ^21^ (O+ 4 2 ^ ( 0 + ^21^(0
(2. J l)
This means that: E = I, O
O O , and A =
Al Al
'h 1 -^22
then (2.11) has a low index ifand only if A2^ is non-singular Proof: rank E - r and
degdet r(.v) = degdet
A.-, is non-singiilar.
si,,
'^u
AJ4, A
71 22 = degK^v/^. - /^ii)/!^-,] equals to r ifonly if2.4. Calculation of the index of a DAE modei
The index of a DAE (2.2) can be calculated more simply with the foUowing Lemma 2.5,2.6.
Lemma 2.5
Consider a DAE (2.2) after KCF transformation is represented as DAE (2.5) then the index of the DAE (2.2) is
max..{O,(power of s of (/^,_,. + sJ) ) +1}
-'- - M^i\- _ ! - _ - 3 _
Proof:
from (2.5) and (2.7b). we see after a KCF transformation that the max power of s in yEs — A^ is s'~^ determined by the largest:
19
Ii
Hl
Chapter 2 : Mathcmatical propeities of first order DAE model
[/ + sJ] y-i j'i 1 - s ••• ( - l ) - ' " ' ^ O l O • • • -s 1
where detF/ + .v^l = 1 and j is the cfegree ofnilpotency or index of DAE (2.2).
The following Lemma 2.6 is dcnvcd to caleulatc the index of DAE (2.2) from its fractional system matrix inverse, since in most cases transfonnation of a DAE (2.2) into a KCF is not easy.
Lemma 2.6.
Consider a DAE (2,2) wilh its system matrix T{s)=yEs— A^ and its fractional polynomial inalrix inverse:
T~\s) = [Es-A] ^ =
^7'M)..
^U r det S,., {s) \ \ d e t r ( i ) y y (2.12) i,J = U...,niwhere sub-matrix S-j(x) of matrix TeW"''''"[s] is inXm matrix ohtained hy pkicing zeros in the i-th row and inj~lh column of T(s) and replacing the ij-fh element with 1
(J.G. BroidaflS], P. Lancaster [72]): / S,(s) =
^„CO
O ^ . ( • 0 \o
1o
Us)
\'m o ••• t,.J^) Jthen the index of DAE (2.2) or the index of T{s) is given by:
m(^eA(r(^}) = max^^lO,(degdet5'^^(iO-degdetn
Pr GO f
From Kroneeker Canonical Form transfonnation we get:
n-L
(2.13)
si.. - A
L
r r OO
ƒ + sJ in—i = N[Es- AY M , 7V-' = N, A/"' = M
Chapter 2 : Mathematical propcrties of first order DAE model
[Es-A\' =N si -A^. O O l +sJ -1-1 r m-r M r det S-, \ d e t ( & - y 4 ) \l = N, (.?/, - . 4 , ) " ' M, + 7 V , ( / _ +..J)"' M._. N = \n, A^,], M = M, det 5^, (.?)), = P ( 5 ) + det(Ky-^)"\;V,(/„,_^+^J)"'Af,
det5,(.))1^[(/.,.(.))^.
-1 + det(£:.9-^)"'.KM"^)
ywhere: deg/7y.(^) < degdet (sE ~ A) itfollows:
deg det Sji {s) = max | deg pij (5), deg det(£'.? - A)-\- deg A'^y {s) \ => deg det 5",, (s) = deg detff'^ - A) + deg kj; (s)
deg kji {s} = deg det LS* .y (s) - deg det{ Es- A)
Since the index of DAE (2.2) is eqiiol to
max,^. {O, (power of >? of (/„,_,. + 57) ) + l} (Lemma 2.5), then itfollows:
index[T{s)) = m a x . 10,(degdet SJs) - deg det T{s) +1)}
2.5. Solving a consistent initialisation and a re-initialisation
problem
Initial conditions and input variables are requircd to solve a DAE (2.2). The following gives a definition of consistent initial conditions and solving of a consistent
rc-initialisation problem.
