observability and identi ability of nonlinear systems with ... · observability and identi ability...

Thesis for the degree of doctor of philosophy

Observability and identiability of nonlinearsystems with applications in biology

Milena Anguelova

Department of Mathematical SciencesDivision of Mathematics

Chalmers University of Technology and Göteborg UniversityGöteborg, Sweden 2007

Observability and identiability of nonlinear systems with applications inbiologyMilena AnguelovaISBN 978-91-7385-035-3

c©Milena Anguelova, 2007

Doktorsavhandlingar vid Chalmers Tekniska högskolaNy serie nr 2716ISSN 0346-718X

Department of Mathematical SciencesDivision of MathematicsChalmers University of Technology and Göteborg UniversitySE-412 96 GöteborgSwedenTelephone: +46 (0)31-772 1000

Printed at the Department of Mathematical SciencesGöteborg, Sweden, 2007

Observability and identiability of nonlinear systems withapplications in biologyMilena AnguelovaDepartment of Mathematical SciencesChalmers University of Technology and Göteborg University

Abstract

This thesis concerns the properties of observability and identiability of non-linear systems. It consists of two parts, the rst dealing with systems ofordinary dierential equations and the second with delay-dierential equa-tions with discrete time delays.

The rst part presents a review of two dierent approaches to study theobservability of nonlinear ODE-systems found in literature. The dierential-geometric and algebraic approaches both lead to the so-called rank test wherethe observability of a control system is determined by calculating the dimen-sion of the space spanned by gradients of the time-derivatives of its outputfunctions. We show that for analytic systems ane in the input variables,the number of time-derivatives of the output that have to be considered inthe rank test is limited by the number of state variables.

Parameter identiability is a special case of the observability problem. Acase study is presented in which the parameter identiability of a previouslypublished kinetic model for the metabolism of S. cerevisiae (baker's yeast)has been analysed. The results show that some of the model parameterscannot be identied from any set of experimental data.

The general features of kinetic models of metabolism are examined andshown to allow a simplied identiability analysis, where all sources of struc-tural unidentiability are to be found in single reaction rate expressions. Weshow how the assumption of an algebraic relation between concentrations inmetabolic models can cause parameters to be unidentiable.

The second part concerning delay systems begins by an introduction tothe algebraic framework of modules over noncommutative rings. We thenpresent both previously published and new results on the problem of ob-servability. New results are shown on the problems of state elimination andcharacterisation of the identiability of time-lag parameters. Their identia-bility is determined by the form of the system's input-output representation.Linear-algebraic criteria are formulated to decide the identiability of thedelay parameters which eliminate the need for explicit computation of theinput-output equations. The criteria are applied in the analysis of biological

models from the literature.

Keywords: Observability, identiability, nonlinear systems, time delay, de-lay systems, state elimination, metabolism, conservation laws, signallingpathways.

ContentsThis thesis consists of two parts, Part I and Part II, and includes Papers I,II, III and IV as follows:Part I. Observability and identiability of nonlinear ODE systems: Generaltheory and a case study of metabolic models• Paper I. Anguelova, M., Cedersund, G., Johansson, M., Franzén, C.J.and Wennberg, B. Conservation laws and unidentiability of rate ex-pressions in biochemical models, IET Systems Biology, 2007, 1(4), pp.230-237.

Part II. Observability and identiability for nonlinear systems of delay-dierential equations with discrete time-delays• Paper II Anguelova, M. and Wennberg, B. State elimination and iden-tiability of the delay parameter for nonlinear time-delay systems, Au-tomatica, 2007, to appear.

• Paper III Anguelova, M. and Wennberg, B. Identiability of the time-lag parameter in delay systems with applications to systems biology, inProc. of FOSBE 2007 (Foundation of Systems Biology in Engineering),Stuttgart, Germany, September 9-13, 2007.

• Paper IV Anguelova, M. and Wennberg, B. State elimination and iden-tiability of delay parameters for nonlinear systems with multiple time-delays, in Proc. of IFAC Workshop on Time Delay Systems, TDS'07Nantes, France, September 17-19, 2007.

AcknowledgementsFirst and foremost, I would like to thank my advisor Bernt Wennberg - forsuggesting that I work on identiability in the rst place, which turned outto be such an interesting topic; for always matching my enthusiasm at everysmall progress; for managing to help me every time I got stuck; for all thegood ideas that he came up with; for reading my manuscripts repeatedly atall times of the day and night.

I would also like to thank my co-advisor Carl Johan Franzén for his helpwith all the biochemistry- and biology-related questions and for spending ahuge amount of time improving our manuscripts. For this, I would also liketo thank my pair-project collaborator, Mikael Johansson.

The National Research School in Genomics and Bioinformatics and itscoordinator Anders Blomberg are gratefully acknowledged for both nancialand moral support of this project.

Thanks to my sister Iana Anguelova (who just happens to be a real math-ematician) for help with the algebra, valuable discussions and comments tothe manuscript.

Big thanks to Borys Stoew who sacriced his time to read and commentthe manuscript. This led to a momentous improvement in its appearance.

Thanks to Karin Kraft, Anna Nyström and Yulia Yurgens for the pleasantcoee-breaks.

I would like to thank my parents for their invaluable help with the logisticsof life with children and for their support at all times.

Finally, I have Vessen, Maria and Emanuella to thank for my perfect life.

PrefaceThe work described in this thesis has been nanced by the National ResearchSchool in Genomics and Bioinformatics under the project title Large ScaleMetabolic Modelling. The work is also related to the more experimentally-oriented project A metabolome and metabolic modeling approach to func-tional genomics, also sponsored by the research school. The aim of the latterhas involved the construction of metabolic models for the understanding ofregulation and signal transduction within cells. The analysis of structuralproperties of metabolic models is the aim of this work and the propertiesthat we have investigated are observability and identiability. The biologicalmodels that we have come across have motivated the study of both ODE anddelay systems and the division of the theoretical results of this thesis intotwo parts. Here follows a brief description of the latter.Part I. Observability and identiability of nonlinear ODE systems: Generaltheory and a case study of metabolic models

This part of the thesis concerns the observability and identiability prob-lem for nonlinear systems of ordinary dierential equations with applicationsin the kinetic modelling of metabolism in yeast. It consists of a monographand one paper, Paper I, see below. The monograph reviews already publishedwork before describing some new results and introduces a case study of a ki-netic model of glycolysis from the literature. Of a particular interest for thestudy are enzymatic rate equations and how they are parameterised. This isdiscussed further in Paper I, which is briey introduced in the monograph.Part II. Observability and identiability for nonlinear systems of delay-dierential equations with discrete time-delays

This part of the thesis concerns some control problems for nonlinear time-delay systems, such as observability, identiability and state elimination andapplication of the results to biological models from the literature. It consistsof a monograph and three papers, Paper II, III and IV. The monograph intro-duces a mathematical framework for control based on modules over noncom-mutative rings before describing both previously published and new resultson the observability problem. A new result on state elimination is shown,which leads to a characterisation of the identiability of the delay parame-ters, the main result of this part of the thesis. The monograph concludesby an application of the result to a model of genetic regulation from theliterature. Application to other models from systems biology can be foundin Paper III. The theoretical results from Papers II and IV are described inthe monograph part, omitting detailed derivations, for which the reader isreferred to the papers themselves.

Part I

Observability and identiability of nonlinear

ODE systems

General theory and a case study of metabolic models

Milena Anguelova

Abstract

Observability is a structural property of a control system dened as thepossibility to deduce the state of the system from observing its input-outputbehaviour.

We present a review of two dierent methods to test the observabilityof nonlinear control systems found in literature. The dierential geometricand algebraic approaches have been applied to dierent classes of controlsystems. Both methods lead to the so-called rank test where the observabilityof a control system is determined by calculating the dimension of the spacespanned by gradients of the time-derivatives of its output functions. It hasbeen shown previously that for rational systems with n state-variables, onlythe rst n−1 time-derivatives have to be considered in the rank test. In thiswork, we show that this result applies for a broader class of analytic systems.

The rank test can be used to determine parameter identiability whichis a special case of the observability problem. A case study is presented inwhich the parameter identiability of a previously published kinetic modelfor the metabolism of S. cerevisiae (baker's yeast) has been analysed. Theresults show that some of the model parameters cannot be identied fromany set of experimental data.

The general features of kinetic models of metabolism are examined andshown to allow a simplied identiability analysis, where all sources of struc-tural unidentiability are to be found in single reaction rate expressions. Weshow how the assumption of an algebraic relation between concentrationsin metabolic models can cause parameters to be unidentiable. A generalmethod is presented to determine whether a conserved moiety renders a givenrate expression unidentiable and to reparameterise it into identiable pa-rameters.

i

ii

Keywords: Nonlinear systems, observability, identiability, observabilityrank condition, metabolic model, kinetic model, metabolism, glycolysis, Sac-charomyces cerevisiae

CONTENTS iii

Contents

1 Introduction 11.1 Motivation for studying observability and

identiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Organisation of the report . . . . . . . . . . . . . . . . . . . . 4

2 The dierential-geometric approach 52.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The observability rank condition (ORC) . . . . . . . . . . . . 62.3 Dierentiable inputs . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Observation space for analytic inputs . . . . . . . . . . 102.3.2 Symbolic notation for the inputs . . . . . . . . . . . . . 12

3 The algebraic point of view 133.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Algebraic setting . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Algebraic observability . . . . . . . . . . . . . . . . . . 143.2.2 Derivations and transcendence degree . . . . . . . . . . 143.2.3 Rank calculation . . . . . . . . . . . . . . . . . . . . . 16

3.3 The observability rank condition (ORC) for rational systems . 173.3.1 The ORC for polynomial systems . . . . . . . . . . . . 173.3.2 The ORC for rational systems . . . . . . . . . . . . . . 18

3.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Sedoglavic's algorithm . . . . . . . . . . . . . . . . . . . . . . 22

4 The number of output derivatives 24

5 Parameter identiability 29

6 Case study of a metabolic model 316.1 The central metabolic pathways . . . . . . . . . . . . . . . . . 316.2 The model of metabolic dynamics by Rizzi et al. . . . . . . . . 336.3 Identiability analysis . . . . . . . . . . . . . . . . . . . . . . 366.4 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.5 General features of kinetic models and identiability . . . . . . 39

7 Discussion 41

iv CONTENTS

8 Appendix I8.1 Why do d and Lf commute when f depends on a control

variable u? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I8.2 A basis for derivations on R〈U〉(x) that are trivial on R〈U〉 . . II8.3 Nomenclature for Rizzi's model . . . . . . . . . . . . . . . . . II

8.3.1 Superscripts . . . . . . . . . . . . . . . . . . . . . . . . II8.3.2 Symbols and abbreviations . . . . . . . . . . . . . . . . III8.3.3 Metabolites . . . . . . . . . . . . . . . . . . . . . . . . III8.3.4 Enzymes and ux indexes . . . . . . . . . . . . . . . . IV

8.4 Other results on the identiability of Rizzi's model . . . . . . IV

1

1 Introduction

1.1 Motivation for studying observability andidentifiability

Consider a culture of yeast cells grown in a reactor and the chemical reactionsthat take place in their metabolism. Thus far, the possibility to observe whatoccurs inside a single cell as far as metabolic uxes are concerned is verylimited. It is therefore not unnatural to consider the cell as a box where wesee what goes in (nutrients) and what comes out (secreted products), butnot what happens inside.

There is, however, extensive knowledge of the chemistry and biology thattakes place within the cell, and based on that, models are made for thetransformation that occurs inside the box. In preparation for a mathematicaldescription of the situation, we transform the above picture as follows:

We now label the part that can be controlled - for example, the amountof food given to the cells - u and call it input, or control variable. Theinput often varies with time and is thus a function u(t). The part that canbe observed over some interval of time - e.g. the dierent secreted productsthe uxes of which can be measured - is denoted by y(t) and called output.What happens inside the cell is accounted for in terms of changes in theconcentrations of the dierent chemical species present with respect to time;these concentrations are referred to as state-variables and denoted by c(t).We also have a number of parameters that come with the model used forcellular metabolism, denoted by p. In this rst part of the thesis, we assumethat the future concentrations of the chemical species c depend only on theirpresent concentrations and those of the inputs. Thus, the history of the celldoes not matter and the changes in the concentrations with respect to time

2 1 INTRODUCTION

can be described by ordinary dierential equations. The following continuousstate-space model can be formulated:

p(t) = 0c(t) = f(c(t), p, u(t))y(t) = g(c(t), p(t)) ,

(1.1)

where c(t) denotes the time-derivative of the state-variables at time t. Ahypothetical setting is assumed where we start feeding an input u(t) to thecell at time zero when the system is at an unknown state c(0) and we observethe cell's behaviour in terms of the outputs produced. It is assumed that uis a function of time that we can choose, and that the values of y and allits time-derivatives at the starting point (time zero) can be measured. Thevariation with time, (t), will not be explicitly written when it is clear fromthe context.

It is often the case that metabolic models contain numerous parameterswith unknown in vivo values. Sometimes, for the purpose of simulation, thelatter are approximated by their in vitro values, see for example (Teusink etal., 2000). Often, however, one is interested in obtaining the values that ta given set of experimental data. Thus, the parameters are estimated basedon observing the input-output behaviour of the system. The property ofidentiability is the possibility to dene the values of the model parametersuniquely in terms of known quantities, that is, inputs, outputs and theirtime-derivatives.

1.2 Problem statement

A generalisation of identiability is the property of observability. Considerthe following control system which generalises the example above:

Σ.=

x(t) = f(x, u)y = h(x, u) .

(1.2)

In this system, x are the state-variables, the inputs are denoted by u and theoutputs by y; all their components are functions of time. Note that parame-ters can be considered state-variables with time-derivative zero. We have noknowledge of the initial conditions for the state-variables (or, respectively, ofthe parameter values). It is assumed that we have a perfect measurement ofthe outputs so that they are known as functions of time in some interval andall their time-derivatives at time zero can be calculated. The observabilityproblem consists of investigating whether there exist relations binding thestate-variables to the inputs, outputs and their time-derivatives and thus

1.2 Problem statement 3

locally dening them uniquely in terms of controllable/measurable quanti-ties without the need for knowing the initial conditions. If no such relationsexist, the initial state of the system cannot be deduced from observing itsinput-output behaviour. In the biological setting above, for instance, thiscan mean that there are innitely many parameter sets that produce exactlythe same output for every input and thus the model parameters cannot beestimated from any experimental measurements.

Before we dene the problem of observability, consider the following ex-ample of a control system taken from Sedoglavic (2002):

x1 = x2

x1

x2 = x3

x2

x3 = x1θ − uy = x1 .

(1.3)

In this system, x1, x2 and x3 are state-variables, θ is a parameter, there is asingle input u and a single output y. In the following we use capital lettersto denote initial values of a function and its derivatives, i.e. u(r)(0) = U (r),y(r)(0) = Y (r) for r ≥ 0. By computing time-derivatives of the output attime zero, we obtain the equations:

Y (1) = x1 =x2

x1

(1.4)

Y (2) = x1 =x2x1 − x1x2

x21

=x3

x2x1 − x2

x1x2

x21

=x3

x1x2

− x22

x31

(1.5)

Y (3) = x(3)1 =

x3x1x2 − x3(x1x2 + x1x2)

x21x

22

− 2x2x2x31 − x2

23x21x1

x61

=

=(x1θ − U (0))x1x2 − x3(

x2

x1x2 + x1

x3

x2)

x21x

22

−2x2

x3

x2x3

1 − x223x

21

x2

x1

x61

=

=θ

x2

− U (0)

x1x2

− x23

x1x32

− 3x3

x31

+3x3

2

x51

. (1.6)

For this simple example, it is actually possible to explicitly calculate theinitial values of the state-variables and the parameters in terms of the inputsand outputs and their time-derivatives at time zero as shown in Sedoglavic(2002):

x1 = Y (0) (1.7)x2 = Y (0)Y (1) (1.8)x3 = Y (0)Y (1)

((Y (1))2 + Y (0)Y (2)

)(1.9)

θ =

((Y (1))2 + Y (0)Y (2)

)2+ Y (0)Y (1)(3Y (1)Y (2) + Y (0)Y (3))− U (0)

Y (0).(1.10)

4 1 INTRODUCTION

A given input-output behaviour thus corresponds to a unique state of thesystem. In general, we are not going to demand a globally unique state. Itis enough that the equations have a nite number of solutions each deninga locally unique state. The observability problem concerns the existence ofsuch relations and not the explicit calculation of the state variables from theequations. Depending on the theoretical approach, dierent denitions ofobservability can be given, as shown in this report.

1.3 Organisation of the report

In this work, a method for investigating the observability of certain classesof nonlinear control systems is described by using dierent theoretical pointsof view, each of which adds to our understanding of the problem.

Sections 2 and 3 present a survey of the theory on nonlinear observabil-ity available in the literature. Observability has been dealt with in both adierential geometrical interpretation, and an algebraical one. The two ap-proaches are introduced and the results in terms of obtaining an observabilitytest are described.

Section 4 attempts to answer the following question that arises duringthe literature surveys. If the derived observability test is to be applied inpractice, a bound must be introduced for the number of time-derivatives ofthe output that have to be considered in obtaining equations for the variables.Such an upper bound is given for rational systems in Section 3. In Section 4this bound is shown to apply for analytic systems.

Section 5 describes the identiability problem as a special case of observ-ability.

In Section 6 we apply the theory discussed in the preceding sections to acase study of a kinetic model for the metabolism of Saccharomyces cerevisiae,also known as bakers yeast. We use an algorithm by Sedoglavic (2002) andits implementation in Maple which performs an observability/identiabilitytest of rational models. We obtain results for the identiability of the kineticmodel and nd the non-identiable parameters. The results are interpretedin terms of the biological structure of the model.

The case study in Section 6 leads us to consider whether the special struc-ture of metabolic models allows for a simplied identiability test, in whichonly individual reaction rate expressions need to be analysed. Assumptionsof conserved moieties of chemical species, often used in kinetic modellingof metabolism, are shown to lead to unidentiable rate expressions, and inturn, to unidentiable parameters in the models. This is discussed in detailin Paper I, where we also show how the models can be reparameterised intoidentiable rate expressions.

5

2 The differential-geometric approach to

nonlinear observability

In this section we present the basics of the theory of nonlinear observabilityin a dierential-geometric approach that we have gathered from the worksof Hermann and Krener (1977); Krener (1985); Isidori (1995); Sontag (1991)and Sussmann (1979).

2.1 Definitions

Throughout this section we will consider control systems ane in the inputvariables which is a valid description of many real-world systems. They havethe form:

Σ

x(t) = f(x(t), u(t)) = g0(x(t)) + g(x(t))u(t)y(t) = h(x(t)) ,

(2.1)

where u(t) denotes the input, x(t) the state variables and y(t) the outputs(measurements). Throughout the text, the time-dependence (t) will not bewritten explicitly where it is understood from the context. We assume thatx ∈ M where M is an open subset of Rn, u ∈ Rm, y ∈ Rp and g0, and them columns of g, denoted by gi for i = 1, . . . ,m, are analytic vector eldsdened on M . We also have to assume that the system is complete, thatis, for every bounded measurable input u(t) and every x0 ∈ M there existsa solution to x(t) = f(x(t), u(t)) such that x(0) = x0 and x(t) ∈ M for allt ∈ R.

Here follow several denitions. Let W denote an open subset of M .

Denition 2.1 A pair of points x0 and x1 in M are W-distinguishable ifthere exists a measurable bounded input u(t) dened on the interval [0,T] thatgenerates solutions x0(t) and x1(t) of x = f(x, u) satisfying xi(0) = xi suchthat xi(t) ∈ W for all t ∈ [0, T ] and h(x0(t)) 6= h(x1(t)) for some t ∈ [0, T ].We denote by I(x0,W ) all points x1 ∈ W that are not W-distinguishablefrom x0.

Denition 2.2 The system Σ is observable at x0 ∈M if I(x0,M) = x0.

If a system is observable according to the above denition, it is still possiblethat there is an arbitrarily large interval of time in which two points ofM cannot be distinguished form each other. Therefore a local concept isintroduced which guarantees that to distinguish between the points of anopen subset W of M , we do not have to go outside of it, which necessarilysets a limit to the time interval as well.

6 2 THE DIFFERENTIAL-GEOMETRIC APPROACH

Denition 2.3 The system Σ is locally observable at x0 ∈M if for everyopen neighborhood W of x0, I(x0,W ) = x0.

Clearly, local observability implies observability as we can set W in Deni-tion 2.3 equal to M . On the other hand, since W can be chosen arbitrarilysmall, local observability implies that we can distinguish between neighbour-ing points instantaneously (since the trajectory is bound to be within W ,setting a limit to the time interval).

Remark: In this section local observability is a stronger property thanobservability because it implies that only local information is needed.

Both the denitions above ensure that a point x0 ∈ M can be distin-guished from every other point in M . For practical purposes though, it isoften enough to be able to distinguish between neighbours inM , which leadsus to the following two concepts:

Denition 2.4 The system Σ has the distinguishability property atx0 ∈M if x0 has an open neighborhood V such that I(x0,M) ∩ V = x0.

In a system having this property, any point x0 can be distinguished fromneighbouring points but there could be arbitrarily large intervals of time[0, T ] in which the points cannot be distinguished. In order to set a limit onthe time interval, a stronger concept is introduced:

Denition 2.5 The system Σ has the local distinguishability propertyat x0 ∈ M if x0 has an open neighbourhood V such that for every openneighbourhood W of x0, I(x0,W ) ∩ V = x0.

Clearly, local observability implies local distinguishability as we can set Vequal to M . Thus, if a system does not have the local distinguishabilityproperty at some x0, it is not locally observable at that point either.

It is the nal property of local distinguishability that lends itself to a test.

2.2 The observability rank condition (ORC)

This subsection describes how to determine if a system possesses the localdistinguishability property by the so-called "observability rank condition" asintroduced by Hermann and Krener (1977).

Throughout this subsection, we will use the following simple example ofa control system:

x1 = 0x2 = u− x1x2

y = x1x2 .(2.2)

2.2 The observability rank condition (ORC) 7

For this system, g0(x1, x2) =

(0

−x1x2

), g(x1, x2) =

(01

)and h(x1, x2) =

x1x2 (according to the notation introduced in the previous subsection) withp = 1, m = 1, and n = 2.

We now introduce dierentiation with respect to time along the systemdynamics. Formally, this is done by so-called Lie-dierentiation. The Liederivative of a C∞ function φ on M by a vector eld v on M is

Lv(φ)(x) :=< dφ, v > . (2.3)

Here <> denotes the scalar product and dφ the gradient of φ.Applying this to our example system, note that g0(x1, x2) and g(x1, x2)

are vector elds on M and we can calculate the Lie derivative of h(x1, x2)along them:

Lg0(h)(x1, x2) =< dh, g0 >= (x2 x1)

(0

−x1x2

)= −x2

1x2 (2.4)

andLg(h)(x1, x2) =< dh, g >= (x2 x1)

(01

)= x1 . (2.5)

The ow Φ(t, x) of a vector eld v on M is by denition the solution of:∂∂t

Φ(t, x) = v(Φ(t, x))Φ(0, x) = x .

(2.6)

Observe that we have the following equality:

Lv(φ)(x) =d

dt

∣∣∣∣t=0

(φ(Φ(t, x)

). (2.7)

The Taylor series of φ(Φ(t, x)) with respect to t are called Lie series:

φ(Φ(t, x)

)=

∞∑l=0

tl

l!Ll

v(φ)(x) . (2.8)

Let us now link the local distinguishability property to these new concepts.First of all, as observed in (Sussmann, 1979), if two points x0 and x1 in Mare W -distinguishable by a bounded measurable input, then they must beW -distinguishable by a piecewise constant input. This is due to uniformconvergence since the outputs depend continuously on the inputs. For aconstant input u, f(x, u) denes a vector eld on M and we can dene theow Φ(t, x) and the Lie series expansion of hi(Φ(t, x)) for i = 1, . . . , p. To see


how this generalises to piecewise-constant inputs, we follow (Isidori, 1995)and consider the input such that for i = 1, . . . ,m,

ui(t) = u1i , t ∈ [0, t1)

ui(t) = uli, t ∈ [t1 + · · ·+ tl−1, t1 + · · ·+ tl), l ≥ 2 ,

(2.9)

where uli ∈ R. With no loss of generality, we can assume that t1 = · · · = tl.

Dene the vector eldsθl = g0 + gul (2.10)

and denote their corresponding ows by Φlt. Under this input, the state

reached at time t1 + · · ·+ tl starting from x0 at t = 0 can be expressed as

x(tl) = Φltl · · · Φ1

t1(x0) . (2.11)

The corresponding output becomes

yi(t1 + · · ·+ tl) = hi

(Φl

tl · · · Φ1

t1(x0)

). (2.12)

This output can be regarded as the value of the mapping

F x0

i : (−ε, ε)l → R(t1, . . . , tl) 7→ hi Φl

tl · · · Φ1

t1(x0) . (2.13)

If two initial states x0 and x1 are such that they produce the same outputfor all possible piecewise-constant inputs, then

F x0

i (t1, . . . , tl) = F x1

i (t1, . . . , tl) (2.14)

for all possible (t1, . . . , tl) with 0 ≤ tj < ε and all i = 1, . . . , p. Thus,

( ∂lF x0

i

∂t1 · · · ∂tl)

t1=···=tl=0=

( ∂lF x1

i

∂t1 · · · ∂tl)

t1=···=tl=0. (2.15)

Since ( ∂lF x0

i

∂t1 · · · ∂tl)

t1=···=tl=0= Lθ1 · · ·Lθl

hi(x0) , (2.16)

we must have,Lθ1 · · ·Lθl

hi(x0) = Lθ1 · · ·Lθl

hi(x1) . (2.17)

Suppose now that there exists an open neighbourhood V of x0 such that allpoints in V are distinguishable from x0 instantaneously (which is the require-ment for local distinguishability). Then, there exists a piecewise-constantinput u such that the map from V to the space spanned by the Lie deriva-tives Lθ1 · · ·Lθl

hi is 1 : 1. Let us formally describe the "observation" space

2.2 The observability rank condition (ORC) 9

spanned by the Lθ1 · · ·Lθlhi which will be denoted by G. It can be shown,

(Isidori, 1995; Sontag, 1991), that

G = spanRLgi1Lgi2 · · ·Lgir (hi) : r ≥ 0, ij = 0, . . . ,m, i = 1, . . . , p .(2.18)

Since we are interested in the Jacobian of the 1 : 1 map mentioned above,the space spanned by the gradients of the elements of G is introduced anddenoted by dG:

dG = spanRxdφ : φ ∈ G , (2.19)

where Rx denotes the eld of meromorphic functions on M .It is the dimension of dG which determines the local distinguishability

property. For each x ∈M , let dG(x) be the subspace of the cotangent spaceat x obtained by evaluating the elements of dG at x. The rank of dG(x) isconstant in M except at certain singular points, where the rank is smaller(this property is due to the system being analytic, see for example (Krener,1985) or Chapter 3 in (Isidori, 1995). Then dimRxdG is dened as the genericor maximal rank of dG(x), that is, dimRxdG = maxx∈M(dimRdG(x)).

We can now formulate the so-called "observability rank condition" intro-duced by Hermann and Krener (1977):

Theorem 2.1 The system Σ has the local distinguishability property forall x in an open dense set of M if and only if dimRxdG = n.

Let us apply this test to the example system. We observe by inspectionthat the space G for this system is spanned by functions of the forms xk

1 andxk

1x2 (the rst two Lie derivatives were calculated above). Thus, the spacedG is spanned by one-forms of the type (kxk−1

1 0) and (kxk−11 x2 xk

1).Therefore we conclude that this example system has the local distinguisha-bility property almost everywhere except on the line x1 = 0.

Consider another example:x1 = u− x1

x2 = u− x2

y = x1 + x2 .(2.20)

For this system, g0(x1, x2) =

(−x1

−x2

), h(x1, x2) =

(11

)and h(x1, x2) =

x1 +x2 (according to the previously used notation). The rst two Lie deriva-tives are

Lg0(h)(x1, x2) =< dh, g0 >= (1 1)

(−x1

−x2

)= −x1 − x2 (2.21)


andLg(h)(x1, x2) =< dh, g >= (1 1)

(11

)= 2 . (2.22)

The space G for this example is spanned by constant functions and the func-tion x1 + x2. Thus, the space dG is spanned by one-forms of the type (1 1)and (0 0). Clearly, this space is of dimension 1, which means that thesystem does not have the local distinguishability property anywhere.

2.3 From piecewise-constant to differentiable in-puts - a different definition of observationspace

2.3.1 Observation space for analytic inputs

In the previous section, the observation space was dened in terms of piecewise-constant inputs to be:

G = spanRLgi1Lgi2 · · ·Lgir (hi) : r ≥ 0, ij = 0, . . . ,m, i = 1, . . . , p .(2.23)

In this subsection it is shown that the observation space can be denedequally well in terms of analytic inputs. We follow the works of Sontag (1991)and Krener (1985).

A time-dependent vector eld v(t, x) denes a time-dependent ow in asimilar way as in the previous section:

∂∂t

Φ(t, x) = v(t,Φ(t, x))Φ(0, x) = x .

(2.24)

Let Φu(t, x) denote the time-dependent ow corresponding to the time-dependentvector eld f(x(t), u(t)), where we now assume that we have a single input uwhich is an analytic function of time (the results in this section can be gen-eralised to apply for vector-valued inputs). Let the initial values of u and itsderivatives be u(r)(0) = U (r) for r ≥ 0 with U (r) ∈ R. For any non-negativeinteger l and any U = (U (0), . . . , U (l−1)) ∈ Rl, dene the functions

ψrm+i(x, U) =dr

dtr

∣∣∣∣t=0

gi(Φu(t, x)) , (2.25)

for 1 ≤ i ≤ p, 0 ≤ r ≤ l − 1. (Observe that the result of this formula isactually the Lie derivation dened earlier, where extra terms appear due tothe time dependence of the input. In fact, ψrp+i(x, U) = Lr

fhi where we deneLf =

∑nj=1 fj

∂∂xj

+∑

l=0 U(l+1) ∂

∂u(l) .) Applying repeatedly the chain rule, we

2.3 Dierentiable inputs 11

see that the functions ψi can be expressed as polynomials in U (0), . . . , U (l−1)

with coecients that are functions of x, (Sontag, 1991).As in Subsection 2.2, we can again dene the Taylor series of g(Φu(t, x))

with respect to t:

hi(Φu(t, x)) =∞∑

r=0

ψrp+i(x, U)tr

r!. (2.26)

Similarly to Subsection 2.2, where we considered the space spanned bythe coecients of the Lie series for hi(Φu(t, x)), we now construct the spacespanned by the ψj:

G = spanRψlp+i(x, U) : U ∈ Rl, l ≥ 0, i = 1, . . . , p . (2.27)

Wang and Sontag (1989) proved that G = G. We can illustrate this with theobservable example from Subsection 2.2:

x1 = 0x2 = u− x1x2

y = x1x2 .(2.28)

The time-dependent ow for the time-dependent vector eld

f(x, u) =

(0

u− x1x2

)becomes Φu(t, x) =

(Φu,1(t, x)Φu,2(t, x)

), where

∂∂t

Φu,1(t, x) = 0∂∂t

Φu,2(t, x) = u(t)− Φu,1(t, x)Φu,2(t, x)Φu,1(0, x) = x1

Φu,2(0, x) = x2 .

(2.29)

The rst few ψi:s can be calculated as follows:

ψ1(x, U) = h(Φu(t, x))|t=0 =(Φu,1(t, x)Φu,2(t, x)

)|t=0

=

= Φu,1(0, x)Φu,2(0, x) = x1x2

ψ2(x, U) =dh(Φu(t, x))

dt

∣∣∣∣t=0

=∂(Φu,1(t, x)Φu,2(t, x)

)∂t

∣∣∣∣t=0

=

=(Φu,2(t, x)

∂Φu,1(t, x)

∂t+ Φu,1(t, x)

∂Φu,2(t, x)

∂t

)|t=0

=

=(Φu,2(t, x) · 0 + Φu,1(t, x)(u(t)− Φu,1(t, x)Φu,2(t, x))

)|t=0

=

= Φu,1(0, x)(U(0) − Φu,1(0, x)Φu,2(0, x)) = x1(U

(0) − x1x2)

ψ3(x, U) =d2h(Φu(t, x))

dt2

∣∣∣∣t=0

=∂2

(Φu,1(t, x)Φu,2(t, x)

)∂t2

∣∣∣∣t=0

=

= x1

(U (1) − x1(U

(0) − x1x2))

. (2.30)


We see now that if the U (i):s are free to vary over R, then the space G forthis example is spanned by the functions xk

1 and xk1x2 for k ≥ 1, exactly as

the space G that we calculated in Subsection 2.2.

