algebraic linear identification modelling and aplications

J O H A N N E S K E P L E R

U N I V E R S I T A T L I N ZN e t z w e r k f u r F o r s c h u n g , L e h r e u n d P r a x i s

Algebraic Linear Identification, Modelling,and Applications of Flatness-based Control

Dissertation

zur Erlangung des akademischen Grades

Doktor der Technischen Wissenschaften

Angefertigt am Institut fur Regelungstechnik und Prozessautomatisierung

Eingereicht von:

Dipl.–Ing. Stefan Fuchshumer

Graben 17, A–4722 Peuerbach

Betreuung:

o.Univ.–Prof. Dipl.–Ing. Dr.techn. Kurt Schlacher

Begutachtung:

o.Univ.–Prof. Dipl.–Ing. Dr.techn. Kurt Schlacher

Univ.–Prof. Dipl.–Ing. Dr.techn. Andreas Kugi

Linz, im Dezember 2005

Johannes Kepler Universitat Linz

A-4040 Linz, Altenberger Str. 69, Internet: http://www.uni-linz.ac.at, DVR 0093696

for Susanne,

for her love, inspiration and patience.

Preface

This thesis aims at reflecting selected issues of my research activities done during the lastyears as a research assistant at the Christian Doppler Laboratory for Automatic Controlof Mechatronic Systems in Steel Industries, installed at the Institute of Automatic Controland Control Systems Technology at the Johannes Kepler University of Linz. Accompaniedand guided by the scientific advice of Professor Kurt Schlacher I had the pleasure to finda very stimulating scientific culture. For his advice, support and for always finding timefor me, even if he was covered up with work, I wish to express my gratitude.

For having a significant contribution on arousing my interests on control theory, whenI was an undergraduate student, via his incomparable style of transporting the key ideasin his lectures, for his advice, his help, and for preparing an expert’s report for this thesis,I wish to address my thanks to Professor Andreas Kugi, chair of System Theory andAutomatic Control, University of Saarbrucken, Germany.

Special thanks I wish to express to our industrial partner Voest-Alpine Industrieanla-genbau GmbH Linz for the cooperativeness, the financial support, and for providing spacefor research. In particular, I wish to acknowledge the contributions of Georg Keintzel,Bruno Lindorfer and Karl Aistleitner.

To my friends and colleagues Werner Haas, Rainer Novak, Gernot Grabmair, Jo-hann Holl, Reinhard Gahleitner, Hannes Seyrkammer, Kurt Zehetleitner, Bernhard Roi-der, Johannes Schrock, Markus Schoberl, Richard Stadlmayr, Martin Staudecker, HelmutEnnsbrunner, Brigitta Peitl and Harald Pachler I wish to convey my thanks for manynon-scientific and scientific discussions, the good atmosphere, and stimulating thoughtexperiments from most various fields of concern.

Besides my occupation as a lecturer, I particularly enjoyed the time I had the opportu-nity and pleasure to spend with my friends and diploma students Klaus Straka, GunnarGrabmair, Marc Polzer, Thomas Rittenschober and Achim Berger. The endeavour ofconjoint research and all the conversations beyond this scope were very valuable for me.

However, the center of all my thoughts and the key reason for enjoying my life like this ismy beautiful, bright and charming wife Susanne. I love you, Susanne!

Last but not least I wish to express my deepest gratitude to my parents Maria Anna andFranz Fuchshumer, who shaped my senses from the very beginning of my especially beau-tiful childhood, and made me admire nature and literature. Liebe Eltern, ich danke Euchfur all Euer Engagement und dafur, dass Ihr mir eine unbeschwerte Zeit des Studiumsgeschenkt habt!

Stefan Fuchshumer

Kurzfassung

Befasst mit der Anwendung moderner linearer und nichtlinearer Regelungstheorie adres-siert diese Arbeit ausgewahlte Themen und Anwendungsbeispiele aus den Schwerpunktender Identifikation linearer zeitdiskreter zeitinvarianter dynamischer Systeme, einem alge-braischen Zugang folgend, der Modellbildung basierend auf physikalischen Betrachtungensowie der modellbasierten nichtlinearen Reglersynthese. Der Beitrag zur Identifikationzeitdiskreter linearer Systeme stellt dabei eine Ubertragung und Weiterentwicklung einerin der gegenwartigen Literatur vorgeschlagenen algebraischen Methodik zur Parameter-identifikation zeitkontinuierlicher linearer Systeme dar. Die Ausfuhrungen sind fur Sy-steme n-ter Ordnung im Operatorenbereich dargestellt. Illustriert anhand eines als La-borexperiment verfugbaren Modells eines 3-Massen-Torsionsschwingers folgt als Resumeschließlich eine Diskussion der insbesondere bei abnehmender Abtastzeit deutlich dif-ferierenden numerischen Beschaffenheit zweier durch die bilineare Tustin-Transformationverbundenen Parametrierungen dieser Identifikationsmethodik.

Zwei industrielle Anwendungsbeispiele aus unterschiedlichen technischen Disziplinen,jedoch verbunden durch ihre gemeinsame Eigenschaft der differentiellen Flachheit, stehenim zweiten Themenbereich der Arbeit im Blickpunkt: eine Anwendung aus dem Kalt-walzbereich der Stahlindustrie sowie ein Vorschlag fur eine Mehrgroßen-Fahrdynamikrege-lung als Beispiel aus der Automobilindustrie. In beiden Beispielen folgt der Reglerentwurf,getragen von einem auf physikalischen Uberlegungen basierenden Modell, der flachheits-basierten Methodik. Angemerkt sei, dass in beiden Fallen der gewahlte flache Ausgangjeweils eine sehr einfache physikalische Bedeutung tragt, was sich im Zuge der Regler-synthese sowie der Trajektorienplanung als wertvoll zeigt. Die Schwerpunkte der beidenBeispiele sind jedoch sehr kontrar: Wahrend im Falle der Fahrdynamik-Applikation dasmathematische Modell in der Gestalt des vielfach anzutreffenden (holonomen) planarenEinspurmodells der Literatur entnommen werden kann und der zentrale Beitrag der Ar-beit die Beobachtung der differentiellen Flachheit dieses Systems sowie der Auspragungdes flachen Ausganges ist, so liegt der Fokus des Kaltwalz-Beispieles deutlich auf der Ent-wicklung eines fur die Simulation und den Reglerentwurf geeigneten Modells. DieserAufwand in Richtung der Modellierung ruhrt von dem Wunsch, auch jene Kaltwalz-Umformvorgange, welche von signifikanter elastischer Deformation der Arbeitswalzengekennzeichnet sind, adaquat nachbilden zu konnen. Das Auftreten eines deutlich nicht-kreisformigen Kontaktbereichs zwischen Walze und Band soll demnach vom Modell wieder-gegeben werden. Im Hinblick auf eine fur den regelungstechnischen Einsatz handhab-bare rechentechnische Komplexitat wird in dieser Arbeit nun eine fur den Walzprozessgeeignete Approximation des elasto-statischen Walzendeformationsproblems im Sinne derMethode von Rayleigh-Ritz vorgeschlagen. Dieser Zugang, verbunden mit einer aus derLiteratur entnommenen Beschreibung fur das umgeformte Band, ergibt schließlich dasWalzspaltmodell. Mit diesem Modell im Zentrum wird ein flachheitsbasierter Zugang fureine Mehrgroßenregelung einer Kaltwalzanlage vorgestellt.

Abstract

This thesis, dealing with applications of modern linear and non-linear control theory, ad-dresses selected issues and application examples from the fields of identification of lineartime-invariant discrete-time systems following an algebraic approach, mathematical mod-elling based on physical considerations, and model-based non-linear control synthesis. Thecontribution on identification of discrete-time linear systems is essentially stimulated byan algebraic approach to parameter identification of continuous-time linear systems pro-posed in the recent literature. The discussions are given for n-th order systems resortingto the operational domain. On the basis of a 3-mass drive-train model being availableas a laboratory setup the numerical conditions of two different parametrizations of theidentifier, which are linked by means of the bilinear Tustin transform, are investigatedand found to significantly differ with decreasing sampling times.

The second subject area of this thesis is concerned with two industrial applicationsstemming from different engineering disciplines, but connected via their common differ-ential flatness property: a cold rolling mill application from steel industries and a vehicledynamics control application, regarding the steering angle and the longitudinal tire forcesas control inputs, from automotive industries. In both examples the control design, basedon physically motivated models, is performed invoking the flatness-based methodologies.It is worth mentioning that in both cases the flat output is attached with a clear physicalmeaning which is valuable for control synthesis and trajectory planning as well. The focalpoints of these two applications are very different, however: While in the vehicle dynamicsapplication the mathematical model in the shape of the well-known planar (holonomic)bicycle model is taken from the literature and the central new contribution is the observa-tion of the flatness property and the determination of a representative of the flat output,the focus of the rolling mill example is laid on the development of a model being suitablefor the simulation and control purpose. This effort on modelling arises from the objectiveto rendering also those roll gap scenarios adequately which are characterized by significantelastic work roll deformations, and, thus, a non-circular shape of the roll gap. In order todevelop a model being suitable for the control purpose due to manageable computationaleffort, an approximation of the elastio-static work roll deformation problem invoking theRayleigh-Ritz method is proposed, with particular emphasis laid on the choice of a set ofappropriate shape functions for the displacement fields. This approach, combined witha mathematical description of the deformed strip taken from the literature, finally givesthe non-circular arc roll gap model. With this model as a central point, a flatness-basedapproach to multivariable control of a cold rolling mill is proposed.

CONTENTS

1 Introduction 1

2 An Algebraic Approach to Discrete-Time Linear Systems Identification 5

2.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 A z-Domain Approach . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 A q-Domain Approach. . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 The z-Domain Approach – Continued . . . . . . . . . . . . . . . . . 172.2.4 A Note on the Standard LS Identification Method . . . . . . . . . . 182.2.5 A q-Domain Approach – “Direct Method” . . . . . . . . . . . . . . 192.2.6 Applications and Discussion . . . . . . . . . . . . . . . . . . . . . . 21

3 A Mathematical Model of a Rolling Mill 34

3.1 Mill Stand Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 The Hydraulic Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Non-linear Hydraulic Gap Control (HGC) . . . . . . . . . . . . . . . . . . 37

3.3.1 Position Control Mode (PCM) . . . . . . . . . . . . . . . . . . . . . 373.3.2 Force Control Mode (FCM) . . . . . . . . . . . . . . . . . . . . . . 38

3.4 A Novel Non-Circular Arc Roll Gap Model . . . . . . . . . . . . . . . . . . 393.4.1 The Elasto-static Work Roll Deformation Problem . . . . . . . . . . 413.4.2 Two Sub-Problems as Prerequisites. . . . . . . . . . . . . . . . . . . 443.4.3 A Ritz Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.4 The Shape of the Roll/Strip Contact Arc . . . . . . . . . . . . . . . 683.4.5 The Strip Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4.6 The Implicit Non-Circular Arc Roll Gap Model . . . . . . . . . . . 723.4.7 Conclusions on the RGM . . . . . . . . . . . . . . . . . . . . . . . . 79

3.5 Bridle Roll Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.5.1 On the Localisation of the Slip Arcs . . . . . . . . . . . . . . . . . . 833.5.2 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 84

3.6 The Non-linear Dynamics of the Rolling Mill . . . . . . . . . . . . . . . . . 853.6.1 Analysis of the Linearized Mill Dynamics . . . . . . . . . . . . . . . 88

I

4 Flatness-based Rolling Mill Control 90

4.1 A Reduced-Order Model of the Rolling Mill . . . . . . . . . . . . . . . . . 904.1.1 A Quasi-static Mill Stand Model . . . . . . . . . . . . . . . . . . . 904.1.2 A Reduced-Order Model of the Bridle Rolls . . . . . . . . . . . . . 92

4.2 Rolling Mill Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5 Non-linear Vehicle Dynamics Control – A Flatness-based Approach 96

5.1 The Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 The Flatness Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.1 Some Key Observations . . . . . . . . . . . . . . . . . . . . . . . . 1005.2.2 Front- and Rear-Wheel Drive . . . . . . . . . . . . . . . . . . . . . 102

5.3 On Configuration Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.4 On the Non-holonomic Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . 1095.5 A Flatness-based Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Bibliography 122

II

chapter

ONE

Introduction

Addressing selected issues of the field of modern control theory, this thesis spans froman algebraic approach to discrete-time linear systems identification, mathematical

modelling of complex industrial systems based on physical considerations, to the applica-tion of non-linear model-based control theory, in particular flatness-based methods. Bothmain examples of this thesis, namely a cold rolling mill example from steel industries tobe dealt with in the Chapters 3 and 4, and an example from the automotive industryon vehicle dynamics control (to be addressed in Chapter 5) are linked by their commonproperty of differential flatness, with the flat output attached with a clear physical mean-ing. Differential flatness, introduced by Michel Fliess, Jean Levine, Philippe Martin andPierre Rouchon in [FLMR92], [FLMR95], is a structural property that allows for a com-plete, finite and free differential parametrization of a dynamic system by means of the flatoutput. Thus, once chosen the trajectories of the flat output, the associated trajectoriesof the system variables, in particular those of the control inputs, can be derived fromthis flat output’s trajectory and its time derivatives, without the need for integrating thesystem’s differential equations. Moreover, given a flat output there exists a systematicapproach to the control synthesis.

Additionally to their common structural property regarding flatness, both applicationshave in common that their mathematical models are evolved on the basis of physicalconsiderations thus allowing for a parametrization in terms of the geometry data andthe material parameters. This firm reference to physics exploits significant advantagesfor the system analysis and the control synthesis as it provides additional insight intothe system, and, not at least, for the re-usability of the models (and the controllers) fordifferent system configurations.

Finally, the third common ground relating these applications is that in both casesthe trajectory tracking problem is central, an objective well complying with the flatnessbased control approach. Clearly, in the field of rolling mills, for productivity reasons thetime made available to the acceleration and the deceleration of the mill is required to bekept as short as possible, with simultaneously meeting the tight tolerances imposed onthe control variables. However, apart from demands specified on-line by the plant opera-

1

1. Introduction 2

tor, the trajectories of the flat output of the rolling mill, which is found to coincide withthe control variables, might be designed off-line. Contrary, the trajectory planning taskfor the vehicle, when addressing vehicle dynamics control to improve the safety, clearlyrequires a real-time trajectory generation permanently accounting for the inputs suppliedby the driver, i.e. the current angle of the steering wheel and the current position ofthe throttle/brake pedal. Besides the objective of supporting the driver in case of anemergency situation via conjoint actuation of the steering angle and the engine/brakingtorque, i.e. vehicle dynamics control as primarily addressed in Chapter 5, there is a pos-sible second field of application of the vehicle’s (configuration) flatness property, namelythe task of tracking a vehicle along a pre-specified path on a test or race track. Thistask, typically left to test pilots, is arranged in order to investigate e.g. the consequencesof different chassis adjustments. Automatic tracking control for test vehicles (equippedwith global-positioning systems) on test tracks might help in order to produce particulartraceable runs.

While in the vehicle dynamics control example the mathematical model the control designis based upon is taken from the literature and the main new contribution is the observationof the differential flatness property, the focus of the rolling mill example is laid on thederivation of a mathematical model, being suitable for the simulation and the controlpurpose. The impulse for setting up a new rolling mill model, particularly concerningthe modelling of the roll gap, crucially stems from the following observations: In orderto being applicable to cold, thin strip and temper rolling, the roll gap model is requiredto include a detailed description of the elastic work roll deformations which occur underthe action of the rolling load. In the case of cold rolling, typically the assumption ofa circular roll/strip contact shape, though with a larger so-called equivalent roll radius,qualifies to be appropriate, cf, e.g., [BF52]. However, with decreasing strip thickness andin the temper rolling case as well where the thickness reduction of the strip is very small,this simple assumption is to be dropped. There is a huge literature on roll gap modelsincluding a detailed description of the work roll deformations, typically referred to as non-circular arc roll gap models, see, e.g., [JOZ60], [FJ87], [FJMZ92], [DET94], [LS01]. Theyare usually either based on the so-called elasto-static half-space solution or on Jortner’ssolution respectively, i.e., the solution to the problem of an elastic half-space/cylinderloaded with a constant normal stress. Due to the computational effort involved, thesemodels are primarily intended for off-line calculations. So, Chapter 3 aims at introducinga novel non-circular arc roll gap model exhibiting reduced computational effort and beingthus applicable for the control purpose. To this end, the elasto-static work roll deformationproblem is addressed in the sense of the Rayleigh-Ritz method, with special emphasis laidon an appropriate choice of shape functions. For the case study discussed in Chapter 3, atotal number of 56 generalized coordinates (i.e. degrees of freedom) of the proposed Ritzansatz suffices to obtain a very suitable result, compared with a model incorporating adetailed description of the roll deformations. The approximation of an infinite-dimensionalsystem via a suitable Ritz approximation in order to obtain a model the control designcan be based upon is an approach very often encountered in control applications, as, e.g.,

1. Introduction 3

in structural control, for systems including beams and plates, etc. Typically, as also seenin the roll gap modelling, the key problem is to find an appropriate set of shape functions.

As sketched above, the key contribution concerning the vehicle dynamics control appli-cation of Chapter 5 is the observation of the differential flatness property of the planarholonomic bicycle model and the proposition of a flatness-based approach to vehicle dy-namics control regarding the (front) steering angle and the longitudinal forces of thetires as control inputs. Vehicle dynamics control systems as e.g. ESP (electronic sta-bility program) acting on the brakes and the traction control system are implemtentedin series-production vehicles. These systems support the driver in emergency situationsby producing a (counter-) yaw torque due to individually controlled braking of all fourwheels in the case of exceedance of a certain yaw rate. Accordingly, the potential of in-volving the steering system to handle emergency situations is very rich. The automotiveindustry aims at utilizing these possibilities e.g. by means of supporting the driver forthe difficult task of counter-steering by application of a respective artificial torque to thesteering wheel, i.e., as a haptic recommendation for the driver. As a very recent advance,active front steering (AFS) [KB02] has been implemented in series-production vehicles.The basic function of AFS is to mechanically add an additional steering angle (adjustede.g. by an electric drive) to the steering angle given by the driver. Besides the objectiveto enable a velocity-dependent gain of the steering mechanism, the AFS system can beused for vehicle dynamics control, too.

The bicycle model, proposed in [RS40], evolves from the four-wheel car by gluingtogether the front and the rear wheels to a single (mass-less) front and rear wheel, locatedat the longitudinal axis of the car. The contact between the tires and the road is modelledin terms of contact forces, which implies that the tires are enabled to slip and slide onthe road. This planar model, known as a well-established basis for the design of vehicledynamics control systems, see, e.g., [ABO99], [Bun98], [Rit04], is capable of rendering thelongitudinal, lateral and yaw dynamics of the vehicle. The pitch and roll dynamics of avehicle are clearly not involved in the scope of this model.

Within the scope of vehicle dynamics control as an assistance for the driver to copewith emergency situations, the components of a flat output of the bicycle dynamics,with the subsystem related to the global position dropped as not being involved in thecontrol objective, are revealed as the lateral and the longitudinal velocity component ofa distinguished point Ξ located on the longitudinal axis of the vehicle. This property isshown for the front-, rear- and all-wheel driven vehicle, without referring to particularrepresentatives of the functions modelling the lateral tire forces. In case the global positionof the vehicle is involved in the control objective (e.g., tracking of a prescribed path), theconfiguration flatness property of the bicycle dynamics, with the position of the point Ξqualifying as a flat output, can be exploited.

In order to leading over directly to the matter of this thesis, we will proceed by introducinga contribution on discrete-time linear systems parametrical identification in Chapter 2, es-sentially stimulated by the algebraic approach [FSR03] to continuous-time linear systems

1. Introduction 4

identification introduced by Michel Fliess and Hebertt Sira-Ramırez. By now, this alge-braic approach [FSR03] has led to a variety of related research, as e.g. on fault diagnosis[FJ03], on signal processing and signal compression [FMMSR03], and on the estimationof the derivatives of signals [RSRF05].

In accordance to the continuous-time framework of [FSR03], the operational represen-tation of the discrete-time constant linear system (in the z-domain) is considered. Initialconditions are allowed to being ignored by taking derivatives with respect to the oper-ational operator z. To determine the unknown system parameters, subsequent iteratedsummations of the discrete-time counterpart of the resulting operational equation arecarried out to set up a system of linear equations. To cope with measurement noise,besides the possibility of a straight-forward incorporation of linear filters, the setup of anoverdetermined system of linear equations by means of additional iterated summationsqualifies to be suitable. Starting with introductory examples to illustrating the proposedapproach, the discussion then proceeds with the elaboration of a linear identifier for ageneral n-th order discrete-time constant linear dynamics.

On the basis of a fifth order model of a drive-train, which is available as a laboratoryexperiment, the problem of inaccurate estimation of those system zeros, which have onlyminor effect on the system response, and, hence, are difficult to estimate in presence ofnoise, is illustrated. In order to overcome this problem, the idea to discarding (or pre-setting) those non-essential zeros is proposed, first commencing with the application ofthe bilinear Tustin transform, also referred to as q-transform for short, to the z-domainrepresentation. The motivation for discussing this idea for the q-domain case first simplyis that, by virtue of the close similarity to the continuous-time domain, things might beparticularly apparent from the control engineer’s point of view. Clearly, discarding certainzeros of the q-domain transfer function, i.e. shifting those zeros to infinity, correspondsto placing the associated zeros of the z-domain transfer function at z = −1. This idea ofpre-setting those non-essential zeros to obtain a “reduced-order” linear identifier is shownto providing particular attractiveness.

Again referring to the drive-train example, it is found that the numerical condition ofthe linear identifier given in terms of the q-domain parameters shows significant advan-tages compared to the z-domain parametrization, becoming particularly apparent withdecreasing sampling times. This effect is observed independently from whether or not theidea of pre-setting certain zeros is applied. The increasingly poor numerical condition ofthe z-domain approach emerging with decreasing sampling times might become apparentby reflecting the relation zi = exp (siTa) between the poles si of the continuous-time sys-tem and the poles zi of the according discrete-time representation. Hence, with decreasingsampling time Ta, the poles zi approach the point z = 1.

To resume, the algebraic theory on linear identification of [FSR03], which, in turn,has led to the research to be introduced in Chapter 2, provides easy-to-implement on-lineidentifiers, which have shown to perform well in applications.

Data aequatione quotcunque

fluentes quantitae involvente

fluxiones invenire et vice versa.

(Es ist nutzlich, Funktionen zu differenzieren

und Differentialgleichungen zu losen.)