In this scction we will that a DAE model will have a problcm for the solving a consistent (re-) initialisation, when it has a high index or it has non minimum state variables. Morcover, we will see also that a low index DAE model can have non minimum state variables, which results to a problem for solving a consistent (re-) initialisation.
2.5.1. Solving a consistent initialisation problem
Definition 2.7 Consistent Ifiitial Conditions
Given a DAE (2.2) and its input sïgnals w(/„), then {^{t^•^)^z{t^^)) is a Consistent Initial Conditions of (2.2) ifand only ifit saiisfies:
Chaptcr 2 : Mathcmatical proporties of fïrst order DAE model Chapter 2 : Mathcmatical propeities of first order DAE model ti II or a) Ei{Q^Az{t^^)^Bii{t,) w + t ^ I
-b) r(/„) lies on the sulution trajectories of DAE (2.2).
(2.14ü) (2.14b)
The Kronecker Canonieal Farm transforwatinii of DAE model (2.2) (see equation 2.5) decomposes the system into a /--dimensional ordinary state spaee part and a (m-r) dimensional algebraic part, siich that the system has /•-arbitrai7 initial eondilions
f,,, and (m-r) fixed initial eonditions of z^u = O to avoid impulsive behaviour.
In other words, the originai differential variables z{t) are transfonned and dccomposed into r- state variables f, (/) and (ni-r) algebraie variables (non-sfate)
i.2i'').
For given input variables //(7,|) at time / =/(, the m cquations (2.14a) have m+r' variables (K/u)'^(^u))' where /•' is the column rank of E. The column rank of £'gives the number of'differential variables on the DAE (2.2). From the KCF it follows,
CorollaiT 2.(S
For a given DAE model (2.2) with r' differential variables, where r' = co]umn rank (E). There are only r-variables of (£(f))tbal ean be assigned arbitrarily to compute a set of eonsistenl iuitialisation values i£.(fu),z_it^^))of DAE (2.2). where r ^ deg det T{s) and r<r\ This number r is called the 'dvnamic degree offreedom ' or the 'system order' of the DAE (2.2).
Pr 00f
See the KCF transfonnation. where :
r = deg det T{s) = deg det
sf - A,. O
O / + sJ
m-r= r
and column rank of E = r' = column raiik ir O
O J = r + rank{J) >r
For a given low index DAE with a minimum state variables, i.e. Column rank(£') - r ' - r , then this gives: i
22
\
Cnrollan- 2.9
Given a low index DAE uf the f)rm (2.2) with coliitn rcink{E)- rank(E). r' = r , ihen there are r-dijferential variables of {z{t^,)} that ean be assigned arbitrarily to compute a set of consistent inifialisation values {^{t^^)^z_[tf))of DAE (2.2), where
r = deg det T(s).
We see from CoroUary 2.8 and 2.9 that the degree of freedom to assign initial values
z_(t^) arbitrarily is given by r — dcgdGiT(s). This means that not all of the variables
_r(/'y) ean be assigned arbitrarily to calcnlate consistent initial values (z_{t^),z{t^^)) for a high index DAE model or a low index DAE model with non minimum state variables.
Solving Consistent Initialisation Problem (P.I. Barton [71)
Usually a Consistent Initialisation Problem is solved as an algebraic problem to satisfy criterion (2.14a) on definition 2.7, i.e. for the existcnce of a solution. The following givcs a sufficiënt eondition for solving of a consistent initialisation problem.