2.3.2 Symbolic notation for the inputs

Following (Sontag, 1991), let us now consider the ψj:s as formal polyno-mials in U (0), U (1), . . . with coecients that are functions of x. Denote byK = R(U (0), U (1), . . . ) the eld obtained by adjoining the indeterminatesU (0), U (1), . . . to R. Recall that Rx is the eld of meromorphic functions onM . Dene Kx = Rx(U

(0), U (1), . . . ) as the eld obtained by adjoining (a nitenumber of) the indeterminates U (0), U (1), . . . to Rx, where Kx is seen as avector space over K. Let FK be the subspace of Kx spanned by the functionsψj over K, that is,

FK = spanKψj : j ≥ 1 . (2.31)This is now a dierent denition of the observation space. As before, weare also interested in the space spanned by the dierentials of the elements ofFK. The latter can be seen as polynomial functions of U (0), U (1), . . . with co-ecients that are covector elds on M . For the example in Subsection 2.3.1,the dierentials of the ψj:s can be written:

dψ1 = (x2 x1)

dψ2 = (U (0) − 2x1x2 − x21) = (1 0)U (0) + (−2x1x2 − x2

1)

dψ3 = (U (1) − 2U (0)x1 + 3x21x2 x3

1) =

= (1 0)U (1) + (−2x1 0)U (0) + (3x21x2 x3

1) .

Recall from the previous section that the space dG for this example is spannedby one-forms of the type (kxk−1

1 0) and (kxk−11 x2 xk

1). The covectorelds calculated above are clearly of the same form.

Now letOK = spanKx

dψi : ψi ∈ FK . (2.32)Sontag (1991) proved the following result:

Theorem 2.2 For the analytic system (2.1)

dimRxdG = dimKxOK . (2.33)

Thus, the property of local distinguishability can be determined from thedimension of the space OK. The signicance of this result is that u can nowbe treated symbolically in calculating the rank. This observation is used inSection 4 to derive an upper bound for the number of dψj that have to beconsidered in the rank test.

13

3 The algebraic point of view: observ-

ability of rational models

This section introduces the algebraic point of view in the treatment of theobservability problem according to the works of Diop and Fliess (1991a,b);Diop and Wang (1993) and Sedoglavic (2002).

3.1 Example

Before we describe the algebraic setting for our general control problem,consider the following simple example:

x1 = x1x22 + u

x2 = x1

y = x1 .(3.1)

We obtain two equations for the state-variables x1 and x2 from the outputfunction and its rst Lie derivative where we use the notations u(r)(0) = U (r)

and y(r)(0) = Y (r), r ≥ 0 for the time derivatives at zero of the input andoutput, respectively:

Y (0) = x1 (3.2)Y (1) = Lfx1 = x1x

22 + U (0) . (3.3)

By simple algebraic manipulation of these equations, we can obtain the fol-lowing polynomial equations for each of the variables with coecients inU = (U (0), U (1), . . .) and Y = (Y (0), Y (1), . . .):

x1 = Y (0) (3.4)Y (0)x2

2 + U (0) − Y (1) = 0 . (3.5)

There are nitely many (two) solutions of these equations for a given setof inputs and outputs (except on the lines x1 = 0 and x2 = 0). Eachone is locally unique and determines the state of the system completely frominformation on the input and output values. (In the terminology of Section 2,this example system has the local distinguishability property for all x exceptfor those on the lines x1 = 0 and x2 = 0).

This was a very simple example where we could derive (and solve) thesepolynomial equations for the variables explicitly. In general, however, the ob-servability problem concerns the existence of such equations rather than theirexplicit calculation. We now review an algebraic formulation of observabilityfor control systems consisting of polynomial or rational expressions.

14 3 THE ALGEBRAIC POINT OF VIEW

3.2 Algebraic setting

3.2.1 Algebraic observability

Consider now polynomial control systems of the form:

Σ

x = f(x, u)y = h(x, u) ,

(3.6)

where u stands for the m input variables, f and h are for now vectors ofn and p polynomial functions, respectively (we will make the transition torational functions later).

The equations obtained by dierentiating the output functions will nowcontain polynomial expressions only. This allows us to make a new deni-tion of observability based on the following rather intuitive idea - the state-variable xi, i = 1, . . . , n is observable if there exists an algebraic relation thatbinds xi to the inputs, outputs and a nite number of their time-derivatives.If each xi is the solution of a polynomial equation in U and Y , then we knowthat a given input-output map corresponds to a locally unique state of thesystem. We will now prepare for a formal denition of algebraic observability.

Let R〈U, Y 〉 denote the eld obtained by adjoining the indeterminatesU

(0)i , U (1)

i , . . . , i = 1, . . . ,m and Y(0)j , Y

(1)j , . . . , j = 1, . . . , p to R (or any

other eld of characteristic zero). Then we can make the following denitionof algebraic observability:

Denition 3.1 xi, i ∈ 1, . . . , n is algebraically observable if xi is algebraicover the eld R〈U, Y 〉. The system Σ is algebraically observable if the eldextension R〈U, Y 〉 → R〈U, Y 〉(x) is purely algebraic.

3.2.2 Derivations and transcendence degree

The transcendence degree of the eld extension R〈U, Y 〉 → R〈U, Y 〉(x) isnow equal to the number of non-observable state-variables which should beassumed known (i.e. should have known initial conditions) in order to obtainan observable system. Our purpose is now to nd a way to calculate thistranscendence degree. For this, the theory of derivations over subelds asdescribed in (Jacobsson, 1980) and (Lang, 1993) is used.

Denition 3.2 A derivation D of a ring R is a linear map D : R→ R suchthat

D(a+ b) = D(a) +D(b) (3.7)D(ab) = aD(b) +D(a)b , (3.8)

for a, b ∈ R.

3.2 Algebraic setting 15

For example, the partial derivative ∂∂Xi

, i = 1, . . . , n, is a derivation of thepolynomial ring k[X1, . . . , Xn] over a eld k.

Consider now a eld F of characteristic 0 and a nitely-generated eldextension E = F (x) = F (x1, . . . , xk). Can a derivation D of F be extendedto a derivation D∗ of E which coincides with D on F? Consider the idealdetermined by (x) in F [X] and denoted by I, that is, the set of polynomialsin F [X] vanishing on (x). If such a derivation D∗ exists and p(X) ∈ I, thenthe following must hold:

0 = D(0) = D∗0 = D∗p(x) = pD(x) +n∑

i=1

∂p

∂xi

D∗xi , (3.9)

where pD denotes the polynomial obtained by applying D to all the coe-cients of p (which are elements of F ) and ∂p

∂xidenotes the polynomial ∂p

∂Xi

evaluated at (x). If the above is true for a set of generators of the ideal I,then it is satised by all polynomials in I. This is now a necessary conditionfor extending the derivation D to E = F (x). It is also a sucient conditionas shown in (Jacobsson, 1980) and (Lang, 1993):

Theorem 3.1 Let D be a derivation of a eld F . Let (x) = (x1, . . . , xn)be a nite family of elements in an extension of F . Let pα(X) be a set ofgenerators for the ideal determined by (x) in F [X]. Then, if (w) is any setof elements of F (x) satisfying the equations

0 = pD(x) +n∑

i=1

∂pα

∂xi

wi , (3.10)

there is one and only one derivation D∗ of F (x) coinciding with D on F andsuch that D∗xi = wi.

Suppose now that the derivation D on F is the trivial derivation, that is,Dx = 0 for all x ∈ F . Then, pD(x) = 0 in the equation above and thus,0 =

∑ni=1

∂pα

∂xiwi. The wi:s are thus solutions of a homogeneous linear equation

system and there exists a non-trivial derivation D∗ of E = F (x) only if thematrix formed by the ∂pα

∂xi:s is not full-ranked.

Let DerFE denote the set of derivations of E = F (x) that are trivial onF . DerFE forms a vector space over E if we dene (bD)(x) = b(D(x)) forb ∈ E. The dimension of this vector space can be calculated as follows, see(Jacobsson, 1980):

Theorem 3.2 Let E = F (x1, . . . , xn) and let X = p1, . . . , pq be a niteset of generators for the ideal of polynomials p in F [X1, . . . , Xn] such that


p(x1, . . . , xn) = 0 (this set exists due to Hilbert's basis theorem). Then:[DerFE : E] = n− rank(J(p1, . . . , pq)) , (3.11)

where J(p1, . . . , pq) is the Jacobian matrix∂p1

∂x1. . . ∂p1

∂xn... . . . ...∂pq

∂x1. . . ∂pq

∂xn

.

(3.12)

To see how the space DerFE is related to the transcendence degree of theeld extension F → E suppose that E = F (x) and x is algebraic over Fwith minimal polynomial p. If D is a derivation of E which is trivial on F ,then 0 = p′(x)Dx and thus Dx = 0 since p′(x) cannot be zero (the eld Fhas characteristic zero). Therefore D is trivial on E. We have the followinggeneral result Jacobsson (1980):Theorem 3.3 If E = F (x1, . . . , xn), then DerFE = 0 if and only if E isalgebraic over F . Moreover, [DerFE : E] is equal to the transcendence degreeof E over F .

3.2.3 Rank calculation

We now have a way of calculating the transcendence degree of E = F (x)over F by a rank calculation. Suppose that the transcendence degree isequal to r > 0 and thus some of the xi:s are not algebraic over F . Wewish to know if element xj is algebraic over F . Consider the eld ex-tensions F → F (xj) → E. We can calculate the transcendence degreeof the eld extension F (xj) → E by the method described above. SinceE = F (xj)(x1, . . . , xj−1, xj+1, . . . , xn), this will involve a calculation of therank of the following matrix:

∂p1

∂p1. . . ∂p1

∂xj−1

∂p1

∂xj+1. . . ∂p1

∂xn

... . . . ...∂pq

∂x1... ∂pq

∂xj−1

∂pq

∂xj+1. . . ∂pq

∂xn

.

(3.13)

If the transcendence degree of the eld extension F (xj) → E is equal to r (i.e.the above matrix has rank (n− 1)− r), then the variable xj is algebraic overF . This is due to the fact that if we have the eld extensions F → F ′ → E,then (Lang, 1993):

tr.deg.(E/F ) = tr.deg.(E/F ′) + tr.deg.(F ′/F ) . (3.14)We thus have a way of classifying all xi as either algebraic over F or notby eliminating the i:th column in the Jacobian and observing if there is achange of its rank.

3.3 The observability rank condition (ORC) for rational systems 17

3.3 The observability rank condition (ORC) forrational systems

3.3.1 The ORC for polynomial systems

Setting F = R〈U, Y 〉 and E = F (x1, . . . , xn), we can apply the theory fromSubsection 3.2 to our control problem. We have obtained a method fortesting the observability of polynomial control systems by calculating thetranscendence degree of the eld extension R〈U, Y 〉 → R〈U, Y 〉(x). In orderto perform the calculations described above, we need to describe the ideal I ofpolynomials p in k〈U, Y 〉[X] such that p(x1, . . . , xn) = 0. Clearly, Y (0)

j −gj ∈I for all j = 1, . . . , p. Dierentiating the j:th output variable with respect totime at zero we obtain (by Lie-dierentiation where the time-dependence ofthe inputs is taken into account, as in Section 2.3 of the previous section):

Y(1)j = Lfgj =

∑k=0

l∑i=1

∂gj

∂u(k)i

U(k+1)i (3.15)

Y(2)j = L2

fgj =∑k=0

l∑i=1

∂(Lfgj)

∂u(k)i

U(k+1)i , (3.16)

etc. Clearly Y(1)j − Lfgj and Y

(2)j − L2

fgj are elements of R〈U, Y 〉(x) andpolynomials in I. In fact, all such polynomials obtained by Lie-derivationbelong to I. It can be shown that I is generated by the polynomials Y (i)

j −Lifgj

for j = 1, . . . , p, i = 0, . . . , n − 1 by the following argument of Sedoglavic's(Sedoglavic, 2002).

We haveR〈U〉 ⊂ R〈U, Y 〉 ⊂ R〈U〉(x) , (3.17)

since each Y (i)j is a polynomial function of x with coecients in R〈U〉. Thus,

as in 3.2.3,

tr.deg.(R〈U〉(x)/R〈U〉) =

= tr.deg.(R〈U〉(x)/R〈U, Y 〉) + tr.deg.(R〈U, Y 〉/R〈U〉) , (3.18)

and the transcendence degree of the eld extension R〈U〉 → R〈U, Y 〉 istherefore at most n. Thus, for every j = 1, . . . , p, there exists an algebraicrelation qj(Y (0)

j , . . . , Y(n)j ) = 0 with coecients in R〈U〉. Thus the polynomial

Y(n)j − Ln

fgj belongs to the ideal generated by the polynomials Y (i)j − Li

fgj

for i = 1, . . . , n − 1. We therefore conclude that we need only consider theequations obtained by the rst n− 1 Lie-derivatives of the output functions.


Hence, according to Theorem 3.2, in order to calculate the transcendencedegree of the eld extension R〈U, Y 〉 → R〈U, Y 〉(x) we have to nd the rankof the following matrix:

∂L0f g1

∂x1. . .

∂L0f g1

∂xn... . . . ...∂L0

f gp

∂x1. . .

∂L0f gp

∂xn... . . . ...∂Ln−1

f g1

∂x1. . .

∂Ln−1f g1

∂xn... . . . ...∂Ln−1

f gp

∂x1. . .

∂Ln−1f gp

∂xn

.

(3.19)

If this Jacobian matrix is full-ranked, then the transcendence degree is zeroby Theorems 3.2 and 3.3 and we have an algebraically observable system.We have arrived at the observability rank condition that was derived for dif-ferentiable inputs in the dierential geometric approach in Subsection 2.3.2,but this time we have a nite number of Lie derivatives to consider.

If the system is not algebraically observable, we can nd the non-observablevariables by removing columns in this matrix and calculating the rank of thereduced matrices, as described in Subsection 3.2.3.

3.3.2 The ORC for rational systems

We will now generalise this theory to apply for rational systems of type:

Σ

x = f(x, u)y = g(x, u) ,

(3.20)

where now fi = pi(u, x)/qi(x) for i = 1, . . . , n and gj = rj(x, u)/sj(x) forj = 1, . . . , p with pi, qi, rj and sj polynomial functions.

We observe that just as before, Y (i)j − Li

fgj ∈ R〈U, Y 〉(x) for all i =0, . . . , n − 1, j = 1, . . . , p, but they are no longer polynomials. However, asshown by Diop et al. (1993) and Sedoglavic (2002), these rational expressionscan be used in the rank test instead of the polynomials that generate the idealI, and therefore we may use the same Jacobian in this case, as for polynomialsystems.

Remark: Observe that the algebraic interpretation has lead us to theobservability rank condition derived for analytic inputs in Subsection 2.3of the previous section, showing the equivalence of algebraic observabilityand local distinguishability, see (Diop et al., 1993). In fact, the ideal I of

3.4 Symmetry 19

polynomials p in R〈U, Y 〉[X] such that p(x1, . . . , xn) = 0 is generated by thesame functions that span the space FK dened in Section 2. The rank of theJacobian

∂L0f g1

∂x1. . .

∂L0f g1

∂xn... . . . ...∂L0

f gp

∂x1. . .

∂L0f gp

∂xn

. . .∂Ln−1

f g1

∂x1. . .

∂Ln−1f g1

∂xn... . . . ...∂Ln−1

f gm

∂x1. . .

∂Ln−1f gm

∂xn

(3.21)

is exactly the dimension of the space OK which, as we recall, determines thelocal distingushability property according to Theorem 2.2. The result of thealgebraic approach of this section is that we have been able to show that forrational systems the space OK is generated by a nite number of functions.In Section 4, we take a dierent approach to show that this is in fact truefor all analytical systems of the form (2.1).

3.4 Symmetry

Suppose now that by applying the rank test above, we nd that our controlsystem is not algebraically observable and that the transcendence degree isr. This means that DerR〈U,Y 〉R〈U, Y 〉(x) is not empty and has dimension r.The dierential-geometric concept that corresponds to derivations is that oftangent vectors. We can therefore interpret the existence of derivations onR〈U, Y 〉(x) that are trivial on R〈U, Y 〉 as the existence of tangent vectorsto the space of solutions to our control system, such that if we move intheir direction, the output remains the same and we cannot observe thatthe system is in a dierent state. In other words, there are innitely manytrajectories for the control system that cannot be distinguished from eachother by observing the input-output map.

A derivation therefore generates a family of symmetries for the controlsystem - symmetries in the variables leaving the inputs and outputs invari-able. In this section we will show how these can be calculated.

First of all, observe that the partial derivatives ∂∂xi

form a basis for thederivations on R〈U〉(x) that are trivial on R〈U〉 (see Appendix 8.2 for ex-planation). Among these, we wish to nd the ones that are trivial also onR〈U, Y 〉. If v is one of them, recall from Theorems 3.1 and 3.2 that we must


have:

∂L0f g1

∂x1. . .

∂L0f g1

∂xn... . . . ...∂L0

f gp

∂x1. . .

∂L0f gp

∂xn... . . . ...∂Ln−1

f g1

∂x1. . .

∂Ln−1f g1

∂xn... . . . ...∂Ln−1

f gp

∂x1. . .

∂Ln−1f gp

∂xn

· v = 0 . (3.22)

Thus, v belongs to the kernel of the above Jacobian matrix. Suppose thatv = (v1, . . . , vn), where vi ∈ R〈U, Y 〉(x). Then v is the Lie-derivationv =

∑ni=1 vi

∂∂xi

which corresponds to a vector eld v and a ow Φ(ρ, x) of vgiven by (see Section 2):

∂∂ρ

Φ(ρ, x) = v(Φ(ρ, x))

Φ(0, x) = x .(3.23)

The solution of this system of dierential equations evaluated at any ρ > 0corresponds to a new initial state of the system which cannot be distinguishedfrom the original one, (x1, . . . , xn), by observing the input-output it produces.

We now have a strategy for nding the families of symmetries for ourcontrol system. First, we have to dene a basis for the kernel of the Jacobianmatrix. In order to obtain the associated families of symmetries, we have tosolve the system of dierential equations that correspond to each element ofthe chosen basis. To make the calculations simpler, we can use the obser-vations from Subsection 3.2.3 to nd the non-observable variables. Insteadof calculating the kernel of the Jacobian matrix, we can calculate the kernelof its maximal singular minor which is obtained when the columns and rowscorresponding to the observable variables are removed. Then, the systemof dierential equations to be solved will only involve the non-observablevariables.

We will now apply this to a non-observable example:x1 = x2x4 + ux2 = x2x3

x3 = 0x4 = 0y = x1 .

(3.24)

3.4 Symmetry 21

We need to calculate the rst three Lie-derivatives of the output function:

Y (1) = Lfx1 = x2x4 + U (0) (3.25)Y (2) = L2

fx1 = Lf (x2x4 + u) = x4x2x3 + U (1) (3.26)Y (3) = L3

fx1 = Lf (x4x2x3 + u) = x4x3x2x3 + U (2) . (3.27)

Thus the Jacobian matrix becomes:

1 0 0 00 x4 0 x2

0 x3x4 x2x4 x2x3

0 x23x4 2x2x3x4 x2x

23

∼

1 0 0 00 x4 0 x2

0 0 x2x4 00 0 0 0

.

(3.28)

Clearly, this matrix has rank 3 and the non-observable variables are x2 andx4 - removing the second or fourth column does not change the rank of thematrix. We can now eliminate the rst and third rows and columns andconsider the kernel of the remaining minor, which is the matrix

[x4 x2

x23x4 x2x

23

].

(3.29)

This kernel is generated by the vector (x2,−x4). The derivation x2∂

∂x2−x4

∂∂x4

thus corresponds to the system of dierential equations:

Φ2(ρ, x) = Φ2(ρ, x)

Φ4(ρ, x) = −Φ4(ρ, x)Φ2(0, x) = x2

Φ4(0, x) = x4 .

(3.30)

The solution is: Φ2(ρ, x) = x2e

ρ

Φ4(ρ, x) = x4e−ρ .

(3.31)

If we set eρ = λ, we nd that multiplying x2 by λ and dividing x4 by itdenes a new state x that is indistinguishable from the original one for anyλ. Indeed, we see that performing this procedure does not change the output


and its Lie-derivatives:

˙x1 = x2x4 + u = λx2x4/λ+ u = x2x4 + u˙x2 = x2x3 = x2x3 = λx2x3

˙x3 = 0˙x4 = 0Y (0) = x1 = x1 = Y (0)

Y (1) = Lf x1 = x2x4 + U (0) = x2x4 + U (0) = Y (1)

Y (2) = L2fx1 = Lf (x2x4 + u) = x4 ˙x2 + U (1) = x4x2x3 + U (1) =

= 1λx4λx2x3 + U (1) = Y (2)

Y (3) = L3fx1 = Lf (x4x2x3 + u) = x3x4 ˙x2 + U (2) =

= x31λx4λx2x3 + U (2) = Y (3) .

(3.32)We know from Subsection 3.3.1 that we need not consider any further Liederivatives since they depend on the previous ones.

We have now dened a family of symmetries

σλ : x1, x2, x3, x4 → x1, λx2, x3, x4/λ (3.33)

of the control system which leaves the input and output invariant.

3.5 Sedoglavic's algorithm

There is a published algorithm by Sedoglavic (2002) with a Maple implemen-tation which performs an observability test of rational systems and for non-identiable systems, predicts the non-identiable variables with high proba-bility. This is done in polynomial time with respect to system complexity.

The algorithm is mainly based on generic rank computation, for details,see (Sedoglavic, 2002). The symbolic computation of the Jacobian matrixdened in Subsection 3.3 can be cumbersome for systems with many vari-ables and parameters and it cannot be done in polynomial time. Instead,the parameters are specialised on some random integer values, and the in-puts are specialised on a power series of t with integer coecients. To limitthe growth of these integers in the process of rank computation, the cal-culations are done on a nite eld Fp (p refers to a prime number). Theprobabilistic aspects of the algorithm concern the choice of specialisation ofparameters and inputs and also the fact that cancelation of the determinantof the Jacobian modulo p has to be avoided. The calculation of the rank isdeterministic for observable systems, that is, when the process states thatthe system is observable, the answer is correct. For non-observable systems,the probability of a correct answer depends on the complexity of the system

3.5 Sedoglavic's algorithm 23

and on the prime number p. The predicted non-observable variables can befurther analysed to nd a family of symmetries which then can conrm thetest result.

The Maple implementation takes as an input a rational system of dif-ferential equations where parameters, state-variables and inputs have to bestated as such, and also a set of outputs has to be dened. The transcendencedegree of the eld extension associated to the system is calculated and thenon-observable parameters and state-variables are predicted.

We have used this implementation for our case study in Section 6.

24 4 THE NUMBER OF OUTPUT DERIVATIVES

4 The first n − 1 derivatives of the out-

put function determine the observabil-

ity of analytic systems with n state vari-

ables

This section deals with several questions that arise from Sections 2 and 3. Thedierential-geometric approach from Section 2 results in the observabilityrank test for observability of analytic systems. In this test, the rank ofthe linear space containing the gradients of all Lie derivatives of the outputfunctions must be calculated. Since no bound is given for the number ofLie derivatives necessary for the calculation, the practical application of thetest to other than the simplest examples is dicult. Such an upper boundis derived for the case of rational systems in Section 3 using the algebraicalapproach. The following questions now arise. Can an upper bound be givenonly for rational systems? How do such requirements for the class of thesystem arise? In this section, we attempt to extend the upper bound forthe number of time-derivatives of the output function to apply for the classof analytical systems ane in the input variable that are addressed by thedierential-geometric approach in Section 2. We are going to use the resultsby Sontag (1991) described in Subsection 2.3 where the observability rankcondition was dened in terms of dierentiable inputs.

Consider once again the example from the introduction, taken from Se-doglavic (2002):

x1 = x2

x1

x2 = x3

x2

x3 = x1θ − uy = x1 .

(4.1)

Recall that we obtained the following equations for the state-variables andthe parameter from calculating the rst three time-derivatives at zero of theoutput (see Subsection 1.2):

r1(x1, Y(0)) = Y (0) − x1 = 0

r2(x1, x2, Y(1)) = Y (1) − x2

x1= 0

r3(x1, x2, x3, Y(2)) = Y (2) − ( x3

x1x2− x2

2

x31) = 0

r4(x1, x2, x3, θ, Y(3)) = Y (3) − ( θ

x2− U(0)

x1x2− 3x3

x31− x2

3

x1x32

+3x3

2

x51) = 0 .

(4.2)The problem now is to determine whether these equations are enough toensure that a given input-output behaviour corresponds to a locally unique

25

state of the system. From the implicit function theorem it follows that thevariables x1, x2, x3 and the parameter θ can be expressed locally (in the neigh-bourhood of a given point in the space of solutions of the dierential equa-tions) as functions of U (0) and Y (0), Y (1), Y (2), Y (3) if the rank of the followingJacobian matrix evaluated at that point is equal to four:

∂(r1)∂x1

∂(r1)∂x2

∂(r1)∂x3

∂(r1)∂θ

∂(r2)∂x1

∂(r2)∂x2

∂(r2)∂x3

∂(r2)∂θ

∂(r3)∂x1

∂(r3)∂x2

∂(r3)∂x3

∂(r3)∂θ

∂(r4)∂x1

∂(r4)∂x2

∂(r4)∂x3

∂(r4)∂θ

= (4.3)

= −

1 0 0 0−x2

x21

1x1

0 0

− x3

x21x2

+3x2

2

x41

− x3

x1x22

+ 2x2

x31

1x21x2

0

ux2x2

1+ 9x3

x41

+x23

x32x2

1− 15x3

2

x61

− θx22

+ ux1x2

2+

3x23

x1x42

+9x2

2

x51

− 3x31− 2x3

x1x32

1x2

.

Clearly, this matrix has full rank for all values of x1, x2, x3 and θ and thusthe system has a locally unique state for a given input-output behaviour.

Now the following question arises - if the rank of the above matrix is notfull, can we then conclude that the system is not locally observable withoutconsidering further derivatives of the output function which would producenew equations? In other words, is the rank of the Jacobian determined bythe rst n equations, where n is the total number of state-variables andparameters? We will now show that this is true for the analytic systemsane in the input variable that were discussed in Section 2.

Consider again the analytic control system of the form (equation (2.1)):

Σ

x = f(x, u) = g0(x) + g(x)uy = h(x) .

(4.4)

As previously (Section 2), the elements of the n-dimensional vectors g0 andg are analytic functions of x and we assume for the moment that we havea single analytic output h(x) and also a single analytic input u. The nstate-variables x are assumed to occupy an open subset M of Rn.

The rst two equations obtained by dierentiating the output function


with respect to time at zero are:

Y (1) = Lfh(x) = dh · f|t=0 = dh · (g0 + gU (0)) =

= dh · g0 + U (0)(dh · g) (4.5)

Y (2) = L2fh(x) = Lf (dh · f) = (d(dh · f) · f)|t=0 +

∂(dh · f)

∂uU (1) =

= d(dh · g0 + U (0)(dh · g)

)·(g0 + gU (0)

)+

+∂(dh · g0 + u(dh · g)

)∂u

U (1) =

= d(dh · g0 + U (0)(dh · g)

)· h0 + U (0)d

(dh · g0 + U (0)(dh · g)

)· g +

+ U (1)(dh · g) =

= d(dh · g0) · g0 + U (0)(d(dh · g) · g0 + d(dh · g0) · g

)+

+(U (0))2(d(dh · g)

)· g) + U (1)(dh · g) . (4.6)

These calculations conrm the result by Sontag (1991) that the rst n−1 Liederivatives of the output function g(x) for the system (2.1) are polynomialfunctions of U (0), U (1), . . . , U (n−2) with coecients that are analytic functionson M .

Thus we have that L(i)f h ∈ Kx for i = 0, . . . , n − 1; recall from Subsec-

tion 2.3 that Kx = Rx(U(0), U (1), . . . ) is the eld of meromorphic functions

on M to which we add the indeterminates U (0), U (1), . . . and obtain rationalfunctions of U (0), U (1), . . . with coecients that are meromorphic functionson M . See also (Sontag, 1991).

Following the notation from example (4.1) above, the rst n equationsfor the state-variables can now be formulated:

r1(x, Y(0)) = Y (0) − h = 0 (4.7)

r2(x, u, Y(1)) = Y (1) − Lfh = 0 (4.8)

...rn(x, u, . . . , u(n−2), Y (n−1)) = Y (n−1) − Ln−1

f h = 0 . (4.9)

Therefore, the Jacobian that we are interested in is:

−

∂h∂x1

. . . ∂h∂xn... . . . ...

∂Ln−1f h

∂x1. . .

∂Ln−1f h

∂xn

.

(4.10)

Since L(i)f h ∈ Kx for i = 0, . . . , n − 1, the elements of this Jacobian also

belong to Kx. We will now show that if this Jacobian is not full-ranked,

27

that is, the rst n gradients of the output function and its Lie derivatives arelinearly dependent over the eld Kx, then any further Lie derivative producesa gradient which is linearly dependent of the rst n and we can thus concludethat the system is not locally observable. Furthermore, if the rst q gradients,where q ≤ n, are linearly-dependent, then no further gradients are necessaryfor the calculation of the rank, which becomes ≤ q − 1. In fact, we canstop Lie dierentiating the output function at the rst instance of lineardependence.

Remark: To be able to discuss linear dependence, we have to know thatthe gradients of the Lie derivatives produce a linear space over a eld (ora free module over a commutative ring). This was the case for the rationalsystems in Section 3 and this is also the case here for analytic systems of theabove type, because the elements of the Jacobian belong to the eld Kx.

Theorem 4.1 Let Σ be the systemx = f(x, u) = g0(x) + g(x)uy = h(x) ,

(4.11)

where x is a vector of n state-variables occupying an open subset M of Rn,g0 and g are n-dimensional vectors of analytic functions on M , the outputh(x) is an analytic function on M and the control variable u is an analyticfunction of time.

If q is an integer such that dL(i)f h, i = 0, . . . , q are linearly dependent over

the eld Kx, then the dimension of the space OK = spanKxdL(i)

f h, i ≥ 0(see Subsection 2.3) is less than or equal to q − 1. If q < n, the system Σ isnot locally observable.

Proof: Suppose that the rst q gradients are linearly dependent and q isthe least such number (it certainly exits as the rank is ≤ n and a single non-zero vector is linearly independent of itself). Then, there exist coecientski ∈ Kx, i = 0, . . . , q − 1, not all of them zero, such that

q−1∑i=0

kidLifh = 0 . (4.12)

We can take the Lie derivative of both sides (which are co-vector elds) toobtain:

0 = Lf (

q−1∑i=0

kidLifh) =

q−1∑i=0

Lf (kidLifh) =

q−1∑i=0

((Lfki)dL

ifh+ kiLf (dL

ifh)

).

(4.13)


We now observe the following fact (which is simply saying that the d andLf operators commute even when f depends on a control variable u(t), seeAppendix 8.1 for derivation):

Lf (dLifh) = dLi+1

f h , (4.14)

for i ≥ 0.It follows that

0 =

q−1∑i=0

((Lfki)dL

ifh+ kidL

i+1f h

). (4.15)

Recalling the structure of the eld Kx, we know that Lfki ∈ Kx since

Lfki = dki · f +∂ki

∂uU (1) = dki · (g0 + gU (0)) +

∂ki

∂uU (1) , (4.16)

which is clearly a rational function of U (0), U (1), . . . with coecients thatare meromorphic functions of x. Since we know that kq−1 is not zero (weassumed that q was the least number such that the rst q gradients arelinearly dependent), we conclude that dLq

fh is linearly dependent on thepreceding gradients.

Using the same calculations we can prove by induction that any furthergradient is linearly dependent on the previous ones which then means thatdL0

fh, . . . , dLq−1f h form a basis for the space OK which determines local dis-

tinguishability (by Theorems 2.1 and 2.2) and thus local observability. Ifq < n, this space has rank less than n and thus the system is not locallyobservable.

Thus it is enough to consider the rst n− 1 Lie derivatives of the outputfunction in the rank test and also, we can stop calculating further derivativesof the output function at the rst instance of linear dependence among theirgradients.

Remark: We note that in the case of multiple output functions one needsto calculate n− 1 time-derivatives of each.