Newton in einem Brief anLeibniz im Jahre 1676

chapter

TWO

An Algebraic Approach to Discrete-Time Linear Systems

Identification

The following investigations aim at introducing an approach to discrete-time linear sys-tems parametrical identification, essentially stimulated by the theory of M.Fliess and

H.Sira-Ramırez developed for continuous-time linear systems [FSR03]. As a prerequisitefor addressing the discrete-time framework, we will briefly revisit some key ideas of thecontinuous-time identification approach on the basis of a simple example, following thelines of [FSR03]. The underlying theoretical framework utilizing, in particular, moduletheory and Mikusinski’s operational calculus, can also be found in [FSR03].

2.1 Continuous-Time Systems

Example 2.1 Consider the second-order linear time-invariant dynamics [FSR03]

y + a1y + a0y = bu+ γσ (t) , y (0) = y0, y (0) = y0 (2.1)

with the unknown parameters a0, a1, b ∈ R, and a constant, but unknown disturbance ofmagnitude γ ∈ R (σ denotes the step function) acting on the system input. Equivalently,in the operational domain, we have

(

s2 + a1s+ a0

)

y − y0 − (s+ a1) y0 = bu+γ

s. (2.2)

Now, multiply by s and then take derivatives, three times, with respect to s in order to allowfor ignoring the initial conditions y0, y0 and the constant load perturbation. To avoid, inthe according time-domain representation, derivatives with respect to time, divide bothsides of the resulting equation by s3 and re-sort w.r.t. the parameters a0, a1 and b,(

1

s2

d3y

ds3+

3

s3

d2y

ds2

)

a0 +

(

1

s

d3y

ds3+

6

s2

d2y

ds2+

6

s3

dy

ds

)

a1 −(

1

s2

d3u

ds3+

3

s3

d2u

ds2

)

b =

= −d3y

ds3− 9

s

d2y

ds2− 18

s2

dy

ds− 6

s3y. (2.3)

5

2. Identification. . . 2.1. Continuous-Time Systems 6

For calculating the corresponding time-domain representation of (2.3), notice that

dν

dsνf (s) • (−t)ν f (t) , ν ≥ 0 (2.4)

holds. Thus, we obtain

P11 (t) a0 + P12 (t) a1 + P13 (t) b = Q1 (t) (2.5)

with1

P11 (t) = −∫ (2)

t3y + 3

∫ (3)

t2y, P12 (t) = −∫

t3y + 6

∫ (2)

t2y − 6

∫ (3)

ty,

P13 (t) =

∫ (2)

t3u− 3

∫ (3)

t2u, Q1 (t) = t3y − 9

∫

t2y + 18

∫ (2)

ty − 6

∫ (3)

y.

(2.6)

Finally, integrate (2.5), twice,

Pij (t) =

∫ (i−1)

P1j (t) , Qi (t) =

∫ (i−1)

Q1 (t) , i = 2, 3, j = 1, 2, 3, (2.7)

to set up the (square) system of linear equations

P (t)λ = Q (t) , λT =[

a0 a1 b]

(2.8)

for determining the parameters λ.

Remark 2.1 There are no derivatives with respect to time involved, but only integra-tions. This is particularly valuable in the presence of high frequency perturbations ormeasurements corrupted by noise.

Remark 2.2 The incorporation of a filter yf = F (s) y is straight-forward. Clearly, theorder of derivatives w.r.t. s increases by the order of F (s).

Remark 2.3 (Implementation issues). Instead of taking dim (λ)−1 iterated integrals forsetting up a system of dim (λ) linear equations as done in the introductory example, it isuseful also to consider subsequent time integrals thus yielding an over-determined set ofequations, to be solved e.g. by means of the least-squares method. Additionally, droppingthe “first few” equations of the (over-determined) system Pλ = Q, which are clearly morearticulately affected by noise than the subsequent time integrals, has turned out to improvethe performance of this parameter estimation method significantly.

1 The notation∫ (α)

φ (t) =∫ t

0

∫ β1

0· · ·∫ βα−1

0φ (βα) dβα · · · dβ2dβ1,

∫

φ (t) =∫ (1)

φ (t) =∫ t

0φ (β1) dβ1

is arranged.

2. Identification. . . 2.2. Discrete-Time Systems 7

As seen in Example 2.1, the matrices P and Q (represented in the operational do-main) involve expressions of the type s−(γ+1)

(

dkx/dsk)

, γ, k ≥ 0, see (2.3) and (2.7).Alternatively to the representation via iterated integrals, the according time domain rep-resentation (notice that 1/sγ+1 • tγ/γ!) might be written as

1

sγ+1

dkx

dsk• (−1)k

γ!

∫ t

0

(t− τ)γ τ kx (τ) dτ (2.9)

due to the convolution rule of operational calculus. Let f : τ 7→ (t− τ)γ τ kx (τ). Byapplying a change of coordinates τ = g (τ),

∫ t1

t0

f (τ) dτ =

∫ g−1(t1)

g−1(t0)

f (g (τ)) ∂τg (τ) dτ ,

namely τ = τ t, we have∫ t

0

f (τ) dτ =

∫ 1

0

f (τ t) tdτ = tγ+k+1

∫ 1

0

(1 − τ)γ τ kx (τ t) dτ . (2.10)

Example 2.2 (Example 2.1 cont’d) With (2.9) and (2.10) we obtain

Pi1 (t) = − ti+4

(i+ 1)!

∫ 1

0

((i+ 4) τ − 3) (1 − τ)i τ 2y (τ t) dτ

Pi2 (t) = − ti+3

(i+ 1)!

∫ 1

0

(

(i+ 4) (i+ 3) τ 2 − 6 (i+ 3) τ + 6)

(1 − τ)i−1 τ y (τ t) dτ

Pi3 (t) =ti+4

(i+ 1)!

∫ 1

0

((i+ 4) τ − 3) (1 − τ)i τ 2u (τ t) dτ ,

for i ≥ 1, and

Q1 (t) = t3(

y (t) − 3

∫ 1

0

(

10τ 2 − 8τ + 1)

y (τ t) dτ

)

Qi (t) =ti+2

(i+ 1)!

∫ 1

0

(

η42 (i) τ 3 − 9η3

2 (i) τ 2 + 18η22 (i) τ − 6

)

(1 − τ)i−2 y (τ t) dτ ,

i ≥ 2, with ηβα (i) =∏β

j=α (i+ j). Hence, for numerical reasons it is suitable to factor

P (t) and Q (t) as P (t) = Υ (t) P (t), Q (t) = Υ (t) Q (t), with Υ (t) = diag (ti+2/ (i+ 1)!),i = 1, . . . , dimQ.

2.2 Discrete-Time Systems

Now, let us consider linear time-invariant discrete-time SISO systems,n∑

i=0

aiyk+i =m∑

i=0

biuk+i, m ≤ n, an = 1. (2.11)

We will commence from the z-domain representation of (2.11) to evolve an identificationapproach following the light of [FSR03].

2. Identification. . . 2.2.1. A z-Domain Approach 8

2.2.1 A z-Domain Approach

Consider sequences (fk), with fk = 0 for k < 0. Let |fk| ≤ BAk, k ≥ 0, for some A,B ∈ R.Then, the Laurent series

fz (z) =∞∑

i=0

fiz−i (2.12)

is absolutely convergent for all z with |z| > A. The holomorphic function fz (z) is calledthe z-transform of (fk), indicated as fz (z) • (fk) for short. Notice that

(fk+n) • zn(

fz (z) −n−1∑

j=0

fjz−j

)

. (2.13)

The j-th order derivative of fz (z) with respect to z is

dj

dzjfz (z) =

∞∑

i=0

fidj

dzj(

z−i)

= (−1)j z−j∞∑

i=0

(

j−1∏

s=0

(i+ s)

)

fiz−i,

thus, we have

zjf (j) (z) • (−1)j((

j−1∏

s=0

(k + s)

)

fk

)

, (2.14)

arranging the notation (·)(j) = (∂z)j. To start with, let us first illustrate the basic ideas

of the identification method to be discussed with a simple introductory example.

Example 2.3 Consider the system

yk+1 + ayk = buk, (2.15)

a, b unknown, or, equivalently, by applying the z-transform,

(z + a) yz (z) − zy0 = buz (z) . (2.16)

Taking derivatives, twice, with respect to z in order to allow for ignoring the initial con-dition yields

2y(1)z + (z + a) y(2)

z = bu(2)z . (2.17)

Notice thatzf (1) (z) • − (kfk) , z2f (2) (z) • (k (k + 1) fk) ,

see (2.14). Multiply (2.17) by z and re-sort w.r.t. the parameters a, b,

1

z

(

z2y(2)z

)

a− 1

z

(

z2u(2)z

)

b = −2(

zy(1)z

)

−(

z2y(2)z

)

. (2.18)


The according discrete-time domain representation of (2.18), introducing the equationerror sequence (e0

k), reads

e0k = k (k − 1) yk−1a− k (k − 1)uk−1b− 2kyk + k (k + 1) yk =

= k (k − 1) yk + yk−1a− uk−1b . (2.19)

Following the lines of the continuous-time approach, e.g. calculate N ≥ 1 iterated sums ofthe right-hand side of (2.19), i.e., divide (2.18) N times by (z − 1), to set up the systemof (N + 1) linear equations for determining the parameters a and b,2

−k (k − 1) yk−1 k (k − 1) uk−1

−k−1∑

i=1

i (i− 1) yi−1

k−1∑

i=1

i (i− 1) ui−1

......

−(N)∑

iN (iN − 1) yiN−1

(N)∑

iN (iN − 1) uiN−1

[

ab

]

+ e =

k (k − 1) ykk−1∑

i=0

i (i− 1) yi

...(N)∑

iN (iN − 1) yiN

,

eT =[

e0k e1k · · · eNk]

. The least-squares solution to this (under-determined) problem,

i.e., minλ ‖e‖22, λ

T = [a, b], Pλ+ e = Q, reads λ =(

P TP)−1

P TQ.

Though aiming at discarding the initial conditions by taking derivatives w.r.t. z, thenumber of derivatives for achieving that goal is clearly not unique. In Example 2.3, theinitial condition y0 was discarded by differentiating twice, i.e., (n+ 1) times, w.r.t. z.However, as an alternative way, one might think of first dividing (2.16) by z, followedby an n times differentiation in order to allow for ignoring the initial conditions. Thisapproach is illustrated via the following example, and further pursued in Remark 2.5.

Example 2.4 Consider again (2.15) and (2.16). Now, divide (2.16) by z, then differen-tiate once w.r.t. z to obtain

(

1 + az−1)

y(1)z − az−2yz = b

(

z−1u(1)z − z−2uz

)

.

Collecting for the parameters a, b,

1

z

(

y(1)z − 1

zyz

)

a− 1

z

(

u(1)z − 1

zuz

)

b = −y(1)z ,

and subsequent multiplication with z yields

1

z

(

zy(1)z − yz

)

a− 1

z

(

zu(1)z − uz

)

b = −zy(1)z

instead of (2.18). The according discrete-time representation of the equation error se-quence (e0

k) readse0k = −k yk + yk−1a− uk−1b . (2.20)

Compare this expression with (2.19). The iterated summation and the calculation of theparameters proceeds as discussed in Example 2.3.

2 The notation(α)∑

fiα=

k−1∑

i1=0

i1−1∑

i2=0

· · ·iα−1−1∑

iα=0

fiαis arranged.


Remark 2.4 Clearly, by construction, both error sequences (e0k) and (e0

k) do not refer tovalues of u and y of the past (i.e., k < 0). While both equation errors are “forced” tozero at the initial time instant k = 0 (which would, in absence of the factor k, involvey−1 and u−1), the value of (e0

k) is additionally set to zero at k = 1. Both ideas are furtherdeveloped in the following, to finally conclude in Remark 2.11, on the basis of a 5th-ordermodel of a drive-train, that no clear preference between them regarding performance andnumerical condition was located.

Let us now address the general case (2.11), with the associated z-domain representation

n∑

i=0

aizi

(

yz (z) −i−1∑

j=0

yjz−j

)

=m∑

i=0

bizi

(

uz (z) −i−1∑

j=0

ujz−j

)

and the transfer function

G (z) =yz (z)

uz (z)=

∑mi=0 biz

i

∑ni=0 aiz

i=Bz (z)

Az (z), an = 1, (2.21)

hence,

Azyz −n∑

i=0

aizi

i−1∑

j=0

yjz−j = Bzuz −

m∑

i=0

bizi

i−1∑

j=0

ujz−j. (2.22)

Taking derivatives on (2.22), n times, n ≥ n+1, w.r.t. z in order to allow for ignoring the

initial conditions yj, j = 0, . . . n− 1, and uj, j = 0, . . .m− 1, yields (Azyz)(n) = (Bzuz)

(n),and, by virtue of Leibniz’ product rule,

n∑

j=0

(

n

j

)

A(j)z y(n−j)

z =m∑

j=0

(

n

j

)

B(j)z u(n−j)

z , (2.23)

noticing that A(j)z (z) = 0, j > n, and B

(j)z (z) = 0, j > m. With the j-th order derivatives

of the polynomials Az and Bz reading

A(j)z =

n∑

s=j

s!

(s− j)!asz

s−j, B(j)z =

m∑

s=j

s!

(s− j)!bsz

s−j,

equation (2.23) takes the form

n!n∑

j=0

z−jy(n−j)z

(n− j)!

n∑

s=j

(

s

j

)

aszs = n!

m∑

j=0

z−ju(n−j)z

(n− j)!

m∑

s=j

(

s

j

)

bszs. (2.24)

To further reveal the structure of (2.24), and, in particular, to facilitate the transformationback to the discrete-time domain, see also (2.14), let

Y (j)z

.=zjy

(j)z

j!, U (j)

z.=zju

(j)z

j!, (2.25)


then, by applying a suitable “shift operation”, i.e., multiplication of (2.24) by zn−n, weobtain

n!n∑

j=0

Y (n−j)z

n∑

s=j

(

s

j

)

aszs−n = n!

m∑

j=0

U (n−j)z

m∑

s=j

(

s

j

)

bszs−n. (2.26)

Re-arrangement of the summations of (2.26) to collecting for the parameters ai, i =0, . . . , n− 1, and bi, i = 0, . . . ,m, finally yields the representation

n−1∑

i=0

zi−n

(

n!i∑

j=0

(

i

j

)

Y (n−j)z

)

ai −m∑

i=0

zi−n

(

n!i∑

j=0

(

i

j

)

U (n−j)z

)

bi = −n!n∑

j=0

(

n

j

)

Y (n−j)z ,

(2.27)noticing that an = 1 by (2.11). Thus, each parameter ai (or bi) is associated with an

expression involving Y(n−j)z (or U

(n−j)z , respectively), 0 ≤ j ≤ i, which is to be shifted

(n− i) times.Now, let us determine the discrete-time representation of (2.27). First of all, from

(2.14) and (2.25) we obtain

F (j)z • (−1)j

j!

(

j−1∏

s=0

(k + s)

)

fk,(

Fz, (fk))

∈(

Yz, (yk))

,(

Uz, (uk))

. (2.28)

Following (2.27), the expressions associated to the parameters ai, bi are

zi−nn!i∑

j=0

(

i

j

)

F (n−j)z • (−1)n

(

i∑

j=0

(

i

j

)

(−1)−j n!

(n− j)!

n−1−j∏

s=0

(k − n+ i+ s)

)

fk−n+i =

= (−1)n(

i∑

j=0

(−1)−j(

i

j

)

n!

(n− j)!

i−j−1∏

s=0

(k − n+ n+ s)

)(

n−1∏

s=i

(k − n+ s)

)

fk−n+i,

and with the identity (notice that n involved in the left-hand side cancels out)

i∑

j=0

(−1)−j(

i

j

)

n!

(n− j)!

i−j−1∏

s=0

(k − n+ n+ s) =i−1∏

s=0

(k − n+ s) ,

we finally find the relation

zi−n

(

n!i∑

j=0

(

i

j

)

F (n−j)z

)

• (−1)n((

n−1∏

s=0

(k − n+ s)

)

fk−n+i

)

. (2.29)

Thus, with (e0k) denoting the equation error sequence of (2.27), we obtain the discrete-time

representation

e0k = (−1)n(

n−1∏

s=0

(k − n+ s)

)

yk +n−1∑

i=0

yk−n+iai −m∑

i=0

uk−n+ibi

=

= (−1)n(

n−1∏

s=0

(k − n+ s)

)

yk −[

−hTa,k hTb,k]

λ

(2.30)


by introducing the abbreviations

hTa,k =[

yk−n · · · yk−1

]

, hTb,k =[

uk−n · · · uk−n+m

]

(2.31)

referred to as the data vectors as containing the measurements. The (n+m+ 1) para-meters are arranged to form the vector λ,

λT =[

a0 · · · an−1 b0 · · · bm]

. (2.32)

Remark 2.5 (following the idea as sketched in Example 2.4) Take (2.22), divide by z,and subsequently differentiate n times w.r.t. z in order to allow for ignoring the initial

conditions. Then, (z−1Azyz)(n)

= (z−1Bzuz)(n)

. By introducing fz = z−1fz, fz ∈ yz, uz,corresponding to an index shift by −1 in the discrete time domain, we have

(Azyz)(n) = (Bzuz)

(n) .

This problem can be immediately traced back to the case discussed above (with n = n),thus yielding

e0k = (−1)n(

n−1∏

s=0

(k − n+ s)

)

yk−1 −[

−hTa,k−1 hTb,k−1

]

λ

,

see (2.30)–(2.32). The application of an index shift by +1 on the right-hand side finallygives the result

e0k = (−1)n

(

n∏

s=1

(k − n+ s)

)

yk −[

−hTa,k hTb,k]

λ

. (2.33)

Notice that (2.30) and (2.33) are related via

e0ke0k

= (−1)n−n (k − n)n−1∏

s=n+1

(k − n+ s) . (2.34)

For calculating the parameters λ, the equation error sequence (2.30) or (2.33), respec-tively, is subject to N ≥ n + m fold iterated summation to assemble the set of (N + 1)linear equations

Pλ+ e = Q, (2.35)

cf. Example 2.3.

Implementation Issues

Referring to Figure 2.1, which depicts the structure of the proposed z-domain onlineidentification approach, the subsequent notes aim at drawing the link to a computer-algebra implementation available at http://regpro.mechatronik.uni-linz.ac.at.


PSfrag replacements

a0 a1 an−1 b0 bm

1z−1

1z−1

1z−1

1z−1

1z−1

1z−1

1z−1

1z−1

1z−1

1z−1

1z−1

1z−1

ξ1

ξn

nn

(k)

. . .. . .

. . .

. . .

. . .

. . .

shiftsshifts

n − 1

ξn−1

ξ2n−m

ξn+1

ξ2n

(uk)

......

...

...

...

......

...

...

...

n − m

(yk)(yk)

z−1

z−1

z−1

z−1

ξµa(0)+1

ξµa(0)+N

ξµa(1)+1

ξµa(1)+N

ξµa(n−1)+1

ξµa(n−1)+N

ξµb(0)+1

ξµb(0)+N

ξµb(m)+1

ξµb(m)+N

ξκ+1

ξκ+N

multiply by (−1)n∏n−1

s=0 (k − n+ s), see (2.30)

P1,1,k P1,2,k P1,n,k P1,n+1,k P1,n+m+1,k Q1,k

yk−n yk−n+1 yk−1 uk−n uk−n+m yk

Figure 2.1: Scheme of the online z-domain identification approach emanating from (2.30)or (2.33), respectively. Arrangement of the shifter and “integrator” states ξ, and link tothe composition of (2.37).

The focal point of Figure 2.1, introducing a possible scheme of a state-space realization(with the state ξ), is the representation of (2.30) or (2.33), respectively, processing themeasurements yk−n+i, i = 0, . . . , n, and uk−n+i, i = 0, . . . ,m, made available via the shifterchains. The expressions of (2.30), (2.33), decomposed with respect to the parametersai, bi, are further processed by means of N -fold iterated summation each, also referredto as (discrete-time) “integrator chains”. With

µa (s) = 2n+ sN, s = 0, . . . , n− 1

µb (s) = 2n+ (n+ s)N, s = 0, . . . ,m,(2.36)

and κ = 2n+ (n+m+ 1)N as the total number of shifters and “integrators” related to

2. Identification. . . 2.2.2. A q-Domain Approach. . . 14

the parameters, the set of linear equations (2.35) reads

P1,1 . . . P1,n P1,n+1 . . . P1,n+m+1

ξµa(0)+1 ξµa(n−1)+1 ξµb(0)+1 ξµb(m)+1

......

......

ξµa(0)+N . . . ξµa(n−1)+N ξµb(0)+N . . . ξµb(m)+N

λ− e =

Q1

ξκ+1

...ξκ+N

. (2.37)

The first row of P and Q, respectively, is given as

P1,i+1,k = χ (k) yk−n+i, P1,n+j+1,k = −χ (k)uk−n+j , Q1,k = −χ (k) yk,

i = 0, . . . , n− 1, j = 0, . . . ,m, with

χ (k) = (−1)nn−1∏

s=0

(k − n+ s) or χ (k) = (−1)nn∏

s=1

(k − n+ s) , (2.38)

depending on the application of (2.30) or (2.33).

2.2.2 A q-Domain Approach, tracing back to the z-Domain Solution

As a first motivation for the application of the bilinear transform C → C,

z =1 + q/Ω0

1 − q/Ω0

, q = Ω0z − 1

z + 1, Ω0 =

2

Ta, (2.39)

referred to as Tustin transform or q-transform for short, let us state a preliminary remarkon subtleties associated to the parametrization of the linear identifier in terms of the z-domain parameters λ as introduced above. These issues will be illustrated in Section 2.2.6on the basis of simulation and measurement results from a 5th order “drive-train” example.

Remark 2.6 Given a continuous-time system G (s) with poles si, then the poles of the ac-cording discrete-time system G (z) are located at zi = exp (siTa). Thus, with the samplingtime Ta decreasing, the poles zi approach the point 1.

The q-domain transfer function, outlined in terms of the “z-domain parameters” ai,i = 0, . . . , n− 1, and bi, i = 0, . . . ,m, is

G# (q) = G (z)|z=

1+q

Ω01−

qΩ0

=

(

1 − qΩ0

)n−m (

bm

(

1 + qΩ0

)m

+ . . .+ b0

(

1 − qΩ0

)m)

(

1 + qΩ0

)n

+ an−1

(

1 − qΩ0

)(

1 + qΩ0

)n−1

+ . . .+ a0

(

1 − qΩ0

)n ,

with the prime on G# (q) arranged to indicate the representation in terms of ai, bi.Clearly, an (n−m)-fold zero at q = Ω0 occurs. Next, arrange a change of the (n+m+ 1)parameters,

(a0, . . . , an−1, b0, . . . , bm) 7→ (A1, . . . , An, B0, . . . , Bm) , (2.40)


to obtain the representation

G# (q) =

(

1 − q

Ω0

)n−m m∑

i=0

Biqi

1 +n∑

i=1

Aiqi=

(

1 − q

Ω0

)n−m

B# (q)

A# (q). (2.41)

Notice that, in general, the map (2.40) is non-linear due to the scaling chosen for the de-nominator, A# (0) = 1. The following (closely related) remarks state some facts on G# (q)to motivate the ideas leading to the q-domain identification method to be introduced.