Consider a DAE of the forra:
i , ( O - 4 1 ^ ( 0 + 4 2 ^ ( 0 + 5,16 (O
0 = ^2,r,(/) + 4 . z . ( / ) + S-,i^(/) (2.11)
Solving of a general consistent initialisation problem of (2.11) at a given time
/ = /(, e I c R and given input variables /£(/„) e M ' is fomiulated as a solving set of algebraic non-linear cquations for the unknown veetor (^|(?n),:^i(^o)^^2(^)) •
ii(/o) = 4i.a(^o) + 42i(A,) + ^ii6(^o)
O = 4 , z, (/„) + .422 z^ (/J + 5,», (^0)
^3„Z,(/J = 4 i ^ ( ' o ) + ^32i(^o)+ 5|..ü(^n)+5:3Ü2(^n)
(2.15a,b,c)
Where: A^^z^ {t^) = A^^z^ (t^) + A^^z^(/„) + 5,,^(/(,) + B-,2ti2{IQ) are r independent initial cquations.
If the matrix
- / A^ [ A^2
O AT. A^-,
4 u ^3 i ^32
is non singular then a set of initial
conditions(ij (/j,),r, (/„),ZT (/„)) ean be caieulated.
23
-Chapter 2 : Mathematical properties of Hrst order DAE model Chapter 2 : Mathematical properties of first order DAE model
Specific case 1
Usüally one gives a set of ^i,('u) =-i,u= / = !,...,/' to computc ihitial condition vector (2i(^u)'i^i^O'"^2(^ü)) • '^^''^^ '^ ^ special case of an initialisation cquations set
^ 4
+
(2.15c) and the consistent initialisation problem is formulated as algcbraic soiving of the foUowing cquations:
ÈiUo) = A^^^Uu)^ A2^(0 + ^^^(Q
o = ^„Zj (/J + .^,,2, (/J + ö,i^ (/J
^„•(^o) = z M ) ' ƒ = 1 , . . . , / '
if the matri.x
- ƒ 4 , A,,
0 / 0
or Aj^ is non-sincular, thcn a set of initial
conditions(f|(/„),r,{/jj),r-,(/jj)) can bc calculated.
We see here (as given on Lemma 2.4 and Corollary 2.9) that solving a consistent initialisation problem by specifying r-indcpendcnt initial conditions (z_(/^)) can be only applicd fora iow index DAE model with minimum state variables.
Specific case 2:
Corol/an' 2.10. Steadv state solution is a consistent initial condition (Kroner et.al [68]):
Given a DAE (2.10) with a given input variables ï/^. The DAE (2.10) will have a steady state solution o/'(i, ,2^.) = ( 0 , z j if and only if:
[Ti-')l-. -
A\ Al
21 ''^22
is nun-singiilar
Since a steady state solution of a DAE lies on the solution trajectofy. then
a steady state solution. (:^j,^_j.) = (O, j j , is a consistent initial condition of a given DAEfor the given input variatie values ZY .
2.5.2. Solving a consistent re-initialisation problem
Obviously system behaviour can be subjccted to discontinuities of non-state variables or input signals. Below we defme a consistent re-initialisation condition. '
' F
•
Defmilion 2.11. Consistent Re-initialisation Condition Given a DAE (2.11) oftheform :
z,(t) = A,,z,{l) + A,.z^{t)-\- B,u^{t) O = A,^=,{f) + A.._z.{t) + B.i±,(f)
at f. = + lim/„ + £. for given non-state variahie (cHscontinuities) as:
W)^
liOul
liit.i
t = t
ot = t^=(,+£
and/or z . (O -liO^l
t = tt = t. =
+ Ot,+e
(note iiil),^2i0 ^^<^ ""^ need to be continuous and differentiatie)
then vector {^iU+X^[(l+).lAl+)) is called Consistent Re-initialisation Condition of (2.11) for given input signals (»(/+)) under the condition of C conlinuity of the state variables, i.e.:
iM^i^iO
ifand only ifit satisfies:
i,
(/j
=
^„^
(r j +
4 , ^ (/j
+
5,z^ (/j
0 = A.,z,{t^) + A,,z^{Q^ B.ij^{tJ
From defmition 2.11, it foUows :
Corollary 2.12. Consistent Re-initialisation Criterion - I
DAE (2.11) has the property of consistent re-initialisation ifand only if:
A^j is non-singular or the DAE model (2.11) has a Iow index with minimum state variables.