29

5 Parameter identifiability

In this relatively short section we will present the problem of parameter iden-tiability of nonlinear control systems as a special case of the observabilityproblem.

Identiability is the possibility to identify the parameters of a controlsystem from its input-output behaviour. By considering parameters as state-variables with time derivative zero, one can use the observability rank testto determine identiability. The property of local observability is then in-terpreted as the existence of only nitely many parameter sets that t theobserved data, each of them locally unique. The use of the rank test for de-termining the identiability of nonlinear systems dates back to at least 1978when Pohjanpalo (1978) used the coecients of the Taylor series expansionof the output to determine the parameter identiability of a class of nonlin-ear systems applied in the analysis of saturation phenomena in pharmacoki-netic studies. A more recent example is the work by Xia and Moog (2003)where dierent concepts of nonlinear identiability are studied in an algebraicframework. They apply the theory to a four-dimensional HIV/AIDS model,and show that their theoretical results can be used to determine whetherall the parameters in the model are determinable from the measurement ofCD4+ T cells and virus load, and if not, what else has to be measured.The minimal number of measurements of the variables for the complete de-termination of all parameters and the best period of time to make suchmeasurements are calculated. Another example with biological applicationis the work by Margaria et al. (2004) where the identiability of some highlystructured biological models of infectious disease dynamics is analysed bothusing the rank method and Sedoglavic's algorithm, (Sedoglavic, 2002) andalso by the constructive method of characteristic set computation describedby Ollivier (1990); Ljung and Glad (1990) and others. Due to the fact thatits computational complexity is exponential in the number of parameters,the latter method can only be applied to relatively small control systems.

We will now describe how the observability rank test can be used todetermine parameter identiability. Consider a physical/chemical/biologicalmodel:

Σ

x = f(x, p, u)y = h(x, p) ,

(5.1)

where as before, x denote the n state-variables, u the m inputs and y the pobserved quantities. The l model parameters are denoted by p and f(x, p, u)and h(x, p) are vectors of analytical functions. We may or may not be given

30 5 PARAMETER IDENTIFIABILITY

a set of initial conditions for the state-variables:

x(0) = x0 . (5.2)

In order to be able to use the theory from the previous sections, we observethat the above model can be represented by the following control system:

Σ

p = 0x = f(x, p, u)y = h(x, p) ,

(5.3)

where x and p can now be considered as the same type of variables. We canapply the rank test to this system in exactly the same way as discussed inthe previous sections.

Without initial conditions for x, the non-observable variables can be bothin x and in p. Suppose now that we are given a full set of initial conditionson x:

x(0) = x0 . (5.4)

The problem of the observability of the x variables now disappears as theinitial state is already uniquely dened. What is left, is exactly the problemof identiability for the parameters - is the set of parameters that realises agiven input-output map unique, at least locally?

This can be determined by the rank test described in the previous chap-ters. For analytical systems (see Section 4) the rank test amounts to calcu-lating the rank of the following Jacobian matrix:

∂L0f g1

∂p1. . .

∂L0f g1

∂pl... . . . ...∂L0

f gm

∂p1. . .

∂L0f gm

∂pl... . . . ...∂Ln−1

f g1

∂p1. . .

∂Ln−1f g1

∂pl... . . . ...∂Ln−1

f gm

∂p1. . .

∂Ln−1f gm

∂pl

.

(5.5)

If the rank of this matrix is l, then the model is identiable. If not, thenon-identiable parameters can be found using the same procedure we usedearlier for nding non-observable variables, see Subsection 3.2.3.

31

6 Case study: Identifiability analysis of

a kinetic model for S. cerevisiae

In this case study, we have investigated the identiability of a publishedmodel of the metabolic dynamics in S. cerevisiae by Rizzi et al. (1997). Webegin by a short description of the biochemistry of the central metabolicpathways.

6.1 The central metabolic pathways

Metabolism is the overall network of enzyme-catalysed reactions in a cell.Its degradative, or energy-releasing phase is called catabolism. The centralcatabolic pathways which are more or less universal among organisms consistof glycolysis, the pentose phosphate pathway and the citric acid cycle. Inglycolysis sugars are degraded to a three-carbon compound called pyruvate.In the absence of oxygen pyruvate is then reduced to lactate, ethanol or otherfermentation products. In aerobic conditions, it is instead oxidised via thecitric acid cycle in the process of cellular respiration. A simplied scheme ofsome of the most important reactions in the central metabolic pathways isshown in the gure on the next page.

The dierent species in the boxes are called metabolites. The reactionsmarked by arrows are catalysed by enzymes which determine their reactionrate or ux, that is, the speed with which the reaction occurs.

Reaction rates are often modelled by using so-called Michaelis-Mentenor Hill kinetics where an equation is derived for the reaction rate based ona biochemical description of the general way in which enzymatic reactionsoccur. For example, a reaction in which a single substrate (reactant) A istransformed to a single product B under the catalysis of a single enzyme Ehas the following rate equation, (Lehninger, 2000):

r =rmaxA

Km + A, (6.1)

where the constants rmax and Km are specic for this reaction. rmax is themaximal rate of the reaction and Km is the substrate concentration at whichthe reaction rate is half rmax.

An enzyme can have several binding sites for the substrate in which casea so-called Hill equation is used. For the above reaction where we allow nbinding sites for the enzyme, the equation becomes (see for example Chapter5 in (Stephanopoulos et al., 1998)):

r =rmaxAn

Km + An. (6.2)

32 6 CASE STUDY OF A METABOLIC MODEL

These equations can take much more complicated forms depending on thenumber of substrates and products and other factors such as reversibilityof the reaction, inhibition and cooperation eects on the enzymes, etc., see(Lehninger, 2000; Segel, 1975) and (Rizzi et al., 1997) for details.

6.2 The model of metabolic dynamics by Rizzi et al. 33

6.2 The model of metabolic dynamics by Rizzi etal.

We now proceed to describe a mathematical model of the dynamics of thechemical reactions in the central metabolic pathways formulated by Rizzi etal. (1997).

The authors propose a kinetic model for the reactions of glycolysis, thecitric acid cycle, the glyoxylate cycle and the respiratory chain in growingcells of S. cerevisiae. The model aims to predict the short-term changes inthe metabolic states of the cells under in vivo conditions after a change inthe glucose feed rate. A schematic picture adapted from (Rizzi et al., 1997)describing the metabolites and uxes included in the model is shown below.


For each metabolite in the scheme, a mass balance is written where thechange of its concentration in time is expressed accounting for the incomingand outcoming uxes as well as the eect of dilution. The following systemof dierential equations is obtained for the concentrations of the dierentspecies (see Appendix 8.3 for nomenclature and parameter description):

dceGLC

dt= D(c0GLC − ceGLC)− cX

ρrCPERM (6.3)

dceGLYC

dt=

cXρrCRES,1 −DceGLYC (6.4)

dceAC

dt=

cXρrCALDH −DceAC (6.5)

dceETOH

dt=

cXρrCADH −DceETOH (6.6)

dceCO2

dt=

cXρ

(VM

VC

rMCO2

+ rCPDC + aCO2,1r

CSYNT,1 + aCO2,2r

CSYNT,2) +

+SCO2 (6.7)dceO2

dt= −VM

VC

CX

ρrMO2

+ SO2 (6.8)

dcCGLC

dt= rC

PERM − rCHK − µcCGLC (6.9)

dcCG6P

dt= rC

HK − rCPGI − rC

SYNT,1 − µcCG6P (6.10)

dcCF6P

dt= rC

PGI − rCPFK − µcCF6P (6.11)

dcCFBP

dt= rC

PFK − rCALDO − µcCFBP (6.12)

dcCDHAP

dt= rC

ALDO − rCTIS − rC

RES,1 − µcCDHAP (6.13)

dcCGAP

dt= rC

ALDO + rCTIS − rC

RES,2 − µcCGAP (6.14)

dcCPEP

dt= rC

RES,2 − rCPK − µcCPEP (6.15)

dcCPYR

dt= rC

PK −VM

VC

rMPDH − rC

PDC − rCSYNT,2 − µcCPYR (6.16)

dcCALDE

dt= rC

PDC − rCADH − rC

ALDH −VM

VC

rMACETYL − µcCALDE (6.17)

6.2 The model of metabolic dynamics by Rizzi et al. 35

dcCADP

dt= rC

HK + rCPFK + aATP,1r

CSYNT,1 + aATP,2r

CSYNT,2 +mATP −

− 2rCADK − rC

RES,2 − rCPK − rC

TR,ADP − µcCADP (6.18)dcCATP

dt= rC

RES,2 + rCPK +

VM

VC

rMTR,ATP + rC

ADK − rCHK − rC

PFK −

− aATP,1rCSYNT,1 −mATP − aATP,2r

CSYNT,2 − µcCATP (6.19)

dcCAMP

dt= rC

ADK − µcCAMP (6.20)

dcCNADH

dt= rC

RES,2 + aNADH,2rCSYNT,2 − rC

RES,1 − rCADH − rC

ALDH −

− rCNADHDH − µcCNADH (6.21)

dcMATP

dt= rM

ATP,R + rMATP,T − rM

TR,ATP − µcMATP (6.22)

dcMNADH

dt= rM

NADH,T − rMNADHQR − µcMNADH . (6.23)

Remark: After comparison with the original version of the model, see(Baltes, 1996), some minor modications have been made to the model de-scription in (Rizzi et al., 1997) due to what appears to be typing errors inthe latter, see (Johansson, 2007).

The uxes r have rate equations based on Michaelis-Menten, Hill or othertypes of enzyme kinetics gathered from the literature or proposed by theauthors. For example, the triosephosphate isomerase reaction convertingdihydroxyacetone phosphate to glyceraldehyde-3-phosphate has the rate ex-pression

rCTIS = rmax

TIS

cCDHAP −cCGAP

Keq,6

KDHAP,6(1 +cCGAP

KGAP,6) + cCDHAP

. (6.24)

Most of the uxes in the model are rational expressions with the exception ofthose uxes where the Hill coecients are not integers. This fact is importantfor the identiability analysis and will be discussed later.

The above model was evaluated in (Rizzi et al., 1997) on the basis of ex-perimental observations previously described in (Theobald et al., 1997). Themodel predictions were compared to the experimental results and the param-eters were estimated from the data (Rizzi et al., 1997). We are now going touse the theory of identiability to nd out whether the kinetic parametersof this model can be uniquely determined from a perfect set of experimentaldata. We must rst formulate a control system for the model. For this,the appropriate set of inputs and outputs must be chosen from the descrip-tion of the experimental setting in (Theobald et al., 1997). In the latter, a


methodology was developed where the changes in metabolite concentrationsafter a glucose feed pulse (a fast injection of a certain volume of glucose inthe medium, (Theobald et al., 1997), were measured over time. The initialconditions were the priorly-known values of the metabolite concentrationsunder so-called steady-state growth - a condition when biomass concentra-tion (and other factors) has stabilised to a constant value for the culture, see(Theobald et al., 1997) for details.

In order to translate the information in the above paragraph into math-ematical language, we include a perfect measurement of all metabolite con-centrations c (thus including the given initial conditions) in the outputs ofthe control system:

p = 0c = f(c, p, u)y = c(c(0) = c0) ,

(6.25)

where c is the vector of metabolite concentrations, f is the right-hand sideof the equation array 6.3 and we denote all the model parameters by p.The initial conditions are in parenthesis as the information they provide isincluded in the output set. The input u is assumed to be the glucose feed.

6.3 Identifiability analysis

We performed an identiability analysis of the above control system. Forthis, Sedoglavic's implementation (see Subsection 3.5) was used as the modelhas around a hundred parameters which makes calculations by hand verydicult. As the algorithm works only for rational control systems, we ap-proximated any non-integer values of the Hill coecients by integers. Ofcourse, in general, such approximations can have an important eect on theidentiability of the system. This turns out not to be the case for Rizzi'smodel, as shown in the next section.

Sedoglavic's algorithm produced the following results - the control systemwas not identiable with transcendence degree 2 and the non-identiableparameters were the kinetic parameters in two of the rate expressions - theexpression for the ux rC

RES,2 and the one for rMPDH which have the following

form:

rCRES,2 = rmax

RES,2

cCNAD+

KNAD,7An1,7−1 + L0,7

cCNAD+

K′NAD,7

Bn1,7−1

An1,7 + L0,7Bn1,7

cCGAPn2,7

KGAP,7 + cCGAPn2,7

, (6.26)

6.4 Symmetry 37

whereA = 1 +

cCNAD+

KNAD,7+

cCNADH

KNADH,7

B = 1 +cCNAD+

K′NAD,7

+cCNADH

K′NADH,7

(6.27)

and

rMPDH = rmax

PDHcCPyrc

MNAD+/(KNAD,13c

CPyr +KPyr,13c

MNAD+ + (6.28)

+KI−Pyr,13KNAD,13

KI−NADH,13

cMNADH + cCPyrcMNAD+ +

KNAD,13

KI−NADH,13

cCPyrcMNADH).

Observe that the concentrations CCNAD+ and CM

NAD are used in the rate equa-tions although no dierential equations are formulated for them in Rizzi'smodel. Instead, these are dened in (Rizzi et al., 1997) as:

cCNAD+ = p1 − cCNADH (6.29)cMNAD+ = p2 − cMNADH , (6.30)

where p1 and p2 are known constants.More results from the identiability analysis are shown in Appendix 8.4.

6.4 Symmetry

The results obtained from the Sedoglavic implementation are probabilistic- their validity must be ascertained by the actual nding of symmetries inthe model. From the theory described in Section 3, we know that it isnecessary to nd two derivations that each of them give rise to a symmetryin the model. The fact that the non-identiable parameters can be separatedinto two groups, each belonging to a rate equation, suggests the possibilitythat the symmetries may be found within each rate expression (since thekinetic parameters of the uxes rC

RES,2 and rMPDH are not used anywhere else

in the model). If this is true, the calculation of the symmetries may begreatly simplied - the formal procedure from Subsection 3.4 for the 11 non-identiable parameters can otherwise be rather cumbersome. In order toverify this hypothesis, we rst used Sedoglavic's algorithm on our controlsystem where we added measurements of the uxes rC

RES,2 and rMPDH to the

set of outputs: p = 0c = f(c, p, u)

y =

crCRES,2

rMPDH

.(6.31)


The idea behind this test is that if this system turns out to result in atranscendence degree of 2 and the same non-identiable parameters as before,then there exist two families of symmetries in these parameters that leaveboth c and rC

RES,2 and rMPDH invariant (see Subsection 3.4). This means that

the rate expressions for rCRES,2 and rM

PDH themselves have symmetries in theirkinetic parameters.

We tested the above system in the Sedoglavic implementation and foundour hypothesis to be true. The rate expressions for rC

RES,2 and rMPDH were

then analysed further to nd the symmetries in the parameters. The de-tailed analysis can be found in Paper I. We found the following families ofsymmetries (see Subsection 3.4) for rC

RES,2:

σλ :

rmaxRES,2

KNAD,7

K ′NAD,7

KNADH,7

K ′NADH,7

L0,7

→

rmaxRES,2p1

KNAD,7(1−λ−1/n1,7 )+p1

KNAD,7λ−1/n1,7p1

KNAD,7(1−λ−1/n1,7 )+p1

(K′NAD,7−KNAD,7(1−λ−1/n1,7 ))p1

KNAD,7(1−λ−1/n1,7 )+p1

KNADH,7λ−1/n1,7p1

KNADH,7(1−λ−1/n1,7 )+p1

(K′NAD,7−KNAD,7(1−λ−1/n1,7 ))p1

KNAD,7(1−λ−1/n1,7 )+K′

NAD,7

K′NADH,7

p1

λL0,7(K′NAD,7−KNAD,7(1−λ−1/n1,7 ))n1,7

Kn1,7NAD,7

(6.32)

and for rMPDH:

σλ :

rmaxPDH

KNAD,13

KPYR,13

KI−PYR,13

KI−NADH,13

→

λrmaxPDH

λKNAD,13 + p2(λ− 1)λKPYR,13

λKNAD,13KI−PYR,13

(1−λ)KI−NADH,13+λKNAD,13KI−NADH,13(λKNAD,13+p2(λ−1))

(1−λ)KI−NADH,13+λKNAD,13

.

(6.33)

We see that the constants p1 and p2 appear in the symmetries. In fact, itis exactly the equations (6.29) and (6.30) corresponding to the conservedmoiety assumptions in the model that cause the system to be unidentiable.This is discussed in detail in paper I.

We have now veried that the kinetic parameters rmaxRES,2, KNAD,7, K ′

NAD,7,KNADH,7, K ′

NADH,7, L0,7, rmaxPDH, KNAD,13, KPyr,13, KI−NADH,13, and KI−Pyr,13

cannot be identied from any experimental data. If parameter estimationis to be performed on Rizzi's model, for example by a numerical procedurewhere the error between model predictions and experimental results is min-imised, then one of the parameters in each of the groups rmax

RES,2, KNAD,7,

6.5 General features of kinetic models and identiability 39

K ′NAD,7, KNADH,7, K ′

NADH,7, L0,7 and rmaxPDH, KNAD,13, KPyr,13, KI−NADH,13,

KI−Pyr,13 must be xed to a value, while varying the rest of the parame-ters.

Remark: Using Sedoglavic's algorithm, we investigated the identiabil-ity of this model with other sets of outputs than the ones discussed above(some of the results are shown in Appendix 8.4). As well as including allpossible outputs - all concentrations cj and all uxes rj, we also tried tolimit the number of measurements by nding a smaller set of outputs whichproduced the same transcendence degree for the system. One such exampleis the set Ce

GLYC, CeAC, Ce

ETOH and CeCO2

. Measuring these concentrationsshould in theory (with perfect error-free measurements) produce the sameinformation on the parameter values as measuring all concentrations and alluxes. Sedoglavic's algorithm can thus be used in the practical planning ofan experiment for the purpose of parameter estimation.

6.5 General features of kinetic models and iden-tifiability

Kinetic models of metabolism can often be described by the following struc-ture, see Chapter 8 of the book on metabolic engineering by Stephanopouloset al. (1998):

cj = uj +∑

i νij · ri(c, pi)− µcj ∀j , (6.34)

for each metabolite j. The coecients νij are the stoichiometric coecientsassociated to each reaction.

In Rizzi's model, the unidentiable parameters can be separated into twogroups each associated to a single rate equation. The question is whetherthis is true for all non-identiable kinetic models of the above form. Wewill show that this is true when a model is not identiable from a full-statemeasurement, that is, from an output set y = c, as in the Rizzi case study.

The rst step in the construction of a kinetic model for cellular metabolismis usually formulating a network of uxes which, at steady-state, obey Kirch-ho's law at every node. This underlying steady-state model must be suchthat the stationary values of all uxes ri can be calculated uniquely from theset of linear equations for r obtained at steady-state. Mathematically, thismeans that the stoichiometric matrix [νij] is of full rank.

If we rewrite the rst equation in (6.34), we have:

cj − uj + µcj =∑

i νijri(c, pi) . (6.35)

Since y = c, all the quantities on the left-hand side are known, if we assumethat we know the value of the specic growth rate µ. The matrix [νij] is


assumed to be of full rank and we thus have unique values of ri(c, pi) com-pletely determined by a given set of inputs and outputs. This means that ifthe system is not identiable and there exist non-identiable parameters suchthat there is a symmetry in them leaving the inputs and outputs invariant,then this symmetry will also leave the ri:s invariant. Since each ux ri onlyinvolves the parameters in the subset pi of p, then any symmetry in p leavingri invariant must involve only the parameters in pi. In result, a kinetic modelof metabolism of the form (6.34) is not identiable only if some of the rateexpressions are symmetric in their respective kinetic parameters.

The fact that unidentiability can in such case be found locally in singlereaction rate expressions can be used to simplify the identiability analy-sis for metabolic models. This is discussed in detail in Paper I, where thesimplied analysis is applied to several well-cited kinetic models of glycoly-sis. The sources of unidentiable parameters in these models can be tracedto the introduction of conserved moiety assumptions. In Paper I, we de-velop a general method for determining whether a conserved moiety rendersa rate expression unidentiable, as well as a method for reparameterisationinto identiable parameters. The reparameterisation shows which combina-tions of the original parameters can be uniquely estimated from the data.We provide all symmetry transformations, which leave the output invariant.Furthermore, we show that identiable rate expressions are enough to ensureidentiability of the entire model, provided that a sucient set of measure-ments is available.

41

7 Discussion

The subject of this report is the problem of investigating the observabilityof nonlinear control systems, a special case of which is the identiabilityproblem. We have presented a review of two rather dierent theoreticalapproaches to the problem used in literature - the dierential-geometric ap-proach and the algebraic one. Each of the two approaches leads to a testfor the observability of a class of control systems. The dierential-geometricapproach covers analytical control systems of the form

Σ

x = f(x, u) = g0(x) + g(x)uy = h(x) ,

while the algebraic one treats rational systems:

Σ

x = f(x, u)y = h(x, u) .

Both approaches lead to the so-called observability rank test where the rankof the space spanned by gradients of the Lie-derivatives of the output func-tions is calculated. In the algebraic approach there is an upper bound derivedfor the number of Lie-derivatives that have to be considered in the test for ra-tional systems. In Section 4 we derive the same upper bound for the numberof Lie-derivatives of the output functions for the class of analytical systemsane in the input variables that are considered in the dierential-geometricapproach. It remains as future work to investigate the validity of such anupper bound for other classes of control systems.

The identiability problem is a special case of observability and we canthus use the previously derived rank test in the investigation of identiability.This has been done in a case study of a dynamic model of the metabolismof S. cerevisiae. By nding symmetries in the model, we show that certainmodel parameters cannot be identied from any set of experimental data.The results from the treatment of this model are generalised to show how thespecial structure of kinetic models of metabolism considerably simplify theanalysis of their identiability and especially the derivation of symmetries.We show that using conservation laws in metabolic modelling can have aneect on the parameter identiability of the models.

I

8 Appendix

8.1 Why do d and Lf commute when f depends ona control variable u?

This appendix contains some calculations that were deferred from Section 4.Consider a control system of the form:

Σ

x = f(x, u) = h0(x) + h(x)uy = g(x) ,

(8.1)

where the elements of the n-dimensional vectors h0 and h are analytic func-tions of x and u, where u is the single control variable. Denote, as in Section 2,the ow corresponding to the time-dependent vector eld f(x, u) by Φu(t, x)which is then an n dimensional vector. Let ψ(x, u) be an analytic functionof x,u and u's time derivatives. We will show that:

Lfdψ = dLfψ . (8.2)

Take the i-th element of dLfψ. It is:

∂

∂xi

Lfψ =∂

∂xi

( n∑j=1

fj∂ψ

∂xj

+∑l=0

∂ψ

∂u(l)U (l+1)

)=

=n∑

j=1

∂

∂xi

(fj∂ψ

∂xj

) +∑l=0

∂

∂xi

( ∂ψ

∂u(l)U (l+1)

)=

=n∑

j=1

∂fj

∂xi

∂ψ

∂xj

+n∑

j=1

∂2ψ

∂xi∂xj

fj +∑l=0

∂

∂xi

( ∂ψ

∂u(l)

)U (l+1) .

Now consider the i-th element of the covector Lfdψ. By the denition of Liederivative we nd,

d

dt

(∂ψ(Φu(t, x), u(t))

)∂xi

)∣∣∣∣t=0

=d

dt

( n∑j=1

∂ψ

∂Φu,j

∂Φu,j

∂xi

)∣∣∣∣t=0

=

=n∑

j=1

d

dt

( ∂ψ

∂Φu,j

∂Φu,j

∂xi

)∣∣∣∣t=0

=

=n∑

j=1

(d

dt

( ∂ψ

∂Φu,j

)∣∣∣∣t=0

∂Φu,j

∂xi

∣∣∣∣t=0

+d

dt

(∂Φu,j

∂xi

)∣∣∣∣t=0

∂ψ

∂Φu,j

∣∣∣∣t=0

)=

II 8 APPENDIX

=n∑

j=1

(( n∑k=1

∂

∂Φu,k

(∂ψ

∂Φu,j

)

∣∣∣∣t=0

dΦu,k

dt

∣∣∣∣t=0

+

+∑l=0

∂

∂u(l)(∂ψ

∂Φu,j

)

∣∣∣∣t=0

U (l+1)))δij +

∂ψ

∂xj

∂fj

xi

)=

=n∑

j=1

(( n∑k=1

∂2ψ

∂xk∂xj

fk +∑l=0

∂

∂u(l)(∂ψ

∂xj

)U (l+1))δij +

∂ψ

∂xj

∂fj

xi

)=

=n∑

k=1

∂2ψ

∂xk∂xi

fk +∑l=0

∂

∂u(l)(∂ψ

∂xi

)U (l+1) +n∑

j=1

∂ψ

∂xj

∂fj

xi

.

One can now see that the i-th elements of the covectors Lfdψ and dLfψ arethe same, which shows the above equality.

8.2 A basis for derivations on R〈U〉(x) that aretrivial on R〈U〉

This appendix refers to Section 3.4 and is based on pp.371-372 in Lang (1993).First of all, observe that the xi form a transcendence basis for R〈U〉(x)

over R〈U〉. The transcendence degree of the eld extension is thus n whichis also the dimension of DerR〈U〉R〈U〉(x). Consider the n derivations Di =∂

∂xi. Clearly, Dixj = δij. Let D be a derivation in DerR〈U〉R〈U〉(x) and let

Dxi = wi (a derivation is dened by its action on a set of generators of thetranscendence basis). Then D =

∑iwiDi and thus the Di:s form a basis for

DerR〈U〉R〈U〉(x). Therefore the partial derivatives ∂∂xi

form a basis for thederivations on R〈U〉(x) that are trivial on R〈U〉.

8.3 Nomenclature for Rizzi's model

This appendix refers to Section 6.For the details of the kinetic model by Rizzi et al we refer to (Rizzi et al.,

1997) but in an attempt to make this report somewhat self-sucient we hereprovide nomenclature for the abbreviations and parameters used in Section 6.

8.3.1 Superscripts

e extracellularC cytoplasmicM mitochondrial

8.3 Nomenclature for Rizzi's model III

8.3.2 Symbols and abbreviations

aj stoichiometric coecientSG gas supply rateVM volume of mitochondriaVC volume of cytoplasmD dilution rateρ specic volumeµ specic growth ratecX biomass concentration

mATP maintenance coecient for ATP

8.3.3 Metabolites

AC acetic acidADP adenosine diphosphateALDE acetaldehydeAMP adenosine monophosphateATP adenosine triphosphateDHAP dihydroxyacetone phosphateETOH ethanolFBP fructose 1,6-bisphosphateF6P fructose 6-phosphateGAP glyceraldehyde 3-phosphateGLC glucoseGLYC glycerolG6P glucose 6-phosphateNAD+/NADH nicotinamide adenine dinucleotidePEP phosphoenol pyruvatePYR pyruvate

IV 8 APPENDIX

8.3.4 Enzymes and flux indexes

ACETYL acetate synthetaseADH alcohol dehydrogenaseADK adenylate kinase

ALDH acetaldehyde dehydrogenaseALDO fructose bisphosphate aldolaseATP,R ATP formation via respiratory chainATP,T ATP formation via the citric acid cycle

HK hexokinaseNADHDH NADH-dehydrogenaseNADHQR NADH-Q-reductase

PDC pyruvate decarboxylasePDH pyruvate dehydrogenase

PERM hexose transporterPFK phosphofructo-1-kinasePGI phosphoglucose isomerasePK pyruvate kinase

RES,1 combination of glycerol-3-phosphatedehydrogenase and glycerol-3-phosphatase

RES,2 combination of glyceraldehyde 3-phosphatedehydrogenase and other enzymes

SYNT,1;SYNT,2 resulting rates for the formation ofmonomeric building blocks

TIS triose phosphate isomeraseTR,ADP and TR,ATP translocases for ADP and ATP respectively.

8.4 Other results on the identifiability of Rizzi'smodel

This appendix shows some of the results obtained from the identiabilityanalysis of Rizzi's model. They can be used in choosing a set of measurementsin a hypothetical experiment.

Since the parameters cX and ρ always appear together as cXρone of them

must be known - otherwise it is clear that they will not be identiable. Wetherefore assume that the value of one of them is measured in any hypothet-ical experiment when we perform the identiability analysis, although we donot explicitly write them as outputs. The same applies for the parametersVM and VC appearing in VM

VC.

8.4 Other results on the identiability of Rizzi's model V

Outputs Transcendence degreeceGLYC, c

eAC, c

eETOH, c

eCO2

2ceGLYC, c

eAC, c

eETOH, c

eCO2

, rmaxRES,2, r

maxPDH 0

all cj:s 2all cj:s, rC

RES,2, rMPDH 2

all cj:s, all rj:s 2

One conclusion is that measuring ceGLYC, ceAC, ceETOH and ceCO2produces

as much information for the theoretical identication of parameter values asmaking all possible measurements altogether.

VI REFERENCES

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Paper I

Conservation laws and unidentifiability of rateexpressions in biochemical models

M. Anguelova, G. Cedersund, M. Johansson, C.J. Franzen and B. Wennberg

Abstract: New experimental techniques in bioscience provide us with high-quality data allowingquantitative mathematical modelling. Parameter estimation is often necessary and, in connectionwith this, it is important to know whether all parameters can be uniquely estimated from availabledata, (i.e. whether the model is identifiable). Dealing essentially with models for metabolism, weshow how the assumption of an algebraic relation between concentrations may cause parameters tobe unidentifiable. If a sufficient data set is available, the problem with unidentifiability ariseslocally in individual rate expressions. A general method for reparameterisation to identifiablerate expressions is provided, together with a Mathematica code to help with the calculations.The general results are exemplified by four well-cited models for glycolysis.

1 Introduction

Throughout the years, a large number of mathematicalmodels have been published that describe smaller or largerparts of the system of chemical reactions in living cells.Such models usually consist of systems of ordinary differen-tial equations describing the reaction rates, and algebraicrelations describing, for example conserved moieties.Models of this kind naturally contain a large number of

parameters that must be determined before the model canbe used for simulation. For example, in a simple reactionof Michaelis–Menten type, the reaction rate is given by

r ¼VmaxcSKM þ cS

where cS denotes the concentration of a substrate, and Vmax

and KM are two parameters. The parameter values may beknown, but in other cases they must be determinedthrough parameter estimation: by comparing simulatedvalues for some concentrations with an experimentallyobtained time series, it may be possible to determine aunique set of parameter values.On the other hand, infinitely many parameter sets may

produce exactly the same simulated values, and henceagree equally well with experimental data. The problemof determining a priori whether the parameters of a modelcan be uniquely determined from a given experiment ispart of the general theory of identifiability analysis, and isthe subject matter of this paper.

More precisely, we consider a family of kinetic modelsfor cell metabolism, where some of the dynamic mass bal-ances are replaced by algebraic relations. Although theresults are not restricted to this case, we focus on exampleswhere the algebraic relations express the conservation of[NADþ]þ [NADH]. This is a commonly used assumptionin biochemical modelling and is only valid under short-termstudies. In long-term studies, the synthesis or degradation ofthe components must be accounted for. We study three well-cited models of yeast metabolism as a in detail [1–3]. Somesets of parameters in these models are unidentifiable as aresult of conserved [NADþ]þ [NADH] moiety.That such assumptions may lead to unidentifiability of par-

ameters in some rate expressions is not new to the mathemat-ical modelling community. A tool of the trade is to recover anidentifiable rate expression by non-dimensionalisation or‘divide-through’. However, neither a mathematically stringentanalysis of the reasons for such unidentifiability, nor a generalmethod for treating the problem seems to have been published.We provide a mathematical framework to allow for such

unified treatment of a large class of rate expressions, includingthe particular examples from [1–3]. Thus, we develop ageneral method for determining whether a conserved moietymay render a rate expression unidentifiable, and a method forreparameterisation into identifiable parameters. The reparame-terisation showswhich combinations of the original parametersmay be uniquely estimated from the data. We provide all sym-metry transformations, which leave the output invariant.Furthermore, we show that identifiable rate expressions areenough to ensure identifiability of the entire model, providedthat a sufficient set of measurements is available. In otherwords, all sources of structural unidentifiability can in suchcase be found locally in single reaction rate expression.The general method is well adapted for implementation

in symbolic programming languages, and we provide aMathematica notebook that can be used for analysing aclass of rate expressions, including all rational ones.