Remark 2.7 Given the continuous-time system G (s), the approximation

G# (jΩ) ≈ G (jω) , Ω =2

Tatan

(

ωTa2

)

, |ωTa| 1

holds. In particular, this fact gives rise to discard, except for the (n−m)-fold zeros atΩ0, those zeros of G# (q) located significantly beyond the locus of the system poles, as theyaffect the system response inessentially.

Remark 2.8 Let s = −a be a pole of G (s), then the according pole of G# (q) is q =Ω0 tanh (−a/Ω0). According relations hold for complex conjugate poles. This observationdraws a close link between the q-domain and the s-domain identification.

Now we are ready to state the point of departure for the q-domain parametrizationof the identification approach. Following Remark 2.7, we propose to (optionally) discardthose zeros located significantly beyond the domain of the poles of G# (q) to obtain

G# (q) =

(

1 − q

Ω0

)n−m m∑

i=0

Biqi

1 +n∑

i=1

Aiqi=

(

1 − q

Ω0

)n−m

B# (q)

A# (q), m ≤ m, (2.42)

with the tilde indicating the approximation of G# (q). Next, we will trace back to thez-domain solution, and re-cast those results ((2.30) and (2.33), respectively) in terms ofthe q-domain parameters Λ,

ΛT =[

A1 · · · An B0 · · · Bm

]

. (2.43)

To this end, let us first determine the parameter transformation. The z-domain transferfunction (with an 6= 1 in general) according to the specified q-domain representationG# (q) reads

G (z) =

2n−mm∑

i=0

BiΩi0 (z − 1)i (z + 1)m−i

(z + 1)n +n∑

i=1

AiΩi0 (z − 1)i (z + 1)n−i

=

m∑

s=0

bszs

n∑

s=0

aszs. (2.44)


By virtue of the binomial theorem,

(x+ y)n =n∑

i=0

(

n

i

)

xiyn−i, x, y ∈ C, n ∈ N0+, (2.45)

we have

(z − 1)i (z + 1)n−i = (−1)in−i∑

j=0

i∑

l=0

(−1)l(

i

l

)(

n− i

j

)

zj+l =

= (−1)in∑

s=0

(

(−1)ss∑

j=0

(−1)j(

i

s− j

)(

n− i

j

)

)

zs,

and, finally,

G (z) =

2n−mm∑

s=0

(

m∑

i=0

(−Ω0)i

s∑

j=0

(−1)s+j(

i

s− j

)(

m− i

j

)

Bi

)

zs

n∑

s=0

(

(

n

s

)

+n∑

i=1

(−Ω0)i

s∑

j=0

(−1)s+j(

i

s− j

)(

n− i

j

)

Ai

)

zs. (2.46)

Hence, the (n+m+ 2) z-domain coefficients as, s = 0, . . . , n, and bs, s = 0, . . . ,m, arerelated to the (n+ m+ 1) q-domain coefficients Ai, Bi by means of the linear/affinemappings

bs =m∑

i=0

Πbs,iBi, s = 0, . . . ,m (2.47)

and

as =

(

n

s

)

+n∑

i=1

Πas,iAi, s = 0, . . . , n, (2.48)

with the (m+ 1) × (m+ 1) dimensional matrix[

Πbs,i

]

and the (n+ 1) × n dimensional

matrix[

Πas,i

]

,

Πbs,i = 2n−m (−Ω0)

is∑

j=0

(−1)s+j(

i

s− j

)(

m− i

j

)

,

Πas,i = (−Ω0)

is∑

j=0

(−1)s+j(

i

s− j

)(

n− i

j

)

.

(2.49)

Plugging the transformation (2.47)–(2.49) into the z-domain solution (2.30) or (2.33)(notice that these equation have to be slightly adapted so as to allow for arbitrary an),we finally find

e0k = χ (k)

n∑

r=0

(

n

r

)

yk−n+r +n∑

i=1

(

n∑

r=0

yk−n+rΠar,i

)

Ai −m∑

i=0

(

m∑

r=0

uk−n+rΠbr,i

)

Bi

2. Identification. . . 2.2.3. The z-Domain Approach – Continued 17

with χ (k) due to (2.38). In a more compact matrix notation, we have

e0k = χ (k)

[

hTa,k yk]

(

n0

)

...(

nn

)

−[ [

−hTa,k −yk]

Πa,[

hTb,k]

Πb]

Λ

, (2.50)

with the data vectors ha,k, hb,k as introduced in (2.31).

Remark 2.9 The procedure of setting up the set of linear equations by means of iteratedsummation of (2.50), and the solution for Λ as well, proceeds as discussed above.

2.2.3 The z-Domain Approach – Continued

The idea introduced in the previous Section 2.2.2 to discarding those zeros of the transferfunction, which only have minor influence on the system response, and, hence, are difficultto estimate in presence of noise (to be illustrated in Section 2.2.6), will now be equivalentlyapplied to the z-domain approach. The reason for discussing this idea for the q-domaincase first simply was that, by virtue of the close similarity to the continuous-time domain,things might be particularly apparent from the control engineer’s point of view.

Clearly, by (2.39), discarding certain zeros ofG# (q), i.e. shifting those zeros to infinity,corresponds to placing the associated zeros of the z-domain transfer function G (z) atz = −1 (see again (2.44) involving the occurrence of an (m− m)-fold zero at z = −1).Now, introduce the counterpart of (2.42),

G (z) =

(z + 1)m−mm∑

i=0

bizi

n∑

i=0

aizi=

m−m∑

s=0

m∑

i=0

bi(

m−ms

)

zs+i

n∑

i=0

aizi=

m∑

i=0

bizi

n∑

i=0

aizi, (2.51)

with the tilde indicating the approximation of G (z). There should be no confusionby labeling the coefficients of the numerator again as bi, i = 0, . . .m, though nowreferring to G (z) instead of G (z). The injective linear map relating the parametersbT =

[

b0 . . . bm]

to the numerator coefficients bT =[

b0 . . . bm]

is represented as

b = Ξbb. LetλT =

[

a0 · · · an−1 b0 · · · bm]

, (2.52)

then

λ = Ξλ with Ξ =

[

En×n 00 Ξb

]

,

and the linear identifier (2.37), re-casted in terms of the parameters λ, reads

P λ+ e = Q, P = PΞ. (2.53)

2. Identification. . . 2.2.4. A Note on the Standard LS Identification Method 18

2.2.4 A Note on the Standard LS Identification Method

Consider (2.11) again,

yk +n−1∑

i=0

aiyk−n+i =m∑

i=0

biuk−n+i, m ≤ n,

and introduce the (equation-) error (ek),

ek = yk −(

−n−1∑

i=0

yk−n+iai +m∑

i=0

uk−n+ibi

)

= yk −[

−hTa,k hTb,k]

λ, (2.54)

see (2.31) and (2.32) for the definitions of ha,k, hb,k and λ. Compare this definition of (ek)to (2.30) and (2.33).

Referring to the z-domain representation, with G (z) due to (2.21), this error ez (z),also referred to as the “generalized error” as linearly depending on the parameters λ,reads

ez = M−12 yz −M1uz, M1 (z) =

∑mi=0 biz

i

zn, M2 (z) =

zn

zn +∑n−1

i=0 aizi.

Taking N + 1 measurements, ek = yk −[

−hTa,k hTb,k]

λ, k = 0, . . . , N , we end up withthe (under-determined) system of N + 1 linear equations

e0...eN

=

y0...yN

−

−hTa,0 hTb,0...

...−hTa,N hTb,N

λ, (2.55)

or e = Q− Pλ for short. The least-squares solution to (2.55), λ =(

P TP)−1

P TQ, finally

gives the estimate for λ in the sense minλ ‖e‖22.

This is a standard procedure for discrete-time linear systems identification. Analo-gously to the previous discussion, we will recast this method in terms of the accordingq-domain parameters Λ, see (2.43). To this end, commence from (2.54), adapted to allowfor an arbitrary, and apply the transformation (2.47), (2.48) to obtain

ek =n∑

r=0

(

n

r

)

yk−n+r +n∑

i=1

(

n∑

r=0

yk−n+rΠar,i

)

Ai −m∑

i=0

(

m∑

r=0

uk−n+rΠbr,i

)

Bi. (2.56)

In a more compact notation, the resulting set of (N + 1) linear equations evolving from(2.56) reads

e0...eN

=

hTa,0 y0...

...hTa,N yN

(

n0

)

...(

nn

)

−

−hTa,0 −y0...

...−hTa,N −yN

Πa,

hTb,0...

hTb,N

Πb

Λ. (2.57)

Compare this result to (2.50).

2. Identification. . . 2.2.5. A q-Domain Approach – “Direct Method” 19

Remark 2.10 Each equation of (2.55) or (2.57), evaluated at k, i.e., (2.54) or (2.56),involves the measurements yk−s, uk−s, s = 0, . . . , n, but, in contrast to the approachintroduced in the Sections 2.2.1 and 2.2.2, clearly not the prior history.

2.2.5 A q-Domain Approach – “Direct Method”

Though not revealing new results compared to the results from Section 2.2.2, this para-graph, thought of as an addendum, provides a different way of addressing the objective ofSection 2.2.2, namely to commence from the q-domain representation of a discrete-timelinear system to follow the light of [FSR03]. Instead of tracing back to the z-domain solu-tion as discussed in Section 2.2.2, we will now carry out the calculations in the q-domainexclusively. To this end, let us define the q-domain representation of sequences (fk) as

fq (q).= fz

(

1 + qΩ0

1 − qΩ0

)

=∞∑

i=0

fi

(

1 + qΩ0

1 − qΩ0

)−i

, fz (z) = fq

(

Ω0z − 1

z + 1

)

, (2.58)

indicated as fq (q) ↔ fz (z) • (fk) for short. For notational convenience the abbrevia-tion z = ϕ (q) is arranged for (2.39), thus fq = fz ϕ. The identification procedure to beoutlined will involve the repeated derivatives of signals fq w.r.t. q,

∂qfq (q) =2/Ω0

(1 − q/Ω0)2

(

f (1)z ϕ

)

,

(∂q)2 fq (q) =

(

2/Ω0

(1 − q/Ω0)2

)2(

f (2)z +

(

1 − q

Ω0

)

f (1)z

)

ϕ.(2.59)

Analogously to the abbreviation f(i)z = (∂z)

i fz, the notation f(i)q = (∂q)

i fq is introducedfor q-domain signals. From (2.59) we can directly deduce the relations to the discretetime domain, i.e.,

−Ω0

2

(

1 −(

q

Ω0

)2)

f (1)q • (kfk)

and(

Ω0

2

(

1 −(

q

Ω0

)2))2

f (2)q • (k (k + 1) fk) − 2 (−1)k ? (kfk) ,

cf. (2.14), with ? as the convolution operator. However, for convenience, within thesubsequent discussions we will not carry out the transformations up to the discrete timedomain, but evolve from the representations

f (1)q =

2/Ω0

(1 − q/Ω0) (1 + q/Ω0)

(

F (1)z ϕ

)

f (2)q =

(

2/Ω0

(1 − q/Ω0) (1 + q/Ω0)

)2(

2F (2)z +

(

1 +q

Ω0

)

F (1)z

)

ϕ(2.60)

see (2.25) for the definition of F(j)z .

2. Identification. . . 2.2.5. A q-Domain Approach – “Direct Method” 20

Example 2.5 (Example 2.3 cont’d) Application of (2.39) to (2.16) yields the q-domainrepresentation of (2.15),

(

1 + a+ (1 − a)q

Ω0

)

yq −(

1 +q

Ω0

)

y0 =

(

1 − q

Ω0

)

buq. (2.61)

The transfer function, obtained via the re-parametrization a = (1 − AΩ0) / (1 + AΩ0),b = 2B/ (1 + AΩ0), is

G# (q) =(1 − q/Ω0)B

1 + Aq,

see (2.40) and (2.41). Then, (2.61) takes the form

(1 + Aq) yq −1

2(1 + AΩ0)

(

1 +q

Ω0

)

y0 =

(

1 − q

Ω0

)

Buq,

establishing the point of departure. Taking derivatives, twice, with respect to q in order toallow for ignoring the initial condition yields

(1 + Aq) y(2)q + 2Ay(1)

q = B

((

1 − q

Ω0

)

u(2)q − 2

Ω0

u(1)q

)

.

Next, apply (2.60), multiply byΩ2

0

2

(

1 − qΩ0

)2 (

1 + qΩ0

)

, and re-sort with respect to the

parameters A,B,

2

(

2Y(2)z q

1 + qΩ0

+ Ω0Y(1)z

)

A− 41 − q

Ω0

1 + qΩ0

U (2)z B = −2

(

2Y(2)z

1 + qΩ0

+ Y (1)z

)

. (2.62)

(For notational brevity, the composition of Y(j)z , U

(j)z with ϕ has been dropped). It is easy

to verify that the discrete-time representation of (2.62) coincides with (2.50).

To (briefly) address the general case, commence from the q-domain representation of

(2.11), outlined in terms of the parameters

Ai, Bi

, see (2.42). Taking derivatives w.r.t.

q, n = n+ 1 times, to discard the initial conditions, we obtain

(

A#yq)n+1

=

(

(

1 − q

Ω0

)n−m

B#uq

)n+1

,

and, by virtue of Leibniz’ rule,

n∑

j=0

(

n+ 1

j

)

A#(j)y(n+1−j)q −

(

(

1 − q

Ω0

)n−m

B#

)(j)

u(n+1−j)q

= 0. (2.63)

2. Identification. . . 2.2.6. Applications and Discussion 21

To proceed, let us first introduce a suitable representation of f(i)q , revisiting (2.59).

We find

f (i)q =

(

2/Ω0

(1 − q/Ω0)2

)i(

f (i)z +

i−1∑

j=1

ci,j

(

1 − q

Ω0

)i−j

f (j)z

)

ϕ (2.64)

i ≥ 0, with

ci,j = 2j−ii!

j!

(

i− 1

j − 1

)

. (2.65)

Furthermore, to obtain the generalization of (2.60), extend the expressions involving f(j)z

with z−jzj, hence, finally,

f (i)q =

(

2/Ω0

(1 − q/Ω0) (1 + q/Ω0)

)i(

i!F (i)z +

i−1∑

j=1

ci,j

(

1 +q

Ω0

)i−j

j!F (j)z

)

ϕ. (2.66)

Clearly, by virtue of this equation, the relation of f(i)q to the discrete-time domain is easily

established.

Again, the coincidence of (2.63), multiplied by 12

(

1 + qΩ0

)

Ωn+10

(

1 − qΩ0

)n+1

, with

(2.50) can be verified by straight-forward, though tedious, computations.

2.2.6 Applications and Discussion

To illustrate the behavior of the presented algebraic approach to discrete-time linearsystems identification, and, in particular, to reveal the subtleties involved, we will finallydiscuss a selected application available as a laboratory experiment.

Consider the lab model of a drive train as depicted in Figure 2.2. The parameters aregiven as follows: LA = 896µH (armature inductance), RA = 6.38Ω (armature resistance),km = 41 ·10−3Nm/A (torque constant), c1 = c2 = 1.72 ·10−3Nm/rad (spring coefficients),Θ1 = 25.65 · 10−6kgm2, Θ2 = 6.44 · 10−6kgm2, Θ3 = 5.1 · 10−6kgm2 (moments of inertia ofthe rotors), and d1 = 3.98·10−6Nms, d2 = 0.92·10−6Nms, d3 = 2.4·10−6Nms (coefficientsof viscous friction, related to the bearings of the respective rotors). By discarding thedynamics related to the electrical subsystem in the sense of the singular perturbationpoint of view, the dynamics of the drive-train are obtained as

G (s) =ω

u=

23.7(

1 +s

8.1

)

(

1 + 2 × 0.1s

12.4+( s

12.4

)2)(

1 + 2 × 0.0083s

27.7+( s

27.7

)2) ,

thus n = 5, m = 4 for the according discrete-time system G (z).The following examples provide different case scenarios regarding the choice of the

parameter m ≤ m (for both the q- and z-domain approach; see Sections 2.2.2 and 2.2.3)and the sampling time Ta. Comments on the results are provided in the respective figurecaptions. All calculations are carried out with the following settings: The representation(2.30) of the equation error sequence is chosen, i.e. χ (k) = (−1)n

∏n−1s=0 (k − n+ s), see


PSfrag replacements

Θ1 Θ2 Θ3

c1 c2

ω

permanent-magnetdc motor

u

Figure 2.2: The lab experiment “drive-train”.

(2.38), with n = n+1. See also Remark 2.11 for a comment on this issue. The “first” α = 7equations of the scheme of Figure 2.1 are discarded from the parameter calculation as theyare more articulately affected by noise (chosen as Gaussian distributed for the simulations)than the subsequent ones, see also Remark 2.3. Additonally, an over-determined set oflinear equations is set up, N = n + m + α + β, with β = 10 chosen as the number ofadditional iterated summations added. Hence, from the (1 +N) equations, see Figure 2.1,the equations no. (α + 1) up to (1 +N), i.e., a number of 1 + N − α = 1 + n + m + βequations, are used for calculating the (1 + n+ m) unknown parameters (in the least-squares sense). All equations are normalized (by dividing by the maximum absoluteentry of P of the respective row) to improve the numerical condition. The linear on-lineidentifiers start at t = 0, and the root loci and Bode diagrams of the identified z- andq-domain transfer functions given in the figures are due to λ and Λ, evaluated at the finaltime tend = 1.35s of the simulation.

Example 2.6 Let m = m and choose the sampling time Ta = 10ms. The simulation re-sults given in the Figures (2.3, 2.4) are associated with the observation that, in presence ofnoise, the estimation of the zeros which have minor influence on the system response (seethe top right subplot of Figure 2.3) is very poor. Additionally, it is found that, emergingwith decreasing sampling times, the numerical condition of the linear identifier parame-trized in terms of the z-domain parameters λ becomes increasingly worse, in contrast tothe parametrization of the q-domain approach. To illustrate this, the condition numbersof P (i.e. the ratio of the largest singular value of P to the smallest) for the z- and q-domain identifier, evaluated at the final time tend, for different sampling times, are given:(10ms, 3.0e10, 1.7e10), (5ms, 5.1e11, 8.0e9), (2ms, 2.9e13, 1.2e10), (1ms, 5.4e14, 1.4e10),(0.5ms, 9.0e15, 1.5e10). In order to show that these condition numbers are only weaklyaffected by the noise added to the output signal, the following triplets give the conditionnumbers obtained by noise-free simulations: (10ms, 3.0e10, 1.7e10), (5ms, 5.2e11, 7.8e9),(2ms, 2.9e13, 1.2e10), (1ms, 5.4e14, 1.4e10), (0.5ms, 8.9e15, 1.5e10).

The first observation of Example 2.6 (inappropriate estimation of the zeros in presenceof noise) is the motivation for applying the approximations of the numerators as proposedin the Sections 2.2.2 and 2.2.3, which is addressed in the following example.

Example 2.7 Let m = 0 and Ta = 10ms to obtain the simulation results given in theFigures (2.5, 2.6). The approach to discarding those zeros having negligible effect on


the system response is seen to be appropriate, see in particular the top right subplotof Figure 2.5. The estimation of the poles is again accurate. Additionally, due to thereduced number of parameters, this approach is associated with the advantage of hav-ing better numerical condition compared to the case m = m of the previous example,illustrated by the following condition numbers: (10ms, 1.8e7, 2.3e7), (5ms, 6.0e8, 2.7e7),(2ms, 3.5e10, 3.0e7), (1ms, 6.4e11, 3.1e7), (0.5ms, 1.1e13, 3.2e7). Again, the numericalcondition of the z-domain approach suffers with decreasing sampling times, whereas theparametrization in terms of Λ does not experience these problems. The according conditionnumbers obtained by noise-free simulations are: (10ms, 1.9e7, 2.3e7), (5ms, 5.8e8, 2.7e7),(2ms, 3.5e10, 3.0e7), (1ms, 6.4e11, 3.1e7), (0.5ms, 1.1e13, 3.2e7). To close this example,Figure 2.7 finally depicts the Bode diagrams of the z- and q-domain identification results,obtained with Ta = 1ms.

Example 2.8 (measurement results) Let m = 0 and Ta = 10ms. The Figures (2.8, 2.9)depict the identification results obtained from measurements of the lab model “drive-train”.See again the figure captions for comments.

Remark 2.11 Regarding the performance of the identifiers (and the numerical conditionas well), investigated on the basis of the drive-train example, no clear preference wasfound between the different approaches (2.30), and (2.33) to setting up the equation errorsequences.

Remark 2.12 Though this algebraic approach, following the light of [FSR03], providespromising results, it is worth mentioning that, clearly, the linear identifier cannot incor-porate a-priori knowledge on stability. More concretely, for the drive-train example, thepole-pair related to the “fast eigenfrequency” is located very close to the stability mar-gin, hence, in presence of noisy signals, this pole-pair might be found to shift beyond thestability margin.

To resume, let us finally accent some characteristics of the proposed algebraic z- andq-domain identification approach. As illustrated in the Figures 2.3–2.9, the idea to dis-carding those zeros of G (z) and G# (q) having negligible effect on the system dynamics(regarding the “interesting” frequency domain), qualifies as attractive. Additionally, refer-ring to the discussed example, the numerical condition of the linear identifier parametrizedin terms of λ (or λ, respectively) is found to suffer with decreasing sampling times, whichis not the case for the q-domain parametrization.

The problem of determining the system order n and to a-priori deciding whether ornot to discard certain zeros to find a suitable setting for 0 ≤ m ≤ m is preferably tackledby deriving a mathematical model based on physical considerations.

Finally, this chapter closes with examples on invoking the standard least-squares (LS)identification method of Section 2.2.4 to the drive-train example, with the z- and q-domainrepresentation entitled as “LS-z” and “LS-q” for short. These following Examples 2.9,2.10 represent the counterparts of the Examples 2.6, 2.7.


Example 2.9 (LS counterpart of Example 2.6, i.e., m = m, Ta = 10ms). The simulationresults given in the Figures (2.10, 2.12) indicate poor performance, even for the estimationof the pole-pair related to the “slow eigenfrequency”. The numerical condition of the LSidentification problem casted in terms of Λ, cf. (2.57), which amounts to 3.8e10, is verypoor compared to the respective parametrization in terms of λ, cf. (2.55), which is 593. Re-

scaling of the parameters Λ to Λsc =[

Asci , Bscj

]

, Asci = Ωi0Ai, i = 1, . . . , n, Bsc

j = Ωj0Bj,

j = 0, . . . , m, (no summations on the repeated indices) yields a significantly improvedcondition number, 54, however, the identification results obtained coincide with those ofthe “un-scaled parametrization” from above, i.e., the results depicted in the Figures (2.10,2.12).