From corollary 2.9, it follows:
Corollary 2.13. Consistent Re-initialisation Criterion - 2
Consider a DAE with given matrices E and T{s) has the property of consistent re-initialisation ifand only if:
The column rank of E (or numter ofdifferential variaties) =
degdet T{s)
V
Chapter 2 : Mathematica! propertics of first order DAE model Chapter_2: Mathematica! propertics of Hrst order DAE mode
II
For a high index DAE the number of differential variables (the colun-in rank (E)) is greater than degdet T{s). This mcans that a high index does not have a consistent re-initialisation property.
No{e:
2.6. Index reduction of a DAE model
In some cases an index one DAE model does not have a consistent re-initialisation property, since the column rank of E for number of dijferentitil variables) > degdet r(.v). This is called a low index DAE model with non minimum state variables.
Process modelling bascd on the balance and constitutivc equations always gives a DAE model which has the inteeral ferm, i.e.:
df{=)
dï
(2.16a)
These equations are usually represented as non minimal state variables and can bc transformed/writtcn into a minimal state representation as follows:
0 = x-/(r)
(2.16b)An index one DAE model given with non minimum states representation can be transformed into an index one DAE model with minimum state representation, without performing additional differcntiation.
BriiU and Paliaske [20] give an cxample of an index one DAE, which does not satisty
consistent re-initialisation criterion, given in the following:
Consider an index one DAE of the form;
'^u{x^.X2)x^ + T^2(^^,X2)x2=f{x^,X2,li)
{2.17a)
can be written on an index one DAE , and satisfies consistent re-initialisation criterion as follovv-s; 0 = g(x^,x2aJ.) 0^ p{x^,x.)-z (2.17b) where: P(^M^2) = ^ i ( ; I p l 2 ) i + 7;-,(x,,^.)i2 and ^P ~T f V dx, dp
- ^ = 7 ; . ( i i , ^ 2 )
ÖX-. LThcre are 3 main difficulties in solving a high index DAE: 1. it may require differcntiation operators {see 2.8.a,b),
2. the algebraic solving consistent (re-) initialisation problem (see eorollary 2.13),
3. the break down of en"or contro! on Standard multistep implicit numerical methods (Brenan. et.al. [I7[)
Besides that. the degrec of frecdom of the state variables having C continuity in a high index model is less than the primitive state variable or 'storage' variables as assumed by the modeller.
Bujakiewicz [21] proposed a modified multistep implicit Backward Differcnce
Fonnulae (BDF), where 'numerical differcntiation' is implemented for the error contrei. This is done by 'scaling" of error control by the Information from the matrix
[Es- Ay^ . The drawbacks of this method are:
1. We do not know the numerical stability and performance of the method for large scale DAE models.
2. This numerical solving technique may not be easy and can be time consuming for a large scale DAE model.
3. Solving algebraic consistent (re-) initialisation problem remains problematic.
Another method to solvc high index DAE models is through symbolic index reduction to find a low index DAE model representation. Several advantages why we want to get low index DAE models, are:
1. the problem on algebraic sol ving of a consistent (re-) initialisation does not exist
2. to be able to use a Standard (multistep) implicit numerical solving method 3. to know information about the state variables in the model
The extra information is obtaincd from the index reduction steps, namely: 1. the (hidden) states eonstraint equations and
2. to alert the modeller in case the chosen 'inputs' variables iniply non eausal behaviour
In the following wc give two methods of index reduction namely; modified row strict system equivalence and modified column strict system equivalence operation. but first we are going to examine the strict system equivalence operation for an input output linear DAE model that is given as follows:
Ez= Az + Bu
y-Cz
(2.18a)or in Laplace domain with zero inifial values gives
T(s)z(s) = Bii(s) y(s) = C={s)
where: T{s) = iEs-A)
(2.18b)