2 Identifiability of kinetic models

Kinetic models of metabolism are usually systems of differ-ential equations, representing mass balances for each

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-syb:20060081

Paper first received 13th March 2006 and in revised form 9th May 2007

M. Anguelova and B. Wennberg are with Mathematical Sciences, ChalmersUniversity of Technology, Gothenburg SE-412 96, Sweden and also with theMathematical Sciences, Goteborg University, Gothenburg SE-41296, Sweden

G. Cedersund is with the Department of Cell Biology, Linkoping University,Linkoping SE-581 85, Sweden

M. Johansson and C. J. Franzen are with Chemical and Biological Engineering,Chemical Reaction Engineering, Chalmers University of Technology,Gothenburg SE-412 96, Sweden

E-mail: [email protected]

IET Syst. Biol., 2007, 1, (4), pp. 230–237230

metabolite j [4]. They can normally be written in the form

dcj

dt¼

Xinji ri(c, k) (1)

where cj denotes the intracellular concentration of metab-olite j, and c is the vector containing all such concentrations.The symbol ri denotes the rate of reaction i, nji denotes thestoichiometric coefficient of metabolite j in reaction i and kdenotes the kinetic parameters in the rate expressions of themodel.The property of structural identifiability guarantees the

uniqueness of the parameters k for a given input–outputstructure corresponding to a set of measurements for thepurpose of parameter estimation. This set of measurementsusually consists of some metabolic concentrations and/orfluxes or combinations of these, denoted by the vector y.The latter can then be written as a vector-valued functiong of the variables in c and the parameters k. Equation (1)together with

y ¼ g(c, k) (2)

defines the input–output structure.For realistic situations, sources of unidentifiability can be

an insufficient number of measurements as well as a struc-tural property of the model, which is independent of theinput–output structure and renders it unidentifiable evenfrom a perfect data set. In this paper, we focus on thelatter and analyse structural identifiability when a perfectdata set is available. Clearly, if a model is unidentifiablefor such an input–output structure, it will not be identifiablefor any realistic measurement set-ups. On the other hand, itshould be noted that measuring a single (noise free) meta-bolic flux can be enough for the identification of all the par-ameters of an identifiable model.A perfect data set for a model with a full stoichiometric

matrix can be attained by measurements of all metabolicconcentrations, as shown by the following calculation.Suppose that a model is not identifiable for an input–outputstructure of the form

dcj

dt¼

Pi nji ri(c, k)

y ¼ c

((3)

that is, there are at least two sets of parameters, k and K,which produce the same output [the same solution c(t) tothe system of differential equations above with giveninitial conditions]. This can be expressed as

Xinji ri(c, k) ¼

dcj

dt¼

Xinji ri(c, K) (4)

If the underlying steady-state model has a stoichiometricmatrix (the matrix with elements nji) of full rank, we musthave ri (c, k) ¼ ri (c, K) for all the rates ri. Thus, measure-ments of metabolic fluxes cannot be used to distinguishbetween the two sets k and K. Consequently, no amountof additional flux measurements will suffice to make (3)identifiable. If the stoichiometric matrix is not of fullrank, then the perfect output set must also contain measure-ments of some metabolic fluxes ri.If a model is not identifiable from a perfect output set,

then it must contain some over-parameterised rate expres-sion, i.e. the reason for the unidentifiability can be tracedback to isolated rate expressions. Indeed, since the para-meter sets k and K are by assumption different, theequalities ri(c, k) ¼ ri(c, K) imply that there is at least oneri with a rate expression that has the same value for different

combinations of its kinetic parameters. This means thatstructural unidentifiability has its source within individualkinetic expressions.Consider first a simple example taken from Segel [5]

r ¼VmaxcS

KS(1þ (cP=KP))þ cS(5)

where cS and cP are the concentrations of the substrate andproduct respectively. The expression contains three par-ameters: Vmax, KS, and KP. By measuring r(t‘), cS (t‘),and cP (t‘) at three different times t‘, one finds threeequations for the three unknowns, if the dynamic variablesare varying sufficiently independently, and the problem canin principle be solved.However, if there are algebraic constraints, it may be

impossible to find a sufficient number of independentequations for the unknown rate parameters. In the Segelexample, if cSþ cP ¼ a, where a is a constant, then weobtain

r ¼VmaxcS

KS(1þ (ða cSÞ=KP))þ cS

¼cS

(KS[1þ (a=KP)]=Vmax)þ ([1 (KS=KP)]=Vmax)cS

¼cS

k1 þ k2cS(6)

where

k1 ¼KS[1þ (a=KP)]

Vmax

and k2 ¼[1 (KS=KP)]

Vmax

(7)

It now becomes clear that the original three parametersKS, KP and Vmax are effectively only two independentones. One can find infinitely many combinations of KS,KP and Vmax that result in exactly the same rate expressionvalue. For example, if one multiplies Vmax by any constant land then adjusts the other two parameters according to thescheme below, the value of the rate expression does notchange at all

Vnewmax ¼ lVmax (8a)

KnewS ¼ lKS a(1 l) (8b)

KnewP ¼

lKS a(1 l)

1 lþ l(KS=KP)(8c)

In other words, there is a one-parameter family of trans-formations of the set of parameters, which leave the reactionrate invariant. Consequently, if such a rate equation isincluded in a metabolic model, not all of its kinetic par-ameters will be identifiable from experimental data. In(6), one can identify only the parameter combinations k1and k2, and before attempting to fit a model to experimentaldata, one should first rewrite the model in an identifiableform.We note here that algebraic constraints may be intro-

duced explicitly in the model, or may be hidden in the for-mulation of (1). When this is the case, the stoichiometricmatrix is not of full rank.The theory of identifiability is well studied (see for

example [6–14] and the references therein), and tools forautomated identifiability analysis are available. They aregenerally based on differential algebraic techniques, andthe computational time grows exponentially with thenumber of variables and parameters [9, 11–13]. However,a probabilistic algorithm that is also useful for largesystems has been constructed by Sedoglavic [14]. The

IET Syst. Biol., Vol. 1, No. 4, July 2007 231

methods have been applied to models for yeast glycolysis asdiscussed, for example in [15].There are also published algorithms that efficiently look

for conserved quantities in models of the form (1), arecent example being [16]. Three of the models used asexamples in this paper are accessible in the JWS Onlinedatabase [17], where models can be searched for dependen-cies for the variables, (i.e. conserved moieties).A final comment here is that methods for practical iden-

tifiability must also handle noise and other measurementlimitations, and that the actual parameter estimationusually is based on some kind of maximum likelihoodoptimisation [18].

3 Examples

The main biochemical interactions of glycolysis werecharacterised over 100 years ago. Some of the earlier mod-elling attempts occurred in the 1960s [19–21], but thesewere mostly minimal models trying to explain, forexample, the temporal oscillations found in 1957 [22].Since then the models have grown in size and comprehen-sion, and there have been several attempts at constructingquantitative models. Examples are found, for instance, inthe work of Rizzi et al. [1], Teusink et al. [2], Hynneet al. [3] and Lambeth et al. [23].It is from [1, 2] and [3] that we take examples of reaction

rate expressions which are unidentifiable because of a con-served pool of, for instance, NADHþ NAD þ andATPþADPþ AMP. If the model parameters are to beidentified, these rate expressions should be reparameterised.Analysing these models with the method of Sedoglavic

[14] would give a list of unidentifiable parameters.However, it will become apparent that in these cases theproblem of unidentifiability is located to individual reactionrate expressions. Therefore, the methods explained inSection 4 can be used directly on each and every rateexpression and would give all the needed information.In this section, we discuss the unidentifiable reaction rate

expressions from the models in [1, 2] and [3] and propose ageneral form for reparameterisation. For completeness, wealso give examples of similar rate expressions from [23],where the introduction of a conserved quantity does notlead to unidentifiability.For the four models above, all instances of rate

expressions where the variables from a conservation lawappear together have been analysed. The results are statedbelow and the details of the treatment are provided in thesupplementary material available online.

3.1 Rizzi model

The first example is a kinetic model of the central metabolicpathways in Saccharomyces cerevisiae formulated in [1].The model includes the reactions of glycolysis, the citricacid cycle, the glyoxylate cycle and the respiratory chainand a simplified description of biosynthesis in growingcells. Subcellular localisation of reactions and metabolitesare taken into account by including cytosolic and mitochon-drial compartments. We refer to [1] for the complete modeldescription and focus here only on the aspects that are rel-evant for our analysis.The concentrations of cytoplasmic and mitochondrial

NADþ are calculated under the assumption that the totalconcentrations of nicotinamide nucleotides are constant ineach compartment. This results in the following equations

for cytoplasmic and mitochondrial NADþ

cCNADþ ¼ p1 cCNADH (9a)

cMNADþ ¼ p2 cMNADH (9b)

where p1 and p2 denote the constant total concentrations ofnicotinamide nucleotides in the cytoplasm and mitochon-dria, respectively.There are two reaction rates in this model that contain

unidentifiable parameters: the pyruvate dehydrogenase(PDH) reaction, and the lumped reactions (RES2) from gly-ceraldehyde 3-phosphate to phosphoenolpyruvate; these arethe only two kinetic expressions that contain all variablesfrom the conserved moieties.The pyruvate dehydrogenase complex catalyses the reac-

tion

Pyruvateþ CoAþ NADþ! AcCoAþ CO2

þ NADHþ Hþ (10)

In the Rizzi model, its rate equation is formulated

rMPDH ¼

rmaxPDHc

CPyrc

MNADþ

KNAD,13cCPyr þ KPyr,13c

MNADþ

þ(KIPyr, 13KNAD,13)

(KINADH,13)

cMNADH þ cCPyrc

MNADþ

þKNAD,13

ðKINADH,13)

cCPyrc

MNADH

0BBBBB@

1CCCCCA(11)

where superscripts C and M refer to cytoplasmic and mito-chondrial concentrations, respectively. Inserting (9b) into(11) gives

rMPDH ¼rmaxPDHc

CPyr(p2c

MNADH)

(KNAD,13cCPYRþKPYR,13(p2 c

MNADH)

þðKIPYR,13KNAD,13Þ

ðKINADH,13Þ

cMNADHþ cCPyr(p2 cMNADH)

þKNAD,13

ðKINADH,13ÞcCPyrc

MNADH

0BBBBB@

1CCCCCA

(12)

After rearranging terms, the right hand side becomes

cCPyr(p2c

MNADH)

(p2KPYR,13=rmaxPDH)þ ((KNAD,13þp2)=r

maxPDH)c

CPyr

þ[(KIPyr,13KNAD,13)=(KINADH,13)]KPyr,13

rmaxPDH

cMNADH

þ[KNAD,13=(KINADH,13)]1

rmaxPDH

cCPyrc

MNADH

0BBBBB@

1CCCCCA(13)

Thus, (11) can be rewritten as

rMPDH ¼

cCPyr(p2 cMNADH)

k1þ k2cCPyrþk3c

MNADHþk4c

CPyrc

MNADH

(14)

The five original parameters are thus combined into fournew parameters, that can be identified. Solving for rmax

PDH,KNAD,13, KPyr,13, KI2NADH,13 and KI2Pyr,13, in terms of k1,k2, k3 and k4, result in a one parameter family of solutions.The RES2 reaction is a lumped reaction of the glycolytic

steps from glyceraldehyde 3-phosphate (GAP) to

IET Syst. Biol., Vol. 1, No. 4, July 2007232

phosphoenolpyruvate, with the rate equation

rCRES,2 ¼ rmaxRES,2

(cCNADþ=KNAD,7)An1,71

þ L0,7(cCNADþ=K 0

NAD,7)Bn1,71

An1,7 þ L0,7Bn1,7

cCGAP

n2,7

KGAP,7 þ cCGAPn2,7

(15)

where

A ¼ 1þcCNADþ

KNAD,7

þcCNADHKNADH,7

B ¼ 1þcCNADþ

K 0NAD,7

þcCNADHK 0NADH,7

ð16Þ

This expression differs from the other reaction rates con-sidered in this paper in that it is not a rational expression.However, it can be expressed as a (homogeneous) functionincluding the rational expression A/B, and therefore thisreaction can be handled in essentially the same way.The conserved moiety, cC

NADþ ¼ p1 cCNADH appears onlyin the first part of the expression, and this can be reparame-terised in the following way

rmaxRES,2

cCNADþ

KNAD,7

An1,71þL0,7

cCNADþ

K 0NAD,7

Bn1,71

An1,7þL0,7Bn1,7

¼

(p1cCNADH) 1þk3

(1þk4cCNADH)"

k5k3

þ

k2k5k1k3

cCNADH

#8>>>><>>>>:

9>>>>=>>>>;

n1,710BBBB@

1CCCCA

(k1þk2cCNADH) 1þ

ð1þk4cCNADHÞ"

k5k3

þ

k2k5k1k3

cCNADH

#8>>>><>>>>:

9>>>>=>>>>;

n1,70BBBB@

1CCCCA(17)

where we set k1 ¼ (KNAD,7 þ p1)=rmaxRES,2, k2 ¼ (KNAD,7

KNADH,7)=KNADH,7rmaxRES,2, k3 ¼ ðKNAD,7=K

0NAD,7Þ L

1n

0,7, k4 ¼

(K 0NAD,7 K

0NAD,7)=K

0NADH,7(K

0NAD,7 þ p1) and k5 ¼ (KNAD,7

þp1)=K0NAD,7 þ p1: The original six kinetic parameters in

this part of the rate equation can thus be replaced by thefive independent ones, to k1–k5. The combinations of theoriginal parameters that they represent are identifiable.By using the Sedoglavic algorithm [14], it was confirmed

that the assumption of conserved moieties of nicotinamidenucleotides was the only source of structural unidentifiabil-ity for the Rizzi model.

3.2 Hynne, Teusink and Lambeth models

The Hynne model, originally presented in [3], describes gly-colytic oscillations in S. cerevisiae during cultivation in acontinuous stirred tank reactor. Two conserved moietiesare implicitly present in its stoichiometric matrix, one forthe nicotinamide nucleotides and one for the adeninenucleotides. It is only the first that leads to unidentifiability.The rate expressions for GAPDH and lpGlyc can both bereparameterised in fewer parameters, as shown in theSupplementary material. In the rate expression for the AK

reaction, all the variables from the adenine nucleotidemoiety appear together, but its kinetic parameters are never-theless identifiable.The Teusink model [2] also describes yeast glycolysis,

and just like the Rizzi model, it has two conserved moietiesdescribed by cNADHþ cNADþ ¼ p1 and cATPþ cADPþcAMP ¼ p2. There is no kinetic expression containing allthree adenine nucleotides, and thus the second assumptiondoes not lead to unidentifiability. However, the three dehy-drogenase reactions GAPDH, ADH and G3PDH, whichinclude both cNADH and cNADþ in their rate expressions,were all found to be unidentifiable. The calculation iscarried out in the Supplementary material, where also thereparametrisation into identifiable parameters is given. Forthis model, there are two additional conservation relationsbecause of equilibrium assumptions for [GraP] and [glycer-onephosphate] and for the adenine nucleotides. The vari-ables in the first equilibrium assumption, [GraP] and[glyceronephosphate], both appear in the ALD reactionrate expression that was therefore analysed for a potentialsource of unidentifiability. By applying the generalmethod presented in the next section, it was found that thekinetic parameters in the ALD rate expression were identifi-able even after introducing the equilibrium assumption.The last model that has been tested describes muscle

metabolism, and was originally presented by Lambethet al. [23]. It contains the same conserved moieties as theprevious models implicitly present in the stoichiometricmatrix. The two dehydrogenase reactions included,GAPDH and LDH, are modelled by reversible kineticexpressions that include both cNADþ and cNADH, and theycould potentially have the reported problem with uniden-tifiability. So could the PFK and the ADK reactions, astheir rate expressions include all the variables from theadenine nucleotide moiety. However, by applying thegeneral method, we were able to see that even after introdu-cing the conserved moieties, all parameters in these rateexpressions remained identifiable.

4 Unidentifiability as a result of conservedquantities: a linear algebra formalism

Rate expressions like the ones considered here are oftenrational functions of a number of concentrations. What actu-ally happens in the examples discussed in the previoussection is that the presence of a conserved quantityreduces the number of independent coefficients in therational expression. It is possible to compute an upperbound of the number of independent (and thus identifiable)rate parameters in terms of, for example, the number of con-centrations and the degree of the terms in which they appearin the rate expression. In this section, we formalise this pro-cedure, giving a formula for determining cases of uniden-tifiability and showing how to obtain an identifiable set ofparameters with a known relation to the original parameters.We also present a method for constructing the group oftransformations of the original parameters that leaves therate expression invariant.

4.1 Initial example revisited

To introduce the notation, we first demonstrate the pro-cedure on the example of (5). Recall that there were threeparameters, Vmax, KS and KP, and that these were collectedin a vector k

k ¼ Vmax KS KP

TIET Syst. Biol., Vol. 1, No. 4, July 2007 233

Introduce the following two symbols ci, and the followingfour symbols aij and bij

c1 ¼ cS, , c2 ¼ cP, a10 ¼ Vmax ,

b00 ¼ KS , b10 ¼ 1 , b01 ¼KS

KP

(18)

With this notation, (5) becomes

r ¼a10c1

b00 þ b10c1 þ b01c2(19)

This is a rational expression, being the quotient of poly-nomials in the two variables c1 and c2. Collect the coeffi-cients in a vector a

a ¼ a10 b00 b10 b01 T

(20)

Note that even if c1 and c2 were independent, we could onlyestimate a up to a multiplicative factor l0 [ Rn0, sincer(a) ¼ r(l0a). Using (18), we could, of course, directlyeliminate l0, but it is convenient to keep it, because itreveals the essentially linear structure of the coming trans-formations. At this point, we see that by inserting therelation c1þ c2 ¼ p in (19) we obtain

~r ¼~a1c1

~b0 þ~b1c1

(21)

where ~a1 ¼ a10,~b0 ¼ b00 þ pb01 and ~b1 ¼ b10 b01. This

is a rational expression of only one variable, c1, and withthree coefficients. The relation between the coefficients in(19) and (21) is linear and is most naturally expressed by

~a1~b0~b1

264

375 ¼

1 0 0 0

0 1 0 p

0 0 1 1

24

35

a10b00b10b01

2664

3775 (22)

which can also be written as

~a ¼ Aa: (23)

Note that the linearity of this transformation is not associ-ated with a property of the underlying dynamical system,but only of the relation between the coefficients a and ~a.There is therefore no approximation associated with thistransformation, as is the case, for example, for a linearisa-tion of a dynamical system.The rank of A shows the number of coefficients ~a that are

independently mapped from a. In this example, A is of full(row)rank. However, there is still a common factor thatneeds to be removed. Let the identifiable parameters bedenoted ~k. By dividing all coefficients ~a by ~a1, we obtainthe following identifiable parameters

~k ¼ ~k1~k2

h i¼

KS(1þ ðp=KPÞ)

Vmax

(1 ðKS=KPÞ)

Vmax

(24)

Note that this is the same result we obtained in (6) inSection 2. Finally, we now know that the conservationlaw has introduced an unidentifiability since the resultingnumber of parameters is less than the original number ofparameters (i.e. 2 , 3).Equation (24) expresses ~k as a function of k; we now turn

to the opposite problem, i.e. expressing k as a function of ~k.Let a superscript 0 denote a specific estimate, and consider aspecific estimate of the identifiable parameters ~k0. The first

step is from ~k0 to ~a0; in this example it is simply given by

~a0 ¼ [~a01~b00

~b01]T¼ [l0 l0

~k1 l0~k2]

T (25)

where l0 [ R n f0g is an unknown scalar. Going from ~a0 toa0, we use that A is of full rank. This means that there is a

vector a0, which satisfies (23), for all estimates ~a0. One

solution is a00 ¼ [~a01

~b00~b01 0]T; however, this is not

unique because A has a one-dimensional null-space, ker(A),spanned by w1 ¼ [0 p 1 1]T. The freedom tochoose a multiplicative factor l0 remains, and hence forany l0, l1 [ R,

a ¼ a(l0, l1) ¼ l0a00 þ l1w1 (26)

yield a set of coefficients that is consistent with theestimates ~a0. To translate all the way to the originalkinetic parameters k, we need to solve (18). In thisexample, we can explicitly obtain

Vmax ¼ l0 ~a01 (27a)

KS ¼ l0~b00 l1p (27b)

1 ¼ l0~b01 þ l1 (27c)

KP ¼l0

~b00 l1p

l1: (27d)

Note that there is only one real degree of freedom since, forexample, the third of these equations can be used to fix l0,

i.e. l0 ¼ (1 l1)=~b0

1, which gives

Vmax ¼ (1 l1)~a01=~b0

1, KS ¼ (1 l1)~b0

0=~b0

1 l1p

KP ¼ (1=l1 1)~b0

0=~b0

1 p, (28)

and finally

Vmax ¼ ð1 l1Þ=~k2, KS ¼ ((1 l1)

~k1)=~k2 l1p

KP ¼ ((1=l1 1)~k1)=~k2 p: (29)

Let us now see how all these calculations can be general-ised, following the same procedure as summarised in thisdiagram

4.2 Reparameterisation into identifiableparameters

Consider a reaction rate r depending on N concentrations c1,. . . , cN and m parameters k1, . . . , km, and assume that it canbe written as a rational function,

r ¼P(c1, . . . , cN ; k1, . . . , km)

Q(c1, . . . , cN ; k1, . . . , km)(31)

where P and Q are polynomials in the concentrationsc1, . . . , cN


P ¼XnPj¼0

Xr1þþrN¼j

ar1... rN cr11 ; . . . ; c

rNN (32)

Q ¼XnQj¼0

Xr1þþrN¼j

br1...rN cr11 ; . . . ; c

rNN (33)

A general polynomial P in three concentrations, c1, c2, andc3, of second degree would with this notation be written

P ¼ a000

þ a100c1 þ a010c2 þ a001c3

þ a200c21 þ a020c

22 þ a002c

23 þ a110c1c2

þ a101c1c3 þ a011c2c3 (34)

Note that the sum of the indices of a coefficient ar1, . . . , rNis equal to the degree of the corresponding term, and thateach coefficient can be a function of the m original kineticparameters k1, . . . , km. We denote the transformation,which gives the coefficients ar1, . . . , rN and br1, . . . , rN asfunctions of the parameters by G, i.e.

G(k1, . . . , km) ¼ ( . . . , ar1,...,rN , . . . , br1,...,rN , . . . )T (35)

so that G is a function from the m-dimensional parameterspace to the M-dimensional space of rational coefficients.We assume here that G parameterises an m-dimensionalmanifold, because otherwise the parameters would be uni-dentifiable in the original rate expression.If there is a constant linear combination of the concen-

trations Xajcj ¼ p (36)

one of the concentrations cj can be eliminated from therational expression. With no loss of generality, we mayassume that the eliminated concentration is cN. Replacingall occurrences of cN in (31) withp

Pj,N ajcj gives a new rate expression

~rX ¼~P(c1, . . . , cN1; k1, . . . , km)

~Q(c1, . . . , cN1; k1, . . . , km)(37)

The new set of coefficients ~ar1...rN1and ~br1...rN1

are linearcombinations of the coefficients ar1...rN and br1...rN , becauseeach monomial containing cN is transformed into a poly-nomial without cN, which is added to those terms of P (orQ) that do not contain cN. This relation can again beexpressed as

a ¼ Aa (38)

If the matrix A is not of full rank, as in the rADH examplefrom the Teusink model (cf. the Supplementary material),one can proceed by choosing a set of independent rowsfrom A.It follows that the number of identifiable parameters is

given by the rank of (A2 1), and we have the following for-mulas for calculating a lower bound on the number of uni-dentifiable parameters.

1. [dim k] 2[rank A]þ 1 (e.g. 3 2 3þ 1 ¼ 1), or2. [dim k]2 [dim a]þ [dim ker(A)]þ 1 (e.g. 32 4þ 1þ 1 ¼ 1).

The numbers within the parenthesis refer to the example inSection 1. Since a non-zero number of unidentifiable par-ameters implies an unidentifiable rate expression, the

above formulas are a simple way of detecting unidentifiablerate expressions.

4.3 Back-translation and calculation of thesymmetry families

The first step in the back-translation is from k to a. Consider

a specific estimate k0. Assume, without loss of generality,that it is the first coefficient ~a1 that was used for the normal-isation to the identifiable parameters. We then have the fol-lowing back-translation formula

a0 ¼ ~a01 a02 . . .

¼ l0 1 ~a2(k0) . . .

h i(39)

where ~ai(k0) is equal to a single ~k

0

j if ai was one of the

non-eliminated rows, and equal to the linear combinationsmentioned above otherwise (these linear combinationsonly occur if A is of less than full rank).The next step is to translate from a to a. Let J ¼ dim

ker(A) denotes the dimension of the null space of A, andlet w1, . . . ,wJ be a set of vectors that span the null space.Let a specific solution of (23) be given by a0 (such a sol-ution always exists). Then the set of all a that are consistentwith k0 is spanned by l0 [ R0 and li [ R according tothe following formula

a ¼ l0a0þ l1w1 þ þ lJwJ (40)

The final step in the back-translation is from a to k. This isthe reversal of the nonlinear mapping G, and it is thereforedifficult to treat in the general case. Nevertheless, in manycases, explicit solutions should be available through sym-bolic software packages such as Mathematica and Maple.However, because of these difficulties, we now alsopresent a geometrical interpretation of these results, andan alternative way of calculating the symmetry transform-ations, i.e. those transformations of k that leave the rateexpression v invariant.Recall that the function G parameterises an m-dimensional

submanifold ofRM (whichwe denote byG aswell). Ifwewishto determine whether a parameter set k0 ¼ (k01 , . . . , k

0m) is

uniquely defined or whether there is a family of parametersk(s) that give the same rate expression, we compute thedifferential G0(k0), and we let u1(k

0), . . . , uj(k0) be a set

of vectors so that G0(k0)u1(k0), . . . , G0(k0)uj(k

0) span theintersection of the range of G0(k0) and la0, l [ R<

ker(A). The number of vectors u(k0) is the dimension ofthe set of equivalent parameter combinations. With a properchoice of the vectors ui (in particular, they should be continu-ous as functions of k), the family of transformations can befound by solving

dk(s)

ds¼ ui½k(s)

k(0) ¼ k0

The original parameters that are identifiable correspond tothose components that are zero in all vectors ui, i ¼ 1, . . . ,j. This calculation is similar to the one in [11].


For the example shown in Section 4.1, the matrix G0 isgiven by

G0¼

@a01@Vmax

@a01@KS

@a01@KP

@b00@Vmax

@b00@KS

@b00@KP

@b10@Vmax

@b10@KS

@b10@KP

@b01@Vmax

@b01@KS

@b01@KP

2666666666664

3777777777775¼

1 0 0

0 1 0

0 0 0

0 1=KP KS=K2P

26664

37775

(41)

and the degree of freedom for the choice of parameters isalso given by the dimension of the intersection of therange of G0 with spana00,w1. Again this gives a one-dimensional set, which is spanned by w> ¼ a0 b10w1.The one-dimensional set of parameters that are consistentwith a given set of measurements can be obtained bysolving the ordinary differential equation

d[Vmax, KS, KP]T

ds¼ (G0)1w> (42)

where w> depends on Vmax, KS and KP, and (G0)21 can becomputed because w> is in the range of G0.The Supplementary material contains a Mathematica

implementation of the calculations both for the identifi-cation of the identifiable parameters and for the calculationof the symmetry relations. The Mathematica implemen-tation includes all the reaction rate expressions that wehave discussed in this paper, but it is also easily extensibleto other problems.

5 Discussion and conclusions

The problem with unidentifiability as a result of conservedmoieties has been observed before. However, there does notseem to exist a systematic treatment of the problem, and thecommon explanation for the problem does not hold. Thisexplanation is based on the observation that the insertionof conserved moieties in rate expressions usually leads tothe situation of one parameter, or combination of par-ameters, in front of each term. This means that the coeffi-cients in front of the terms are unidentifiable, but thatdoes not necessarily imply unidentifiability of the originalparameters. Consider for instance the following rateexpression

r ¼p1c

ðp2=p1Þ þ p2c

with two parameters, p1 and p2, and one concentration, c.This rate expression has one coefficient in front of eachterm, but the parameters are nevertheless identifiable. It istherefore clear that the situation of coefficients in front ofeach term is not sufficient in itself to guarantee unidentifia-bility of the underlying parameters (i.e. additional con-ditions are necessary).On the other hand, the intuitive method of using the coef-

ficients as new identifiable parameters is a valid method.One coefficient must, however, be removed through div-ision of both the numerator and denominator, and thismethod is therefore sometimes referred to as the ‘dividethrough’ method. The ‘divide through’ method is similarto our method proposed. However, the ‘divide through’method does not involve an analysis of the transformationmatrix A, and this has some drawbacks. As is shown in

the vADH example in the Supplementary material, the analy-sis of the linear dependencies in A allows for fewer identifi-able parameters than the ‘divide through’ method wouldyield. Likewise, it is the analysis of the null-space of Athat allows for easy formulas for the translation back tothe original parameters [e.g. (27)]. There are also othermethods that might be useful when trying to manuallyrewrite the expression in identifiable parameters, forinstance, non-dimensionalisation [24]. However, non-dimensionalisation does not guarantee identifiability in theresulting expressions.There are also some frameworks to deal with structural

unidentifiability for the general situation. One such frame-work is differential algebra. However, many of itsmethods are not applicable to realistically large systems.One exception is the probabilistic algorithm bySedoglavic [14]. Advantages with our method comparedwith Sedoglavic’s algorithm is that our results are exact(i.e. not probabilistic), and that we provide a reparametrisa-tion to identifiable parameters that may be expressed in theoriginal parameters. A general advantage with our methodcompared with all methods based on differential algebra[11, 14] is that our method is built on a much simplertheory, something which, e.g. yields a more intuitive under-standing of the origin of the unidentifiability.Given all these options, it is also important to discuss how

to proceed in different circumstances. In some cases, theproblem with unidentifiability in the rate expressions maybe left untreated altogether. This is the case if one, forexample, is only interested in whether a given model struc-ture is capable of explaining the data. One can also disre-gard identifiability issues if the parameter values areobtained elsewhere (e.g. from in vitro characterisations),and the model is used as a pure forward simulationmodel. Finally, if one is only interested in the value of theactual flux, these specific identifiability problems can bedisregarded because the flux is identifiable even thoughsome of the parameters describing it are not.There are, on the other hand, also several scenarios where

it is necessary to deal with problems of unidentifiability. Ifone would, for example, seek to compare the in vivo withthe in vitro kinetics for a given enzyme, it is necessary todeal with these issues [18]. Such comparisons are importanttools for the understanding of the general differencesbetween in vivo and in vitro kinetics, and this is a centralissue for the general understanding of life. Furthermore, ifone wants to determine the quality of various model predic-tions, that is, how well they are characterised by the avail-able data, it is also necessary to handle identifiabilityproblems. For instance, if one wants to analyse the controlcoefficients in a model (i.e. the sensitivity of a specificmodel output with respect to perturbations in the model’sparameters), then such an analysis will give differentresults depending on which value of the unidentifiabilitymanifolds one chooses, even though all such choices leadto an identical agreement with the data. In such situations,it is therefore more advantageous to characterise the ident-ifiable core in a model, and only consider the results withrespect to this core [18, 25]. One way to find those par-ameter combinations that describe this core model is theintuitive ‘divide through’ method. However, as explainedabove our proposed method has advantages both in termsof yielding fewer parameters and in terms of back-translation of the result.In conclusion, this article has presented a new framework

to treat structural unidentifiability in single, rate expressionscaused by conservation relations. The framework provides avalid explanation of the reasons for the problem, a


straightforward method for the detection of it, and a way tochoose identifiable parameters if an unidentifiability shouldbe detected. Furthermore, the analysis provides a translationfrom the identifiable parameters back to the originalparameters. The ideas are illustrated by the analysis of somewell-cited models of glycolysis, and the proposed method iseasy to use via the provided Mathematica implementation.

6 Acknowledgment

We thank all involved reviewers for constructive criticism.Financial support from the National Research School in

Genomics and Bioinformatics, Goteborg University,Sweden is gratefully acknowledged. B. W. was partiallysupported by the Swedish Research Council and by theSwedish Foundation for Strategic Research via theGothenburg Mathematical Modelling Center. G. C. carriedout most of the work at Fraunhofer-Chalmers Centre forIndustrial Mathematics, and was, during this time, sup-ported by the BioSim Network of Excellence and by theSwedish Foundation for Strategic Research.

7 Supplementary data

Supplementary data are available on IET Systems Biologyonline.