Remark 2.13 By applying the scaling of the q-domain parameters Λ as given in Exam-ple 2.9 to the algebraic identification method, the previously clear advantage regarding thenumerical condition of the q-domain parametrization compared to the z-domain formula-tion, cf. Examples 2.6, 2.7, is foiled (in both cases m = m and m = 0). The tendency ofthe q-domain condition numbers associated to the parametrization in terms of Λsc followstightly those of the z-domain identifier to getting worse with decreasing sampling times.

Example 2.10 (LS counterpart of Example 2.7, i.e., m = 0, Ta = 10ms). The Figures(2.11, 2.13) show the simulation results, associated with the condition numbers 270 (LS-z) and 50 (LS-q; using the scaled representation in terms of Λsc, cf. Example 2.9).Again, similarly as encountered in the case m = m of Example 2.9, the estimation ofthe “dominant” pole-pair associated to the “slow eigenfrequency” is inaccurate, the otherestimated poles are even far away from the nominal ones.

Remark 2.14 (on the performance of the LS method) In order to obtain suitable resultswith the LS identification method (in the case of the considered drive-train example),it is advisable to increase the sampling time, say, e.g., to Ta = 50ms, additionally toarticulately increasing the “observation time span”.


−2

−1

0

1

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

u[V

]

t [s]

−120

−80

−40

0

40

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω[rad

/s]

t [s]

ω sim.q-ident.z-ident.

−40

−30

−20

−10

0

10

20

30

40

−600−400−200 0 200 400 600

Im

Re

q-ident. (red)

−30

−20

−10

0

10

20

30

−10 −8 −6 −4 −2 0

Im

Re

q-ident. (detail)

−1

−0.5

0

0.5

1

−25 −20 −15 −10 −5 0 5

Im

Re

z-ident. (red)

−1

−0.5

0

0.5

1

−0.5 0 0.5 1 1.5

Im

Re

z-ident. (detail)

Figure 2.3: (cf. Example 2.6) Simulation results, m = m, Ta = 10ms. Both z−and q−domain approach provide good results (regarding the system dynamics in the“interesting” frequency domain, see the top right subplot, and also Figure 2.4 for the Bodediagrams). However, the estimation of the zeros of both approaches is very inappropriateindeed.


−120

−80

−40

0

40

0.1 1 10 100 1000 10000

∣ ∣

G#

(jΩ

)∣ ∣[d

B]

q-ident.z-ident.

nom.

−540

−450

−360

−270

−180

−90

0

0.1 1 10 100 1000 10000

argG

#(j

Ω)

[deg

]

Ω = Ω0 tan (ω/Ω0) [rad/s]

q-ident.z-ident.

nom.

Figure 2.4: (Figure 2.3 cont’d; m = m, Ta = 10ms) Bode diagrams.


−2

−1

0

1

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

u[V

]

t [s]

−120

−80

−40

0

40

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω[rad

/s]

t [s]

ω sim.q-ident.z-ident.

−30

−20

−10

0

10

20

30

−600−400−200 0 200 400 600

Im

Re

q-ident. (red)

−30

−20

−10

0

10

20

30

−10 −8 −6 −4 −2 0

Im

Re

q-ident. (detail)

−1

−0.5

0

0.5

1

−25 −20 −15 −10 −5 0 5

Im

Re

z-ident. (red)

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

Im

Re

z-ident. (detail)

Figure 2.5: (cf. Example 2.7) Simulation results, m = 0, Ta = 10ms. The usefulnessof the idea of approximating the numerator in the sense of Sections 2.2.2 and 2.2.3 isverified, see, in particular, the top right subplot. See also Figure 2.6 for the associatedBode diagrams.


−240

−200

−160

−120

−80

−40

0

40

0.1 1 10 100 1000 10000

∣ ∣

G#

(jΩ

)∣ ∣[d

B] q-ident.

z-ident.nom.

−540

−450

−360

−270

−180

−90

0

0.1 1 10 100 1000 10000

argG

#(j

Ω)

[deg

]


q-ident.z-ident.

nom.

Figure 2.6: (Figure 2.5 cont’d; m = 0, Ta = 10ms) Bode diagrams.

−240

−200

−160

−120

−80

−40

0

40

0.1 1 10 100 1000 10000

∣ ∣

G#

(jΩ

)∣ ∣[d

B] q-ident.

z-ident.nom.

−540

−450

−360

−270

−180

−90

0

0.1 1 10 100 1000 10000

argG

#(j

Ω)

[deg

]


q-ident.z-ident.

nom.

Figure 2.7: (cf. Example 2.7) Simulation results, m = 0, Ta = 1ms. Bode diagrams.


−2

−1

0

1

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

u[V

]

t [s]

−60

−40

−20

0

20

40

60

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω[rad

/s]

t [s]

ω meas.q-ident.z-ident.

−30

−20

−10

0

10

20

30

−600−400−200 0 200 400 600

Im

Re

q-ident. (red)

−30

−20

−10

0

10

20

30

−10 −8 −6 −4 −2 0

Im

Re

q-ident. (detail)

−1

−0.5

0

0.5

1

−25 −20 −15 −10 −5 0 5

Im

Re

z-ident. (red)

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.92 0.94 0.96 0.98 1

Im

Re

z-ident. (detail)

Figure 2.8: (cf. Example 2.8) Measurement results, m = 0, Ta = 10ms. See alsoFigure 2.9 for the Bode diagrams.


−240

−200

−160

−120

−80

−40

0

40

0.1 1 10 100 1000 10000

∣ ∣

G#

(jΩ

)∣ ∣[d

B]

q-ident.z-ident.

nom.

−540

−450

−360

−270

−180

−90

0

0.1 1 10 100 1000 10000

argG

#(j

Ω)

[deg

]


q-ident.z-ident.

nom.

Figure 2.9: (Figure 2.8 cont’d; Ta = 10ms) Bode diagrams (measurement results).


−2

−1

0

1

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

u[V

]

t [s]

−120

−80

−40

0

40

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω[rad

/s]

t [s]

ω sim.LS-qLS-z

−300

−200

−100

0

100

200

300

-3e5 -2e5 -1e5 0e0 1e5

Im

Re

LS-q (red)

−30

−20

−10

0

10

20

30

−10 −8 −6 −4 −2 0

Im

Re

LS-q (detail)

−3

−2

−1

0

1

2

3

−25 −20 −15 −10 −5 0 5

Im

Re

LS-z (red)

−1

−0.5

0

0.5

1

−0.5 0 0.5 1 1.5

Im

Re

LS-z (detail)

Figure 2.10: (cf. Example 2.9) LS Simulation results, m = m, Ta = 10ms. (LS counter-part of Figure 2.3). See also Figure 2.12 for the associated Bode diagrams.


−2

−1

0

1

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

u[V

]

t [s]

−120

−80

−40

0

40

80

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω[rad

/s]

t [s]

ω sim.LS-qLS-z

−300

−200

−100

0

100

200

300

-9e3 -6e3 -3e3 0e0 3e3

Im

Re

LS-q (red)

−30

−20

−10

0

10

20

30

−10 −8 −6 −4 −2 0

Im

Re

LS-q (detail)

−1

−0.5

0

0.5

1

−25 −20 −15 −10 −5 0 5

Im

Re

LS-z (red)

−1

−0.5

0

0.5

1

−1 −0.5 0 0.5 1

Im

Re

LS-z (detail)

Figure 2.11: (cf. Example 2.10) LS Simulation results, m = 0, Ta = 10ms. (LS counter-part of Figure 2.5). See also Figure 2.13 for the associated Bode diagrams.


−120

−80

−40

0

40

0.1 1 10 100 1000 10000

∣ ∣

G#

(jΩ

)∣ ∣[d

B] LS-q

LS-znom.

−810−720−630−540−450−360−270−180−90

0

0.1 1 10 100 1000 10000

argG

#(j

Ω)

[deg

]


LS-qLS-znom.

Figure 2.12: (Figure 2.10 cont’d; m = m, Ta = 10ms) LS counterpart of Figure 2.4.

−160

−120

−80

−40

0

40

0.1 1 10 100 1000 10000

∣ ∣

G#

(jΩ

)∣ ∣[d

B] LS-q

LS-znom.

−540

−450

−360

−270

−180

−90

0

0.1 1 10 100 1000 10000

argG

#(j

Ω)

[deg

]


LS-qLS-znom.

Figure 2.13: (Figure 2.11 cont’d; m = 0, Ta = 10ms) LS counterpart of Figure 2.6.

chapter

THREE

A Mathematical Model of a Rolling Mill

Based on physical considerations, a mathematical model of the rolling mill schematicallydepicted in Figure 3.1 is to be evolved in the scope of this chapter, intended as the

basis for the controller design to be addressed in Chapter 4.

PSfrag replacements

entrybridle rollsbridle rollsexit

hydraulicactuator

Feco Fxco

Mebr,1

Mebr,2

Mxbr,1

Mxbr,2

vebr vxbrσen σexωR

upperbackup rollupper

work roll

Figure 3.1: Configuration of the rolling mill under consideration.

The forming of the strip is performed in the roll gap of a four-high mill stand underthe action of the force exerted by the hydraulic actuator and the strip entry (backward)tension σen and the exit (forward) tension σex.

Commencing with a brief outline of a lumped-parameter mill stand model and a non-linear position/force controller for the hydraulic actuator proposed by Kugi [Kug01], thischapter proceeds with the detailed discussion of a novel non-circular arc roll gap modelessentially based on an approximation of the displacement fields of the elastic work rolls

34

3. A Rolling Mill Model. . . 3.1. Mill Stand Dynamics 35

in the sense of the Rayleigh–Ritz method. The implicit algebraic equations of the roll gapmodel together with the mill stand and bridle roll dynamics finally yield a DAE systemwith index 1, with the system’s motion constrained to the manifold given by the roll gapmodel equations.

3.1 Mill Stand Dynamics

For the purpose of simulation and control a simple lumped-parameter mill stand modelas depicted in Figure 3.2, cf. [KSK99], [KSN01], [KHS+00], is used.

PSfrag replacements

d1

Fh

m0

Fr Fr h0h0

x1

d0

xpA1

A2

m1

cgexit bridle rolls

roll gap

σex

base-line

Q

Figure 3.2: Simple lumped-parameter mill stand models.

The elastic stretching of the mill stand due to the force Fh is taken into account via thespring coefficient cg, which is obtained in a calibration process during the mill setup. Thebase-line of the strip is assumed to coincide with the inertial frame. The two approaches ofFigure 3.2 differ in the representation of the masses actuated by the hydraulic actuator. InFigure 3.2 (left) the total mass of all moving parts (i.e., the piston, upper work roll, upperbackup roll, chocks) is represented by m0, while in Figure 3.2 (right) a decomposition intothe piston plus backup roll mass and the mass of the work roll is done. The 3-mass modelis also capable of modelling eccentricity effects caused by the work rolls and/or backuprolls, see, e.g., [KSK99], [KHS+00], which is not addressed in the scope of this thesis. Wewill further use the model of Figure 3.2 (left), whose equations of motion read

x1 = v1, m1v1 = Fh − cg (x1 − l1) − d1v1 −m1g

h0 = v0, m0v0 = Fr − Fh − d0v0 −m0g,(3.1)

where g denotes the constant of gravity, Fr the roll force, d0 and d1 coefficients of viscousfriction, and l1 the length of the unloaded mill spring cg. The piston position is xp =x1 − h0 − l2 with some offset length l2.

3. A Rolling Mill Model. . . 3.2. The Hydraulic Actuator 36

3.2 The Hydraulic Actuator

The force Fh is provided by a hydraulic actuator in a single-acting piston configuration asshown in Figure 3.3. The return chamber is loaded with a constant pressure p2 acting onthe effective piston area A2. The following considerations concerning the modelling andthe control design can be extended to a double-acting double-ended piston configuration,as discussed in detail in [Kug01].

PSfrag replacements

p1

p2 = const.

Q

Qleak

xpA1

A2

forward chamber

return chamber

Figure 3.3: Scheme of the hydraulic actuator in a single-acting piston configuration.

The pressure in the forward chamber with the effective piston area A1 is denotedby p1, and V0 is the volume of the pipe. The flow from the servo valve to the forwardchamber is denoted by Q, and the leakage flows are taken into account by Qleak. Under theassumptions that the supply pressure pS is constant, the servo valve is rigidly connectedto the supply pressure pump, the temperature T of the oil is constant, and the oil isisotropic with ρoil (p1) as the mass density, we have

d

dt(ρoil (p1) (V0 + A1xp)) = ρoil (p1) (Q−Qleak) (3.2)

from the continuity equation. Using the definition of the isothermal bulk modulus Eoil ofthe oil,

1

Eoil=

1

ρoil

(

∂ρoil∂p1

)

T=const.

the mass balance equation (3.2) can be rewritten as

p1 =Eoil

V0 + A1xp(−A1vp +Q−Qleak) , vp = xp.

Hence, the differential equation for the hydraulic force Fh = p1A1 − p2A2 reads

Fh =EoilA1

V0 + A1xp(−A1vp +Q−Qleak) . (3.3)

The dynamics of the servo valve are neglected, because they are very fast compared to thedynamics of the hydraulic actuator. Thus, the displacement xv of the servo valve pistonis regarded as the control input. In order to describe the (quasi-static) valve behavior,

3. A Rolling Mill Model. . . 3.3. Non-linear Hydraulic Gap Control (HGC) 37

the cases xv ≥ 0 with the supply pressure pS being connected to the forward chamber,and xv < 0 with the forward chamber connected to the tank (with pressure pT ) have tobe distinguished. With Kd denoting the valve coefficient, the behavior of the servo valveis modelled as

Q = Kdxv√pS − p1, xv ≥ 0

Q = Kdxv√p1 − pT , xv < 0.

(3.4)

3.3 Non-linear Hydraulic Gap Control (HGC)

A key element of the control concept for the rolling mill is the hydraulic gap control(HGC) proposed by Kugi [KSN01], [Kug01], i.e., the control of the piston position xp, or,alternatively, the control of the hydraulic force Fh. A central feature of this HGC is theobservation that the output

z = Fh − EoilA1 ln

(

V0

V0 + A1xp

)

(3.5)

entails an exact input/output linearization (with relative degree 1) with the control lawnot involving the velocity vp of the piston. Indeed, this property is particularly valuableas the velocity signal vp is not directly measureable in the considered configuration, andan approximate numerical differentiation of the position signal xp, which is corruptedby considerable transducer and quantization noise, is practically unsuitable. A generaltreatment of the exact linearization problem for explicit systems with constrained mea-surements can be found in [SKN01].

With the leakage flow Qleak set to zero in the scope of the control design, the controllaw entailing the exact linearization is obtained from

z = Fh +EoilA1

V0 + A1xpA1vp =

EoilA1

V0 + A1xpQ

.= Eoilu

with the new input u as

Q =V0 + A1xp

A1

u. (3.6)

The controllers for the position control mode (PCM) as well as for the force control mode(FCM) to be discussed subsequently are based on (3.6).

3.3.1 Position Control Mode (PCM)

For controlling the piston position xp the control law

u = αpcmA1 ln

(

V0 + A1xdp

V0 + A1xp

)

(3.7)

3. A Rolling Mill Model. . . 3.3.2. Force Control Mode (FCM) 38

and, thus,

Q = αpcm (V0 + A1xp) ln

(

V0 + A1xdp

V0 + A1xp

)

(3.8)

is proposed in [KSN01], [Kug01], where xdp is the desired piston position and αpcm servesas a parameter to adjust the desired closed loop dynamics. From (3.3) and (3.8) theclosed-loop dynamics

Fh = EoilA1

(

αpcm ln

(

V0 + A1xdp

V0 + A1xp

)

− A1vp +Qleak

V0 + A1xp

)

(3.9)

are obtained. By including the compensation of the servo valve function of (3.4), wefinally get the non-linear state feedback control law

u > 0 : xv =V0 + A1xp

A1Kd

√pS − p1

u

u ≤ 0 : xv =V0 + A1xp

A1Kd

√p1 − pT

u

(3.10)

with u from (3.7). The proof of the asymptotic stability for the parameter 0 < αpcm <αmax with a suitable αmax is based on the Popov criterion and can be found in [KSN01],[Kug01], where also the more general case of a double-acting double-ended piston config-uration is discussed.

3.3.2 Force Control Mode (FCM)

In order to control the force Fh, the control law

u = αfcm(

F dh − Fh

)

(3.11)

with the parameter αfcm is applied. Thus, with (3.6) the control law for the force controlmode reads

Q = αfcm (V0 + A1xp)F dh − FhA1

. (3.12)

From (3.3) and (3.12) the closed-loop dynamics

Fh = EoilA1

(

αfcmF dh − FhA1

− A1vp +Qleak

V0 + A1xp

)

(3.13)

for the force-controlled actuator are obtained. The compensation of the servo valve func-tion proceeds as given above.

3. A Rolling Mill Model. . . 3.4. A Novel Non-Circular Arc Roll Gap Model 39

3.4 A Novel Non-Circular Arc Roll Gap Model

Under the action of the roll force Fr and the entry and exit strip tensions σen, σex thestrip is deformed in the roll gap elasto-plastically in order to achieve the desired outputthickness hex or the desired elongation coefficient ε = (vex − ven) /ven, respectively, seeFigure 3.4. The strip entering the roll gap moves slower than the work roll surface(backward slip), such that due to the frictional and normal stresses arising in the roll/stripinterface plastic deformation of the strip occurs (zone B) after a short elastic compressionzone A, see also Figure 3.5. After passing the so-called neutral point, where the stripspeed coincides with the velocity of the roll, the frictional stresses change their directiondue to the occurrence of forward slip. This implies a decrease of the contact stresses (zoneC) unless the plastic deformation of the strip stops and an elastic recovery zone D occursat the exit domain of the roll gap.

PSfrag replacements

hen

ωr

ωr

hex

σen σex

Fr

Fr

σxxσxx

upper work roll

lower work roll

vex

ven −σ−τ

x

Figure 3.4: Scheme of the roll gap (the indicated directions of the roll/strip contact loadσ, τ are related to the strip).

As the dynamics of the processes taking place in the roll gap are considerably fasterthan the dynamics of the mill stand and the hydraulic actuator, it is appropriate to followa quasi-static point of view.

Besides the forming of the strip, the normal and frictional stresses σ, τ acting inthe roll/strip interface also invoke elastic deformations of the work rolls, see again theFigures 3.4 and 3.5. In the case of cold rolling, the assumption of a circular roll gapshape, though with a larger so-called equivalent radius, qualifies as appropriate, see e.g.,[BF52]. However, for the case of temper and thin strip rolling, this approximation is nolonger valid. Thus, the elastic work roll deformations have to be accounted for in detail.

Manifold research results on the temper and thin strip rolling case have been reportedin the metal forming literature, see, e.g., [JOZ60], [FJ87], [FJMZ92], [DET94], [LS01],commonly referred to as non-circular arc roll gap models. They all have in commonthat a detailed description of the work roll deformations is incorporated in the model,either based on the elastic half-space solution (see, e.g., [Joh85]) or on Jortner’s solution[JOZ60] for the radial displacements of an elastic cylinder caused by a diametricallyapplied constant normal load (to be briefly revisited in Remark 3.3). As these modelstypically involve considerable computational effort, they are primarily used to calculate

3. A Rolling Mill Model. . . 3.4. A Novel Non-Circular Arc Roll Gap Model 40

desired set-points of a rolling mill in an off-line manner.

0.986

0.988

0.990

0.992

0.994

0.996

0.998

1.000h

[mm

]

-400

-200

0

200

400

600

800

1000

1200

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

σ,τ,σxx

[N/m

m2]

distance from the roll centerline, x [mm]

elasticcompressionzone A

plastic reductionzone B(backward slip)

plastic reductionzone C(forward slip)

elasticrecoveryzone D

σxx−στ

Figure 3.5: A non-circular arc roll gap with the strip thickness h and the stresses σ,τ and σxx. R = 0.22m, E = 2.1 × 1011N/m2, ν = 0.3, hen = 1mm, hex = 0.99mm,σen = 70N/mm2, σex = 70N/mm2, yield stress kf = 750N/mm2, and Coulomb frictioncoeff. µ = 0.1.

Contrary to these approaches, the proposed non-circular arc roll gap model addressesthe work roll deformation problem in the sense of the Rayleigh–Ritz method. This methodamounts to defining a finite number of shape functions for the radial and circumferentialdisplacement fields u and v, which are to be associated with generalized coordinates q, inorder to approximate the exact solutions, see, e.g., [Zie92]. Thus, the elasto-static workroll deformation problem is approximated via a finite-dimensional one. The motivationfor this approach emanates from a control point of view: The objective is to design amathematical model which contains the main physical effects and at the same time in-volves a manageable computational effort that makes it applicable for simulation and

3. A Rolling Mill Model. . .3.4.1. The Elasto-static Work Roll Deformation Problem 41

control. Clearly, besides the number of shape functions, the accuracy of this approxima-tion intrinsically depends on the choice of such functions. This key issue of defining asuitable set of ansatz functions for the displacement fields of the roll is the focal point ofSection 3.4.3, closing with a comparison of the displacements obtained with the proposedRitz approach and the numerical solution to this elasto-static problem.

The stresses building up in the formed steel strip are calculated following the classicalstripe model of metal forming, see, e.g., [PP00] or the literature cited above. This model,briefly revisited in Section 3.4.5, is essentially based on the assumption that plane sec-tions of the strip remain plane throughout the roll gap. This yields the spatial ordinarydifferential equations (ODEs) of the strip model for the elastic and the plastic regions ofdeformation. The roll gap model introduced in Section 3.4.6 is finally given as a set ofimplicit algebraic equations, whose evaluation involves the numerical integration of thestrip ODEs. The feasibility of the proposed approach is discussed on the basis of a com-parison with numerical results obtained from a roll gap model based on Jortner’s solutionto the roll deformation problem. Conclusions finally close the discussions on the roll gapmodel.

Besides the advantage of involving significantly reduced computational effort comparedto the roll gap models based on the elastic half-space concept or on Jortner’s solution,respectively, the proposed approach offers additional flexibility to adjust to different rollingconditions. With increasing indentations of the work rolls, it is advisable to refine theRitz ansatz by means of introducing additional degrees of freedom. Section 3.4.3 providesa discussion on suitable possibilities for such a refinement. For rolling conditions near theclassical cold rolling case, however, only a few degrees of freedom will suffice to obtaina reasonable accuracy. Thus, given a desired choice of the shape functions, the setupand the processing of the Ritz ansatz, and in particular the code-generation as well, ispreferably left to a computer algebra program.