8 References

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2 Teusink, B., Passarge, J., Reijenga, C.A., Esgalhado, E., van derWeijden, C.C., Schepper, M., Walsh, M.C., Bakker, B., van Dam, K.,Westerhoff, H.V., and Snoep, J.L.: ‘Can yeast glycolysis beunderstood in terms of in vitro kinetics of the constituent enzymes?testing biochemistry’, Eur. J. Biochem., 2000, 267, (17), pp. 5313–5329

3 Hynne, F., Danø, S., and Sørensen, P.G.: ‘Full-scale model of glycolysisin Saccharomyces cerevisiae’, Biophys. Chem., 2001, 94, pp. 121–163

4 Stephanopoulos, G.N., Aristidou, A.A., and Nielsen, J.: ‘MetabolicEngineering – principles and methodologies’ (Academic Press, 1998)

5 Segel, I.H.: ‘Enzyme kinetics’ (John Wiley and Sons, New York, 1975)6 Pohjanpalo, H.: ‘System identifiability based on the power series

expansion of the solution’, Math. Biosci., 1978, 41, (1–2), pp. 21–337 Vajda, S., Godfrey, K.R., and Rabitz, H.: ‘Similarity transformation

approach to identifiability analysis of nonlinear comportementalmodels’, Math. Biosci., 1989, 93, (2), pp. 217–248

8 Walter, E.D.: ‘Identifiability of parametric models’ (Pergamon BooksLtd, 1987)

9 Ollivier, F.: ‘Le probleme de l’identifiabilite structurelle globale:approche theorique, methodes effectives et bornes de complexite’,PhD thesis, Ecole polytechnique, France, 1990

10 Diop, S., and Fliess, M.: ‘On nonlinear observability’. Proc. FirstEuropean Control Conf., Grenoble, France, July 1991, pp. 152–157

11 Chappell, M.J., Evans, N.D., Godfrey, K.R., and Chapman, M.J.:‘Structural identifiability of controlled state space systems:a quasi-automated methodology for generating identifiablereparameterisations of unidentifiable systems’, SymbolicComputation for Control (Ref. No. 1999/088), IEE Colloquium on,1999, pp. 8/1-812

12 Margaria, G., Riccomagno, E., Chappell, M.J., and Wynn, H.P.:‘Differential algebra methods for the study of the structuralidentifiability of rational polynomial state-space models in thebiosciences’, Math. Biosci., 2001, 174, pp. 1–26

13 Audoly, S., Bellu, G., D’Angio, L., Saccomani, M.P., and Cobelli, C.:‘Global identifiability of nonlinear models of biological systems’,IEEE Trans. Biomed. Eng., 2001, 48, (1), pp. 55–65

14 Sedoglavic, A.: ‘A probabilistic algorithm to test local algebraicobservability in polynomial time’, J. Symbolic Comput., 2002, 33,pp. 735–755

15 Anguelova, M.: ‘Nonlinear observability and identifiability: generaltheory and a case study of a kinetic models for S. cerevisiae’,Licentiate thesis, Chalmers University of Technology andGothenburg University, Sweden, 2004

16 Vallabhajosyula, R., Chickarmane, V., and Sauro, H.: ‘Conservationanalysis of large biochemical networks’, Bioinformatics, 2006, 22,(3), pp. 346–353

17 Olivier, B.G., and Snoep, J.L.: ‘Web-based kinetic modelling usingJWS Online’, Bioinformatics, 2004, 20, pp. 2143–2144

18 Cedersund, G.: ‘Core-box modelling – theorical contributions andapplications to glucose homeostasis related systems’. PhD thesis,Chalmers University of Technology, Sweden, 2006

19 Higgins, J.: ‘A chemical mechanism for oscillation of glycolyticintermediates in yeast cells’, Proc. Natl Acad. Sci. USA, 1964, 51,pp. 989–994

20 Sel’kov, E.E.: ‘Self-oscillations in glycolysis. A simple kineticmodel’, Eur. J. Biochem., 1968, 4, pp. 79–86

21 Goldbeter, A.: ‘Biochemical oscillations and cellular rhythms’(Cambridge University Press, Cambridge, UK, 1996)

22 Danø, S., Sorensen, P.G., and Hynne, F.: ‘Sustained oscillations inliving cells’, Nature, 1999, 402, (6759), pp. 320–322

23 Lambeth, M., and Kushmerick, M.J.: ‘A computational model forglycogenolysis in skeletal muscle’, Ann. Biomed. Eng., 2002, 30,pp. 808–827

24 Lin, C.C., and Segel, L.A.: ‘Mathematics applied to deterministicproblems in the natural sciences’ (SIAM, 1988)

25 Cedersund, G., Danø, S., Sørensen, P.G., and Jirstrand, M.: ‘From invitro biochemistry to in vivo understanding of the glycolyticoscillations in Saccharomyces cerevisiae’. Proc. BioMedSim’2005,Linkoping, Sweden, May 2005, pp. 105–114


Conservation laws and non-identifiability of rate

expressions in biochemical models

Milena Anguelova∗,∗∗, Gunnar Cedersund§, Mikael Johansson#,

Carl Johan Franzen#, Bernt Wennberg∗,∗∗

∗ Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg,

Sweden∗∗ Mathematical Sciences, Goteborg University, SE-412 96 Gothenburg, Sweden

§ Department of Cell Biology, Linkoping University, SE-581 85 Linkoping, Sweden

# Chemical and Biological Engineering, Chemical Reaction Engineering, Chalmers

University of Technology, SE-412 96 Gothenburg, Sweden

Keywords

identifiability, structural identifiability, conserved moieties, rate expressions

Correspondence

Milena Anguelova

Address: Mathematical Sciences, Chalmers University of Technology, S-412 96 Gothen-

burg, Sweden

Email: [email protected]

Phone: +46-(0)31-7725323

1

SUPPLEMENTARY

MATERIAL

2

A Detailed derivations from Section 3

A.1 The Hynne model

The Hynne model was originally presented in [3]. It consists of 22 states and60 parameters. The model describes glycolytic oscillations in Saccharomyces

cerevisiae during cultivation in a continuous stirred tank reactor. There are twocompartments in the model, one for the extracellular metabolites and one forthe intracellular ones. In the intracellular compartment (corresponding to thetotal cytosol volume of all the cells) there are two conserved moieties, one forthe nicotinamide nucleotides and one for the adenine nucleotides

cNADH + cNAD+ = p1 (A1a)

cATP + cADP + cAMP = p2 (A1b)

These conserved moieties are not explicitly formulated in the model, but areimplicitly present in the stoichiometric matrix.

There are three reactions that include the inter-conversion of the nicoti-namides: glyceraldehyde-3-phosphate-dehydrogenase (GAPDH), a lumped re-action forming glycerol from dihydroxyacetonephosphate (lpGlyc), and the al-cohol dehydrogenase (ADH). The ADH reaction is formulated as an irreversiblereaction, and only cNADH appears in the kinetic expression. However, bothGAPDH and lpGlyc have cNADH and cNAD+ in their kinetic expressions whichwere therefore analysed. Both were found to contain unidentifiable parametersby the criterium presented in Section 4. The reaction rate expressions can bereparameterised in fewer identifiable parameters as follows:

The first rate expression is

rGAPDH =V8m(cGAPcNAD+ −

cBPGcNADH

K8eq)

K8GAPK8NAD(1 + cGAP

K8GAP+ cBPG

k8BPG)(1 +

cNAD+

K8NAD+ cNADH

K8NADH)

,

(A2)

where there are three unidentifiable kinetic parameters, V8m,K8NAD and K8NADH,due to the assumption cNADH + cNAD+ = p1. These three parameters can bereplaced by two new ones k1 and k2 to obtain:

vGAPDH =cGAP(p− cNADH)− cBPGcNADH

K8eq

K8GAP(1 + cGAP

K8GAP+ cBPG

k8BPG)(k1 + k2cNADH)

, (A3)

where k1 = K8NAD

V8m(1 + p1

K8NAD) and k2 = K8NAD

V8m( 1

K8NADH−

1K8NAD

).

The second reaction rate is

rlpGlyc =V15mcDHAP

K15DHAP

(

1 +K15INADH

cNADH

(1 +cNAD+

K15INAD

)

)

+

cDHAP

(

1 +K15NADH

cNADH

(1 +cNAD+

K15INAD

)

)

,

(A4)

3

with five unidentifiable parameters V15m,K15DHAP,K15INADH,K15INAD and K15NADH.It can be rewritten as

rlpGlyc =cDHAPcNADH

k1 + k2cNADH + k3cDHAP + k4cDHAPcNADH

, (A5)

where

k1 =K15DHAPK15INADH

V15m

(1 +p

K15INAD

) (A6a)

k2 =K15DHAP

V15m

(1−K15INADH

K15INAD

) (A6b)

k3 =K15NADH

V15m

(1 +p

K15INAD

) (A6c)

k4 =1

V15m

(1−K15NADH

K15INAD

) . (A6d)

In the rate expression for the adenylate kinase (AK) reaction, the three vari-ables from the second conserved moiety, cATP, cADP and cAMP appear together,and AK has therefore been analysed for a potential source for unidentifiability.The rate expression is

rAK = k24f cAMPcATP − k24rc2ADP (A7)

which after insertion of cAMP = p2 − cATP − cADP becomes

rAK = k24f (p2 − cATP − cADP)cATP − k24rc2ADP (A8)

= k24fp2 − k24f c2ATP − k24fcADPcATP − k24rc

2ADP . (A9)

The two kinetic parameters k24f and k24r are thus identifiable.The Sedoglavic algorithm showed in this case that the presence of the con-

served moiety (A1a) was the only source of unidentifiability if all metabolitescan be measured.

In the experimental setup used by Hynne et al. only NADH was measuredwith high time resolution. It is therefore interesting to note that even withy = k[NADH], the conservation law (A1a) is the only source that leads tostructural unidentifiability among the reactions in the cytosol.

A.2 The Teusink model

The Teusink model [2] also describes yeast glycolysis, and just like the Rizzi andHynne models, it has two conserved moieties described by (A1a) and (A1b).Again there is no kinetic expression containing all three adenine nucleotides.There are three dehydrogenase reactions: GAPDH, ADH, and glycerol-3-phosphatedehydrogenase (G3PDH). All these reactions involve the inter-conversion of thecofactors NAD+ and NADH, and the corresponding rate equations contain bothcNAD+ and cNADH in the denominator. For this model, there are two addi-tional conservation relations due to equilibrium assumptions for [GraP] and[glyceronephosphate] and for the adenine nucleotides. The variables in the firstequilibrium assumption, [GraP] and [glyceronephosphate], both appear in theALD reaction rate expression which must therefore be analysed for a potentialsource of unidentifiability.

4

All three dehydrogenase reactions were found to be unidentifiable, due tothe conserved moiety. Here we show this by a direct calculation, but the Math-ematica code in part B of this supplementary material gives the same result(except that there are, of course, many ways of reparameterising the expres-sions in order to obtain an identifiable model). The rate equation for the firstdehydrogenase reaction, GAPDH, is

rGAPDH =

VmaxfcGAPcNAD+

KGAPKNAD−

VmaxrcBPGcNADH

KBPGKNADH

(1 + cGAP

KGAP+ cBPG

KBPG)(1 +

cNAD+

KNAD+ cNADH

KNADH)

. (A10)

The constraint cNADH + cNAD+ = const = p leads to the parameters Vmaxf ,Vmaxr, KNAD and KNADH being unidentifiable. These four parameters can bereplaced by three new ones to obtain the following re-parameterisation of therate expression

rGAPDH =pcGAP − cGAPcNADH −

k1cBPGcNADH

KBPG

(1 + cGAP

KGAP+ cBPG

KBPG)(k2 + k3cNADH)

(A11)

where

k1 =KGAPKNADVmaxr

Vmaxf

(A12a)

k2 =KGAPKNAD(1 + p

KNAD)

Vmaxf

(A12b)

k3 = (1

KNADH

−1

KNAD

)KGAPKNAD

Vmaxf

(A12c)

The rate equation for the second dehydrogenase reaction, G3PDH, is:

rG3PDH =

Vmax(cDHAPcNADH

KDHAPKNADH

−cGlycerolcNAD+

KDHAPKNADHKeq

)

(1 + cDHAP

KDHAP+

cGlycerol

KGlycerol)(1 + cNADH

KNADH+ cNAD+

KNAD)

(A13)

The unidentifiable parameters are Vmax,KNAD and KNADH. These three uniden-tifiable parameters can be replaced by two new ones to obtain the followingre-parameterisation of the rate expression:

rG3PDH =

cDHAPcNADH

KDHAP−

cGlycerol(p− cNADH)KDHAPKeq

(1 + cDHAPKDHAP

+cGlycerol

KGlycerol)(k1 + k2cNADH)

(A14)

where

k1 = (1 +p

KNAD

)KNADH

Vmax

(A15a)

k2 = (1

KNADH

−1

KNAD

)KNADH

Vmax

(A15b)

5

The rate equation for the last dehydrogenase reaction, ADH, is:

rADH,num = Vmax(cEtOHcNAD

KEtOHKINAD

−cACAcNADH

KEtOHKINADKeq

) (A16)

rADH,denom = 1 +cNAD+

KINAD

+cEtOHKNAD

KINADKEtOH

+cACAKNADH

KINADHKACALD

+

cNADH

KINADH

+cEtOHcNAD+

KINADKEtOH

+cNAD+cACAKNADH

KINADKINADHKACALD

+cEtOHcNADHKNAD

KINADKINADHKEtOH

+

+cACAcNADH

KACALDKINADH

+cEtOHcNAD+cACA

KINADKIACALDKEtOH

+

cEtOHcACAcNADH

KIEtOHKINADHKACALD

(A17)

rADH =rADH,num

rADH,denom

(A18)

The unidentifiable parameters are Vmax, KEtOH, KIEtOH, KNAD, KINAD, KNADH,KINADH, KACALD, KIACALD. These nine unidentifiable parameters can be re-placed by eight new ones to obtain the following re-parameterisation of the rateexpression:

rADH,num = cEtOH(p− cNADH)−cACAcNADH

Keq

(A19a)

rADH,denom = k1 + k2cEtOH + k3cNADH + k4cACA + k5cEtOHcNADH

+k6cEtOHcACA + k7cNADHcACA + k8cEtOHcNADHcACA

(A19b)

where

k1 = (1 +p

KINAD

)KEtOHKINAD

Vmax

(A20a)

k2 = (KNAD + p

KINADKEtOH

)KEtOHKINAD

Vmax

(A20b)

k3 = (1

KINADH

−1

KINAD

)KEtOHKINAD

Vmax

(A20c)

k4 = (KNADH

KINADHKACALD

+pKNADH

KINADKINADHKACALD

)KEtOHKINAD

Vmax

(A20d)

k5 = (KNAD

KINADKINADHKEtOH

−1

KINADKEtOH

)KEtOHKINAD

Vmax

(A20e)

k6 =p

KIACALDVmax

(A20f)

k7 = (1

KINADHKACALD

−KNADH

KINADKINADHKACALD

)KEtOHKINAD

Vmax

(A20g)

k8 = (1

KIEtOHKINADHKACALD

−1

KINADKIACALDKEtOH

)KEtOHKINAD

Vmax

(A20h)

The ALD rate expression containing the variables [GraP] and [glyceronephosphate]from one of the equillibrium assumptions was analysed using the Mathematica

6

implementation and found to be identifiable. The details can be seen by runningthe Mathematica code for the ALD rate expression.

7

B Application examples with the general method

B.1 The rADH reaction rate from the Teusink model

In this section we consider Teusink’s rate expression for rADH in the frame-work of Section 4. This is the same reaction as in equation (A18). By use ofSedoglavic’s algorithm we learned that nine of the ten parameters are proba-bly not identifiable, whereas Keq is, and hence it is essentially enough to studythe denominator of the expression. Below we do not assume any knowledgeabout which of the parameters that are identifiable, but rather we consider thecomplete expression, and then have 10 parameters:

c1 = NAD+ c2 = ACA c3 = NADHc4 = EtOH k1 = Vmax k2 = KINAD

k3 = KEtOH k4 = Keq k5 = KNADH

k6 = KINADH k7 = KACALD k8 = KNAD

k9 = KIACALD k10 = KIEtOH

(B1)

With this notation we may write

rADH =k1

k3k2c1c4 −

k1

k3k2k4c2c3

1 + 1k2

c1 + k5

k6k7c2 + 1

k6c3 + k8

k2k3c4 + k5

k2k6k7c1c2 + 1

k7k6c2c3 +

1k2k3

c1c4 + k8

k2k6k3c3c4 + 1

k2k9k3c1c2c4 + 1

k10k6k7c2c3c4

(B2)

We next adapt the notation to that of Section 4, and write

a1001 = k1

k3k2a0110 = −

k1

k3k2k4b0000 = 1

b1000 = 1k2

b0100 = k5

k6k7b0010 = 1

k6

b0001 = k8

k2k3b1100 = k5

k2k6k7b1001 = 1

k2k3

b0110 = 1k7k6

b0011 = k8

k2k6k3b1101 = 1

k2k9k3

b0111 = 1k10k6k7

(B3)

to obtain

rADH =a1001c1c4 + a0110c2c3

b0000 + b1000c1 + b0100c2 + b0010c3 + b0001c4 + b1100c1c2 + b0110c2c3 +b1001c1c4 + b0011c3c4 + b1101c1c2c4 + b0111c2c3c4

(B4)

Introducing the conserved quantity c1 = p− c3 gives

rADH =a001c4 + a110c2c3 + a011c3c4

b000 + b100c2 + b010c3 + b001c4 + b110c2c3

+b101c2c4 + b011c3c4 + b111c2c3c4

.

(B5)

8

The new coefficients are obtained from the old ones by the transformation

a001

a110

a011

b000

b100

b010

b001

b110

b101

b011

b111

=

p 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0−1 0 0 0 0 0 0 0 0 0 0 0 00 0 1 p 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 p 0 0 0 0 00 0 0 −1 0 1 0 0 0 0 0 0 00 0 0 0 0 0 1 0 p 0 0 0 00 0 0 0 0 0 0 −1 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 p 00 0 0 0 0 0 0 0 −1 0 1 0 00 0 0 0 0 0 0 0 0 0 0 −1 1

a1001

a0110

b0000

b1000

b0100

b0010

b0001

b1100

b1001

b0110

b0011

b1101

b0111

(B6)

We write a = Aa for the expression (B6), and as in Section 4, we denote by thesuperscript 0 an estimated set of coefficients. The rank of A is 10. The numberof original kinetic parameters is 10. That means that the counting argumentfrom Section 4

1. [no of parameters] − [rank A] +1 (i.e., 10− 10 + 1 = 1)

indicates that there is a degree of freedom in choosing the kinetic parameters.Since the matrix A is not of full rank, we need to follow the procedure

described for the general case in Section 4, if we want to obtain the identifiableparameters. The first row in A can be obtained by multiplying the third rowby −p. Therefore: replace a001 with −pa011 and divide both the numerator anddenominator with a011 in equation (B5). Identifying the resulting coefficientsin the denominator as the new identifiable parameters, gives an expression thatis structurally identical to (A19).

We will end by giving a geometrical picture of the kinetic parameters givingidentical reaction rates. Consider k = (k1, ...k10). Each coefficient a ∈ R

13 is afunction of k, and we have

k 7→ a(k) ∈ Γ ⊂ R13 (B7)

where Γ typically would be a 10-dimensional sub-manifold of R13. The three

dimensional null space of A is spanned by

w1 = [ 0 0 0 0 0 −p 0 1 0 1 0 0 0 ]tr

w2 = [ 0 0 0 −p 0 0 1 0 1 0 0 0 0 ]tr

w3 = [ 0 0 −p 1 0 1 0 0 0 0 0 0 0 ]tr

(B8)

Hence, given a0 such that a

0 = Aa0, every a of the form

a = λ0a0 + λ1w1 + λ2w2 + λ3w3 (B9)

is also a coefficient vector that agrees with the estimated a0 (where λ0 is the

degree of freedom due to the coefficients being determined only up to a constant

9

factor). This is a four-dimensional subset of R13. The set of kinetic parameters

that agree with the estimated a0 is given by

K =

k ∈ R10 such that a(k) ∈ span(a0,w1,w2,w3)

(B10)

and if k 7→ a(k) is an injective map, K is at least one-dimensional.

B.2 The RES2-reaction in the Rizzi model

The RES2-reaction rate from the Rizzi model (equation (13, 14) involves theconcentrations of NAD+, NADH and GAP. However, for the purpose of thisdiscussion, it is enough to consider the following part of the expression, involvingonly NAD and NADH:

rmaxRES,2

1

KNAD,7

An1,7−1 + L0,7

1

K ′

NAD,7

Bn1,7−1

An1,7 + L0,7Bn1,7

(B11)

where

A = 1 +cC

NAD+

KNAD,7+

cCNADH

KNADH,7

B = 1 +cC

NAD+

K′

NAD,7

+cCNADH

K′

NADH,7

(B12)

This can be expressed as (we set n1,7 = η for notational convenience, and forthe argument here, we assume that η is a constant)

γ1

(

b00 + b10c1 + b01c2

)η−1+ γ2

(

a00 + a10c1 + a01c2

)η−1

(b00 + b10c1 + b01c2)η + γ3

(

a00 + a10c1 + a01c2

)η , (B13)

where

c1 = cCNAD+ c2 = cC

NADH

γ1 = rmaxRES,2/KNAD,7 γ2 = rmax

RES,2L0,7/K′

NAD,7 γ3 = L0,7 ,

a00 = 1 a10 = 1/KNAD,7 a01 = 1/KNADH,7

b00 = 1 b10 = 1/K ′

NAD,7 b01 = 1/K ′

NADH,7

(B14)

While a rational expression is left invariant by the multiplication of all coeffi-cients by the same constant, this does not hold here.

Replacing c2 by p− c1 gives the new expression

γ1

(

b0 + b1c1

)η−1+ γ2

(

a0 + a1c1

)η−1

(b0 + b1c1)η + γ3

(

a0 + a1c1

)η

=γ1

(

1 + b1c1

)η−1+ γ2

(

1 + a1c1

)η−1

(1 + b1c1)η + γ3

(

1 + a1c1

)η , (B15)

where

b1 =b10 − b01

1 + pb01

a1 =a10 − a01

1 + pa01

γ3 = γ3

(

1 + pa01

1 + pb01

)η

γ1 =γ1

1 + pb01

γ2 = γ2

(1 + pa01)η−1

(1 + pb01)η

(B16)

10

In fact, the actual form of the coefficients may not be very important, but it isinteresting to see that there are only five of them, and hence the six physicalparameters cannot be determined uniquely; the expression is not identifiable.This example cannot be as easily treated with the Mathematica code as theother ones.

In the Segel example in Section 4, the linear transformation of the coeffi-cients have a one-dimensional null-space, which can then be identified with thedegree of freedom in choosing physical parameters. Here the null-space of thetransformation

(γ1, γ2, γ3, a10, a01, b10, b01) → (γ1, γ2, γ3, a1, b1) (B17)

has a two-dimensional null-space. There is, however, still just a one-dimensionalset of kinetic parameters that give identical rate values. This set can be geo-metrically understood as the intersection of the two-dimensional null-space to(B17) with the six-dimensional image of the physical parameters in the spaceof coefficients (γ1, γ2, γ3, a10, a01, b10, b01).

11

C A brief description of the Mathematica im-

plementation

Although the mathematics used in this paper is elementary, the calculationssoon become cumbersome, and some software for symbolic calculation may behelpful. We have created a very simple Mathematica-notebook of the computa-tions described in this paper.

The implementation consists of three notebooks, which should be run se-quentially:

• symmBerData.nb

• symmBerStart.nb

• symmBer.nb

Before starting, the notebook symmBerData.nb should be edited to containinformation about the actual rate equation, as shown in Figure 1. The providedfile contains data for all the rate expressions discussed in the article, and itshould be easy to see how to modify for any rational rate expression. A littlemore work is needed to deal with meromorphic rate expressions.

For reasonably large expressions, all calculations are carried out symboli-cally, and in this case one can normally execute symmBerStart without anymodifications. There are some parameters that can be changed, in particularwith respect to how much of the intermediary results one wishes to see.

Very complicated rate expressions cannot be treated directly without a verylarge amount of memory in the computer. Hence part of the calculation can becarried out in a semi-numerical fashion. This is much faster, but in that case thepresent code can only verify that an expression is identifiable. The parametersthat decide whether a full symbolic calculation should be attemted or not, canalso be set in symmBerStart, together with special parameters related to thesemi-numerical computaiton.

After executing the notebook symmBer.nb, the result is presented as shownin Figure 3.

A remark is in its place here: in terms of which parameters are identifiable,and the degree of freedom in choosing the unidentifable ones, the results ob-tained with the Mathematica code agree with the calculations done by handand presented in the main part of the paper, and in Supplement B. However,due to the implentation, the suggested reparametrisations are not necessarilythe same, nor are the corresponding matrices A.

The code works well in the cases we have tried, but of course we cannotguarantee that there are no cases where it would fail. There is certainly muchspace for improvement of the code, both in terms of efficiency and readability.

The Mathematica notbooks are also accessible from the web-pagehttp://www.math.chalmers.se/~wennberg/Code

We will attempt to keep that code up to date. Also, new versions of thecode will be available from that web page.

12

Figure 1: Data input to Mathematica code for analysing the identifiable of arate expression.

13

Figure 2: Result from Mathematica code for analysing the identifiable of a rateexpression, part1.

14

Figure 3: Result from Mathematica code for analysing the identifiable of a rateexpression, part2.

15

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)

a± $& L´ 0 I JI L dOQL L dceL !$ POQIφ(δ, y, . . . , y(l) u, . . . , u(k)) = 0

%% !$ 9O JI Ò $

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φ(. . .)) !Ì IUO QD^ 'I ! %c(δ]φ(y, . . . , y(l), u

. . . , u(k))

]c(δ] ∈ K(δ]

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φ(δ, y, . . . , y(l), u, . . . , u(k)) = 0 ,J]+! JIPÒ $ %

δJI !Ì % %''I !$ ! # K !Ì OQI # Kf l-OQ^f l Q O? !`^ % !$ %

∂h(j)i (t)

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1 ≤ i ≤ p0 ≤ j ≤ si

s ≥ 1

1 ≤ r ≤ m

!Ì

k ≥ 0 ! "!$# &JI L !`^ ! Q

u(k)r

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x1 = h1xs1 = h

(s1−1)1

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(s2−1)2

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xs1+···+sp+1 = g1xn = gn−K

. 2V -D

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˙x1 = x2

˙x2 = x3˙xs1 = f1(x)˙xs1+1 = xs1+2˙xs1+s2 = f2(x)˙xs1+···+sp

= fp(x)y1 = x1

y2 = xs1+1yp = x1+s1+···+sp−1 .

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x1 = x2(t) + x2(t− τ)x2 = x2(t)y(t) = x1(t)ϕ1(t) = et

ϕ2(t) = t+ 1 .

2S#C3

dcutsd

y(t) = x2(t) + x2(t− τ)2V

y(t) = x2(t) + x2(t− τ) = y(t) 2S

uv|g%lc!gpD£g?nxlic9|ghvl/uvlinohs|

x1 = h1

x2 = h(1)1 ,

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˙x1(t) = x2(t)˙x2(t) = x2(t)y(t) = x1(t)x1(t0) = x1(t0)x2(t0) = x2(t0) + x2(t0 − τ) .

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x

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0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30original output of the delay systemoutput of the ODE−system

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!Ì y(t)

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∂(S,hi,...,h(j)i )

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∂x= si + 1 > si = rankK(δ]

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x(t) = f(


y(t) = h(

x(t), x(t− τ1), . . . , x(t− τ`))

x(t) = ϕ(t), t ∈ [−maxiτi, 0]u(t) = u0(t), t ∈ [−T, 0] . 2S&"JD

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τi£i = 1, . . . , `

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˙x1 = x2˙xs1−1 = xs1

˙xs1 = f1( ˙xs1(t− τ ), x, x(t− τ ), u, . . . ,u(γ), u(t− τ ), . . . , u(γ)(t− τ ))

˙xs1+1 = xs1+2˙xs1+s2−1 = xs1+s2

˙xs1+s2 = f2( ˙xs1+s2(t− τ ), x, x(t− τ ), u,. . . , u(γ), u(t− τ ), . . . , u(γ)(t− τ ))

˙xs1+···+sp= fp( ˙xs1+···+sp

(t− τ ), x, x(t− τ ), u,. . . , u(γ), u(t− τ ), . . . , u(γ)(t− τ ))

y1 = x1

y2 = xs1+1yp = x1+s1+···+sp−1

x(t) = ϕ(t), t ∈ [t0 −maxj τj, t0] .

2VH((

$|yeuvnof(y(t), u(t))

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x1(t) = x22(t− 1)

x2(t) = x1(t−√

2)x2(t)y(t) = x1(t)

x1(t) = ϕ1(t) = et, t ∈ [−√

2, 0]

x2(t) = ϕ2(t) = t+ 2, t ∈ [−√

2, 0] .

2VH( )

dcgutsd

y = (δ1(x2))2 2VH(J2

y = 2δ1(x2)δ1δ2(x1)δ1(x2) = 2(δ1(x2))2δ1δ2(x1) .

2VH( D

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˙x1(t) = x2(t)˙x2(t) = 2(x2(t))

2x1(t− 1−√

(2))y(t) = x1(t)

x(t) = ϕ(t), t ∈ [t0 − 1−√

2, t0] .

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t ≥ 1 +√

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licgdpujd

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

Output from the original systemOutput from the single−delay system

nosgfmd 2 ] OQL L %y(t)

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x1(t) = −x2(t− τ1)x2(t) = x1(t− τ2)y1(t) = x1(t)y2(t) = x2(t− τ2)x(t) = ϕ(t), t ∈ [−T, 0]

. 2VH(J3

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y1 = −δ1x2

2V&) y1 = −δ1δ2x1

2V&) uj|g

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dy1

dy2

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1 00 δ1δ1δ2 00 δ2

[

dx1

dx2

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.

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dy1 = −δ1δ2dy1

2V&)G"

δ1dy2 = −δ2dy1 ⇔ dy2 = −δ−11 δ2dy1 .

2V&) (dlc!gpAcutsd

∆10

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x1(t) = x22(t− τ1) + u

x2(t) = x1(t− τ2)x2

y(t) = x1(t)x(t) = ϕ(t), t ∈ [−τ, 0] .

2V&)G2

dcgutsd

y = (δ1(x2))2 + u

2V&)D

y = 2δ1(x2)δ1δ2(x1)δ1(x2) + u = 2(δ1(x2))2δ1δ2(x1) + u

2S:) F

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y(3) = h(3)([δ2( K !

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x(t) = f(x(t), x(t− τ), u, u(t− τ))

y(t) = h(x(t), x(t− τ), u, u(t− τ))

x(t) = ϕ(t), t ∈ [−τ, 0]

u(t) = u0(t), t ∈ [−T, 0]

,ôgöaù

æQçyÖ~×dÖx ∈ R

nßcÖ~ÞyãÜ:ÖaÕ#Ü:çyÖBÕgÜdÙÜ:Ö+ØÙ×dÚYÙÔyÛYÖWÕ~ð

u ∈ RmÚYÕÜ:çqÖÚÞqàyìcÜÙÞqß

y ∈ RpÚYÕÜ:çqÖãìcÜdàyìcÜaéÞÝõà.Ù×Ùá1äÖÜ:ÖW×dÕpå_ã×Ü:çyÖÕgÝcÕgÜ:Ö~á èWÙÞ ÔlÖUæQ×dÚ£Ü:Ü:ÖWÞ ÙÕÕgÜdÙÜ:ÖUØÙ×dÚsÙÔyÛYÖWÕæQÚ£ÜdçÜ:ÚYá1Ö~äwßcÖW×:ÚYØÙÜ:ÚYØÖzWÖ~×dãqéê4çyÖDìyÞ%LÞyãæQÞ#èãÞqÕ<ÜÙÞÜ¶Ü:ÚYá,ÖäßcÖ~ÛsÙaÝÚsÕmßcÖ~ÞyãÜ:Öaß+ÔLÝ

τ ∈ [0, T )ðT ∈ R

é>ê4çyÖÖWÞÜd×:ÚYÖWÕRãåfÙÞ.ß

hÙ×dÖQá,Ö~×dãá,ã×:àyçyÚsè4ÚÞÜ:çyÖWÚ×²Ù×dëìyá,ÖWÞÜdÕôeíìyãÜ:ÚYÖ~ÞÜÕãåmèãÞØÖ~×dëÖWÞÜàlãæFÖ~×UÕgÖW×:ÚYÖWÕæQÚ£Üdç×:ÖWÙÛè~ãLÖ`èÚYÖ~ÞÜdÕùÙÞ.ß

ϕ : [−τ, 0] → RnÚsÕQÙÞUìyÞ ÞqãæQÞèãÞÜdÚÞLìyãìqÕ²å_ìqÞqè³ÜdÚãÞ¼ãåÚÞyÚÜ:ÚsÙÛèãÞqßcÚÜ:ÚYãÞqÕWé³ê4çqÖkÕgÖ~ÜãåÚÞyÚÜ:ÚsÙÛå_ìyÞqèÜ:ÚYãÞqÕå_ã×Ü:çyÖkØÙ×:ÚäÙÔyÛYÖWÕ

xÚsÕQßcÖ~ÞyãÜ:Öaß#ÔLÝ

C := C([−τ, 0], Rn)é-á,ÖW×:ãá1ã×gäàyçyÚYèRå_ìyÞqèÜ:ÚYãÞ

uÚYÕèWÙÛYÛÖaßÙÞÙßcá,ÚYÕdÕgÚYÔyÛYÖRÚYÞyàyìyÜDÚ£åkÜ:çyÖßcÚå]äå]ÖW×:ÖWÞÜdÚYÙÛÖWíìqÙÜ:ÚYãÞÙÔlãØÖRÙßyá1ÚÜdÕÙ1ìqÞyÚYíìyÖpÕgãÛYìcÜ:ÚYãÞéqê4çyÖÕgÖÜ²ãå>ÙÛYÛlÕgì.èdçÚYÞyàyìcÜ²å]ìqÞqè³ÜdÚãÞqÕFÚYÕFßyÖ~ÞyãÜ:ÖWßÔLÝ

CU

éÓDÞyÖmãå2ÜdçyÖãÔcû<ÖWè³ÜdÚØÖaÕãåkÜ:çyÚsÕæ²ã×$UÚYÕÜ:ã#ÚÞLØÖWÕgÜ:ÚYëÙÜ:ÖRÜdçyÖày×:ãàlÖ~×:ÜzÝãåFÛYãcèWÙÛ2ÚYßcÖWÞÜdÚ£â.ÙÔyÚYÛÚÜzÝãå¶Ü:çyÖßcÖWÛYÙÝà.Ù×Ùá,ÖÜdÖ~×τðæQçyÚsèdç#æFÖDßcÖâ.ÞyÖDÙÕkå_ãÛÛYãæÕa 2ËÆÒeÌWÒeÎlË É !#"%$&('$*)+$,-#./#)

τ 02131 $ 0 4.654789":5;< $"=">& 0 ?A@B. 0 C $!7A":D$*. τ0 ∈ (0, T ) 0 E .FG#)+D+H 021 . 1 $*@I58'A@1 A. W 3 τ0 J W ⊂ [0, T ) J 1LK < 9.=$*. ∀τ1 ∈ W : τ1 6= τ0 J∀ϕ0, ϕ1 ∈ C J .=A)+M8H 021 . t0 ≥ 0

$*@Nu ∈ CU 1LK < .FG$.

y(t0, ϕ1, u, τ1) 6= y(t0, ϕ0, u, τ0) JPO G#)+ y(t, ϕ, u, τ)?A;

@N5./ 1 .=Q'G$)+$,-A.6A) 0:R ST5 K .>' K . E 5*)U.= 0 @ 0 . 0 $" ESK @ < . 0 5@ ϕ J.FV$!, 02181L0 7A": 0 @' K . u $*@NW!#":$*& τ X NGV!A":$&'$)+$*,MA./#)τ 0211 $ 0 Y.65Y7SZ"%5 < $*"F">& 0 !#@B. 0 C $?7A": 0 E[0 . 021 ":5 < $"F"\& 0 ?A@G. 0 C $?7A":E 5*)P$"=" τ0 ∈ (0, T ) Xï ãÛYÛYãæQÚYÞyëõÜ:çyÖÞyãÜdÙÜ:ÚYãÞqÕÙÞ.ßáÙÜ:çqÖ~áÙÜdÚYèWÙÛmÕ:ÖÜ:Ü:ÚYÞyë5ãå ãLãëDÖ~Ü ÙÛ¬écôù³ð ×íìyÖ~ä Ù×gÜ YÞyÖFÖÜ ÙÛéyôù³ð1DÚYÙÖÜÙÛ¬éôù¶ÙÞ.ß]lçqÙÞyë1ÖÜÙÛéôùðÛYÖÜ

KÔlÖ*Ü:çqÖ*âqÖ~Ûsßãåqá,Ö~×dãá,ã×:àyçyÚsèkå_ìyÞqèÜ:ÚYãÞqÕ2ãå.Ùâ.ÞyÚ£ÜdÖ²ÞLìyápÔlÖ~×2ãå.ØÙ×dÚsÙÔyÛYÖWÕå]×dãá

x(t− kτ), u(t− kτ), . . . , u(l)(t− kτ), k, l ∈ Z+ÙÞqßóÛYÖÜ

δßcÖWÞyãÜdÖ ÜdçyÖõÜdÚá,ÖäzÕ:çyÚ£å]Ü ãàlÖW×dÙÜ:ã×Wð

δ(ξ(t)) =ξ(t−τ)

ðξ(t) ∈ K

éÖ~ÜK(δ]

ßyÖ~ÞyãÜdÖ2ÜdçyÖ¶Õ:ÖÜ>ãåLàlãÛÝLÞyãá1ÚsÙÛsÕãåÜ:çyÖå_ã×dá

a(δ] = a0(t) + a1(t)δ + · · ·+ ara(t)δra ,

ô!ùæQçyÖ~×dÖ

aj(t) ∈ Ké;_ÚÜ:ç ÙßyßyÚ£ÜdÚãÞ ßcÖ~âqÞyÖaß ÙÕpìqÕgì.ÙÛ²ÙÞqß

ápìyÛ£ÜdÚYàyÛÚsè~ÙÜ:ÚYãÞ#ëÚYØÖ~ÞÔLÝ

a(δ]b(δ] =

ra+rb∑

k=0

i≤ra,j≤rb∑

i+j=k

ai(t)bj(t− iτ)δk ,ô ù

K(δ]ÚsÕIÙÞyãÞqèãá1ápìyÜdÙÜdÚØÖF×dÚÞqëqðaæQçyÚYèçpÚsÕ 'ãLÖÜdçyÖ~×dÚsÙÞpÙÞ.ßÙRÛYÖå]ÜÓD×dÖßcãáÙÚÞéê4çqÖÛYÙÜgÜdÖ~×FÚá,àyÛYÚYÖWÕ¶Ü:çqÙÜ¶Ü:çqÖ×ÙÞ%1ãå>Ùá1ãyßcìyÛYÖpãØÖ~×

K(δ]ÚsÕDæFÖWÛÛäwßyÖâqÞyÖaß ô FãçyÞ>ðIöW÷LøLöaùé^Ö~Ü

MßcÖ~ÞyãÜ:ÖQÜ:çyÖá1ãcßyìyÛÖspanK(δ]dξ : ξ ∈ K

éê4çyÖèÛYãÕ:ìy×dÖ4ãåÙpÕ:ìyÔyá,ãcßcìyÛYÖ

NÚÞM

ÚYÕ¶ÜdçyÖ*Õ:ìyÔyá,ãcßcìyÛYÖN = w ∈ M :

∃a(δ] ∈ K(δ], a(δ]w ∈ Né"wÜ ÚsÕ2Ü:çyÖÛYÙ×:ëÖWÕgÜkÕgìyÔqá1ãyßcìyÛYÖãå

Mè~ãÞÜÙÚYÞyÚYÞyë

NæQÚÜ:çU×dÙÞ%1ÖaíìqÙÛ.Ü:ã

rankK(δ]NôLDÚYÙ

ÖÜ4ÙÛ¬éð&ù³é\Ú:]lÖW×:ÖWÞÜdÚYÙÜ:ÚYãÞãåQå_ìyÞ.è³Ü:ÚYãÞ.Õ

φ(x(t − iτ), u(t − jτ), . . .ð

u(l)(t − jτ))ð0 ≤ i, j ≤ k

ðl ≥ 0

ÚYÞKÙÞ.ßãÞyÖ~äå_ã×:áÕ

ω =∑

i κixdx(t−iτ)+

∑

ij νidu(j)(t−iτ)ÚYÞM

ÚYÕßcÖâqÞyÖaßÚÞÜ:çyÖDÞqÙÜdìy×ÙÛqæ4ÙaÝ a

φ =∑k

i=0∂φ

∂x(t−iτ)δif+

+∑l

r=0

∑kj=0

∂φ

∂u(r)(t−jτ)u(r+1)(t− jτ)

ω =∑

i κixdx(t− iτ) +

∑

ij νidu(j)(t− iτ)+

+∑

i κixdδif +

∑

ij νidu(j+1)(t− iτ).

ô^ù

\Ö~âqÞyÖ

X = spanK(δ]dx

Yk = spanK(δ]dy, dy, . . . , dy(k−1)

U = spanK(δ]du, du, . . . .

ô!+ù

ê4çyÖ~Þ(Yk + U) ∩ X = (Yn + U) ∩ X

å_ã×k ≥ n

ÙÞqßrankK(δ](Yn + U) ∩ X ≤ n

ôLDÚYÙpÖÜ²ÙÛéYð&ù³é

_ `ÌyÌMeÒ ÒeËcÌWÒÎ.Ë"zÞÜdçyÚsÕ4Õ:ÖWè³ÜdÚãÞ#æFÖ@èãÞqÕgÚsßcÖ~×FÜdçyÖ*ày×dãÔyÛYÖ~á ãåÆãÔcÜÙÚYÞyÚÞqë1ÙÞÚÞyàqìcÜgäwãìcÜdàyìcÜm×dÖ~àq×:ÖaÕgÖWÞÜdÙÜ:ÚYãÞ)å_×dãá ÜdçyÖÕ<ÜÙÜ:Ö~äwÕ:àqÙèÖpå_ã×:áãå¶Ù,ÜdÚá,ÖäzßcÖ~ÛsÙÝ¼è~ãÞÜ:×dãÛÆÕ:ÝcÕ<ÜdÖ~áélê4çyÚYÕàq×:ãÔyÛÖWá çqÙÕÔlÖ~Ö~ÞÜ:×dÖWÙÜdÖWß@å_ã×ÆàlãÛYÝLÞyãá,ÚsÙÛÕgÝcÕgÜ:ÖWá,ÕÆÚÞ ï ã×ÕgáÙÞRÖ~Ü2ÙÛ¬éôgöW÷÷*ùé"wÜ²æQÚYÛYÛlÔlÖ@ÕgçyãæQÞÜdçqÙÜ²å_ã×4ÙÕ:ÝLÕgÜ:ÖWá ãåÜdçyÖDå_ã×dá ô<öùkÜdçyÖ~×dÖÙÛYæ²ÙÝLÕ.Ö cÚYÕgÜÕ~ð³ÙÜlÛYÖWÙÕ<ÜÛãyè~ÙÛYÛYÝðÙFÕ:ÖÜãåÚYÞyàyìcÜ:ä¬ãìcÜ:àqìcÜßcÖ~ÛsÙaÝäßcÚ\]Ö~×dÖ~ÞÜ:ÚsÙÛlÖWíìqÙÜdÚãÞqÕ ãåÜdçyÖå_ã×dá

F (δ, y, u) = 0ðæQçqÖ~×dÖa

F (δ, y, u) := F (δ, y, . . . , y(k), u, . . . , u(γ)) =

= F (y, . . . , y(k), . . . , δl(y), . . . , δl(y(k)),

u, ...u(γ), . . . , δ(β)(u), . . . , δ(β)(u(γ))) ôùÕgìqèçÜdçqÙÜÙÞLÝQàqÙÚY×(y(t), u(t))

æQçyÚYèç*Õ:ãÛYØÖWÕ2ô<öaùðÙÛsÕgã²ÕdÙÜdÚYÕgäâqÖWÕkôùðå_ã×tÕ:ìqèdçDÜdçqÙÜ>ÙÛYÛßcÖ~×dÚYØÙÜdÚØÖaÕ.ÚYÞØãÛYØÖWßDÙ×:Ö2èãÞÜdÚÞcäìyãìqÕWéê4çyÖå_ìqÞqè³Ü:ÚYãÞ

FÚsÕ¶á,Ö~×dãá,ã×dàyçyÚsè4ÚYÞÚÜdÕ²Ù×dëìyá,Ö~ÞÜdÕWé

LÎ.Í 'É 0 A@($ 1 & 1 ./#, 5 E .= E 5*)L, ôgöaù J .=A)+Y8H 021 .0 @B.6#) 1 γ J β J k$*@N

l$*@N $@ 58'G#@ ?A@ 1 1LK 7 1 #. V

5 EC × C

γ+1U J 1LK < 4.=$. 0 @ .=3@ 0 7S5)+G55 45 E $@G& '5 0 @G.5 E V J .FA)+ 8H 021 . 1 $@ 0 @*' K . ; 5 K .\' K . )+/'N)+ 1 A@G./$. 0 5@45 E .FG1 & 1 ./#, 5 E .= E 5)L, ôù X

)+55 E a.ê4çyÖRàq×:ãLãåÆÚsÕÙÞÙßyÙàcÜÙÜdÚãÞUãåÆÜdçyÖmày×dãLãåIãåkê4çyÖäã×dÖ~á cé céYöé.ÚÞbFãÞÜ:Öð ãLãë#ÙÞqßú2Ö~×ßcãÞôgöW÷÷÷ù²å_ã×ÜdçyÖÙÞqÙÛYãëãìqÕk×dÖWÕ:ìyÛ£Ü¶å_ã×4Ó \käzÕgÝcÕgÜ:ÖWá,ÕWéÖÜ

fÔlÖ,ÙÞ

räwßcÚYá,Ö~ÞqÕ:ÚYãÞqÙÛIØÖWèÜ:ã×DæQÚÜ:ç+Ö~ÞÜd×:ÚYÖWÕ

fj ∈ Ké

ÖÜ ∂f∂x

ßcÖWÞyãÜdÖÜ:çyÖr × n

áÙÜ:×dÚ[ 1æQÚÜ:ç#Ö~ÞÜ:×dÚÖaÕ(

∂f

∂x

)

j,i

=∑

`

∂fj

∂xi(t− `τ)δ` ∈ K(δ] .

ô¬øù

\Ö~ÞqãÜ:ÖDÔÝs1ÜdçyÖÛYÖWÙÕ<ÜFÞyãÞyÞyÖ~ëÙÜdÚØÖÚYÞÜdÖ~ëÖW×FÕ:ìqèçÜ:çqÙÜ

rankK(δ]∂(h1, ..., h

(s1−1)1 )

∂x= rankK(δ]

∂(h1, ..., h(s1)1 )

∂x.ôýù

"wå ∂h1

∂x≡ 0

Ü:çqÖ~Þæ²ÖÕ:ÖÜs1 = 0

é"zÞqßcìqèÜ:ÚYØÖ~ÛYÝðå_ã×1 < i ≤ pßcÖ~ÞyãÜ:ÖÔLÝ

si

Ü:çqÖÛYÖWÙÕ<ÜFÞyãÞqÞyÖ~ëÙÜ:ÚYØÖÚYÞÜ:ÖWëÖ~×²Õ:ìqèdçÜdçqÙÜ


(s1−1)1 , . . . , hi, . . . , h

(si−1)i )

∂x=

= rankK(δ]∂(h1, ..., h

(s1−1)1 , . . . , hi, . . . , h

(si)i )

∂x.

ô÷ù

ÖÜS = (h1, ..., h

(s1−1)1 , . . . , hp, . . . , h

(sp−1)p )

ðIæQçyÖW×:ÖhißcãLÖWÕ¶ÞyãÜ4ÙàyàlÖWÙ×¶Ú£å

si = 0éLê4çyÖ~Þ

rankK(δ]∂S

∂x= s1 + · · ·+ sp = K ≤ n .

ôgöù"wå

K < nðÜdçyÖ~×dÖÖ cÚYÕgÜpá,Ö~×dãá,ã×dàyçyÚsèå_ìyÞ.è³Ü:ÚYãÞ.Õ

g1(δ, x)ð

. . .ðgn−K(δ, x)

ÕgìqèçÜ:ç.ÙÜrankK(δ]

∂(S,g1,...,gn−K)∂x

= né

ï ã×²Õ:Úá,àyÛYÚYè~Ú£Ü<ÝðÚYÞÜd×:ãyßcìqèÖÜ:çyÖDÞyãÜdÙÜ:ÚYãÞ

x1 = h1

. . .

xs1= h

(s1−1)1

xs1+1 = h2

. . .

xs1+s2= h

(s2−1)2

. . .

xs1+···+sp= h

(sp−1)p

xs1+···+sp+1 = g1

. . .

xn = gn−K

.ôgööù

ê4çyÖ*ãÞyÖä¬å_ã×dáÕdxi

ði = 1, . . . , n

Þqãæå_ã×dáÙÔqÙÕgÚsÕ²ãåXéê4çyÖDßcÖ~âqÞyÚÜ:ÚYãÞãå

si

ðôe÷ùkÚYá,àyÛÚYÖWÕ¶Ü:ç.ÙÜ¶å_ã×FÖWÙèçi

∂h(si)i

∂x∈ spanK(δ]

∂(h1, . . . , h(s1−1)1 , . . . , h

(si−1)i )

∂x

.ô<ö1ùê4çìqÕ4ÜdçyÖ~×dÖ@Ö cÚYÕgÜQÞyãÞ%~ÖW×:ã,àlãÛYÝÞqãá,ÚYÙÛYÕbi(δ] ∈ K(δ]

ði =

1, . . . , pÕgìqèçÜ:ç.ÙÜ

bi(δ]∂h

(si)i

∂x∈ spanK(δ]

∂(h1, . . . , h(s1−1)1 , . . . , h

(si−1)i )

∂x

.ô<ö ùê4çyÖ~×dÖå_ã×:Ö

bi(δ]dh(si)i +

∑mr=1

∑γj=0 cj,r(δ]du

(j)r ∈

spanK(δ]dx1, . . . , dxs1+···+siå_ã×Õ:ãá,Ö

γ ≥ 0ðqæQçyÖ~×dÖ

γÚYÕ ÜdçyÖçqÚëçyÖWÕgÜ²ßcÖ~×dÚØÙÜdÚØÖãåuÙàyàlÖaÙ×dÚÞyëmÚÞÜdçyÖå_ìqÞqè³ÜdÚãÞqÕÚÞ

SðLÙÞ.ß

cj,r(δ] ∈ K(δ]éLñÖ~ÞqèÖð

bi(δ]dh(si)i +

m∑

r=1

γ∑

j=0

cj,r(δ]du(j)r −

s1+···+si∑

j=1

aj(δ]dxj = 0 ,

ô<öALùå_ã×*Õ:ãá,Öaj(δ] ∈ K(δ]

éYLÚÞ.èÖ1ÙÛYÛÆå_ìyÞqè³ÜdÚãÞqÕ*Ù×:ÖÙÕdÕ:ìyá,ÖWßá1ÖW×:ãá,ã×dàyçyÚsè#ÙÞqßæFÖçqÙaØÖèãÞÜdÚÞLìyãìqÕßcÖ~àlÖ~Þ.ßcÖ~Þqè~ÖUå_ã×Ü:çyÖRãìcÜdàyìcÜãÞ#ÜdçyÖRÚÞyàqìcÜÙÞqßUÚYÞyÚÜ:ÚsÙÛå_ìyÞqèÜ:ÚYãÞðcÜdçyÖmÙÔlãØÖÖWíìqÙÛYÚÜzÝ¼çyãÛYßqÕãÞ)ÙÞãàlÖ~Þ)ßyÖ~ÞqÕ:ÖÕgÖ~ÜDãå

C × Cγ+1U

élê4çyÖÛÖ~å]ÜçqÙÞ.ßÕgÚsßcÖãåÖWíìqÙÜdÚYãÞîôgöAùð4ÔlÖ~ÚYÞyëõÖaíìqÙÛ*Ü:ã ~Ö~×dãqðÚYÕ+ÙèÛYãÕ:ÖWß/ãÞyÖä¬å_ã×dá ãÞMð*ÙÞqß/Ü:çyÖW×:Ö~å_ã×dÖðDÙàyàyÛÝLÚYÞyëÜ:çyÖõú2ãÚYÞqè~Ù× ÛÖWá1áÙæFÖ ãÔcÜdÙÚÞ å_ìyÞqèÜ:ÚYãÞqÕ

ξi(t) ∈ KÕgìqèçÜ:ç.ÙÜdξi = bi(δ]dh

(si)i +

∑mr=1

∑γj=0 cj,r(δ]du

(j)r −

∑s1+···+si

j=1 aj(δ]dxj

ÙÞqßξi(δ, h

(si)i , x, u, . . . , u(γ)) = 0

ðå_ã×>ÖWÙèdç

i = 1, . . . , péWê4çqÖkå_ìyÞqèÜ:ÚYãÞ

ξi

ßcãLÖaÕÞyãÜÆßcÖ~àlÖWÞqßRãÞxj

å_ã×j > s1 + · · ·+si

ðÕ:ÚÞ.èÖdξi = 0

æFãìqÛYßRÜdçyÖ~Þ1è~ãÞÜdÙÚYÞÜ:Ö~×dáÕdxj

ðj > s1 + · · ·+ si

æQçqÚYèçÚYÕ¶ÚYá,à.ãÕ:Õ:ÚYÔyÛÖÔÝ)ô<öALùßcìyÖDÜ:ãÜ:çyÖ*ØÙ×dÚYÙÔyÛÖaÕdx

ÔlÖWÚÞyë1ÛÚYÞyÖWÙ×:ÛYÝ,ÚYÞqßcÖ~àlÖWÞqßcÖ~ÞÜ4ãØÖ~×K(δ]

ÔLÝ,ßcÖ~âqÞyÚÜ:ÚYãÞéLê4çLìqÕ~ðæFÖçqÙØÖãÔcÜÙÚYÞyÖWßÙR×dÖ~ÛsÙÜdÚYãÞ

ξi(δ, h(si)i , x1, . . . , xs1+···+si

, u, . . . , u(γ)) = 0,ôgö1+ù

æQçyÚYèç ÜdãëÖÜ:çqÖ~×æQÚÜ:ç ô<ööùày×:ãyßcìqèÖaÕÙÞÚÞqàyìcÜgäwãìcÜdàyìcÜÖWíìqÙÜdÚãÞ

ξi(δ, y(si)i , y1, . . . , y

(s1−1)1 , . . . , y

(si−1)i , u, . . . , u(γ)) = 0.ô<öùê4çyÚYÕ*ÚsÕDÜ:×dìyÖå_ã×@ÖWÙèç

i, 1 ≤ i ≤ p×dÖWÕgìqÛ£ÜdÚÞyë¼ÚÞ

pÚÞqàyìcÜgäãìcÜdàyìcÜÖaíì.ÙÜ:ÚYãÞ.ÕRæQçyÖW×:ÖæFÖUè~ÙÞ ÕgÖ~Ü

k = maxi si i =1, . . . , p

ð>ÙÞqßÛÖ~ÜlÙÞqß

βèã×:×dÖWÕ:àlãÞqß)Ü:ãÜ:çyÖÛYÙ×:ëÖWÕgÜRßcÖä

ÛYÙÝcÕ−lτ

ÙÞqß−βτ

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Ê~ÏLËÌWÒ=[2Ò^Ò_ÌSÎÆÌ ,Ï! cÍ Ì Í"zÞ¼Ü:çyÚsÕÕ:ÖWèÜ:ÚYãÞðyæFÖmÕ:çyãæÜ:çqÙÜQÜ:çyÖmÛYãcè~ÙÛÚsßcÖWÞÜ:Úâ.ÙÔyÚÛYÚÜzÝUãåτÚÞô<öùQßcÖWàlÖ~ÞqßyÕãÞæQçqÖÜ:çqÖ~×ÚÜDÚYÕày×dÖWÕ:Ö~ÞÜÚYÞÜ:çyÖpÚYÞyàyìcÜ:äãìcÜdàyìcÜ×dÖ~ày×dÖWÕ:Ö~ÞÜdÙÜ:ÚYãÞ¼ãåIÜdçyÖpÕ:ÝLÕgÜ:ÖWáéqê4çyÖmÛsÙÜ:Ü:Ö~×Dè~ÙÞÔlÖ

ßcÖWèÚsßcÖaßmÖ~ÚÜdçyÖ~×2ÔÝ@Ü:çyÖ²ãyè~èìy×d×dÖ~Þqè~Ö ãå.ÙDßyÖ~ÛsÙaÝÖaßRÚYÞyàyìcÜkØÙ×:ÚäÙÔyÛYÖ1ÚÞ+ÜdçyÖ1Ü:ÚYá,ÖäwßyÖ~×dÚØÙÜ:ÚYØÖWÕ*ãå¶ÜdçyÖ,ãìcÜdàyìcÜ@å_ìyÞ.è³Ü:ÚYãÞ.Õ@ã×ÔÝRÙÛYÚÞqÖWÙ×:äwÙÛëÖ~Ôy×dÙÚsèkè×dÚÜ:Ö~×dÚYãÞRÚYÞØãÛYØLÚYÞyë×ÙÞ%@èWÙÛsèìyÛsÙÜdÚãÞå]ã×@Ü:çyÖÕ:ÖÜmãå²ë×ÙßyÚÖWÞÜdÕ@ãåFÜ:çqÖ,ãìcÜ:àyìyÜdÕmÙÞ.ßÜ:çyÖWÚ×RÜ:ÚYá,ÖäßcÖ~×dÚØÙÜ:ÚYØÖWÕ ãØÖ~×KÙÞqßãØÖW×

K(δ]é

ê4çyÖày×dÖWÕ:Ö~Þqè~ÖãåτÚYÞÜ:çqÖÚYÞyàyìcÜ:ä¬ãìcÜ:àqìcÜ²×dÖ~ày×dÖWÕ:Ö~ÞÜdÙÜ:ÚYãÞ,ãåÜ:çyÖ*ÕgÝcÕgÜ:ÖWá è~ÙÞÔlÖDßyÖâqÞyÖaß,å_ã×dáÙÛYÛYÝÚYÞÜ:ÖW×:áÕ¶ãå

δa

2ËÆÒeÌWÒeÎlË P@ 0 @' K .; 5 K .>' K .Z K $. 0 5@ φ(δ, y, . . . , y(l) Ju, . . . , u(k)) = 0 021 1 $ 0 -.65 0 @ 5" δ 0 @ $@ 181 A@G. 0 $" O $&0 E .=W,-A)+5,-5) ' 0 < ESK @ < . 0 5@ φ(...) < $@B@N5*.7S O ) 0 . ./#@ $ 1c(δ]φ(y, . . . , y(l), u J . . . , u(k)) O 0 .= c(δ] ∈ K(δ] X_+ÖDçqÙØÖQÜ:çqÖå_ãÛÛYãæQÚÞyëp×:ÖWÕ:ìyÛÜa LÎ.Í 0 A@ $ 1 & 1 ./#, 5 E .= E 5)L, ôgöaù $*@N .= 1 #.S? C @NS 0 @ .FG'N)+ 0 5 K1 1 < . 0 5@ J .FG#)+ 8H 021 . 1 $@ 0 @' K .;5 K .>' K . K $. 0 5@ φ(δ, y, . . . , y(l), u, . . . , u(k)) = 0 J .FG$. 0 @B;

5*" 1 δ 0 @M$*@M 181 A@G. 0 $" O $& 0 E $*@N 5@B">& 0 E60 5*) 60F0 7SA":5 O$)+ 1 $. 021=C 8 X

0^X ∂h(j)i

(t)

∂u(k)r (t−sτ)

6= 0 E 5) 1 5,- 1 ≤ i ≤ p J 0 ≤ j ≤ si Js ≥ 1 J 1 ≤ r ≤ m

$@Nk ≥ 0 J 0=X X $I?A":$&?S

0 @' K .; $) 0 $!7#": u(k)r5 <8< K ) 1 0 @ 1 5*,M 5 E .= ESK @ < . 0 5@ 1

0 @ S, h(s1)1 , . . . , h

(sp)p

0F0^X rankK(δ]∂S∂x6= rankK

∂(S,h(s1)

1 ,...,h(sp)p )

∂x

)+55 E a ï ×dãáîÜdçyÖ*ßcÖâqÞyÚÜdÚãÞãåKÙÞqß

K(δ]ðæFÖçqÙØÖ

rankK(δ]∂S

∂x≤ rankK

∂S

∂x≤ rankK

∂(S, h(s1)1 , . . . , h

(sp)p )

∂x.ôgöøù

ÕdÕgìyá,Öâ.×dÕgÜ¶Ü:çqÙÜ 0^X ÚsÕ Ü:×dìyÖðÚ¬é Öé ∂h(j)i

(t)

∂u(k)r (t−sτ)

6= 0å_ã×²Õ:ãá,Ö

1 ≤ i ≤ pð0 ≤ j ≤ si

ðs ≥ 1

ð1 ≤ r ≤ m

ÙÞqßk ≥ 0

éZFÝßyÚ\]Ö~×dÖ~ÞÜ:ÚsÙÜ:ÚYãÞãÞyÖ#âqÞqßyÕÜ:çqÙÜ ∂h

(si)

i(t)

∂u(k+si−j)r (t−sτ)

6= 0é

_5ÚÜdçyãìcÜkÛYãÕdÕÆãå.ëÖWÞyÖ~×ÙÛYÚÜzÝ@æFÖ4èWÙÞ1ÙÕdÕgìyá,Ö¶ÜdçqÙÜu

(k+si−j)rÚYÕQÜ:çqÖmçyÚYëçyÖWÕgÜDßcÖ~×dÚØÙÜ:ÚYØÖRãå

ur

ÙàyàlÖaÙ×dÚÞqë1ÚYÞh

(si)i

é.ê4çyÖ~ÞÚÞÜdçyÖDÖWíìqÙÛYÚÜzÝ)ô<öù

bi(δ]dh(si)i +

m∑

r=1

γ∑

j=0

cj,r(δ]du(j)r −

s1+···+si∑

j=1

aj(δ]dxj = 0 ,

ôgöWýùck+si−j,r(δ]

ÚYÕmÙàlãÛYÝLÞyãá,ÚsÙÛkãåQßcÖWë×dÖ~Ö,ë×dÖWÙÜdÖ~×@ã×RÖWíìqÙÛÜ:ãs ≥ 1

ÚÞδéê4çLìqÕWðÚÞ-ÜdçyÖ èã×:×dÖWÕ:àlãÞqßcÚYÞyëå_ìqÞqè³ÜdÚãÞ

ξi(δ, h(si)i , x, u, . . . , u(γ))

ðÆÕgì.èdçBÜ:çqÙÜdξi = bi(δ]dh

(si)i +

∑mr=1

∑γj=0 cj,r(δ]du

(j)r −

∑s1+···+si

j=1 aj(δ]dxj

ðqÜ:çqÖpØÙ×:ÚäÙÔyÛYÖ

u(k+si−j)r

ÙàyàlÖWÙ×dÕ5Õgãá1ÖWæQçyÖ~×dÖæQÚÜ:ç Ù ßcÖ~ÛsÙaÝsτèãá,àqÙ×:Öaß)Üdã

h(si)i

ÙÞ.ßÜdçìqÕδè~ÙÞyÞyãÜmÔlÖãá,ÚÜgÜdÖWß+å_×dãá

ξi(δ, y(si)i , y1, . . . , y

(s1−1)1 , . . . , y

(si−1)i , u, . . . , u(γ)) = 0

é

ÕdÕgìyá,Ö Þyãæ ÜdçqÙÜ 0F0^X ÚYÕÜd×:ìqÖðÜ:ç.ÙÜÚsÕ~ð rankK(δ]∂S∂x

<

rankK∂(S,h

(s1)

1 ,...,h(sp)p )

∂x

é "wåÙÛYÛÚYÞyàyìcÜ:ä¬ãìcÜ:àyìcÜ-ÖWíìqÙÜ:ÚYãÞqÕφ(δ, y, . . . , y(l) ð u, . . . , u(k)) = 0