3.4.1 The Elasto-static Work Roll Deformation Problem

Consider a (hollow) cylinder with radii R r0 and width B, R B, Young’s modulusE and Poisson number ν. Assume, that a diametrically equivalent loading σ (θ), τ (θ),θen ≤ θ ≤ θex, zero elsewhere, see Figure 3.6 (left), is applied to the surface of this roll.The loading σ, τ is assumed not to vary along the axial direction, and θen, θex denote theroll gap entry and the exit angle, respectively. Thus, a plane strain problem is considered,with E∗ = E/ (1 − ν2) and ν∗ = ν/ (1 − ν) denoting the plane strain equivalents of theYoung modulus E and the Poisson ratio ν. To enforce the static equilibrium, the constanttangential stress τ0,

τ0 =R

πr20

∫ θex

θen

τ (θ)Rdθ, (3.14)

is applied at the inner boundary.This problem statement actually differs from the real situation found in rolling mills.

Thus, let us first comment on the arrangement of these modelling assumptions, keeping


PSfrag replacements

θ

B

θen θex

r0

∂r

deformedshape

on the effect of work roll bending(not addressed in this thesis)

strip

∂θ

σ

τG

r

R

τ0z

2r0

Figure 3.6: The plane strain elasto-static work roll deformation problem (left) with theaccentuation of a roll gap vicinity G to be defined.

in mind that the focus of interest is at last laid on the roll deformations evolving in asuitable “roll gap domain”, say G.

Due to the finite width of the strip (and the rolls), the rolling load is not exactlyconstant along the axial direction of the roll entailing a deformation mode known as workroll bending, and, thus, “non-flat” products, see Figure 3.6 (right). In order to reducethis undesired bending of the rolls, additional hydraulic actuators (so-called “bendingcylinders”) are usually installed acting on the bearings of the rolls. As this thesis isnot concerned with the issue of work roll bending control, the presence of a bendingmode attenuation control is assumed. Thus, the assumption of a plane strain work rolldeformation problem in the scope of the roll gap modelling qualifies as a reasonableapproximation.

Of course, the load acting on the work rolls is not exactly point-symmetric with respectto the roll center in the case of rolling mills, as the shape of the loading evolving in theroll gap clearly differs from the work roll / backup roll contact stress. In particular,this applies to the tangential load τ . The following argumentation essentially utilizesSt.Venant’s principle, which states that statically equivalent forces, when applied to asmall part of the body, render approximately the same stresses and deformations at asufficiently large distance from this loaded region, [Zie92].

Regarding the normal load σ, the point-symmetry assumption is seen as appropri-ate in view of St.Venant’s principle: The normal forces being statically equivalent tothe actual rolling stress and work roll / backup roll contact stress, respectively, implystatic equilibrium, neglecting the body force due to gravity and the (slight) deflectionbetween the respective lines of action. The applicability of St.Venant’s principle to theproblem under consideration is to be revisited and illustrated in the following paragraphs.


The arrangement of the point-symmetry condition on the tangential loading τ , and, inparticular, the arrangement of a hollow cylinder, with the load τ0 applied at the innerboundary, is effectively more involved. Clearly, the elasto-plastic forming of the strip inthe roll gap requires the appearance of a tangential contact load τ , and, thus, a respectiverolling torque which is to be supplied by the main mill drive. Now, the task is to define astatically equivalent setup, with the drive torque and the tangential stresses arising in thework roll / backup roll contact domain balancing the rolling stress τ . Again emphasizingSt.Venant’s principle and the objective of finding a setting delivering appropriate resultsfor the displacements in a domain G particularly, the loading depicted in Figure 3.6 (left)has been chosen, see [Mei01]. These symmetry approximations will simplify the followinganalysis significantly, as the solutions known for the problems depicted in Figure 3.7,given by [JOZ60] and [Mei01], respectively, can be used advantageously.

Outlined in polar coordinates (r, θ), the equilibrium conditions and the compatibilityrelation imposed on the stresses σrr, σθθ, τrθ of the plane strain problem of Figure 3.6(left) read

∂rσrr +1

r(∂θτrθ + σrr − σθθ) = 0,

1

r∂θσθθ + ∂rτrθ +

2τrθr

= 0 (3.15)

and(

(∂r)2 +

1

r∂r +

1

r2(∂θ)

2

)

(σrr + σθθ) = 0 (3.16)

see, e.g., [Zie92], with (∂r)i = ∂i/∂ri, (∂θ)

i = ∂i/∂θi. The strains εrr, εθθ, εrθ are relatedto the radial and the circumferential displacement fields u, v in terms of the geometricallylinearized equations

εrr = ∂ru, εθθ =1

r(u+ ∂θv) , εrθ = εθr =

1

2

(

1

r∂θu+ ∂rv −

v

r

)

. (3.17)

The constitutive equations of the linear-elastic material are due to Hooke’s law,

εrr =1

E∗(σrr − ν∗σθθ) , εθθ =

1

E∗(σθθ − ν∗σrr) ,

εrθ =1

E∗(1 + ν∗) τrθ =

1

2Gτrθ, G =

E

2 (1 + ν)=

E∗

2 (1 + ν∗),

(3.18)

with G denoting the shear modulus. The compatibility equation (3.16) stems from theintegrability requirement of (3.17).

Remark 3.1 An alternative formulation of the linear elasticity problem (3.15), (3.16) isobtained by introducing the Airy stress function φ (r, θ), see e.g., [Joh85],

σrr =1

r∂rφ+

1

r2(∂θ)

2 φ, σθθ = (∂r)2 φ, τrθ = −∂r

(

1

r∂θφ

)

(3.19)

3. A Rolling Mill Model. . . 3.4.2. Two Sub-Problems as Prerequisites. . . 44

in accordance with the PDEs (3.15). Then, the elasticity problem amounts to solving fora solution of the PDE

(

(∂r)2 +

1

r∂r +

1

r2(∂θ)

2

)(

(∂r)2 +

1

r∂r +

1

r2(∂θ)

2

)

φ = 0, (3.20)

which stems from the compatibility equation (3.16).

As a prerequisite for addressing the elasto-static problem (3.15)–(3.18), with the point-symmetric loading σ, τ, and the balancing tangential load τ0 applied at the inner bound-ary due to (3.14), cf. Figure 3.6 (left), let us first focus on two “sub-problems” which willqualify to play a vital role for the setup of a suitable Ritz ansatz.

3.4.2 Two Sub-Problems as Prerequisites for the Choice of AppropriateRitz Shape Functions

The key idea for choosing appropriate shape functions for the work roll displacementfields u, v is to use the solutions to the problems depicted in Figure 3.7 to approximatethe exact solutions outside a suitable roll gap domain vicinity G, referring to St.Venant’sprinciple. The solution to the problem of the elastic cylinder loaded by diametricallyapplied equivalent normal forces F is well known in the literature, see, e.g., [JOZ60],[TG70]. In the metal forming literature, this solution is usually referred to as Jortner’ssolution, and, therefore, within the scope of this contribution, all variables associated withthis solution will be augmented with a superscript J . Typically, non-circular arc roll gapmodels, see, e.g., [DET94], [FJMZ92], [JOZ60], [LS01], are based on this solution or onthe closely related elastic half-space solution, see, e.g., [TG70].

PSfrag replacementsF

F

problem “J ” problem “M”

P

P

τ02r0

τ0 =PR

πr20

Figure 3.7: Two sub-problems of Figure 3.6 (left) attached with particular meaning forthe proposed Ritz approach.

Let q1 = F/E∗. The displacement fields uJ (q1, r, θ), vJ (q1, r, θ) : R× [0, R]×S1 → R

of the problem depicted in Figure 3.7 (left) are given as

uJ =q1

πΛJ (r, θ) , vJ =

q1

πΓJ (r, θ) (3.21)


with

ΛJ (r, θ) = α1 cos θ + 2α3

(

R2 + r2)

sin2 θ + (1 − ν∗)( r

R− α2 sin θ

)

,

ΓJ (r, θ) = −α1 sin θ − (1 − ν∗)α2 cos θ + α3

(

R2 − r2)

sin (2θ) .(3.22)

The functions α1 (r, θ), α2 (r, θ) and α3 (r, θ) read

α1 (r, θ) = ln

(

R2 + r2 − 2Rr cos θ

R2 + r2 + 2Rr cos θ

)

,

α2 (r, θ) = arctan

(

r sin θ

R− r cos θ

)

+ arctan

(

r sin θ

R + r cos θ

)

,

α3 (r, θ) =(1 + ν∗)Rr

(R2 + r2)2 − 4R2r2 cos2 θ.

(3.23)

The according strain and stress fields are obtained as

εJrr = −2q1

π

(

α1 (r −R cos θ)2 + α2 (r +R cos θ)2 − ν∗ (α1 + α2) (R sin θ)2 − 1 − ν∗

2R

)

εJθθ =2q1

π

(

ν∗(

α1 (r −R cos θ)2 + α2 (r +R cos θ)2)− (α1 + α2) (R sin θ)2 +1 − ν∗

2R

)

εJrθ = −2q1

π(1 + ν∗) (α1 (r −R cos θ) − α2 (r +R cos θ))R sin θ

and

σJrr = −2q1E∗

π

(

α1 (r −R cos θ)2 + α2 (r +R cos θ)2 − 1

2R

)

σJθθ = −2q1E∗

π

(

(α1 + α2) (R sin θ)2 − 1

2R

)

τJrθ = −2q1E∗

π(α1 (r −R cos θ) − α2 (r +R cos θ))R sin θ,

using the abbreviations

α1 (r, θ) =R− r cos θ

(R2 + r2 − 2Rr cos θ)2 , α2 (r, θ) =R + r cos θ

(R2 + r2 + 2Rr cos θ)2 .

In contrast to (3.21)–(3.23), the solution to the problem of calculating the displace-ments evolving in the (hollow) cylinder loaded by tangentially acting forces P , see Fig-ure 3.7 (right) is a more recent result and was given by [Mei01]. All quantities related tothis solution will be augmented with the superscript M. In order to enforce the staticequilibrium, the circumferential loading τ0 = PR/ (πr2

0) is applied at the inner boundaryof the hollow cylinder. The solution

uM, vM

of Meindl to be given below is an approx-imation of the exact solution to this problem, however, for r0 R, this approximation is


very accurate, see [Mei01] for a detailed investigation. With D = [r0, R]×S1, q2 = P/E∗,the (approximate) solution uM (q2, r, θ), vM (q2, r, θ) : R×D → R to the problem depictedin Figure 3.7 (right) is

uM =q2

πΛM (r, θ) , vM =

q2

πΓM (r, θ) (3.24)

with

ΛM = −α1 sin θ − (1 − ν∗)α2 cos θ − α3

(

R2 − r2)

sin (2θ) ,

ΓM = −α1 cos θ + 2α3

(

R2 + r2)

sin2 θ − (1 + ν∗)R

r+ (1 − ν∗)

(

α2 sin θ − πr

2R

)

.(3.25)

The according strain fields read

εMrr =2q2

π

(

(1 + ν∗)(

α23 − α2

4

)

R2r sin3 θ − (α3 − α4) r sin θ)

εMθθ =2q2

π

(

− (1 + ν∗)(

α23 − α2

4

)

R2r sin3 θ + ν∗ (α3 − α4) r sin θ)

εMrθ =2q2

π(1 + ν∗)R

(

(R (α1 + α2) − (α3 + α4)) sin2 θ +1

2r2

)

with

α3 (r, θ) =1

R2 + r2 − 2Rr cos θ, α4 (r, θ) =

1

R2 + r2 + 2Rr cos θ.

Remark 3.2 The coordinates q1 and q2 are to be attached with the meaning of generalizedcoordinates of the elasticity problem of Figure 3.6 (left) outside the roll gap vicinity G inthe scope of the proposed Ritz ansatz.

Example 3.1 Let R = 0.22m, ν = 0.3 and q1 = q2 = 1. The Figures 3.9 and 3.10 illus-trate the shape of the displacement fields

uJ , vJ

and

uM, vM

, respectively, evaluatedat five equidistant slices (w.r.t. r) of the sub-domain [R/10, R] × [−π/2, π/2].

Given the distributed loading σ, τ, cf. Figure 3.6 (left), and τ0 due to (3.14), theassociated displacement fields are obtained by carrying out Green’s integral,

u (r, θ) =R

πE∗

∫ θex

θen

(

ΛM (r, θ − β) τ (β) − ΛJ (r, θ − β) σ (β))

dβ,

v (r, θ) =R

πE∗

∫ θex

θen

(

ΓM (r, θ − β) τ (β) − ΓJ (r, θ − β) σ (β))

dβ.

(3.26)

Example 3.2 Let R = 0.22m, E = 2.1×1011N/m2, ν = 0.3 and suppose that the loadingσ, τ as given in Figure 3.5 is applied to the cylinder. Figure 3.11 illustrates the resultingradial and circumferential displacements u, v due to (3.26) (solid lines), as well as thedisplacements due to the equivalent forces (dashed) using (3.21) and (3.24).

3. A Rolling Mill Model. . . 3.4.3. A Ritz Approximation. . . 47

Figure 3.11 also clearly illustrates the fact that the influence of the particular shapeof the distributed loading on the shape of the displacement fields rapidly decreases withincreasing distance from the loaded region (St.Venant’s principle). On the one hand, thisobservation justifies the assumption of a point-symmetric loading with respect to the rollcenter. On the other hand, it serves as the key motivation for using (3.21) and (3.24) asansatz functions for the work roll displacement fields outside a suitable roll gap vicinityG to be specified in the subsequent section.

Remark 3.3 (“Jortner’s influence functions”) Let σ (β) = σ0, β ∈ [−α, α] + kπ, k ∈ Z,zero elsewhere, and τ = 0. To calculate the radial displacements of the roll surface, firstnotice that

ΛJ (R, θ) = 2 + cos θ ln

(

tan2

(

θ

2

))

− (1 − ν∗)π

2sin θ sign (sin θ) .

From (3.26) we have

u (R, θ) = − σ0R

πE∗

∫ α

−α

ΛJ (R, θ − β) dβ = − σ0R

πE∗Ξ (θ, α) ,

introducing the function Ξ. To cope with the singularity of ΛJ (R, θ) at θ = 0, let usdistinguish the cases |θ| ≤ α and |θ| > α, to be associated with the indices i and o,respectively, indicating whether θ is located inside or outside the loaded region. For |θ| > α,we find

Ξo (θ, α) =

sin (ϕ) ln(

tan2(ϕ

2

))

+ (1 − ν∗)π

2cos (ϕ) sign (sin (ϕ))

∣

∣

∣

ϕ=θ+α

ϕ=θ−α, (3.27)

and the case |θ| ≤ α gives the result

Ξi (θ, α) = limε→0

(∫ θ−ε

−α

ΛJ (R, θ − β) dβ +

∫ α

θ+ε

ΛJ (R, θ − β) dβ

)

=

= Ξo (θ, α) − (1 − ν∗)π. (3.28)

In the metal forming literature, (3.27) and (3.28) are typically referred to as Jortner’sinfluence functions. They are numerously applied to cope with the work roll deformationproblem, associated with the very common approximations v (R, θ) ≈ 0 and τ = 0, see,e.g., [JOZ60], [DET94].

3.4.3 A Ritz Approximation of the Work Roll Deformation Problem

The objective of this section is to introduce a suitable finite-element approximation of theelasticity problem (3.15)–(3.18), subject to diametrically equivalent loading as given inFigure 3.6 (left), in the sense of the Ritz method. To this end, suitable shape functionsu∗, v∗ for the radial and circumferential displacement fields, parametrized by a finite setof variables q (– the generalized coordinates –), are introduced in order to approximatethe exact solutions u, v.


Shape of the Ritz Ansatz

Due to symmetry reasons following from the arrangement of the point-symmetric loadingof Figure 3.6 (left), it suffices to consider the domain B = [r0, R] × [−π/2, π/2]. Now, letus introduce the subdomain G ⊂ B referred to as the “roll gap vicinity”,

G = [r1, R] × [−θ1, θ1] ⊂ B, (3.29)

max (|θen| , |θex|) < θ1 < π/2, parametrized in terms of r1 ∈ ]r0, R[ and θ1 to be chosenappropriately. Note that the domain [−θ1, θ1] of G is not required to coincide with theactual extent of the roll/strip contact arc [θen, θex].

Following the idea sketched above, we propose to arrange the superposition of thedisplacement fields

uJ , vJ

and

uM, vM

, associated with the coordinates q1 and q2,respectively, as approximations for u and v on the domain F = B\G.

Remark 3.4 Due to the use of

uJ , vJ

and

uM, vM

as ansatz functions on thedomain F , the boundary conditions on ∂F\∂G are naturally fulfilled.

As a prerequisite for the design of the shape functions

uG, vG

to be defined on G,which are required to be (at least) C1 coupled to the functions uJ , uM and vJ , vM onthe boundary ∂F ∩ ∂G, let us first introduce the decomposition

uG = uJ + uM + u, vG = vJ + vM + v. (3.30)

Here, uα and vα, α ∈ J ,M, denote suitable functions enforcing the C1 concatenationto uα and vα on ∂F ∩ ∂G. The functions u : R

nu × G → R, v : Rnv × G → R are

chosen as 2-dimensional polynomials on G, associated with nu and nv coordinates qiuand qiv, respectively. They are required to meet the homogeneous boundary conditions(∂r)

i u (qu, r1, θ) = 0, θ ∈ [−θ1, θ1], and (∂θ)i u (qu, r,±θ1) = 0, r ∈ [r1, R], i = 0, 1, with

analogous expressions for v.Now, we will introduce suitable choices for the functions

uJ , vJ

and

uM, vM

utilizing the following proposition.

Proposition 3.1 Let ξ (r, θ) : B → R, B = [r0, R] × [−π/2, π/2], ξ ∈ C2 (B), and defineoperators f and g as

f (ξ) (r, θ) = ξ (r1, θ) − ξ (r1, θ1) + ξ (r, θ1)−

− [∂θξ (r, θ1) − ∂θξ (r1, θ1) − ∂r∂θξ (r1, θ1) (r − r1)]2θ1

πcos

(

θ

θ1

π

2

)

+

+ [∂rξ (r1, θ) − ∂rξ (r1, θ1)] (r − r1)

(3.31)

and

g (ξ) (r, θ) =ξ (r1, θ)

ξ (r1, θ1)ξ (r, θ1) +

(

∂rξ (r1, θ) −ξ (r1, θ)

ξ (r1, θ1)∂rξ (r1, θ1)

)

(r − r1) +

+∂θξ (r1, θ1)

ξ (r1, θ1)[ξ (r, θ1) − ∂rξ (r1, θ1) (r − r1)]

θ1

πsin

(

θπ

θ1

)

+

+ (∂r∂θξ (r1, θ1) (r − r1) − ∂θξ (r, θ1))θ1

πsin

(

θπ

θ1

)

.

(3.32)


Then, f (ξ) : G →R, G = [r1, R] × [−θ1, θ1] ⊂ B, is a C1 continuation to ξ at ∂F ∩ ∂G ifξ is symmetric in θ. If ξ is skew-symmetric in θ, then g (ξ) : G →R is a C1 continuationto ξ at the boundary ∂F ∩ ∂G.

Proof. straight-forward.

Notice that ΛJ , ΓM are symmetric in θ, and ΓJ , ΛM are skew-symmetric in θ. Thus,the functions

uJ =q1

πf(

ΛJ)

, vJ =q1

πg(

ΓJ)

(3.33)

and

uM =q2

πg(

ΛM)

, vM =q2

πf(

ΓM)

(3.34)

qualify as appropriate choices.

Example 3.3 Let R = 0.22m, ν = 0.3 and q1 = q2 = 1. The Figures 3.12 and 3.13depict the shape of the functions

uJ , vJ

and

uM, vM

, respectively, evaluated at fiveequidistant slices (w.r.t. r) of G. Additionally, the functions

uJ , vJ

,

uM, vM

, eval-uated at the respective slices with θ ∈ [−2θ1, 2θ1], are shown.

Remark 3.5 (on the structure of

uJ , vJ

and

uM, vM

). Let

(ξ, α) ∈ (Λ,J ) , (Γ,M) , (ζ, α) ∈ (Γ,J ) , (Λ,M) .

Given the parameters r1, θ1, it is suitable to introduce the constants

cα0 = ξα (r1, θ1) , cα1 = ∂rξα (r1, θ1) , cα2 = c∂θξ

α (r1, θ1) − cα3 r1, cα3 = c∂r∂θξα (r1, θ1) ,

cα4 = (ζα)−1 (r1, θ1) , cα5 = ∂rζα (r1, θ1) , cα6 = ∂θζ

α (r1, θ1) , cα7 = ∂r∂θζα (r1, θ1) ,

with c = 2θ1/π, and define the functions

aα0 (r) = ξα (r, θ1) , bα0 (θ) = ξα (r1, θ) , aα1 (r) = c∂θξα (r, θ1) , bα1 (θ) = ∂rξ

α (r1, θ) ,

aα2 (r) = ζα (r, θ1) , bα2 (θ) = ζα (r1, θ) , aα3 (r) = ∂θζα (r, θ1) , bα3 (θ) = ∂rζ

α (r1, θ) .

Then, we have

f (ξα) = aα0 (r) + bα0 (θ)− (aα1 (r) − cα3 r − cα2 ) cos

(

θ

c

)

+ (bα1 (θ) − cα1 ) (r − r1)− cα0 (3.35)

and

g (ζα) = cα4aα2 (r) bα2 (θ) + (bα3 (θ) − cα4 c

α5 bα2 (θ)) (r − r1) +

+ (cα4 cα6a

α2 (r) + cα8 (r − r1) − aα3 (r))

c

2sin

(

2θ

c

)

, (3.36)

with cα8 = cα7 − cα4 cα5 cα6 . This representation qualifies as useful to carry out the subsequent

symbolic/numeric computations.


Remark 3.6 (on the notation) In the following, variables of the work roll deformationproblem referring to the roll surface r = R are indicated by means of a superscript bar,e.g., u (q, θ)

.= u (q, R, θ).

Remark 3.7 (Remark 3.5 cont’d; evaluation of f (ξα), g (ζα) at the roll surface). Byintroducing the constants

Cα0 = aα0 (R) − cα0 − cα1C2, Cα

1 = aα1 (R) − cα3R− cα2 ,

Cα3 = cα4 (aα2 (R) − C2c

α5 ) , Cα

4 =c

2(cα4 c

α6a

α2 (R) + cα8C2 − aα3 (R)) ,

and C2 = R− r1, we have

f (ξα) = Cα0 + bα0 (θ) + C2b

α1 (θ) − Cα

1 cos

(

θ

c

)

,

g (ζα) = Cα3 b

α2 (θ) + C2b

α3 (θ) + Cα

4 sin

(

2θ

c

)

.