Ù×dÖÖWíìyÚYØÙÛYÖ~ÞÜÜ:ãÙÞÖWíìqÙÜdÚãÞ

φ(y, . . . , y(l)ðu, . . . , u(k)) = 0

æQÚÜ:çyãìcÜδð.Ü:çyÖWÞÜ:çyÖ

pÚYÞyàyìcÜ:ä¬ãìcÜ:àyìcÜ²Öaíì.ÙÜ:ÚYãÞ.ÕkÚYÞÜ:çyÖDày×:ãcãåãåê4çyÖ~ã×:ÖWá öÙ×dÖ²ÖaíìyÚØÙÛYÖ~ÞÜkÜ:ãRÖWíìqÙÜ:ÚYãÞqÕ2ÞyãÜ¶èãÞÜdÙÚÞyÚYÞyëRÙÞÝßcÖ~ÛsÙÝLÕ2ãåÜ:çyÖDØÙ×:ÚsÙÔyÛYÖWÕa

ξi(y(si)i , y1, . . . , y

(s1−1)1 , . . . , y

(si−1)i , u, u, . . . , u(k)) = 0,ô<öW÷ù

å_ã×i = 1, . . . , p

éê4çyÖ~×dÖå_ã×:ÖðrankK

∂(S,h(s1)

1 ,...,h(sp)p )

∂x≤

s1 + · · · + sp = rankK(δ]∂S∂x

ðæQçqÚYèçè~ãÞÜ:×ÙßcÚsè³ÜÕÜ:çqÖ,ÙÕ<äÕgìyá,àcÜdÚãÞéLê4çyÚsÕFèãá1àqÛÖ~Ü:ÖWÕ ÜdçyÖâq×dÕgÜFàqÙ×:ÜFãåÜ:çyÖDày×dãLãå<éLìyàyàlãÕgÖ¼ÞqãæÜ:ç.ÙÜ,Ü:çqÖ~×:ÖÖ cÚsÕ<ÜÕ,ÙÜÛÖaÙÕgÜ,ãÞyÖá1ÖW×:ãá,ã×:äàyçyÚsèÆå_ìyÞqè³ÜdÚãÞ

φÕgìqèçÜdçqÙÜ

φ(δ, y, . . . , y(l), u, . . . , u(k)) = 0å_ã×*Õ:ãá,ÖpÞqãÞyÞyÖWëÙÜdÚØÖmÚYÞÜ:ÖWëÖW×dÕkÙÞqß

lðlæQçqÖ~×dÖ

δèWÙÞyÞyãÜÔlÖUãá,ÚÜgÜdÖWßé _ÖUæQÚYÛÛQÕ:çyãæ ÜdçqÙÜ 0^X ã× 0F0=X ápìqÕgÜ1ÔlÖUÜ:×dìyÖéÕdÕgìyá,ÖRÜdçyÖmãàyàlãÕ:ÚÜ:Öé.ê4çyÖ~ÞÜdçyÖmÖaíì.ÙÛYÚ£Ü<Ý

rankK(δ]∂S∂x

=

rankK∂(S,h

(s1)

1 ,...,h(sp)p )

∂x

ÙÞqß+ô<öøù Úá,àyÛYÝ,ÜdçqÙÜrankK

∂S∂x

=

rankK∂(S,h

(s1)

1 ,...,h(sp)p )

∂x

éñÖ~Þqè~Öðå_ã×FÖWÙèçi = 1, . . . , p

∂h(si)i

∂x∈ spanK

∂(h1, . . . , h(s1−1)1 , . . . , h

(si−1)i )

∂x

.ôùZFÝDÙÕdÕ:ìyá1àyÜ:ÚYãÞðÜdçyÖ2å_ìyÞqè³ÜdÚãÞqÕÚYÞ

S, h(s1)1 , . . . , h

(sp)p

ßcã4ÞyãÜèãÞÜdÙÚÞÙÞÝmßcÖWÛYÙÝÖWßmÚYÞyàyìyÜgäwØÙ×dÚYÙÔyÛYÖWÕWðaÙÞqßmçyÖWÞqèÖðaå_ã×2ÖWÙèdçi = 1, . . . , p

æFÖçqÙØÖÙm×dÖ~ÛsÙÜ:ÚYãÞ

dh(si)i +

m∑

r=1

γ∑

j=0

cj,rdu(j)r −

s1+···+si∑

j=1

ajdxj = 0 ,ô!cöù

æQçyÖ~×dÖaj , cj,r ∈ K

éê4çìqÕlÜdçyÖ2ÚÞyàqìcÜgäwãìcÜdàyìcÜÖWíìqÙÜdÚãÞqÕ2ô<ö1+ùå_ã×²Õgì.èdç#ÙpÕgÝcÕgÜ:ÖWá Ù×dÖãåÜ:çqÖå_ã×:á

ξi(h(si)i , h1, . . . , h

(s1−1)1 , . . . , h

(si−1)i , u, u, . . . , u(γ)) = 0,ôùæQÚ£ÜdçyãìcÜ1ÙÞÝßcÖWÛYÙÝcÕmÚÞÜ:çyÖ#ØÙ×:ÚsÙÔqÛÖaÕRÕ:ÚÞqè~Ö#Ù)ßcÖ~ÛsÙÝ+ÚÞÙØÙ×dÚsÙÔyÛYÖFæFãìyÛsßpá,ÖaÙÞÜ:çqÙÜ Õ:ãá,Ö4ãå.Ü:çyÖ

aj , cj,r

æFÖ~×dÖ²àlãÛÝäÞyãá,ÚsÙÛsÕpãåDßcÖWë×dÖ~ÖUÙÜ1ÛYÖWÙÕ<Ü1ãÞyÖUÚYÞδæQçyÚsèdçæ²ãìyÛsß ÔlÖ¼ÙèãÞÜ:×ÙßcÚsè³ÜdÚãÞéZ²ÝpÜ:çyÖz"zá,àyÛYÚYè~Ú£Ü ï ìyÞqèÜ:ÚYãÞê4çyÖ~ã×:ÖWá¼ðÜdçyÖ~×dÖÖ cÚYÕgÜFá,Ö~×dãá,ã×dàyçyÚsè²å_ìqÞqè³ÜdÚãÞqÕ

fi

Õ:ìqèçÜ:çqÙÜ

h(si)i = fi(h1, . . . , h

(s1−1)1 , . . . , h

(si−1)i , u, . . . , u(γ)) = 0,ô ù

å_ã×i = 1, . . . , p

écê4çyÚYÕFÚYá,àyÛÚYÖWÕ¶ÜdçqÙÜFå_ã×²ÖWÙèçki ≥ si

ðy(ki)iè~ÙÞÔlÖæQ×:ÚÜgÜdÖ~ÞÛYãcè~ÙÛYÛYÝ+ÙÕmÙå_ìyÞqè³ÜdÚãÞãå

y1, . . . , y(s1−1)1

ð. . . , y

(si−1)i , u, u, . . . , u(γ)

æQÚÜ:çqãìcÜkÙÞÝRßyÖ~ÛsÙaÝcÕÆÚYÞmÜdçyÖ²ØÙ×:ÚäÙÔyÛYÖWÕWé àyàyÛYÝLÚYÞyëÜ:çqÚYÕÜdãÜ:çyÖÖWíìqÙÛÚÜ<Ý

φ(δ, y, . . . , y(l), uð

. . . , u(k)) = 0æFÖDãÔcÜÙÚYÞ

φ(δ, y1, . . . , y(s1−1)1 , . . . , y(sp−1)

p , u, . . . , u(γ)) = 0ô!Lù

å]ã×+Õ:ãá,Ö á,Ö~×dãá,ã×dàyçyÚsèφé*ê4çyÚsÕ~ðDçyãæFÖWØÖ~×aðèãÞÜd×dÙßcÚYèÜdÕ

Ü:çyÖ ÛYÚYÞyÖWÙ×)ÚÞ.ßcÖ~àlÖ~Þ.ßcÖ~Þqè~Ö ãådh1, . . . , dh

(s1−1)1 , . . . , dhp

ð. . . , dh

(sp−1)p

ãØÖW×K(δ]

ÙÞqßèãá,àyÛÖ~Ü:ÖaÕRÜ:çyÖUÕ:ÖWèãÞqßà.Ù×:ÜãåÜ:çyÖDày×dãLãå<é¥

LÎ.Í _ 0 A@3$ 1 & 1 ./A, 5 E .FG E 5)L, ôgöaù J τ 021 ":5 < $*"F">&0 !A@G. 0 C $?7A": 0 E $@N 5*@G"\& 0 E .=A)+ 8H 021 . 1 $*@ 0 @' K .; 5 K .>' K . K $. 0 5@ φ(δ, y, . . . , y(l), u, . . . , u(k)) = 0 J .FG$. 0 @ 5" 1 δ0 @M$@M 1S1 #@B. 0 $*" O $& X=E τ 021 @N5*.B":5 < $"=">& 0 ?A@B. 0 C $?7A": J 1 & 1 .6A,ô<öaù < $*@(":5 < $*"F">& 7SV)+8$*" 021 SM$ 1 $*@ ; 1 & 1 ./#, X )+55 E Lìyàqà.ãÕgÖâq×Õ<ÜmÜ:çqÙÜmÜ:çyÖW×:ÖÖ cÚsÕ<ÜÕmÙÞÚÞqàyìcÜgäwãìcÜdàyìcÜÖWíìqÙÜdÚãÞ

φ(δ, y, . . . , y(l), u, . . . , u(k)) = 0ðaÜ:ç.ÙÜ2ÚYÞØãÛYØÖWÕ

δÚÞÙÞ1ÖaÕ:Õ:Ö~ÞÜ:ÚsÙÛyæ²ÙÝéLÚYÞqèÖyÙÞqß

uÙ×:Ö LÞyãæQÞô_á,Ö~×dãá,ã×:äàyçyÚYèaù4å_ìyÞqè³ÜdÚãÞqÕãå2Ü:ÚYá,Öðqå_ã×Ù,â% cÖWß

tðφÚsÕÙ,å]ìqÞqè³ÜdÚãÞãå

τæQÚÜ:çè~ãìyÞÜdÙÔqÛÝpáÙÞLÝ#~Ö~×dãÕWéê4çLìqÕWð

τÚsÕkÛYãcè~ÙÛYÛYÝmÚYßyÖ~ÞÜdÚ£âqäÙÔyÛYÖé

ÕdÕgìyá,ÖÜdçyÖ~×dÖ>ÚYÕ.Þyã¶ÚÞqàyìcÜgäwãìyÜ:àyìcÜÖWíìqÙÜdÚãÞφ(δ, y, . . . , y(l) ð

u, . . . , u(k)) = 0ðÜ:çqÙÜÚYÞØãÛYØÖaÕ

δÚYÞ*ÙÞDÖWÕdÕgÖ~ÞÜ:ÚsÙÛæ²ÙÝéê4çqÖ~Þ

Ü:çyÖ@ÚÞqàyìcÜgäwãìyÜ:àyìcÜÖWíìqÙÜ:ÚYãÞqÕξi(h

(si)i , h1, . . . , h

(s1−1)1 , . . .

ðh

(si−1)i , u, . . . , u(γ)) = 0

ði = 1, . . . , p

ßcã ÞyãÜUèãÞÜdÙÚÞßcÖ~ÛsÙaÝÖaß#ØÙ×dÚYÙÔyÛÖaÕ~écê4çìqÕWðyÛãyè~ÙÛYÛYÝðLÖaÙèç

h(si)i

èWÙÞ¼ÔlÖ@æQ×dÚ£Ü:äÜ:Ö~ÞóÙÕÙ5á,Ö~×dãá,ã×:àyçqÚYèå_ìyÞqè³ÜdÚãÞ

fi

ãåh1, . . . , h

(s1−1)1

ð. . . , h

(si−1)i , u, . . . , u(γ) ðaå_ã×2ëÖWÞyÖ~×dÚYèFèçyãÚsèÖaÕÆãå.ÚÞqÚ£ÜdÚYÙÛcèãÞcäßcÚ£ÜdÚãÞqÕ

ϕ ∈ Cé&_ÚÜ:çUÜdçyÖ*ÞqãÜdÙÜ:ÚYãÞUìqÕgÖaß#ÚÞ]cÖWè³ÜdÚãÞ ðLæFÖãÔcÜÙÚYÞ+ÜdçyÖå_ãÛYÛYãæQÚYÞyë+Ó \ käzÕ:ÝLÕgÜdÖ~á å_ã×RÜ:çqÖØÙ×dÚsÙÔyÛYÖWÕ

xi

ði = 1, . . . , s1 + · · · + sp

æQÚ£ÜdçÜ:çyÖÕdÙá,ÖUãìcÜ:àyìcÜÙÕpÜdçyÖã×dÚëÚÞqÙÛqÕ:ÝcÕ<ÜdÖ~á ôgöaùô_á,ãcßcìyÛYãmÜ:çqÖãìcÜ:àqìcÜFå_ìyÞqè³ÜdÚãÞqÕhi

å_ã×æQçyÚYèçsi = 0

ðçyÖWÞqèÖÜ:çyÖpù7a

˙x1 = x2

. . .

˙xs1= f1(x)

˙xs1+1 = xs1+2

. . .

˙xs1+s2= f2(x)

. . .

˙xs1+···+sp= fp(x)

y1 = x1

y2 = xs1+1

. . .

yp = x1+s1+···+sp−1

.ô!+ù

_+ÖRæQÚÛYÛÕ:çyãæÜdçqÙÜτÚYÞBô<öùFÚsÕ4ÞyãÜQÛYãcè~ÙÛÛYÝÚsßcÖ~ÞÜ:Úâ.ÙÔyÛYÖ@ÔLÝãÔqÕ:Ö~×dØÚYÞyë)ÜdçyÖUå_ãÛÛYãæQÚYÞyëÕ:ÝLá1á,Ö~Ü:×dÝBÚYÞØãÛØLÚYÞyëÜ:çyÖßcÖ~ÛsÙÝàqÙ×Ùá,ÖÜdÖ~×IÙÞqßRÜ:çqÖFå_ìyÞqè³ÜdÚãÞãåqÚYÞyÚÜ:ÚsÙÛcè~ãÞqßcÚÜdÚãÞqÕ~é ï Ú[ RÜdçyÖßcÖ~ÛsÙaÝRà.Ù×Ùá,ÖÜdÖ~×ÆÜ:ã

τ0 ∈ (0, T )ÙÞqßmÜdçyÖ²ÚYÞyÚÜ:ÚsÙÛyèãÞqßcÚÜ:ÚYãÞqÕÜ:ã

ϕ0 ∈ Cé%Ö~Ü

τ1 ∈ (0, T )ðτ1 6= τ0

ÙÞqßUèçyãLãÕgÖ*Ù1ßcÚ:]lÖW×gäÖ~ÞÜDè~ãÞÜ:ÚYÞìyãìqÕ4å_ìyÞqèÜ:ÚYãÞãå2ÚYÞyÚ£ÜdÚYÙÛÆèãÞqßcÚÜ:ÚYãÞqÕϕ1(t)

Õ:ìqèdç

Ü:çqÙÜkÜ:çqÖQØÙÛìqÖWÕ2ãåh

(l)k (0)

Ù×:Ö²Ü:çqÖÕdÙá,ÖRô_Ü:çyÚsÕ¶è~ÙÞ,ÔlÖßcãÞqÖÕgÚYÞqèÖ

h(l)k (t)

ðk = 1, . . . , p

ðl = 0, . . . , si − 1

ßcãÞyãÜ@ÚYÞcäØãÛYØÖßcÖ~ÛsÙÝÖWßÚYÞyàyìcÜ:ä¬ØÙ×dÚYÙÔyÛYÖWÕ, ÖWÖ~àyÚYÞyë

h(l)k (0)

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x(t) = f(x(t), x(t− τ), u, u(t− τ))

y(t) = x(t).

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yðy = f(y, y(t − τ), u, u(t − τ))

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x1 = x3x2(t− τ) + u

x2 = x4x2(t− τ)

x3 = 0

x4 = 0

y = x1

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y = x3x4δ2x2 + u

y(3) = x3x24δ

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∂(S,h(s1)

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∂x=

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0 x3x24δ

3 x24δ

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(y− u)(δy− δu)− (y(3)− u)(δy−δu) = 0

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(y(t0) −u(t0))(y(t0 − τ)− u(t0 − τ))− (y(3)(t0)− u(t0))(y(t0 −τ)− u(t0 − τ)) = 0

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ZH$,V'N": FãÞqÕ:ÚsßcÖ~× ÜdçyÖ*ÕgÝcÕgÜ:Ö~á

x1 = x22(t− τ)

x2 = x2

y = x1

ô ù

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2

y = 2δx2δx2 = 2(δx2)2

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∂(S,h(s1)

1 )

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1 0

0 2δ(x2)δ

0 4δ(x2)δ

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ϕ1(t) = cðc ∈ R

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ð~ÛÖaÙßyÕÜdã4Ü:çyÖ¶ÕgãÛYìcÜ:ÚYãÞx1(t) = e2t

2 +c− 12

ðx2(t) = et+τ

ð∀t ≥ 0

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W(_w¨W¤¯¨Wª|º,2Dº A S7X6T6E:ºZ8«´w¥< ± ¡£¢ ¤¥<¡§¦c¨~©¡|ª|¡|¥«¼©¨W´z ³¢¼a¤¼¥< ¯c³µ2 d°I´z :°<¡| Í d½¯¨W¤´z¡|W¤RW®c¥< ´zWª|¾¥<¡|a¤qº SUTWVYXZ:[RTWV]\_^<Taà` \]uah:^d\eZ:fc^gZ:h<iqc F&USd¹R1Rº

Paper III

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Ä ¦ ¿qÅÆ~Ç ÀµÈ¥É ¿ÇÅª~ ¤ U U £M Y ¹ 83ª£ ª ª ¤ Gq £D5g 8 GsÁL £ µ8½Q¢ J ª Q£ § l¹ 1 £«¢ 1 ¢ x¢gª 5 7M Gg °G £g¤Ê U ¨ §Y £g Q2 £ ª£ G ª ª ¤ © ª£ ª¢YGª£g¤ ª¤£ £ g¢ ª £·ËnsMM µ ¦ª©aÌUÍQÍUÎ?Ïd© M U Qs s£ ¢gªs£ ª¤U£ sª£g¤5Ë Â 1 ¤§ U ¦ª©ÌÍÍIÌÏd© ¨ §g ¨ §g J s £gG ¨ §g ¨ §g J s £ G ª Ë Â? ¹ ªg§ Qg©ÌÍÍ?ÐUÏd© §g^sSQ] ¨ £ËnÑ § Á 5 £Y ÂGl¹ ¹L© Ä8ÒÒÒ Ïd© §g G £ ¨ ¯ Ë<Ó* 2 ¦s©AÌUÍQÍÍYÔ(Â Qs1£Õ

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bM¹U ª¢g 1 £ &x Q¢g£Y ? ¨ Ub M ª J ª £W °§g §! £ M §g G ²G q8©QYQ 8M ¦ ¤ ¦ Q£ª Qq ½I¢ 9 ¨g¨ l²?ª J s Q£Ë 1 §g ? ¯ ? £g¯ ¦ Ë Ä8Ò õ Ò ÏÏ¦

ÌG¦ Ñ ò ¼ Ñ ¿ É9 Ó ³ Æ Ñ ¼¸ ÇÆ~¾

¿£ §g ð ª £°7°sªM¤ ¹ g s £Y Q °§ J/ ª ¨ ªhÝ8ÊG ¨ ª £4 §YM §Y G £G ¶1 §gðG £ s´ U ªª q G l ¨ x £g £gªª£g UG 1© g ¹ ª ¨ 8 ª£ £g¤Q¢g ª ¹J7 £Y¸ 1£g£ ¤2Ë ÌUÍÍ?ÐUÏ¦ ¸ x * §g J x A M ª¤ ¢ U §g1 U U M £ 1¦

zWy QX`DTn[ pbmJ[W`?t4TWVYXQX^sSUmgSJiSJZX TX i~pbioJZ XL^[<][WT[àGT\voJGTaGT9X YSJT\[<oJ`Yc* ª£ gG¢ §g §Y © ° UNI/ £² ¨ ª U3 ª ¨ ªµG1 l gG a° § 3 ¨ £YGª£g¤s£ ¨ ¢ ¢ ¨ ¢ ·q ¢ d ¢ -x §g ¨ ¢ ¨ 7 U ¨ Ub M q ª U ª £ ¦ §gM GG1D§ ° U ¹J U ª 1©

x1 £Yx2© ° 1¤¢g U ¨ k1

£Dk2©9 £Y

£g ª \ ¤ ¨ M 1© τ¦ §g Q£ Qsª1Õs£ ¨ ¢ ¹JU ªM G £g 1 I

u £Y §gM ¢ 1ð J

Iy¦ ¿£g £YG s Q£ § l¹ G Ý£g1x §g

s£ ¨ ¢ 1© u0(t) £Y qbJ ¹J U s ©

ϕ(t)© °§g §ª£ §g

2 U9G 8 GsÁL £ ¡1½I¢ U ª £ 3x¢g£ d ª £ ©G ª£g¤ §gÕ§g q §g G 1 J ª Qq

τ¢g£g bÕY ¯s£ ªM ¦3 ª ¨ Ub M b £Y+s£g £Yg s Q£ x & §g qbJ ¹UU ª ¢g£g¯?£g l°£ © ¢ °9 Q ¢gM § J

τ3¢ q ¯ ¹J ª¢g ª£ §g

s£ ¹J [0, T )

© °§g T ¯?£g J°£ ¦

u!?SJZ"a^sXMy#

x1(t) = k1x2(t− τ) + u(t)x2(t) = k2x2(t− τ)y(t) = x1(t)x(t) = ϕ(t), t ∈ [−τ, 0]u(t) = u0(t), t ∈ [−T, 0]

Ë Ä Ï

¯?ª£g¤ sM9G ¹JJ ¹ U §g ¢ ¨ ¢ ¡J¡ ¤Q ¹ 1£¨ Qs£ ª£ sM © °9 Gb ª£·8½Q¢ J ª £ x 2 §Y U ¹JU ª £Y ¨ M 1

y(t) = k1x2(t− τ) + u(t)y(t) = k1k2x2(t− 2τ) + u(t)

y(3)(t) = k1k22x2(t− 3τ) + u(t)

Ë ÌQÏ

³g §g ¹ 1½I¢ U ª £ 1© ° £² b £²? £ (ª£ ¨ ¢ ¢ ¨ ¢ ¨ £ U ª £& U §g Gq ©¤ ¹ £ I! §Y¥ª£ ¨ ¢ ¢ ¨ ¢ 1½I¢ J s Q£

(y(t)− u(t))(y(t− τ)− u(t− τ))−

−(y(3)(t)− u(t))(y(t− τ)− u(t− τ)) = 0

Ë ïIÏ

Â s£ §gs£ ¨ ¢ 2 £Y Q¢ ¨ ¢ U M¯?£g J°£ x¢g£ ª £ U ªM © ¹U ª¢ J s£Y¤ §g U ¹ M1½I¢ J s Q£ J3 §g U £ ¨ Qs£ t0

ª£ sM £ s ¢ 1 ¢g J §g ª \ ¤ ¨ Ub M τ

© x ² ¨ s ©I Ë<£I¢gM s ª GÏ Ý£YGª£g¤ §g)¶ U §gx¢g£ d ª £ξt0(τ) =

(y(t0)− u(t0))(y(t0 − τ))− u(t0 − τ)− (y(3)(t0)−u(t0))(y(t0−τ)−u(t0−τ))

¦Q £Msªs¢ qU ª £ © °§ l¹ 9¢ 8 §gGsÁa 1£ G8½Q¢ J ª £ Q ¹ ª£7Ñ J U Ë Â § ¨ s£g £Y-§Y ¨ £ ©ÌUÍQÍ Ä Ï/ s3¢g J £3 ¢ ¨ ¢ x N §g ¹ G ? §Y ? ª£g¤

k1 = −2, k2 = −3, ϕ1(t) = t + 1, ϕ2(t) = t2 +1, u(t) = t

£Yτ = 1

£Y ¨ ª 1 §g)x¢g£ ª £ξ6(τ) = (y(6)−u(6))(y(6−τ)−u(6−τ)−(y(3)(6)−u(6))(y(6−τ)−u(6−τ))

x τª£ §g9ª£ ¹J

[0, 2]¦

² ¨ 8 ©J §g x¢g£ d ª £ ¯ ¡ §g ¹J s¢Y¶ x τ = 1

¦

0 2 4 6 8−30

−20

−10

0

10

20

30

40

50A plot of y(t)

t

y(t)

0 0.5 1 1.5 2−3000

−2000

−1000

0

1000

2000

3000

4000

5000

6000

A plot of the function ξ6(τ)

τ

ξ

³ s¤ ¦ Ä ¦ r VgXMoJGTaGT y(t) SJà!TxVgXpbG`L]T\[<oJ` ξ6(τ)

Çµ£ §g §g § £D © !1 £ §Y 1 § UM §gª£ ¨ ¢ \ Q¢ ¨ ¢ 1½I¢ J s Q£ U µGq G £g £ ª£τUA ª ¦ §g1£ §g ª£GÝ£g £ ¥¹J ª¢g ( § U

¨ GG¢ §g M ¢ ¨ ¢ £Y τ1 £g£g ¥G £ s

Ý82 §g l¹J s s UY¦ §g G M £ U 1I! §g¥x ªs J°ª£g¤ ² ¨ ª u!?SJZ"a^ªX3z#

x1 = x22(t− τ)

x2 = x2

y = x1

x(t) = ϕ(t), t ∈ [−τ, 0]

Ë ÎIÏ

É9 ¢Y U ª£g¤ ª G ¹JU ¹ U §g Q¢ ¨ ¢ x¢g£ ª £ U ¹ © °/ Gb ª£y = x2

2(t− τ)y = 2(x2(t− τ))2

Ë Ï £7 Q¢ ¨ ¢ 1½I¢ U ª £- Uµª l°9 G ¤ 1 Ëx G ¹J ª £ Ï x ! §g ¹ G

y(t) − 2y(t) = 0©

°§g §!G ? £g ª£ ¹ ¹ ~G l? §g ¹UU ª 1¦l¸ £ § U

τ £g G £ h´ U ªx §g ² ¨ ª ©I Y ¹ ª£g¤ §gx ªs J°s£Y¤ MM ª£ ¹ Q ¹ ª£g¤ §g

x¢g£ ª £ Uª£g Q£YG ª £ ϕ £D

τ¦³ ! £

§g ¥ Uτ©Y ª£g¤

ϕ1(t) = cϕ2(t) = et+τ , t ∈ [−τ, 0] ,

Ë IÏ°§g

c M £ £ 8© s 9 §g Qs¢ ª £

x1(t) =e2t

2+ c−

1

2x2(t) = et+τ

, ∀t ≥ 0 .Ë ÐÏ

Â ª£ y(t) = x1(t)

© ! ª M § Uτ1 £g£g

G £ sÝ1 §g/ Q¢ ¨ ¢ 1¦

zªz ¡[W`LXSUidvSJ^ tIX1 iSJ[n][<X `YT\[ SQ[W^[xTnm&] id[WTX id[<S§g U ¹ &° 8 U/G ªGª£g¤ §YG £ s´ U ªª q U §g G 8 ¨ M 1© U ¢ © G ¢g x ¨ ª J 1®M GG1 ° §Ü £ +¹UU s £Y ¨ Ub M b © °§g1£ §g4² ¨ ª ª£ ¨ ¢ \ Q¢ ¨ ¢ 1½I¢ J s Q£ £g£g µ 3 Gb ª£g1 ¦D¸ 3§ l¹ §Y x W 3¢Y U 1Êªg £ h´ U ªs qÊ Y 1Ê £Êªª£g U ª¤ gbM ¢Y U ª £ °§g § £ ¨ x 8 W M ¨ ª²M Gg U §g¥x

x(t) = f(x(t), x(t− τ), u, u(t− τ))y(t) = h(x(t), x(t− τ), u, u(t− τ))x(t) = ϕ(t), t ∈ [−τ, 0]u(t) = u0(t), t ∈ [−T, 0]

,Ë õIÏ

°§g x ∈ R

nG1£g §g qbJ ¹UU ª 1©

u ∈R

m §g s£ ¨ ¢ 5 £Y y ∈ R

p - §g Q¢ ¨ ¢ 1¦

§g!¢g£g¯?£g J°£ £ £ / sM G1 l G1£g 1 Iτ ∈ [0, T ), T ∈ R

¦ §g £ ª f £Y

h

Q ¨ §g s£ §g ¡U ¤¢gM1£ N £Y ϕ : [−τ, 0] →R

n £·¢g£Y¯I£Y l°£ £ ª£I¢g Q¢ x¢Y£ d s Q£ Us£g

£Yg s Q£ ¦ §Y s£g x¢g£ d ª £ x §g¹JU ª

x G1£g 1 I

C := C([−τ, 0], Rn)¦(

Q ¨ §g x¢g£ d s Q£ u(t) 1 ªs8 £ GM s

s£ ¨ ¢ h §gGsÁa £ 1½I¢ J s Q£ U ¹ GM 3¢g£gª½I¢g ª¢ ª £ ¦ §Y ª ¢ §/ª£ ¨ ¢ x¢g£ ª £ G £g 1 I

CU

¦ Ç D ¹ § U9 §g U ¹ ? 1W sª l° x gG ¨ Ub M b1© °§Y § £ ª ¨ £ G 1 J ¹J U ª ° § sMG ¹UJ ¹ ¶1 ¦³ ªª l°ª£g¤ §g£g bJ ª £ £YM U §g1 J ª£g¤ U3Ñ ? ¤ ) ¦ Ë ÌUÍQÍÍQÏ© Ñ ½Q¢Y ¶Ñ U £g ¶ ) ¦Ë ÌUÍQÍÍIÏd© µ ¦ Ë ÌUÍQÍQÌQÏ £Y § £g¤M ¦ Ë ÌÍÍIÏd©s

K §YMÝ1ª 9M M ¨ §Y x¢Y£ d s Q£

Ý£g £I¢g U ¹J U ª Qx(t − kτ), u(t −

kτ), . . . , u(l)(t−kτ), k, l ∈ Z+

£YMs δG1£g

§g ªM §gs ¨ bJ 1© δ(ξ(t)) = ξ(t − τ)©ξ(t) ∈

K¦ Ó(

K(δ]G1£g §g ¨ £g M U §g

W

a(δ] = a0(t) + a1(t)δ + · · ·+ ara(t)δra ,

Ë Ò Ï°§g

aj(t) ∈ K¦P¸ § gG ª £ G Ý£g1 Q ¢ ¢

£Y3¢g ¨ ª 1J s Q£&¤Q ¹ 1£ ?

a(δ]b(δ] =

ra+rb∑

k=0

i≤ra,j≤rb∑

i+j=k

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£ £Y ª Ç Êg ª£ ¦ §g J ª ¨ sª § UA §g b £Y¯¥ U M Gg¢gs9 ¹ K(δ]

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φ(x(t− iτ), u(t−jτ), . . .

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©0 ≤ i, j ≤ k

©l ≥ 0

ª£K £Y

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∑

i κixdx(t−iτ)+

∑

ij νidu(j)(t−iτ)s£M

GÝ£Y1&s£ §Y/£ J ¢ L° 8L

φ =

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+l∑

r=0

k∑

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∂φ

∂u(r)(t− jτ)u(r+1)(t− jτ)

ω =∑

i

κixdx(t− iτ) +

∑

ij

νidu(j)(t− iτ)+

+∑

i

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∑

ij

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f £

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U s²° §) £ ª (

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Ë Ä ÌÏ

À £g Is1

§gs £Y £g£g1¤ U ¹ !s£ ¤Q ¢ § § J


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si

§gMs £Y £g£g1¤ J ¹ ª£ 1¤ ¢ § § J


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dx2

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x1 = −k1x1u/k7 + 2k4x3(t− τ)x2 = k1x1u/k7 − k2x

22

x3 = −k3x3 + 0.5k2x22

x4 = k3x3 − k4x3(t− τ)y1 = k5(x2 + 2x3)y2 = k6(x1 + x2 + 2x3) ,

Ë Ä1Ò Ï

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y2 = 2k6(k4δx3 − k3x3)Ë ÌUÍIÏ

y2 = 2k6(−k3k4δx3 + 0.5k2k4(δx2)2 + k2

3x3 −

−0.5k2k3x22)

y(3)2 = k6(−2k3k2x2k1x1u + 2k3k

22x

32k7 −

−2k33k7x3 + k2

3k7k2x22 + 2k4k2δx2k1δx1δu−

−2k4k22δx

32k7 + 2k4k

23k7δx3 −

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¸ £4£g J° § Jδu

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τ G £ s´ U ª Q gs£g¤

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P £Y

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P = αpM − µpPy1 = My2 = P ,

Ë Ì Ä Ï

¸ / Gb ª£ §g/1½I¢ U ª £

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(3)1 (δ2P, δP, δM,P,M, . . . )

Ë ÌUÎIÏ

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Ë Ì Ï. . .