(3.37)

Thus, ˇuJ (q1, θ) = uJ (q1, R, θ) = q1

πf (ΛJ ) (θ), ˇvJ = q1

πg (ΓJ ), and ˇuM = q2

πg (ΛM),

ˇvM = q2

πf (ΓM), cf. (3.33), (3.34).

In order to achieve the required accuracy of the solution obtained with the Ritzapproach it is appropriate to further decompose the domain G into subdomains Gk =[

rka, rkb

]

×[

θka, θkb

]

,

G =⋃

kGk, Gk ∩ Gl = , k 6= l, (3.38)

each of them equipped with 2-dimensional polynomials uk, vk : Gk → R. The coefficientsof these polynomials are denoted as cuk,ij and cvk,ij. By eliminating the linear equationsevolving from the Cp concatenation, p ≥ 1, of these polynomials at the boundaries of thesubdomains and at ∂F ∩ ∂G,

Rucu = 0, Rvcv = 0, cu =[

cuk,ij]

, cv =[

cvk,ij]

, (3.39)

the set of generalized coordinates, collected to the nu and nv-dimensional vectors qu andqv is obtained. To this end, compute bases of kerRu, kerRv, i.e.,

kerRu = span wu,1, . . . , wu,nu , kerRv = span wv,1, . . . , wv,nv

, (3.40)

withnu = dim (cu) − rank (Ru) , nv = dim (cv) − rank (Rv) , (3.41)

and then introduce the coordinates qu, qv by means of

cu =∑nu

s=1qsuwu,s, cv =

∑nv

s=1qsvwv,s. (3.42)


Finally, the vector q representing the generalized coordinates of the displacement fieldson B is

qT =[

q1 q2 qTu qTv]

. (3.43)

Thus, the linear elasticity problem depicted in Figure 3.6 (left) is approximated via afinite-dimensional one of determining 2+nu+nv generalized coordinates q. To summarizethe composition of the Ritz ansatz u∗, v∗, we have

u∗ (q, r, θ) =

uJ (q1, r, θ) + uM (q2, r, θ) + uk (qu, r, θ) , (r, θ) ∈ GkuJ (q1, r, θ) + uM (q2, r, θ) , (r, θ) ∈ J

(3.44)

for the radial displacement field on B, and, accordingly,

v∗ (q, r, θ) =

vJ (q1, r, θ) + vM (q2, r, θ) + vk (qv, r, θ) (r, θ) ∈ GkvJ (q1, r, θ) + vM (q2, r, θ) (r, θ) ∈ J

(3.45)

for the circumferential displacement field. Note that, by construction, the functionsu∗, v∗ are linear with respect to the coordinates q.

The Elastic Potential of the Work Roll

Generally, in the 3-dimensional linear-elastic strain state the potential energy U per unitvolume, outlined in polar coordinates (r, θ, z), reads

U =E

2 (1 + ν)

(

1 − ν

1 − 2νe2 − 2 (εrrεθθ + εθθεzz + εzzεrr) + 2

(

ε2rθ + ε2

θz + ε2rz

)

)

, (3.46)

with the first invariant e = εrr + εθθ + εzz of the strain tensor, see, e.g., [Par88]. In thecase of a plane strain problem, i.e., εzz = εrz = εθz = 0, we obtain

U =E

2 (1 + ν)

(

1 − ν

1 − 2ν

(

ε2rr + ε2

θθ

)

+2ν

1 − 2νεrrεθθ + 2ε2

rθ

)

=

=E∗

2 (1 − ν∗2)

(

ε2rr + ε2

θθ + 2ν∗εrrεθθ + 2 (1 − ν∗) ε2rθ

)

. (3.47)

Thus, the potential energy, also referred to as the elastic potential, stored in the deformedbody with domain B is given as

V =

∫ π2

−π2

∫ R

r0

Urdrdθ =

∫

B

Urdrdθ. (3.48)

By virtue of the linearity of u∗, v∗ w.r.t. q, the elastic potential V (q) can be rewrittenas

V = E∗1

2qTAq, A > 0, E∗ =

E∗

2 (1 − ν∗2)(3.49)


with the positive definite matrix A = [Aij],

Aij =1

E∗

∫

B

∂i∂jUrdrdθ =

∫

B

∂i∂j(

ε2rr + ε2

θθ + 2ν∗εrrεθθ + 2 (1 − ν∗) ε2rθ

)

rdrdθ, (3.50)

and ∂i = ∂/∂qi. In the following, we will focus on some issues regarding the computationof the matrix A.

PSfrag replacements

θ

θen θex

r0

coords. q1, q2

π2

−π2 −θ1 θ1

r

r1

Fa

Fb

F

GR

Figure 3.8: On the decomposition of the domain B, see also Figure 3.6 (left).

Remark 3.8 (computation issues; domain F = B\G) Let Aij = AFij + AG

ij, i, j ∈ 1, 2,with the superscripts F ,G referring to the respective domains of integration, see alsoFigure 3.8. Let εFkl (q

1, q2, r, θ) : R2 × F → R, k, l ∈ r, θ, denote the strain fields

obtained with (3.17), (3.21), (3.24) and (3.44), (3.45), and introduce the abbreviationsεFkl = εJkl + εMkl = q1eJkl + q2eMkl , e

Jkl, e

Mkl : F → R, εFkl,i

.= ∂iε

Fkl = δ1

i eJkl + δ2

i eMkl with the

Kronecker symbols δnm. Then,

AFij =

∫

F

∂i∂j

(

(

εFrr)2

+(

εFθθ)2

+ 2ν∗εFrrεFθθ + 2 (1 − ν∗)

(

εFrθ)2)

rdrdθ =

= 2

∫

F

(

εFrr,iεFrr,j + εFθθ,iε

Fθθ,j + ν∗

(

εFrr,iεFθθ,j + εFrr,jε

Fθθ,i

)

+ 2 (1 − ν∗) εFrθ,iεFrθ,j

)

rdrdθ,

hence, e.g.,

AF11 =

∫

F

4r (1 − ν∗2) (r2 −R2)2((1 − ν∗) r2 (r2 + 2R2) + (5 + 3ν∗)R4)

π2R2 (R4 + r4 − 2R2r2 cos (2θ))2 drdθ.

Notice that AF12 = 0. Let Fa = [r0, r1] × [0, π/2] and Fb = [r1, R] × [θ1, π/2], see Fig-

ure 3.8 again, then AFii = 2

(

AFa

ii + AFb

ii

)

, i = 1, 2, due to symmetry reasons on F . The

integration over the domain Fa yields the result

AFa

11 =(1 − ν∗2) (r2

1 − r20)R

2

π (R2 + r20)

2(R2 + r2

1)2

[

(1 + 3ν∗)R4 − 2 (3 − ν∗) r20r

21−

− (1 − ν∗)

(

(r0r1R

)4

+ r40 + r4

1 + 2

(

R2 +(r0r1R

)2)

(

r20 + r2

1

)

)]

−

−(

1 − ν∗2) 2

πln

(

R2 + r20

R2 − r20

R2 − r21

R2 + r21

)

.


Accordingly, a symbolic expression for AFa

22 can be derived. However, symbolic results forthe terms AFb

11 and AFb

22 could not be established, thus, these terms are left to numericintegration.

In view of the evaluation of (3.50) on the domain G, i.e., AGij, 1 ≤ i, j ≤ 2 + nu + nv,

let us first introduce a decomposition of the strain fields,

εGkl (q, r, θ) = εJkl(

q1, r, θ)

+ εMkl(

q2, r, θ)

+ εkl (qu, qv, r, θ) =

= q1eJkl (r, θ) + q2eMkl (r, θ) + εkl (qu, qv, r, θ) ,(3.51)

k, l ∈ r, θ, with εαkl, α ∈ J ,M, and εkl denoting the strains derived from the dis-placement fields uα, vα and the polynomials u, v, respectively. Additionally, let

εrr (qu, r, θ) = ∂ru = qiuerr,i (r, θ) ,

εθθ (qu, qv, r, θ) = (u+ ∂θv) /r = qiueθθ,i (r, θ) + qivfθθ,i (r, θ) ,

εrθ (qu, qv, r, θ) = (r−1∂θu+ ∂rv − v/r) /2 = qiuerθ,i (r, θ) + qivfrθ,i (r, θ) .

We end up with the following observations.

Remark 3.9 (computation issues; domain G) Consider the domain G and let 1 ≤ i, j ≤2 + nu + nv, ε

Gkl,i

.= ∂iε

Gkl. We have

AGij =

∫

G

∂i∂j

(

(

εGrr)2

+(

εGθθ)2

+ 2ν∗εGrrεGθθ + 2 (1 − ν∗)

(

εGrθ)2)

rdrdθ =

= 2

∫

G

(

εGrr,iεGrr,j + εGθθ,iε

Gθθ,j + ν∗

(

εGrr,iεGθθ,j + εGrr,jε

Gθθ,i

)

+ 2 (1 − ν∗) εGrθ,iεGrθ,j

)

rdrdθ

with εGkl,iεGkl,j =

(

eJklδ1i + eMkl δ

2i + ∂iεkl

) (

eJklδ1j + eMkl δ

2j + ∂j εkl

)

, k, l ∈ r, θ, see (3.51).Hence,

εGkl,iεGkl,j =

(

eJklδ1i + eMkl δ

2i

) (

eJklδ1j + eMkl δ

2j

)

i, j ∈ 1, 2(

eJklδ1i + eMkl δ

2i

)2i = j ∈ 1, 2

(

eJklδ1i + eMkl δ

2i

)

∂j εkl i ∈ 1, 2 , j > 2

∂iεkl∂j εkl i, j > 2

Accordingly, the expression εGrr,iεGθθ,j =

(

eJrrδ1i + eMrr δ

2i + ∂iεrr

) (

eJθθδ1j + eMθθ δ

2j + ∂j εθθ

)

isprocessed.

Remark 3.10 Notice that AG12 = 0, thus A12 = AF

12 + AG12 = 0, see also Remark 3.8.

Remark 3.11 (computation issues; Remark 3.9 cont’d.) Let 2 < i, j ≤ 2 + nu + nv,and let R [r, θ] denote the polynomial ring in the variables r and θ with coefficients in R.Then,

εGrr,iεGrr,jr = ∂iεrr∂j εrrr ∈ R [r, θ] , εGrr,iε

Gθθ,jr = ∂iεrr∂j εθθr ∈ R [r, θ]


as ∂iεrr ∈ R [r, θ] and (∂iεθθ) r ∈ R [r, θ]. However, ∂iεGθθ∂j ε

Gθθr /∈ R [r, θ] as well as

∂iεGrθ∂j ε

Grθr /∈ R [r, θ]. In view of the symbolic integration over (the subdomains of) G,

decompositions of the form

2εθθ,iεθθ,jr =fθθ (θ)

r+ pθθ (r, θ) , pθθ ∈ R [r, θ]

2εrθ,iεrθ,jr =frθ (θ)

r+ prθ (r, θ) , prθ ∈ R [r, θ]

are useful. Then, with G ⊇ Gab = [ra, rb] × [θa, θb], we find

∫ θb

θa

∫ rb

ra

∂i∂j(

εGθθ)2rdrdθ = ln

(

rbra

)∫ θb

θa

fθθ (θ) dθ +

∫ θb

θa

∫ rb

ra

pθθ (r, θ) drdθ. (3.52)

Theorem 3.2 (Cholesky factorization) If A ∈ Rn×n is symmetric positive definite, then

there exists a unique lower triangular L ∈ Rn×n with positive diagonal entries such that

A = LLT .

Proof. see, e.g. [GvL96], p.143.

By virtue of a Cholesky factorization of A > 0 of (3.49), (3.50) the elastic potential ofthe work roll can be represented as

V = E∗1

2qTAq = E∗1

2qTLLT q = E∗1

2qT q. (3.53)

Thus, by arranging the coordinates transform q = LT q, a particular simple representationof the potential energy V is obtained.

The Generalized Forces Associated with the Applied Load σ, τReferring to the notation arranged in Remark 3.6, the generalized forces Q1, Q2 associatedwith the loading σ, τ , τ0, cf. Figure 3.6 (left) and (3.14), read

Qi =

∫ θex

θen

(σ∂i ˇuα + τ ∂i ˇv

α)Rdθ − τ0

∫ π2

−π2

∂ivα (r0, θ) r0dθ, (i, α) ∈ (1,J ) , (2,M)

Notice that ∂1vJ (r0, θ) = ΓJ (r0, θ) /π, ∂2v

M (r0, θ) = ΓM (r0, θ) /π, and

∫ π2

−π2

ΓJ (r0, θ) dθ = 0.

Then, with the constant

K =R

r0π2

∫ π2

−π2

ΓM (r0, θ) dθ, (3.54)


we end up with

Q1 = R

∫ θex

θen

(

σ∂1 ˇuJ + τ ∂1 ˇv

J)

dθ

Q2 = R

∫ θex

θen

(

σ∂2 ˇuM + τ

(

∂2 ˇvM −K

))

dθ

Qu,i = R

∫ θex

θen

σ∂qiu˜udθ, i = 1, . . . , nu

Qv,j = R

∫ θex

θen

τ ∂qjv˜vdθ, j = 1, . . . , nv.

(3.55)

Given the generalized forces QT =[

Q1, Q2, QTu , Q

Tv

]

, the deformed state of the roll re-flected by the coordinates q is calculated by solving for the solution of the set of linearequations

(∂qV )T = E∗Aq = Q. (3.56)

Outlined in terms of the coordinates q = LT q, equation (3.56) reads Lq = Q/E∗.

Example 3.4 Let R, E, ν as given in Example 3.2, r0 = R/100, and adjust the domainboundaries of G as r1 = 0.96R, θ1 = 0.01/R, see also Figure 3.11. Let Gk = [r1, R] × Ik,Ik = ((2k − 5) /3 + [0, 2/3]) θ1, k = 1, 2, 3, and define

u1 =

(

3∑

i=1

2∑

j=0

cu1,ij ri+2θj

)

(

1 − R

r1

)2(

1 + θ)2

u2 =

(

3∑

i=1

6∑

j=0

cu2,ij ri+2θj

)

(

1 − R

r1

)2

u3 =

(

3∑

i=1

2∑

j=0

cu3,ij ri+2θj

)

(

1 − R

r1

)2(

1 − θ)2

with r = (r − r1) / (R− r1) and θ = θ/θ1. The polynomials vk (with the coefficients cvk,ij)are chosen analogously with the same orders as for uk. Thus, dim (cu) = dim (cv) = 39,rank (Ru) = rank (Rv) = 12, and, hence, nu = nv = 27. The Figures 3.14, 3.15 and 3.16provide a comparison of the displacements obtained by means of this Ritz approximation( red) and the solutions due to (3.26) (black) applying the loading σ, τ as given inFigure 3.5.

Example 3.5 Given the Ritz ansatz as specified in Example 3.4 and the loading σ dueFigure 3.5, τ = 0, the Figures 3.19 and 3.20 depict the displacement fields obtained via theRitz approximation, as well as the numerical solution obtained by invoking Femlab 3.1[Mul]. The associated mesh of the finite element approximation of Femlab is illustratedin Figure 3.17, and Figure 3.18 depicts the von Mises effective stress [Zie92]. Particularly,Figure 3.18 provides a vivid impression of the extent of the roll gap domain compared tothe dimensions of the work roll.


Special Case: A Reduced Order Model for τ ≡ 0

In the absence of a shear load, i.e., τ ≡ 0, the equations (3.56) evolved above can befurther reduced in size. This particular case is of some interrest as in the recent literaturedealing with the roll gap models for the temper rolling case (see, e.g., [DET94], [FJMZ92],[LS01]) the effect of the tangential loading τ on the shape of the contact arc is neglected(additionally, in the scope of these models only the radial displacement field u is consideredfor determining the contact arc). As Qv vanishes, we have

E∗

A11 0 · · · A1n

A22

. . ....

Ann

q1

q2

quqv

=

Q1

Q2

Qu

0

,

or, more compactly,

E∗

[

A11 A12

AT12 A22

] [

qqv

]

=

[

Q0

]

, q =

q1

q2

qu

, Q =

Q1

Q2

Qu

. (3.57)

Hence, in this particular case the problem reduces to a (2 + nu)-dimensional one, E∗Aq =Q, with A = A11 − A12A

−122 A

T12 > 0.


−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

radia

ldis

pla

cem

entuJ

(q1

=1)

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

circ

um

fere

ntial

dis

pla

cem

entvJ

(q1

=1)

θ [rad]

Figure 3.9: (cf. Example 3.1) Let q1 = 1. Shape of the functions

uJ , vJ

due to (3.21).θ ∈

[

−π2, π

2

]

, r = R10

(

1 + i94

)

, i = 0, . . . , 4. The displacements of the roll surface areindicated with dashed lines.


−0.3

−0.2

−0.1

0

0.1

0.2

0.3

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

radia

ldis

pla

cem

entuM

(q2

=1)

−5

−4

−3

−2

−1

0

1

2

3

4

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

circ

um

fere

ntial

dis

pla

cem

entvM

(q2

=1)

θ [rad]


uM, vM

due to(3.24). θ ∈

[

−π2, π

2

]

, r = R10

(

1 + i94

)

, i = 0, . . . , 4. The displacements of the roll surfaceare indicated with dashed lines.


−80

−70

−60

−50

−40

−30

−20

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

radia

ldis

pla

cem

entu

[µm

]

−5

−4

−3

−2

−1

0

1

2

3

4

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

circ

um

fere

ntial

dis

pla

cem

entv

[µm

]

θ [rad]

Figure 3.11: (cf. Example 3.2) Displacement fields of the roll (θ ∈ [−2θ1, 2θ1], r =r1 + i (R− r1) /4, i = 0, . . . , 4, θ1 = 0.01/R, r1 = 0.96R) caused by the distributed loadσ, τ due to Figure 3.5 (solid), and the equivalent forces (blue dashed).


−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

uJ,uJ,(q

1=

1)

uJ (domain G)uJ

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

vJ,vJ,(q

1=

1)

θ [rad]


uJ , vJ

(red) definedon G, θ1 = 0.01/R, r1 = 0.96R. Additionally,

uJ , vJ

(blue dashed) for θ ∈ [−2θ1, 2θ1],r = r1 + i (R− r1) /4, i = 0, . . . , 4.


−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

uM,uM

,(q

2=

1)

uM (domain G)uM

1.5

2

2.5

3

3.5

4

4.5

5

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

vM,vM

,(q

2=

1)

θ [rad]


uM, vM

(red) definedon G, θ1 = 0.01/R, r1 = 0.96R. Additionally,

uM, vM

(blue dashed) for θ ∈ [−2θ1, 2θ1],r = r1 + i (R− r1) /4, i = 0, . . . , 4.


−60

−55

−50

−45

−40

−35

−30

−25

−20

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

radia

ldis

pla

cem

entu

[µm

]

−4

−3

−2

−1

0

1

2

3

4

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

circ

um

fere

ntial

dis

pla

cem

entv

[µm

]

θ [rad]

Figure 3.14: (cf. Example 3.4) Ritz approx. (red) vs. solution due to (3.26) (bluedashed): Displacements (θ ∈ [−2θ1, 2θ1], r = r1 + i (R− r1) /4, i = 0, . . . , 4) caused bythe load σ, τ due to Figure 3.5.


−60

−55

−50

−45

−40

−35

−30

−25

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

radia

ldis

pla

cem

entu

[µm

]

−4

−3

−2

−1

0

1

2

3

4

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

circ

um

fere

ntial

dis

pla

cem

entv

[µm

]

θ [rad]

Figure 3.15: Detail of Figure 3.14: Domain G, θ ∈ [−θ1, θ1], r = r1 + i (R− r1) /4,i = 0, . . . , 4.


−60

−55

−50

−45

−40

−35

−30

−25

0.211 0.212 0.213 0.214 0.215 0.216 0.217 0.218 0.219 0.22

radia

ldis

pla

cem

entu

[µm

]

−4

−3

−2

−1

0

1

2

3

0.211 0.212 0.213 0.214 0.215 0.216 0.217 0.218 0.219 0.22

circ

um

fere

ntial

dis

pla

cem

entv

[µm

]

r [m]

Figure 3.16: (cf. Example 3.4) Ritz approx. (red) vs. solution due to (3.26) (bluedashed): Displacements (evaluated along “radial slices of G”: r ∈ [r1, R], θ = iθ1/4,i = 0, . . . , 4) caused by the load σ, τ due to Figure 3.5.


−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Figure 3.17: (cf. Example 3.5) The mesh of the finite element approximation via Fem-

lab 3.1 [Mul].

Figure 3.18: (cf. Example 3.5) Illustration of the finite element solution: The von Miseseffective stress.


−60

−50

−40

−30

−20

−10

0

10

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

radia

ldis

pla

cem

entu

[µm

]

−8

−6

−4

−2

0

2

4

6

8

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

circ

um

fere

ntial

dis

pla

cem

entv

[µm

]

θ [rad]

Figure 3.19: (cf. Example 3.5) Ritz approx. (red) vs. FEM solution (blue dashed):θ ∈

[

−π2, π

2

]

, r = R10

(

1 + i94

)

, i = 0, . . . , 4.


−60

−55

−50

−45

−40

−35

−30

−25

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

radia

ldis

pla

cem

entu

[µm

]

−4

−3

−2

−1

0

1

2

3

4

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05

circ

um

fere

ntial

dis

pla

cem

entv

[µm

]

θ [rad]

Figure 3.20: (cf. Example 3.5) Ritz approx. (red) vs. FEM solution (blue dashed):θ ∈ [−θ1, θ1], r = r1 + i (R− r1) /4, i = 0, . . . , 4 (domain G).

3. A Rolling Mill Model. . . 3.4.4. The Shape of the Roll/Strip Contact Arc 68

3.4.4 The Shape of the Roll/Strip Contact Arc

Assume we are given the generalized coordinates q reflecting the state of the deformedroll in the sense of the preceding section. Then, with the surface displacements u, v ofthe work roll, see Remark 3.6 on the notation, the strip thickness h reads

h (θ;ψ, q, h0) = h0 + 2 (R− (R + u (q, θ)) cos (θ − ψ) + v (q, θ) sin (θ − ψ)) , (3.58)

see Figure 3.21. Here, ψ denotes the deflection of the polar coordinate system (r, θ) ofthe upper work roll with respect to the y-axis coinciding with the work rolls center-line,and h0 is the so-called no-load gap, i.e., the roll gap in absence of a rolling load.

PSfrag replacements

Rψ

u (q, θ)v (q, θ)

12h (θ;ψ, q, h0)

x = ϕ (θ;ψ, q)

h0

2

surface of thedeformed roll

stripcenterline

θ

x

y

z

σxxσxx

h

σ

σ

τ

τ

Figure 3.21: On the shape of the roll/strip contact arc (right), and the notion of theclassical stripe model of metal forming (left).

As a prerequisite for the combination of the strip equations (to be discussed in thesubsequent section) with the roll deformation problem, the diffeomorphic change of thespatial coordinates θ and x,

x = ϕ (θ;ψ, q) = (R + u (q, θ)) sin (θ − ψ) + v (q, θ) cos (θ − ψ) (3.59)

is arranged, see again Figure 3.21. Then, the strip thickness in terms of x, which isindicated by an overset bar, is h = h ϕ−1. Differentiation of the identity ϕ ϕ−1 = id,i.e., ϕ (ϕ−1 (x;ψ, q) ;ψ, q) = x, with respect to x yields

∂xϕ−1 (x;ψ, q) = (∂θϕ)−1 ϕ−1 (x;ψ, q)

and, thus,

∂xh (x;ψ, q, h0) =∂θh

∂θϕ ϕ−1 (x;ψ, q) (3.60)

and

(∂x)2 h =

1

(∂θϕ)2

(

(∂θ)2 h− (∂θ)

2 ϕ

∂θϕ∂θh

)

ϕ−1. (3.61)

3. A Rolling Mill Model. . . 3.4.4. The Shape of the Roll/Strip Contact Arc 69

Accordingly, by differentiating ϕ ϕ−1 = id with respect to qi, i = 1, . . . , 2 + nu + nv,∂i = ∂/∂qi, we obtain

∂iϕ−1 (x;ψ, q) = − ∂iϕ

∂θϕ ϕ−1 (x;ψ, q) (3.62)

and, hence,

∂ih =

(

− (∂θh)∂iϕ

∂θϕ+ ∂ih

)

ϕ−1

∂x∂ih = − 1

∂θϕ

(

∂iϕ

∂θϕ(∂θ)

2 h− ∂i∂θh+

(

∂i∂θϕ− ∂iϕ

∂θϕ(∂θ)

2 ϕ

)

∂θh

∂θϕ

)

ϕ−1.

(3.63)

The Velocity Field of the Roll

Let us first consider the Lagrangian point of view and assume that the work rolls rotatewith constant angular velocity ω. Then, a material point X = [r, θ0]

T ∈ [0, R] × S1 issubjected to a (regular) motion Φ composed by the action of a rotation group,

s =

[

rθ

]

= Φrot (t,X) =

[

rθ0 + ωt

]

,

and a subsequent elastic deformation ξ = Φelast (s, ψ, q),

ξ =

[

xy

]

=

[

(r + u (q, r, θ)) sin (θ − ψ) + v (q, r, θ) cos (θ − ψ)h0

2+R− (r + u (q, r, θ)) cos (θ − ψ) + v (q, r, θ) sin (θ − ψ)

]

,

i.e., Φ = ΦelastΦrot. Clearly, the material velocity of the point X is V (t,X) = dtΦ (t,X).However, for the purpose of the roll gap model, the Eulerian point of view is more

suitable, yielding the spatial velocity

v (t, ξ) = V Φ−1.

In particular, the velocities of surface points are of interest.

V (t, θ0) = dt

(

[

(R + u (q, θ)) sin (θ − ψ) + v (q, θ) cos (θ − ψ)h0

2+R− (R + u (q, θ)) cos (θ − ψ) + v (q, θ) sin (θ − ψ)

]∣

∣

∣

∣

θ=θ0+ωt

)

=

= ω

[

(∂θu− v) sin (θ − ψ) + (R + u+ ∂θv) cos (θ − ψ)(R + u+ ∂θv) sin (θ − ψ) + (v − ∂θu) cos (θ − ψ)

]∣

∣

∣

∣

θ=θ0+ωt

Let x ∈ (−R,R). Then, θ = ϕ−1 (x;ψ, q), see (3.59), and θ0 = θ−ωt = ϕ−1 (x;ψ, q)−ωt.Thus,

v (t, x) = ω

[

(∂θu− v) sin (θ − ψ) + (R + u+ ∂θv) cos (θ − ψ)(R + u+ ∂θv) sin (θ − ψ) + (v − ∂θu) cos (θ − ψ)

]∣

∣

∣

∣

θ=ϕ−1(x;ψ,q)

(3.64)

3. A Rolling Mill Model. . . 3.4.5. The Strip Equations 70

3.4.5 The Strip Equations

In order to obtain a mathematical model for the stresses building up in the deformed strip,typically the assumption that vertical plane sections of the strip remain plane throughoutthe roll gap, see Figure 3.21 (left), is arranged. Thus, the stresses and strains in the stripare allowed to vary in the x-direction only. The stresses σxx, σyy and σzz are treated asprincipal stresses and the assumption of plane strain is arranged. This approach is usuallyreferred to as the stripe model of metal forming, see e.g., [PP00] for a detailed discussionon this topic. The strip equations are outlined in Carthesian coordinates (x, y).

The Classical Stripe Model of Metal Forming – Revisited

By considering the balance of forces for a strip element in the x- and y-direction, cf.Figure 3.21, the equilibrium conditions

d

dx

(

σxxh)

− σ∂xh− 2τ = 0,

−σyy + σ − 1

2τ ∂xh = 0

(3.65)

are obtained. These equations are to be associated with a suitable friction model for theroll/strip interface and a constitutive law of the strip.

Remark 3.12 In view of the geometry of the roll/strip contact arc typically the ap-proximation σyy ≈ σ is encounterd in the rolling literature, cf., e.g., [JOZ60], [FJ87],[FJMZ92], [DET94], [LS01]. In this case, (3.65) reads

d

dxσxx =

1

h

(

(σyy − σxx) ∂xh+ 2τ)

, σ = σyy. (3.66)

Remark 3.13 (friction law for the roll/strip interface) An approach usually encounteredin the analysis of rolling is the assumption of the Coulomb friction law in the roll/stripinterface. We will also make use of this assumption. Thus, the friction law is given asτ = γµσ, µ > 0, with γ = +1 in the regions with backward slip, and γ = −1 otherwise,see also Figure 3.5.

With the Coulomb friction law of Remark 3.13, the equilibrium conditions (3.65) read

d

dxσxx =

1

h

((

1 +γµ

2(

1 − 12γµ∂xh

)∂xh

)

σyy − σxx

)

∂xh+2

h

γµσyy(

1 − 12γµ∂xh

)

σ =

(

1 − 1

2γµ∂xh

)−1

σyy.

(3.67)

The subsequent paragraphs provide the strip equations for the elastic and the plasticdeformation zones, obtained by plugging the respective constitutive laws into (3.67).

3. A Rolling Mill Model. . . 3.4.5. The Strip Equations 71

Elastic Compression and Recovery Zone

With E∗S = ES/ (1 − ν2

S) and ν∗S = νS/ (1 − νS) denoting the plane strain material para-meters of the strip, the constitutive equations for the linear-elastic deformation are dueto Hooke’s law, E∗

Sεxx = σxx − ν∗Sσyy, E∗Sεyy = σyy − ν∗Sσxx, GSγxy = τxy. Then, from

(3.67) we obtain the model

d

dxσyy = E∗

S

d

dxεyy +

1

h

(

−(

1 − ν∗S −γµν∗S

2(

1 − 12γµ∂xh

)∂xh

)

σyy + E∗Sεyy

)

∂xh+

+2ν∗Sγµ

h

σyy(

1 − 12γµ∂xh

)

σ =

(

1 − 1

2γµ∂xh

)−1

σyy.

(3.68)

for the elastic deformation of the strip.

Remark 3.14 (Remark 3.12 cont’d.) Substitution of Hooke’s law into (3.66) and theapplication of the Coulomb friction law given in Remark 3.13 yields the equation

d

dxσyy = E∗

S

d

dxεyy +

1

h(− (1 − ν∗S)σyy + E∗

Sεyy) ∂xh+2ν∗Shγµσyy, σ = σyy.

Furthermore, the approximation

d

dxσyy = E∗

S

d

dxεyy +

2ν∗Shγµσyy, σ = σyy (3.69)

is very frequently used in the literature, see e.g. [LS01].

As the strip processed in the roll gap experiences considerable displacements, thelogarithmic strain measure (see, e.g., [Zie92]),

εyy (x) = ln

(

h (x)

h0

)

(3.70)

also referred to as “effective” or “natural” strain measure, is to be applied. Here, h0

denotes the “unloaded strip thickness” obtained by removing the entry tension σen fromthe strip entering the roll gap (in the case of the roll gap entry section A) or by removingthe exit tension σex in the case of the roll gap exit section D, respectively. For the roll gapentry point, Hooke’s law gives εAyy (xen) = −ν∗Sσen/E∗

S, with the superscript A referring tothe elastic compression zone, and, thus,

h0en = hen exp

(

ν∗SE∗S

σen

)

→ εAyy (x) = ln

(

h (x)

hen

)

− ν∗SE∗S

σen (3.71)

by means of (3.70). Analogously, for the elastic recovery zone D the equation

εDyy (x) = ln

(

h (x)

hex

)

− ν∗SE∗S

σex (3.72)

is obtained.

3. A Rolling Mill Model. . . 3.4.6. The Implicit Non-Circular Arc Roll Gap Model 72

Remark 3.15 (Remark 3.14 cont’d). Substitution of (3.71) or (3.72), respectively, into(3.69) yields

d

dxσyy =

1

h

(

E∗S∂xh+ 2ν∗Sγµσyy

)

, σ = σyy, (3.73)

cf. again [LS01].

Plastic Reduction Zones

In the plastic reduction zones B and C the yield criterion of Tresca (see, e.g., [Zie92],[PP00]) is applied, σxx − σyy = kf , with the yield stress kf = const.

Remark 3.16 (Remark 3.12 cont’d.) Substitution of Tresca’s law into (3.66) and theapplication of the Coulomb friction law yields the equation

d

dxσyy =

1

h

(

−kf∂xh+ 2τ)

, σ = σyy, (3.74)

see, e.g., [LS01].

Remark 3.17 (Implementation issues) In view of the numerical implementation (andthe concatenation of the strip equations with the roll deformation problem) it is suitableto introduce a scaling of the strip tensions,

σ = σ/ασ, σ ∈ σxx, σyy, σ, τ , ασ ∈ R. (3.75)

Thus, (3.73) and (3.74) read

d

dxσyy =

1

h

(

E∗S∂xh+ 2ν∗Sγµσyy

)

, ´σ = σyy, E∗S =

E∗S

ασ(3.76)

andd

dxσyy =

1

h

(

−kf∂xh+ 2γµσyy

)

, kf =kfασ. (3.77)

A convenient choice is ασ = E∗R = E∗

R/ (2 (1 − ν∗2R )), see (3.49). Then, with Q = ασQ,(3.56) reads Aq = Q. In the following, we will exclusively refer to the scaled representa-tives of the strip ODEs and the tensions, and skip the primes.

3.4.6 The Implicit Non-Circular Arc Roll Gap Model

Given the deflection ψ (see Figure 3.21), the coordinates q of the roll deformation problemand the no-load gap h0, the strip tensions of the zones A–D can be obtained by solvingfor the solution of the strip ODEs

d

dxσi = f i

(

x, σi, ψ, q, h0

)

, σi(

xi0)

= σi0, i ∈ A,B,C,D (3.78)


given in the previous Section 3.4.5. With the respective domain of integration indicatedby a superscript, these solutions are referred to as

σA( x, ψ, q, h0, xen (ψ, q, h0) , 0 ),σB( x, ψ, q, h0, xAB, ηAB ),σC( x, ψ, q, h0, xCD (ψ, q) , ηCD ),σD( x, ψ, q, h0, xex, 0 ).

(3.79)

Here, the last two arguments indicate the initial values (xi0, σi0). At the roll gap entry point

xen (elastic compression zone A) and at the exit point xex (elastic recovery zone D), thenormal contact stress σ is zero. The coordinates xAB and xCD represent the boundariesbetween the domains A/B and C/D, respectively. The transition between the plasticreduction zone C and the elastic recovery zone D takes place at the point xCD (ψ, q) with∂xh (xCD;ψ, q, h0) = 0. The variables ηAB and ηCD define the initial values for the stressesof the plastic zones B (backward slip) and C (forward slip), i.e.,

(

xB0 , σB0

)

= (xAB, ηAB)and

(

xC0 , σC0

)

= (xCD, ηCD). The coordinate θN = ϕ−1 (xN ;ψ, q) of the neutral point, i.e.,the boundary between B and C, is fixed by means of the condition

σB (xN , ψ, q, h0, xAB, ηAB) = σC (xN , ψ, q, h0, xCD (ψ, q) , ηCD) . (3.80)

The implicit algebraic roll gap model is outlined in terms of the independent coordinates

λT =[

qT xAB ηAB ηCD xex h0

]

. (3.81)

The entry and exit strip thicknesses hen, hex, the backward and forward tensions σen, σex,the yield stress kf , the Coulomb friction coefficient µ, the material parameters of the roll(index R) and the strip (index S) and the parameter ψ are collected to the parametervector ζ,

ζT =[

hen hex σen σex kf µ E∗R ν∗R E∗

S ν∗S ψ]

. (3.82)

In order to keep the following formulas short, the explicit notation of the dependenceon the parameters ζ will be supressed whenever appropriate. The generalized forcesassociated with the loading σi, τ i, i ∈ A,B,C,D, read

Q1 = R

∫ ϕ−1(xAB ,ψ,q)

θen(ψ,q,h0)

((

σA ϕ)

∂1 ˇuJ +

(

τA ϕ)

∂1 ˇvJ)

dθ

+R

∫ θN (ψ,q,h0,xAB ,ηAB ,ηCD)

ϕ−1(xAB ,ψ,q)

((

σB ϕ)

∂1 ˇuJ +

(

τB ϕ)

∂1 ˇvJ)

dθ

+R

∫ θCD(ψ,q)

θN (ψ,q,h0,xAB ,ηAB ,ηCD)

((

σC ϕ)

∂1 ˇuJ +

(

τC ϕ)

∂1 ˇvJ)

dθ

+R

∫ ϕ−1(xex,ψ,q)

θCD(ψ,q)

((

σD ϕ)

∂1 ˇuJ +

(

τD ϕ)

∂1 ˇvJ)

dθ

(3.83)


and, by collapsing the integrals over the domains A to D,

Q2 = R


θen(ψ,q,h0)

(

(σ ϕ) ∂2 ˇuM + (τ ϕ)

(

∂2 ˇvM −K

))

dθ

Qu,i = R


θen(ψ,q,h0)

(σ ϕ) ∂qiu˜udθ , i = 1, . . . , nu

Qv,j = R


θen(ψ,q,h0)

(τ ϕ) ∂qjv˜vdθ , j = 1, . . . , nv,

(3.84)

see (3.55).As the central result of this section, we are now ready to state the implicit roll gap

model Ω (λ, ζ) = 0, Ω : Rdim(λ) × R

dim(ζ) → Rdim(λ),

Ω (λ, ζ) =

Aq −Q (λ)

h (xex) − hex

(1 − ν∗S)σAyy (xAB) − E∗

S ln

(

h (xAB)

hen

)

+ ν∗S (σen − kf )

σAyy (xAB) − ηAB

(1 − ν∗S) σDyy (xCD) − E∗

S ln

(

h (xCD)

h (xex)

)

+ ν∗S (σex − kf )

σDyy (xCD) − ηCD

= 0 (3.85)

The first equation represents the work roll deformation problem (see also Remark 3.17)and the second equation is due to the demand that the deformed strip leaves the rollgap with the desired exit thickness hex. The remaining four equations fix the transitionsbetween the elastic and plastic zones A–B and C–D.

Given the parameters ζ, the solution of the non-circular arc roll gap model (3.85)amounts to invoking the implicit function theorem. The function Ω is C1 w.r.t. λ, ζ onan open set S ⊂ R

dim(λ) × Rdim(ζ). Let (λ0, ζ0) ∈ S, Ω (λ0, ζ0) = 0, ∂λΩ (λ0, ζ0) regular,

then there exist neighborhoods U ⊂ Rdim(λ) of λ0 and V ⊂ R

dim(ζ) of ζ0 such that for eachζ ∈ V the equation Ω (λ, ζ) = 0 has a unique solution λ ∈ U . This solution is denoted asλ = ω (ζ) with the C1 function ω : V → U .

The roll force Fr, the roll torque Mr and the forward and backward slip ςen = ven/vR,ςex = vex/vR, vR = ωRR, cf. Fig. 3.4, are functions of λ and ζ.

On the Calculation of the Jacobian of Ω

Clearly, the calculation of the Jacobian of (3.85) at the point (λ, ζ) involves the knowledgeof the sensitivites of the interface stresses σ, τ with respect to the parameters q, h0,and the initial values of the strip ODEs. From Remark 3.13, we have τ i = µσi, i ∈A,B, and τ i = −µσi, i ∈ C,D, µ > 0. Let us arrange the abbreviation θN (·) =


θN (ψ, q, h0, xAB, ηAB, ηCD), and define the sensitivities of σα, i.e., Sαj = ∂jσα, T α = ∂xα

0σα,

Xα = ∂σα0σα, α ∈ A,B,C,D. Then, we have

∂jQi = R

(


θen(ψ,q,h0)

(

fA∂jϕ+ SAj + TA∂jxen)

(∂iu+ µvi) dθ+

+

∫ θN (·)

ϕ−1(xAB ,ψ,q)

(

fB∂jϕ+ SBj)

(∂iu+ µvi) dθ+

+

∫ θCD(ψ,q)

θN (·)

(

fC∂jϕ+ SCj + TC∂jxCD)

(∂iu− µvi) dθ+

+


θCD(ψ,q)

(

fD∂jϕ+ SDj)

(∂iu− µvi) dθ−

−(

σA (xAB) − σB (xAB)) ∂jϕ

∂θϕ(∂iu+ µvi)

∣

∣

∣

∣

θ=ϕ−1(xAB ,ψ,q)

+

+ 2µ (∂jθN (·))(

σC ϕ)

vi∣

∣

θ=θN (·)+

+ (∂jθCD (ψ, q))((

σC − σD)

ϕ)

(∂iu− µvi)∣

∣

θ=θCD(ψ,q)

)

(3.86)

withvi (θ) = ∂iv (θ) − δ2

iK. (3.87)

Note that (3.62), (3.80) and the conditions(

σA ϕ)

(θen (ψ, q, h0)) = 0, σD (xex) = 0, see(3.79), have been used. By differentiating the identities

h (xen (ψ, q, h0) ;ψ, q, h0) = hen, ∂xh (xCD (ψ, q) ;ψ, q, h0) = 0

with respect to qj, we obtain

∂jxen (ψ, q, h0) = −∂jh∂xh

(xen (ψ, q, h0) ;ψ, q, h0) ,

∂jxCD (ψ, q) = − ∂j∂xh

(∂x)2 h

(xCD (ψ, q) ;ψ, q, h0) .

Accordingly, the functions

∂jθen (ψ, q, h0) =∂jxen (ψ, q, h0) − ∂jϕ

∂θϕ

∣

∣

∣

∣

θ=θen(ψ,q,h0)

= −∂jh∂θh

∣

∣

∣

∣

θ=θen(ψ,q,h0)

and

∂jθCD (ψ, q) =∂jxCD (ψ, q) − ∂jϕ

∂θϕ

∣

∣

∣

∣

θ=θCD(ψ,q)

are determined. Finally, differentiation of (3.80), i.e.,(

σB − σC)

ϕ (θN (·) ;ψ, q) = 0,w.r.t. qj yields

∂jθN (·) = − 1

∂θϕ

(

SCj − SBj + TC∂jxCD (ψ, q)

fC − fB ϕ+ ∂jϕ

)∣

∣

∣

∣

∣

θ=θN (·)

.


Let Sαh0= ∂h0σ

α. Then, taking the derivative of Q with respect to the “no-load gap” h0

yields

∂h0Qi = R

(


θen(ψ,q,h0)

(

SAh0+ TA∂h0xen

)

(∂iu+ µvi) dθ+

+

∫ θN (·)

ϕ−1(xAB ,ψ,q)

SBh0(∂iu+ µvi) dθ +

∫ θCD(ψ,q)

θN (·)

SCh0(∂iu− µvi) dθ+

+


θCD(ψ,q)

SDh0(∂iu− µvi) dθ + 2µ (∂h0θN (·))

(

σC ϕ)

vi∣

∣

θ=θN (·)

)

,

with∂h0xen (ψ, q, h0) = −

(

∂xh)−1

(xen (ψ, q, h0) ;ψ, q, h0)

and

∂h0θN (·) = − 1

∂θϕ

(

SCh0− SBh0

fC − fB ϕ)∣

∣

∣

∣

θ=θN (·)

.

Accordingly, we have

∂xABQi = R

(

∫ θN (·)

ϕ−1(xAB ,ψ,q)

TB (∂iu+ µvi) dθ + 2µ (∂xABθN (·))

(

σC ϕ)

vi∣

∣

θ=θN (·)+

+(

σA (xAB) − σB (xAB))

(∂θϕ)−1 (∂iu+ µvi)∣

∣

θ=ϕ−1(xAB ,ψ,q)

)

,

with

∂xABθN (·) =

1

∂θϕ

(

TB

fC − fB ϕ)∣

∣

∣

∣

θ=θN (·)

.

The sensitivities of Qi with respect to the initial values ηAB, ηCD for the strip ODEs ofthe domains B and C read

∂ηABQi = R

(

∫ θN (·)

ϕ−1(xAB ,ψ,q)

XB (∂iu+ µvi) dθ + 2µ (∂ηABθN (·))

(

σC ϕ)

vi∣

∣

θ=θN (·)

)

,

∂ηCDQi = R

(

∫ θCD(ψ,q)

θN (·)

XC (∂iu− µvi) dθ + 2µ (∂ηCDθN (·))

(

σC ϕ)

vi∣

∣

θ=θN (·)

)

with

∂ηABθN (·) =

1

∂θϕ

(

XB

fC − fB ϕ)∣

∣

∣

∣

θ=θN (·)

, ∂ηCDθN (·) = − 1

∂θϕ

(

XC

fC − fB ϕ)∣

∣

∣

∣

θ=θN (·)

Finally, the derivative of Qi with respect to the roll gap exit point xex is

∂xexQi = R


θCD(ψ,q)

TD (∂iu− µvi) dθ.

The calculation of the sensitivity functions Sαj , Sαh0, T α and Xα amounts to solving for

the solution of the sensitivity ODEs (cf., e.g., [Kha96]) associated to (3.78). See also theAppendix for a geometric point of view on this issue.