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© §Y ∂h(j)1

∂M

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∂P

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j − 1¦ §g

M £ ¡ § JrankK

∂(S,h1,...,h(8)1 )

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Paper IV

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Æ&D¡G$fff L °G¡+GÛTÈ9fw9.

x(t) = f(x(t), x(t− τ ), u, u(t− τ ))y(t) = h(x(t), x(t− τ ))x(t) = ϕ(t), t ∈ [−maxiτi, 0]u(t) = u0(t), t ∈ [−T, 0]

,Í ¿ Ù

¨ f x ∈ RnGfT9f9T«TT9frµ

u ∈ Rm¡9f(ff¦GX

y ∈ Rp¡9f(P¦Gf¦G§¯*f(¦fG

þ?f ¨ âX9P#L °G¡GT9±GXTÚI)9f« Pτ = (τ1, . . . , τ`)

µτi ∈ [0, T ), T ∈ R

§5*X ®Gf99rP

x(t − τ ) TX9f1 ,L«TT9f

x(t − τ1), . . . , x(t − τ`)µ-X³TXrP¢¦Xr.w

u§

*f P9TfTX

hT9 PL9ff¡r,9f

T9¢¦X Pa.TXϕ : [−maxi(τi), 0] → R

n

¦ffþ?f ¨ PPI¦X¦Xw¦fX TXfr¡TG DGg9rPX§*f# T*rXg9¯w¦fDcDwP"f#«J¡Tfx¡

GfT9,?C := C([−maxi(τi), 0], R

n)§O/L9T

L9ff¡£ff¦fw¦fXc9u(t)

¡r"Tf ¡9fff¦Gr*XGr¤59P9¥I¦XTà`k«#PG ra¦ff¡¥I¦f&Pr¦G9rP-§*f& MTfrP¦Xa"ff¦Gw¦fDcD¡GfT9(?

CU

§ÃfT`XP?Ópc«¯`f¡ ¨ 9þ¯"I«¢PTX±f9`p1T"rfT8GP9gDfrrpÈTfG¡X9 9

τi

§À°P9¦fg9r«Pµτi

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W_XZVS[Z<nJ_³x *fG¡(XTaTL 9a τT9"(

`Gr¡GIrDTXrJτ0 ∈ (0, T )`

rG9f9 ®G¡paT `L

W 3 τ0

µW ⊂ [0, T )`

µ¦XaXT ∀τ1 ∈W : τ1 6= τ0

µ ∀ϕ0, ϕ1 ∈ Cµ`9f9 ®G¡p

t ≥ 0TX

u ∈ CU

¦Xa9XJy(t, ϕ1, u, τ1) 6= y(t, ϕ0, u, τ0)

µ¨ f y(t, ϕ, u, τ )

Gf¯X8XTaTL9© ¦f f¦G¯wPA9f*fr¡TGw¦fXc9rP

ϕµJXf ¡9fff¦G

uXG¡k?

τ§

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Í ×TØPØØIÙ Ñ ÞL 8m§ Í ×TØPØP×PÙ&TXß5XTf¢L 8m§ Í ×ØØáIÙcµr K `LXLX³L9LPfXw¦XXc9rPX"*Xfg9?¦f`T*«JT9¡TfwP x(t − iτ ), u(t −iτ ), . . . , u(l)(t− iτ ), i = (i1, . . . , i`), ij , l ∈ Z

+ µ¨ f9 ¨ Xk«8GXT i1τ1 + · · ·+ i`τ`

Iiτ§I¶-

K(δ]GXT.9f± T`IXL¡

δ1, . . . , δ`¨ g9â?#rP9Lw9 K §&*f¡#¡#·fPX P ¦G9T«9f¢Xµ ¨ fPfGr(¡*f Xf¦X¦XT[µ¨ fr"¦fg9rXr¡T.¢«I

a(δ]b(δ] =

ra+rb∑

k=0

i≤ra,j≤rb∑

i+j=k

ai(t)bj(t−iτ )δk ,Í ×PÙ

¨ f9 ¨ 8D«Gf δk11 . . . δk`

`

?δk§?*f89f¢

K(δ]#Á8? 9f9ÚTXÚ³ V.ÃGPL-§*X

T Xr*DJf"aTfþ#£w9" fG¦fJ«K(δ]

¨ g °G Df Í Æ&Pf-µ ¿ÏIÐG¿ Ù §X¶- M GfT9fLGG¦f

spanK(δ]dξ : ξ ∈ K §*f P¦f9T5¦ffLGf¦fr N r M ¡9f¦XfLGG¦f N = w ∈M : ∃a(δ] = a(δ1, . . . , δ`] ∈ K(δ]

µa(δ]w ∈ N §

N ¡f¢P¦fX fG¦f M PaTff¢ NTX Xk«?rX¢"aTfþ¥P¦DTG9rankK(δ]N

Í Þ" &m§µ×TØØI×Ù §Ä8r¤`P¡T àw¦fX X

φ(x(t − iτ ), u(t −jτ ), . . . , u(l)(t − jτ ))

K X f mwPω =

∑

i κixdx(t − iτ ) +∑

j,r νjdu(r)(t − jτ ) M

G Xf9fXJ9¦faT ¨ Í Þ8¡8£m§µJ×TØPØP× Ñ ß`Xf¢ 8T[§rµD×TØPØáPÙ

φ =∑

i

∂φ

∂x(t− iτ )δif +

+∑

j,r

∂φ

∂u(r)(t− jτ )u(r+1)(t− jτ )

Í Ù

ω =∑

i

κixdx(t− iτ ) +∑

j,r

νjdu(r)(t− jτ ) +

+∑

i

κixdδif +

∑

j,r

νjdu(r+1)(t− jτ )Í Ù

Ä8 Df

X = spanK(δ]dx Í ÿ Ù

Yk = spanK(δ]dy, dy, . . . , dy(k−1) Í áIÙU = spanK(δ]du, du, . . . .

Í Ð Ù*f

(Yk +U)∩X = (Yn +U)∩X w k ≥ nX

rankK(δ](Yn + U) ∩ X ≤ nÍ ÞLm§µX×ØØI×Ùc§

§ Ö A¯¶ÀpãÀpÁ8¯*ÀÃÁÀ°Êf¡± ¨ ² D¡G.9f²fPf TLG 9Tff¢·T1ff¦f ¦G9f¦G9f9PaJ9rPàwP fpaJ XwP T9rL Gkà PP9Gp9.§*f·XPfr D³` 9T$wà`?fL¡TG Ì aL1 #T[§ Í ¿ÏÏ Ù §A¬ ¨ f ¨XT(wP)Gp9 TfwP Í ¿ Ù#X·T ¨ G ®G9µTJ¯Pp¯GTµJ ¯Dff¦G [P¦GX¦GGkI Gg¤59I¡T-¥I¦XTX*TM9fw9.

F (δ, y, . . . , y(k), u, . . . , u(J)) :=

= F (y(t− i0τ ), . . . , y(k)(t− ikτ ),

u(t− j0τ ), . . . , u(J)(t− jlτ )) = 0 ,

Í Ô Ù

¦X9âXT(I)X(y(t), u(t)) ¨ f¡9ÝPr«P Í ¿ ÙcµT¡)T¡pD Í Ô Ù µVP

t¦X9ÝXT.r"G9r«TJ9r«

I««. II¦fP¦X§D*f"w¦fXc9rPF¡L9T

L9ff¡r.gaT9¢¦fLI9§

q UfWanJhWY x «#"Gp9 `9fwP Í ¿ ÙcµX ®G¡paP9¢J ≥ 0

TX.T.`.GD¦fX VT

C × CJ+1U

µ¦D9ÈXT#)9ffr¢PI`Pf?fT"I1`PrI#T

VµX9 ®G¡paÚrfX¦G ¦G9f¦G

9f9PaJ9rPMfGp96TMXw9 Í Ô Ùc§

hnno *f,fGT¡ fG9TÚf,X?T*fP×G§ë×G§ ¿ §MrÚÆ&PP( T[§ Í ¿ÏPÏÏ ÙwP9fTDT¢¦D&¦Xg*wPÃÄ Gp9§¶-

f`ÚT

r frLXX «cP ¨ r/P9

fj ∈ K §¶ ∂f∂x

Gf9fr × n

J9r® ¨ g9I9r(

∂f

∂x

)

j,i

=∑

k

∂fj

∂xi(t− kτ )δk ∈ K(δ] .

Í Ï Ù

Äf(Is19f(rXff¢PT«#rI¢P ¦X9

DJ


(s1−1)1 )

∂x= rankK(δ]

∂(h1, ..., h(s1)1 )

∂x.

Í ¿ ØIÙÀ ∂h1

∂x≡ 0

f ¨ ,GXf s1 = 0§À°XG¦Dc«Pµ

w1 < i ≤ p

GfT9 Isi

fLrXff¢PJ9«P¢¦Xa(9XJ


(s1−1)1 , . . . , h

(si−1)i )

∂x=

= rankK(δ]∂(h1, ..., h

(s1−1)1 , . . . , h

(si)i )

∂x.

Í ¿¿ Ù

¶-S = (h1, ..., h

(s1−1)1 , . . . , h

(sp−1)p )

µ ¨ f9 hiG?*fT8Tf`T*rsi = 0

§X*f

rankK(δ]∂S

∂x= s1 + · · ·+ sp = K ≤ n .

Í ¿ ×ÙÀ

K < nµ9f9â®G²L9LPfX)w¦fX X

g1(δ, x), . . . , gn−K(δ, x)¦Xa(9XJ

rankK(δ]∂(S,g1,...,gn−K)

∂x= n

§Ì LfgpµfP9fG¦X 9f"fTaJ9rP

x1 = h1

. . .

xs1= h

(s1−1)1

xs1+1 = h2

. . .

xs1+s2= h

(s2−1)2

. . .

xs1+···+sp= h(sp−1)

p

xs1+···+sp+1 = g1

. . .

xn = gn−K

.Í ¿ Ù

*f1Pf [w9Ldxi, i = 1, . . . , n

f ¨ w9 X X §&*f,G XXg9rPâT si

Í ¥I¦XT Í ¿¿ ÙpÙrLfr*XT ∂h

(si)

i

∂x

¡*

spanK(δ]

∂(h1, . . . , h(s1−1)1 , . . . , h

(si−1)i )

∂x

.

Í ¿ Ù*I¦X)f ®GÈfX©9`IXL¡

bi(δ] ∈K(δ]

µi = 1, . . . , p

¦D969XJbi(δ]

∂h(si)

i

∂x

¡È

spanK(δ]

∂(h1,...,h(s1−1)

1 ,...,h(si−1)

i)

∂x

§X*f9 wPPµ

bi(δ)dh(si)i +

∑mr=1

∑Jj=0 cj,r(δ)du

(j)r

∈ spanK(δ]dx1, . . . , dxs1+···+si µXwL J ≥ 0

µ¨ f9 J

¡9f#fr¢PfG«JJ9«LTuTf`T9rX¢

rfw¦fDcDrSTX

cj,r(δ] ∈ K(δ]§ Î Xµ

bi(δ]dh(si)i +

m∑

r=1

γ∑

j=0

cj,r(δ]du(j)r −

−s1+···+si∑

j=1

aj(δ]dxj = 0 ,

Í ¿kÿ Ù

VPP aj(δ] ∈ K(δ]

§ Ö X )Tw¦fXc9rPXT99¦fLÛLP PXf1TX ¨ )X«P) II¦fP¦XG`DGX&wM9f¦ff¦G¯9fff¦GTXrfr¡TV¦XXc9rP-µfÚT`J«²¥I¦XTgp f¡f T`GX

C×CJ+1U

§P*f VXXLGT5¥I¦XT Í ¿kÿ ÙcµT`f¢"¥I¦X?9©9XµJ¡ PPf [w9 M §ff?f¢.f Ò XLrLLXµ ¨ LPG9rV¦XXc9rPX

ξi(t) ∈ K¦X9,XT

dξi = bi(δ]dh(si)i +

∑mr=1

∑Jj=0 cj,r(δ]du

(j)r −∑s1+···+si

j=1 aj(δ]dxj

TXξi(δ, h

(si)i , x, u, . . . , u(J)) = 0

µ²VPai =

1, . . . , p§-*fLw¦fXc9

ξi

G?fG`XPxj

µj > s1 + · · ·+ si

µIX dξi = 0 ¨ P¦fLf PP9r9

dxj

µj > s1 + · · · + si

¨ f¡a·¡" `IfrI Í ¿kÿ Ù±G¦X²9 f)«J¡frdx

`f¢ÝrfT9rrXG`XfPàJ« K(δ]

?$G Dfg9rP-§"*?¦Xµ ¨ X«PG9rf T

ξi(δ, h(si)i , x1, . . . , xs1+···+si

, u, . . . , u(J)) = 0 ,Í ¿ áPÙ¨ fa 9¢f ¨ g9 Í ¿ Ù f9GG¦D T rff¦f ¦G9f¦G¥I¦XT

ξi(δ, y(si)i , y1, . . . , y

(s1−1)1 , . . . , y

(si−1)i ,

u, . . . , u(J)) = 0.

Í ¿JÐ Ù

*f 9¦fw P9i, 1 ≤ i ≤ p

¦fr9rf¢rprff¦f ¦G9f¦G8¥I¦XJ9rPX*TMXw9 Í Ô Ùc§

¥

§*ÀpÄÁ*À Ì Àp Àp¶-Àp ÜÃ Ì Î âÄ¶M Ò Â8ã Â Ö

*fc9·G ¨ g9³ff9Xr Tf¡Gf rDTfgpàT9f(L °G¡àXTaTLaτi, i =

1, . . . , ` Í ¿ Ùc§¶-fT r¢X9T 9g99w9¦G

T-µX""f&rfTw9ÊTGXff¦G [P¦GX¦G¥I¦XJ9rPX Í ¿ÿ ÙwP9f"f9«?rP¦X* -§

*f"¥I¦XTX Í ¿kÿ Ù&¢«

ai(δ]dy(si)i +

m∑

r=1

γ∑

j=0

cj,r(δ]du(j)r =

=

i∑

l=1

sl−1∑

j=0

ai,l,j(δ]dy(j)l ,

Í ¿Ô Ù

¬âr f)rI¢faTr°Pµ ¨ P¦fL±DJ(9f`?fL¡T¡ai(δ)

P$fÈ V [DTX ¡GÈT#9fT`J«/¥I¦XTXÛT9/9G¦XrXrP§¶

ai(δ] =∑

k aikδkµ

cj,r(δ] =∑

k cj,rkδkX

ai,l,j(δ] =∑

k ai,l,jkδk§£Äfr&9ffg¤59I LXL

δkf`f¢T`J«I

∆i1, . . . ,∆iq

TX±9f Í G ¢aÙ8rfT" PfrXTDT

τ1, . . . , τ`

DJf9f9PI

Ti1, . . . , Tiq

§OÀLf 99Lrai(δ]

µcj,r(δ]

ai,l,j(δ]

¡·`?fL¡Tδ

G¢9.©Dµ£9XJµ¯9f±rXf¦G ¦G9f¦G(¥I¦XTX PPaT¦fXG¡k"«TT9frµ 9f ¨ ∆i0

¥I¦X

δ0µ ¨ f δ0

GXTf¡GIr°`9TµTDX 99`DGrX¢

Ti0¡"©D§OÀ¡9f PfrDJ

DTi0, . . . , Tiq

τ1, . . . , τ`

DJ"f Lf9fG5¡GIrDTXrgp(TMX"T§¶-

∆i0

`XLXLδkr

ai(δ] ¨ rLrXG ®kÍ aGÊTV

k1, . . . , k`

Ù (g,¡g9f¥I¦X9

∆i0PL#TLf¢X

∆i1, . . . ,∆iq

§*fff¦G [P¦Gf¦G*¥P¦DJD9`Xfrf¢" Í ¿Ô ÙGr` ¨ 9g

y(si)i = fi(∆

−1i0

∆i1, . . . ,∆−1i0

∆iq, y1, . . . , y(s1−1)1 ,

. . . , yi, . . . , y(si−1)i , y

(si)i , u, . . . , u(γ)) Í ¿Ï Ù

yi(si)(t) = fi(y1(t), . . . , y

(si−1)i (t), u(t), . . . , u(γ)(t)

y(s1−1)1 (t− Ti1 + Ti0), . . . , y

(si)i (t− Ti1 + Ti0),

u(t− Ti1 + Ti0), . . . , u(γ)(t− Ti1 + Ti0),

. . . ,

y(s1−1)1 (t− Tiq + Ti0), . . . , y

(si)i (t− Tiq + Ti0),

u(t− Tiq + Ti0), . . . , u(γ)(t− Tiq + Ti0)). Í ×TØIÙ

¶-(T11 − T10

, . . . , T1q − T10, . . . , Tp1 − Tp0

,

. . . , Tpq − Tp0)tr = M(τ1, . . . , τ`)

tr ,

Í × ¿ Ù

¨ f M¡

(1q + · · · + pq) × `P9¢J9r®

TDvtr

Gff aTD`ITv§O¬,f ¨w9¦f¡J9fGP9gDfrrp,r¡#wP

τ1, . . . , τ`(9fwr ¨ f¢ f9`IrP-

hna¸fnkbcZVS[Z<nT_àx À Mf Xf,P`k« Í × ¿ Ù8TD

y(si+j)i (t)

¡XTfP9Tr¥I¦XT¯©wI0 ≤ j < iq − 1

µi = 1, . . . , p µ9f τ1, . . . , τ`T9(Gr·¡GIrDTXr.¢f9TrPµAgTX1Xràr

rank(M) = `§

hnno A¶- Xap rank(M) < `µDJµ-9f9T9

GXfrr LIτi

¨ XaÊ¢Pr«³Xà9TL·rfT

fXTXτ := Ti1 − Ti0 , . . . , Tiq − Ti0

µi = 1, . . . , p

X±9fLff¦G [P¦Gf¦f"¥I¦XJ9rPX Í ×TØIÙTfXT` ¦X±¡GPrw±f

τi

§5¬³9Jf

` ≥ 2TD9 w9#8f¢¦Xk«TLX.¬fI`9¢

Í ×TØPØ Ð ÙOw£9fT5f¢*f¡P§JÀrank(M) =

0µf&X?G¡ATXrP¢¦XO8ff9?TfrX¢¦frJ«JTX(¬³f?`9¢ Í ×ØØ Ð Ù&TX(¡*f9 wPrVP¦G§¬ ¨ r£f ¨ DJ τ1, . . . , τ`

fTrfT(¡Gf rDTf²¢Pf¡rÝI X ¨ f¢Ú9XJ Í ¿ ÙTÛ`XPGr X¦G99-G ¨ r9rL¢P

τ§¯*f¡LG ¡f9Tf·w9 X(ff¦f

¦G9f¦G·¥I¦XJ9X Í ×TØIÙ¦Xf¢ÝfÈfTaJ9rPÛwPÖ c9rP

˙x1 = x2

. . .˙xs1−1 = xs1

˙xs1= f1( ˙xs1

(t− τ ), x, x(t− τ ), u, . . . ,

u(γ), u(t− τ ), . . . , u(γ)(t− τ ))˙xs1+1 = xs1+2

. . .˙xs1+s2−1 = xs1+s2

˙xs1+s2= f2( ˙xs1+s2

(t− τ ), x, x(t− τ ), u,

. . . , u(γ), u(t− τ ), . . . , u(γ)(t− τ )). . .˙xs1+···+sp

= fp( ˙xs1+···+sp(t− τ ), x, x(t− τ ), u,

. . . , u(γ), u(t− τ ), . . . , u(γ)(t− τ ))y1 = x1

y2 = xs1+1

. . .

yp = x1+s1+···+sp−1

x(t) = ϕ(t), t ∈ [t0 −maxj τj , t0]

.

Í ×P×ÙILXT

(y(t), u(t)) ¨ Xa#«£f8P¢XfGp Í ¿ ÙT¡pD¯9f`k«f¦faT`G wt0 ≥ maxiτj

¦X9 XT±T"G«JJ9«³G I¦fP¦X§D*?¦XµfXrXf¦G ¦ff¦G`X«?¦fT£fG f?#fT(ff¢¦XÚf,rfXfrÈT?τi

¨ f¡9"¢Pr«£9f9TL¯f frXTD τ TXτ1, . . . , τ`

fTrGTr#¡GIrDTf"¢f9TrP§Ö ¦ff`Pf ¨ XT rank(M) = `

§f¬X9X9«XT ¨ 9f³¦DpXk« q ≥ `

X ¨ ?¦XD«JPp`Gr¤`P

∆iq

§Æ&D¡G#fI±f¥I¦XJ9rPX Í ×TØPÙcµOw ¨ f¡9 iq ≥ 1

§ «TT¦XJ9³Jf®G9rL`PrI

t0 ≥ Tµ Í ×ØPÙM¢Pr«£ ¥I¦XTw

Ti1, . . . , Tiq

y(si)i (t0) = ξ(y1(t0), . . . , y

(si−1)i (t0),

u(t0), . . . , u(γ)(t0),

y(s1−1)1 (t0 − Ti1 + Ti0), . . . , y

(si)i (t0 − Ti1 + Ti0),

u(t0 − Ti1 + Ti0), . . . , u(γ)(t0 − Ti1 + Ti0),

. . . ,

y(s1−1)1 (t0 − Tiq + Ti0), . . . , y

(si)i (t0 − Tiq + Ti0)),

u(t0 − Tiq + Ti0), . . . , u(γ)(t0 − Tiq + Ti0)). Í × Ù

À£fL [`PrIt0¡8afP.¡T9¢"f¦X¢#G

¦f9X ®GX T¯r-L °G9«JJ9r«*I«Pr«

Í ¨ Xa²·`#P9Xr«³I³af?If¢w ®fTLft0 ≥ (maxisi−1)T

Ùcµ9fGr¤5P9Tf¢ Í × Ù ¨ g99`cO9 ¢«Of ¨ ¥I¦XJ9rPX5w Ti1, . . . , Tiq¨ f¡9ÇârXf`XfPµ X dy(j)i , j ≥ 0

rfT9r#XG`XGP8k« K f¦f iq ≥ 1

y(si+j)i (t0) =

dj

dtjξ(y1(t), . . . , y

(si−1)i (t),

u(t), . . . , u(γ+j)(t),

y1(t− Ti1 + Ti0), . . . , y(si−1)i (t− Ti1 + Ti0 ,

u(t− Ti1 + Ti0), . . . , u(γ+j)(t− Ti1 + Ti0), . . . ,

y1(t− Tiq + Ti0), . . . , y(si−1)i (t− Tiq + Ti0),

u(t− Tiq + Ti0), . . . , u(γ+j)(t− Tiq + Ti0))|t0 .Í × Ù

ÅXry(si+j)i (t)

¡GP¡r±©9(wPL0 ≤

j < q − 1µ-9fXap

qT*f#¥I¦XTX"¡GP9rw

Ti1−Ti0 , . . . , Tiq−Ti0

GTr Í 9f9£LP.`²¦DÊÝTDTI©·¢PrPXT¡GP9rDTfgpfÙc§Ö X

rank(M) = `µτi

µi = 1, . . . , `

*¢f9TrGXf±¦ff¡¥I¦f.I9frGT(¡GIrDTfrfT PfXJ9rPX

Ti1 − Ti0 , . . . , Tiq − Ti0

Í T ?«Ir ¦D®f f³f#P * LLD¦faJ9L GkG9Ù §A*I¦XµAr

τi

µi = 1, . . . , `

T9#¢f9TrGr¡GPrDfrPµ ¨ f¡9 P Xr*f"f9?p§

¥

ÿ §Þã Ò ¶ ÖÀ°â9f¡c9rP ¨ ,r¦XaJ99f,X9ÈIâ f³ ®f XrTX f ¨ 9XJ ¨ þÚPX«TTfrrpÍ XDªJLXTaTL 9LGP9gDfrr°1wLf¢¦f¡TLGGX9L 9aÙ)G?ÚXT f 9T9rLfr¡GIrDTfgp/.f G¡k XTaTL 9aµ1«?«a9f§5*I¦Xµ`9fTG.9ff,L 9fGfwPp9rX¢PX9«JTfgpX GP9gDfrr°wP¯fPfrG G¡? Í ÞT[§µ5×ØØP× Ñ ß`XTf¢T[§rµ×TØPØáIÙ8ff"`L¦XGLf 9f¡GP9rDJ frrpTM9fGk#X9L 9§t?RJY8¸`]W x

x1(t) = x22(t− τ1) + u(t)

x2(t) = x1(t− τ2)x2(t)y(t) = x1(t)x(t) = ϕ(t), t ∈ [−τ, 0]

Í × ÿ Ù

¬³"X«y = (δ1(x2))

2 + u

y = 2δ1(x2)δ1δ2(x1)δ1(x2) + u == 2(δ1(x2))

2δ1δ2(x1) + u

Í ×TáIÙ

TD

dy

dy − du

dy − du

=∂(S, h

(s1)1 )

∂x

[

dx1

dx2

]

=

=

1 00 2δ1(x2)δ1

2(δ1(x2))2δ1δ2 4δ1δ2(x1)δ1(x2)δ1

[

dx1

dx2

]

.

Í × Ð ÙÆ&T9µJ9fJ9r® ∂(S,h

(s1)

1 )

∂x

X9fþ ×J« K(δ]TD(9fG6 ¨ þI#X9«JTXrP P9frf¢

9fG XXg9rPÞ¡ T[§ Í ×TØØI×Ù µfgτ1

Xτ2T9¯þ?f ¨ -§ Î ¨ «µ τ1

TXτ2fTMfP9g`Tf§

*fâff¦G [P¦GX¦G1¥I¦XJ9rP/Üf·wP Í ¥D§Í ¿Ô ÙÙ

dy − du + 2δ1δ2(x1)du == 2(δ1(x2))

2δ1δ2dy + 2δ1δ2(x1)dy,

Í × Ô Ù

TX ¨ )²XT,9f9)T9à ¨ LPfL¡T¡µ ∆0TX∆1 = δ1δ2

¨ g9·9`XGf¢ PfrDJDT0 = 0

XT1 = τ1 + τ2

T9f ¨ (9rL GkG§*I¦Xµ

M =

[

0 01 1

]

¨ raTXþ ¿ µ?XfL¢PT9fTfP9g`Tf§Ì ¨ f¢ f"f"X9XTM9f"f9?TATÆ&9r ¿ µ ¨ (T²¦X#9faXTf¢P#«J¡Tf x1 =y = x1

µx2 = y = (δ1(x2))

2 + u· ¨ r9(fGÉ

˙x1(t) = x2(t)˙x2(t) = 2(x2(t)− u(t))x1(t− T1) + u(t)y(t) = x1(t)

.

Í × Ï Ù

t?RJY8¸`]rWy

x1(t) = −x2(t− τ1)x2(t) = x1(t− τ2)y1(t) = x1(t)y2(t) = x2(t− τ2)x = ϕ(t), t ∈ [−T, 0]

Í ØPÙ

¬"Xk«y1 = −δ1x2

y1 = −δ1δ2x1

Í ¿ Ù

TX ∂(S,h(s1)

1 ,h(s2)

2 )

∂x=

1 00 δ1

δ1δ2 00 δ2

§±*XÜrff¦f

¦G9f¦G¥I¦XTX*fT*wPÉT9

dy1 = −δ1δ2dy1

δ1dy2 = −δ2dy1 ⇔ dy2 = −δ−11 δ2dy1

.Í ×Ù

¬±?¦XXk«∆10

µ∆11 = δ1δ2

µ∆21 = δ1

TX∆22 = δ2

TX(9I¦XµT10 = T10

= 0µT11 = τ1 + τ2

µ

T20= T21 = τ1

µT22 = τ2

XM =

1 10 0−1 1

µ

¨ faL¡¯T59fþ×G§*?¦X τ1TX

τ2T9*fP9g`Tf§

Ì LX#Lf. ®fTLfµ ¨ ±TÚ ¦Xr1T¡ ¦f TÈfÚ«TT¦f³9fÈ ¨ Ê â¢Pw9 f ®Gfg²ff¦G [P¦Gf¦f²¥I¦XJ9rPX

y1 = −δ1δ2y1

µy2 = −δ−1

1 δ2y1§Mr¦XaJ9µ ¨ ¦f¡J9¦G9f¦G9VPX"Gp9 ¨ r #w¦fX Í Ö X XrfTX*fP DP-µI×TØPØ ¿ Ù µJ?L9rf¢

τ1 =1µτ2 =

√2µϕ1(t) = et

TXϕ2(t) = t+1

§P¬³Xfr

µ1(T11−T10)|t0 := y1(t0)+y1(t0−T11 +T10)TXµ2(T22 − T21)|t0 := y2(t0) + y1(t0 − T22 +

T21)w

t0 = 4§&8®G`c9Oµfw¦fX XT9

©9.wPT11 − T10 = 1 +

√

(2)TX

T22 − T21 =

−2 0 2 4−8

−7

−6

−5

−4

−3

−2

−1

0

1

2

A plot of the function µ1(T

11−T

10) for t

0=4

T11

−T10

µ 1

−1 −0.5 0 0.5 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

A plot of the function µ2(T

22−T

21) for t

0=4

T22

−T21

µ 2

Ì r¢D§ ¿ § q UfWMoaG_5\ S[Z<nJ_Xb µ1(T11−T10)|4 RJ_5 µ2(T22−T21)|4

√

(2) − 1X rGTrâfàX·Xr 9?Taµ

Ì ¢X§ ¿ § rPXTµ?f"T9T¡ f*9?TawPµ2(T22 − T21)

DJ(TÚ`±Ú Ì r¢D§ ¿ §*X`G99f f¡&? fTf¢

µ2(T22−T21)|t0 := d

dt(y2(t0) + y1(t0 − T22 + T21))|t0

Í TXT9f&¦fD¥P¦XP9rL °G9«JJ9r«PaÙArX

T22−T21¦X`" 9?w*9fµGGX§

áX§ÆÃÁÆ¶Å Ö ÀÃÁ Ö

¬³LXk«LTXr?©9fL¡GIrDTXrr°,T&fLL ¡T¢X9L 9ArLfffTG ¨ rL¦fg9fr PX9TI*LG¡G§Ö aJ³rLXJ9Ý.X ¨ )?¡ÝTÝ ®?9DTff¦G [P¦Gf¦G19f9I9T T(9f ?.µ9fw9 T ¨ f¡9$G ¡Gf²¡GIrDTXrgpÊT 9fGk XTaTL 9a§ Ì ±Lf fG¡ ¨ g9 w ¨«J¡TXr&XDTaTL 9aµP9fff¦G [¦ff¦G¥I¦XJ DT`¦XG9c9r9fP9gw X«Jr¦X¯X [¡T¢I&w96 ¦f9(fT9f§¬³ÈX«P²wP¦XT$f °T¢f9T¡²r¡âafaþ·9f.¡GIrDTfgp1T89fG¡kàX9 9¨ f¡9àrLrXTX#f³w· ®Gf¡ rT¡ ¦f¡J .TM9f"rfX¦G ¦G9f¦G8TX§

Ð §8Æ ¾ ÁÃ¬ ¶ Ä ã Á Ö

*f¡ ¨ 9þ ¨ P¦fX`ÛIÊ9fÈÁ8TX Â Taa Ö af?r fL¡TX rPrGwPT¡µX Ö ¨ G¡ÚÂTaaÚÆ&P¦fX mµTX1X Ö ¨ G¡Ì ¦fXfTwP Ö 99T¢P¡£ÂTa9«?¡f T9fG f¦f9¢ã±J9fJ9T-ãfGrX¢#Æ&P9§

Â Ì ÂÁÆ ÖX¢¦frJ«JXµ`ãÜ¬³fI`¢Dµ § Ö aJ rLXJ9rPTXÝ¡GIrDTfgp T9f·f¡âX9L wPfffL °G¡Gp9§ hnJjZ¡baZ<nT_`RT]V]glRP\ u\9W°¸`S°W9 §

rþP¦fafµ¶§µ Ã9rJ«5µ §§ Í ×TØPØP×PÙc§ÀpGP9rDTfr r°.XTG¡T¯rfTG¡kP °Gr¤`P9?§»cQ½ MQ,RJSwU *enT_XS[hnJ]`» rdxTz µ Ð Ôf¿ §Æ&f-µ Ò §ã²§ Í ¿Ï?Ð?¿ Ù § Ì 9*f¢P¯TXX¯TX§¶XGP-f*GL Ò §

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observability and identi ability of nonlinear systems with ... · observability and identi ability...

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