The Slip Conditions in the Roll Gap

For calculating the slip conditions, and, in particular, the backward (entry) slip and theforward (exit) slip, ςen = ven/vR, ςex = vex/vR, the elastic volume compression occurringin the elastic zones has to be taken into account. In the elastic zones, the mass density isrelated to the stress state via

ρ (x2)

ρ (x1)=

1 + 1−2νS

ES(σxx (x1) + σyy (x1) + σzz (x1))

1 + 1−2νS

ES(σxx (x2) + σyy (x2) + σzz (x2))

=1 +

1−ν∗SE∗

S(σxx (x1) + σyy (x1))

1 +1−ν∗SE∗

S(σxx (x2) + σyy (x2))

,

see [PP00], with the second identity referring to the plane strain case under consideration.By virtue of the mass balance law ρhv = const applied to the elastic compression zone A,the roll gap entry speed ven is related to the strip velocity at xAB by

ven =h (xAB)

hen

ρ (xAB)

ρ (xen)v (xAB) .

At the point xAB, the yield criterion of Tresca, cf. Section 3.4.5, requires σxx (xAB) =kf + ηAB, therefore, we obtain

ρ (xAB)

ρ (xen)=

1 + (1 − ν∗S)σen/E∗S

1 + (1 − ν∗S) (kf + 2ηAB) /E∗S

,

see also (3.79). In the plastic reduction zones the deformation is assumed to be isochoric,an assumption well verified by experiments on plastic forming, see, e.g., [PP00], [Zie92].Hence, for zone B, we have v (xAB) = h (xN) vR/h (xAB). By combining these equations,the entry slip ςen follows as

ςen =h (xN)

hen

1 + (1 − ν∗S)σen/E∗S

1 + (1 − ν∗S) (kf + 2ηAB) /E∗S

. (3.88)

Analogously, by considering the mass balance of the zones C and D,

vex =h (xCD)

hex

ρ (xCD)

ρ (xex)v (xCD) =

h (xN)

hex

ρ (xCD)

ρ (xex)vR =

h (xN)

hex

1 +1−ν∗SE∗

Sσex

1 +1−ν∗SE∗

S(kf + 2ηCD)

vR,

the exit slip is obtained as

ςex =h (xN)

hex

1 + (1 − ν∗S)σex/E∗S

1 + (1 − ν∗S) (kf + 2ηCD) /E∗S

. (3.89)

Finally, the elongation coefficient ε = vex/ven − 1 = ςex/ςen − 1 follows as

ε =henhex

1 +1−ν∗SE∗

Sσex

1 +1−ν∗SE∗

Sσen

1 +1−ν∗SE∗

S(kf + 2ηAB)

1 +1−ν∗SE∗

S(kf + 2ηCD)

− 1. (3.90)


Some Numerical Results

Example 3.6 Let R, E, ν from Example 3.2, and select the Ritz ansatz as given in Exam-ple 3.4. Figure 3.22 depicts the results for a steel rolling case with a non-circular roll/stripcontact arc, obtained given the parameters kf = 750N/mm2, µ = 0.1, hen = 1mm,hex = 0.9906mm, σen = 70N/mm2 and σex = 70N/mm2. According to the models fromthe literature, see again e.g. [JOZ60], [FJMZ92], [DET94], [LS01], the approximationx ≈ Rθ is arranged and the effect of the tangential load τ on the shape of the roll gapis neglected. The results obtained by using (3.26) (yielding the roll force Fr = 3.73MN)are depicted with dashed lines, and the results due to the proposed Ritz approach (yieldingFr = 3.77MN) are shown in solid lines.

Remark 3.18 (on the computational effort) The proposed roll gap model is implementedusing the programming language C. The shape of the Ritz ansatz is specified and symbol-ically processed via Maple 9, thus allowing for a simple and user-friendly adjustment todifferent rolling situations. As a result, following Section 3.4.3, the Maple script automat-ically generates the C code functions needed to calculate the displacement fields and therequired derivatives with respect to r, θ and qi. Accordingly, the specification of the rollgap model function Ω and the (symbolic) calculation of the derivatives w.r.t. λ is donevia Maple and automatically transferred to C code. The numerical integration of the stripODEs and the associated sensitivity ODEs in the distinct zones A − D is performed byan Euler scheme, with a spatial discretization of Ni nodes, i ∈ A,B,C,D. The roll gapmodel C function is called via the mex-funtion gateway from Matlab. The computer plat-form associated to the following timing information is: Mobile Intel Pentium 4-M CPU1.9GHz, 512MB RAM, MS Windows XP prof. SP1, Maple 9 [Map], Matlab 6.5 (R13)[MS], Simulink 5.0 (R13). Let NA = NB = NC = 40 and ND = 100. The computationalcost for one Newton step to solve for Ω (λ, ζ) = 0, i.e., evaluation of Ω, calculation ofthe Jacobian of Ω w.r.t. λ (using the sensitivity functions associated to the strip ODEs)and solution of the linear system Ω (λk, ζ) + (∂λΩ (λk, ζ)) (λk+1 − λk) = 0 (invoking theLAPACK function dgesv, cf. [Pac]) is 32.2ms. The most time consuming task is theevaluation of the displacement fields and their derivatives w.r.t. r, θ and qi at the dis-cretization nodes of the contact arc, whereas the LAPACK operation for solving the set oflinear equations amounts to approximately 0.5ms. Hence, in order to further reduce thecomputation time, the focus should be laid on additional pre-processing of the displace-ment fields evaluation functions. The timing information associated with one Newtonstep is regarded as the relevant number in view of a real-time implementation, as the rollgap model is to be solved for each sampling period. Given a suitable initial guess (takenfrom the previous sampling period) the Newton iteration might typically converge withina very few steps. Notice that, in order to link the C code to the LAPACK package (whenusing the Matlab gateway), the mex compiler command has to include the search path tolibmwlapack.lib, which is contained in the Matlab distribution.

Example 3.7 Consider the same rolling conditions as in Example 3.6, but, in contrast toExample 3.6, apply (3.58) and (3.59) instead of the respective approximations. Figure 3.23

3. A Rolling Mill Model. . . 3.4.7. Conclusions on the RGM 79

shows a comparison between the results, both evolving from the same Ritz ansatz, abtainedwith the approximations due to Example 3.6 (x ≈ Rθ, effect of τ on the shape of thecontact arc neglected) on the one hand, and the detailed model of the contact arc due to(3.58), (3.59) on the other hand.

3.4.7 Conclusions on the proposed Roll Gap Modelling Approach

The focus of this contribution on the roll gap modelling was laid on the introduction ofa suitable Ritz approximation for the elasto-static work roll deformation problem. Theobjective of this approach is to evolve a non-circular arc roll gap model exhibiting reducedcomputational effort compared to the models given in the literature, thus being applicablefor the control purpose. To this end, in order to obtain a reasonable accuracy of the Ritzapproximation and simultaneously a number of coordinates as small as possible, the keytask is the choice of a set of suitable ansatz functions. In rolling mills applications, thelength of the roll gap is very small compared to the geometry of the roll. The objectiveof finding a Ritz ansatz entailing a considerably tight approximation of the displacementfields particularly in that roll gap domain was achieved by utilizing St.Venant’s principle:As the effect of the particular shape of the loading on the shape of the displacementsdecreases rapidly outside a roll gap domain vicinity, it qualified as suitable to arrangethe solutions caused by diametrically applied forces, which are known in the literature,as shape functions outside this domain. The accuracy of the proposed finite-dimensionalapproximation of the work roll deformation problem was illustrated via numerical results.

By virtue of the proposed composition of the Ritz ansatz on the roll gap domain G,i.e., as the superposition of a suitable continuation of Jortner’s and Meindl’s solution and2-dimensional polynomials, flexibility is attained to adjust to different rolling problems:Besides the possibility to (re-)arranging the boundaries of G, the accuracy of the Ritzapproximation can easily be adjusted via the choice of the decomposition of G into suitablesubdomains Gk, and the setup of the polynomials with suitable orders defined on thesesubdomains. To this end, the use of computer-algebra is very attractive as the compositionof the shape functions, the elimination of the constraints emerging on the boundaries ofthe (sub-) domains, and, finally, the code generation can be left to the computer algebrasystem.

Concerning the numerical solution of the implicit algebraic roll gap model, the notionof the sensitivity ODEs qualifies as particularly useful for an efficient computation of theJacobian. This amounts to numerically integrating the sensitivity ODEs associated withthe strip ODEs to obtain the sensitivities of the strip tensions with respect to the roll gapmodel coordinates λ. It is worth emphasizing that the proposed modelling approach isnot confined to a particular choice of the strip model or the friction law for the roll/stripinterface, such that the classical stripe model of metal forming can be replaced by meansof more sophisticated models. Future work will be concerned with the incorporation of aso-called neutral zone into the roll gap model (instead of a single neutral point) referredto as contained plastic flow of the strip, which occurs in cases of rolling particularly thinand hard strip.


0.986

0.988

0.990

0.992

0.994

0.996

0.998

1.000

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

h[m

m]

sol. due to (3.26)Ritz approx.

-400

-200

0

200

400

600

800

1000

1200

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

σ,τ,σxx

[N/m

m2]


elastic

compression

zone A

plastic reduction

zone B

(backward slip)

plastic reduction

zone C

(forward slip)

elastic

recovery

zone D

σxx

−σA

−σB −σC

−σD

τ

Figure 3.22: (cf. Example 3.6) Shape h of the roll/strip contact arc, the associatedrolling load σ, τ and the strip tension σxx: Ritz approx. (red) vs. spatial discretizationof (3.26) (blue dashed). Approx.: x = Rθ, neglection of the effect of τ on the shape ofthe roll gap.


0.986

0.988

0.990

0.992

0.994

0.996

0.998

1.000

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

h[m

m]

Ritz sol. by applying (3.58), (3.59)approx. x = Rθ, τ neglected; cf. Example 3.6

-200

0

200

400

600

800

1000

1200

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

σ,τ,σxx

[N/m

m2]


Figure 3.23: (cf. Example 3.7) Comparison of the results obtained by using an approxi-mate model of the contact arc (red) (x ≈ Rθ, effect of τ on the shape of the contact arcneglected; cf. Example 3.6 and Figure 3.22), and, displayed in black color, by using thedetailed description of the contact arc due to (3.58) and (3.59).

3. A Rolling Mill Model. . . 3.5. Bridle Roll Dynamics 82

3.5 Bridle Roll Dynamics

The bridle rolls in the entry and exit section of the mill, cf. Figures 3.1 and 3.24, areintended to assigning, by application of the bridle torques, a certain difference of the stripforces between the periphery (i.e., adjacent winders or other strip processing lines) andthe roll gap.

PSfrag replacements

Entry bridle rolls Exit bridle rolls

Fext,1Fext,1

Fext,2Fext,2

pulley 1 pulley 1

pulley 2pulley 2

slip arc

slip arc

slip arc

slip arcstick arc

stick arc

stick arc

stick arc

v = v1

v > v1

v > v2

v = v2 = ω2R

v = v2 = ω2R

v = v1 = ω1R

v < v1

v < v2

v1 = ω1R

v1 = ω1R

q

q

q

q

v2 = ω2R

v2

v∗1

v∗1v∗2

v∗2

φ1

φ1

φ2

φ2

ω1,M1

ω1,M1

ω2,M2ω2,M2

R

R

R

R

DF F

Figure 3.24: Configuration of the entry and exit bridle device (including the locations ofthe slip arcs; operating range: Fext,1 > F > Fext,2).

The notation introduced in Figure 3.24 is arranged with the intention to commonlytreat the main characteristics of both bridle devices. When there is danger of confusion,the variables are being attached with the subscript br ∈ ebr, xbr to distinguish betweenthe entry and the exit bridle rolls. Variables carrying the index 1 are associated with thetop pulley of the bridle device, the index 2 identifies variables associated with the bottompulley. The operating range of both bridle devices is defined as Fext,1 > F > Fext,2. Allpulleys have the same radius R, and the vertical distance between the pulley centers isdenoted by D. The following assumptions are arranged:

• The strip processed by the bridle rolls is assumed to be massless, with negligiblebending rigidity, and to exhibit a constant cross-sectional area A, i.e., Aebr = Bhnomen

3. A Rolling Mill Model. . . 3.5.1. On the Localisation of the Slip Arcs 83

and Axbr = Bhnomex , with hnomen (hnomex ) denoting the nominal strip entry (exit) thick-ness and B the strip width;

• in the slip arcs (to be identified), Coulomb friction is assumed, with the frictioncoefficient µbr = const.;

• the operating range of the bridle system is chosen such that the slip arcs do notexceed the maximum wrap angle of the pulleys. This means that the strip does notslide on the pulleys. Note that the term “sliding” must not be confused with thenotion of “creep” occurring in the slip arcs.

The maximum wrap angle φmax not to be exceeded by the slip arc is given as φmax =π − arccos (2R/D). The moments of inertia, i.e., Θ1 for the top roll and Θ2 for thebottom system, summarize the moments of inertia of the pulley, the associated drive-train and the motor, related to the pulley shaft. The torques M1 and M2 are regardedas control inputs. The friction torques occurring in the drive-train of the pulleys aremodelled as −diωi, i = 1, 2, di = const.

3.5.1 On the Localisation of the Slip Arcs

As a prerequisite for localising the slip arcs of the entry and exit bridle pulleys we willfirst examine the case depicted in Figure 3.25 (see also [Joh85] for the discussion of a beltdrive being a closely related example), which clearly corresponds to the top pulley of theentry bridle device. The results can be applied to the other pulleys of the bridle devicesin a straight-forward manner to get the results illustrated in Figure 3.24.

PSfrag replacements

Fa

Fb

va

vb

slip arc

stick arc

v > ωR

v = vb = ωR

ω,M

φ

ϕ

q

R

Figure 3.25: On the discussion of the slip arc and the creep (Fa > Fb).

Referring to Figure 3.25, let Fa and Fb < Fa be the forces at the tight and the slackside of the system. The associated strains in the strip segments given by Hooke’s law areεa = λFa, εb = λFb, with λ = 1/ (ESA) and ES denoting Young’s modulus of the strip.With ρref denoting the mass density of the unloaded strip, the mass density ρ of the

3. A Rolling Mill Model. . . 3.5.2. The Equations of Motion 84

deformed strip is ρ = ρref/ (1 + ε). From the mass balance law, ρava = ρbvb, we obtain

vb =1 + εb1 + εa

va =1 + λFb1 + λFa

va < va. (3.91)

Hence, a material point of the strip experiences an acceleration while passing from theslack side to the tight side. Clearly, the torque to be applied in order to establish thedifference between Fa and Fb is of negative sign (referring to the sign convention of Fig-ure 3.25). Therefore, the Coulomb frictional traction q acting on the strip is oriented asindicated. The calculation of the strip force F (ϕ) evolving in the creep domain is donevia Euler’s equation of cable friction (see, e.g., [Zie92]), dF/dϕ + µF = 0, which admitsthe solution F (ϕ) = F (ϕ0) exp (−µ (ϕ− ϕ0)). Hence, we have Fb = Fa exp (−µφ), and,thus, φ = µ−1 ln (Fa/Fb) for the extent φ of the slip arc. As the frictional traction q mustoppose the direction of the slip, the strip is moving faster than the pulley in the sliparc. Therefore, the slip arc is located where the strip runs off the pulley, va ≥ v > vb,whereas the sticking region occurring in the entry zone is characterized by v = vb = ωR,see Figure 3.25.

The results obtained so far directly apply to the case of the entry bridle device. Con-cerning the exit bridle rolls, the subdivision of the pulley/strip contact arcs into stickingand creep regions (see Figure 3.24) is obtained in a straight-forward manner. The extentsof the slip arcs are given as φ1 = µ−1 ln (Fext,1/F ) and φ2 = µ−1 ln (F/Fext,2).

3.5.2 The Equations of Motion

The Entry Bridle Roll Device

Following the arrangement made above, we will drop the subscript ebr on the variables fornotational brevity. In view of (3.91), the velocity v∗2 of the strip running off the bottompulley is

v∗2 =1 + λF

1 + λFext,2v2 > v2.

With c = ESA/L = 1/ (λL), (L/2)2 = (D/2)2 − R2, denoting the spring coefficient ofthe strip segment connecting both pulleys, the force F is given as F = c (v1 − v∗2). Thus,with Fext,1 = σenBh

nomen , cf. Figure 3.1, the dynamics of the entry bridle device read

ω1 =1

Θ1

(M1 + (σenBShnomen − F )R− d1ω1)

ω2 =1

Θ2

(M2 + (F − Fext,2)R− d2ω2)

F = cR

(

ω1 −1 + λF

1 + λFext,2ω2

)

.

(3.92)

Remark 3.19 The steady-state “creep ratio” [Joh85] of the entry bridle device is ξ =1−ω2/ω1 = λ (F − Fext,2) / (1 + λF ) > 0. Hence, in steady state, pulley 1 moves (slightly)faster than pulley 2.

3. A Rolling Mill Model. . . 3.6. The Non-linear Dynamics of the Rolling Mill 85

The Exit Bridle Roll Device

Due to the slip arc of the top pulley the velocity v∗1 is related to the circumference speedv1 of the pulley via

v∗1 =1 + λF

1 + λFext,1v1 < v1.

With Fext,1 = σexBhnomex , the equations of motion of the exit bridle rolls read

ω1 =1

Θ1

(M1 + (F − σexBhnomex )R− d1ω1)

ω2 =1

Θ2

(M2 + (Fext,2 − F )R− d2ω2)

F = cR

(

ω2 −1 + λF

1 + λσexBhnomex

ω1

)

.

(3.93)

Again, as in the case of the entry bridle system, in steady-state the pulley 1 (tight side)is moving faster than pulley 2 (slack side).

A Simplified Model

Motivated as a step towards the control design, a simplified model of the bridle rolldynamics is obtained by taking the limit E → ∞, which amounts to neglecting the creepeffect, and, thus, the occurrence of slip arcs. Hence, v∗2 ' v2 for the entry bridle device,and v∗1 ' v1 for the exit device. Then, we simply have

Θ1ω1 = M1 + (Fext,1 − F )R− d1ω1

Θ2ω2 = M2 + (F − Fext,2)R− d2ω2

F = cR (ω1 − ω2) ,

(3.94)

exemplary outlined for the entry bridle rolls.

3.6 The Non-linear Dynamics of the Rolling Mill

As a prerequisite for the assembly of the mill stand dynamics and the roll gap model wewill first introduce a re-arrangement of the roll gap coordinates λ, ζ. The reason is thatthe original choice for λ, ζ of (3.81), (3.82), and the equations (3.85), Ω (λ, ζ) = 0, wasproposed in view of finding an appropriate setting given a desired strip output thicknesshex. In particular, hex is fixed by the second equation of (3.85).

As the variable h0 represents the position of the mass m0, see Figure 3.2 (left), it isremoved from the vector λ. Accordingly, the second equation of (3.85) is removed fromthe roll gap model thus yielding the representation

Ω(

λ, ξ, ζ)

= 0, λT =[

qT xAB ηAB xex ηCD]

, ξT =[

h0 σen σex]

(3.95)


with ζ containing the remaining parameters, see (3.82). For notational convenience we will

drop the dependence on ζ whenever suitable. Given a point(

λ0, ξ0, ζ0)

∈ S, S ⊂ Rdim(λ)×

Rdim(ξ) × R

dim(ζ), Ω being C1 on S, Ω(

λ0, ξ0, ζ0)

= 0 and ∂λΩ(

λ0, ξ0, ζ0)

nonsingular.

Then, by virtue of the implicit function theorem, there exist neighborhoods U ⊂ Rdim(λ)

of λ0 and V ⊂ Rdim(ξ) × R

dim(ζ) of(

ξ0, ζ0)

such that for each(

ξ, ζ)

∈ V the equationΩ(

λ, ξ, ζ)

= 0 has a unique solution λ ∈ U . This solution is denoted as λ = ω(

ξ, ζ)

withthe C1 function ω : V → U .

The dynamics of the mill stand of Figure 3.2 (left) the roll gap model (3.95) is to becombined with is given in (3.1), with Fr = Fr

(

λ, h0

)

. The motion of the main mill drive(MMD) is simply modelled as

ΘRωR = MR −Mr

(

λ, h0

)

− dRωR, (3.96)

with ΘR as the moment of inertia of the rolls and the main mill drive transmission line,MR as the torque supplied by the drive, and dR as a viscous friction coefficient.

The strip elements connecting the bridle devices to the roll gap, cf. Figure 3.1, aremodelled as mass-less linear-elastic springs with constant (nominal) cross sections Bhnomen

and Bhnomex , respectively. Thus, by obeying the slip equations of the roll gap, the striptensions in the entry and exit section of the mill are given as

σen =ESLen

(

ςen(

λ, ξ)

ωRR− ωebr,1Rebr

)

,

σex =ESLex

(

ωxbr,1Rxbr − ςex(

λ, ξ)

ωRR)

.

(3.97)

Thus, finally, the differential-algebraic dynamics of the mill are specified by (3.9) (or (3.13)respectively, if the HGC is operating in force control mode), the bridle roll dynamics (3.92),(3.93), and (3.95)–(3.97). Clearly, the system has index 1, with the motion confined tothe manifold rendered by the implicit roll gap model (3.95). The control inputs are givenas uT =

[

xdp,MR,Mebr,i,Mxbr,i

]

, i = 1, 2. The coupling of the system to the periphery,i.e., adjacent winders or other strip processing lines, is taken into account by means ofthe external forces Feco = Febr,ext,2 and Fxco = Fxbr,ext,2 (not explicitely stated in theequations) acting on the lower pulleys of the bridle roll devices, cf. Figure 3.1. To furtherreveal the structure of the mill dynamics, it is convenient to consider the decompositionxT =

[

wT , ξT]

, wT = [x1, v1, v0, Fh, ωR, ωebr,i, Febr, ωxbr,i, Fxbr], i = 1, 2,

w = fw(

λ, ξ, w, u)

ξ = f ξ(

λ, ξ, w)

0 = Ω(

λ, ξ)

(3.98)

Remark 3.20 (simulation issues) Using the function ω as indicated above for the com-puter simulation of the mill dynamics amounts to solving for the solution of the implicit


roll gap model (3.95) at each time step of the numerical integration scheme. An alter-native approach for the numerical integration of this system is given as follows. From˙Ω =

(

∂λΩ) ˙λ+

(

∂ξΩ)

ξ = 0 we have

˙λ = −(

∂λΩ)−1 (

∂ξΩ)

f ξ.

Rendering the constraint manifold attractive can be easily achieved by augmenting this

equation with the term −(

∂λΩ)−1

P Ω, P > 0, to obtain

˙λ =(

∂λΩ)−1 (−

(

∂ξΩ)

f ξ − P Ω)

.

The simulation model reads

w = fw(

λ, ξ, w, u)

ξ = f ξ(

λ, ξ, w)

˙λ =(

∂λΩ)−1 (−

(

∂ξΩ)

f ξ − P Ω)

(3.99)

with the initial value(

λ0, ξ0)

chosen to meet the constraint Ω(

λ0, ξ0)

= 0. The attrac-tiveness of the constraint manifold is easily seen by application of Lyapunov’s theory. Let

V = 12ΩT Ω, then V = ΩT ˙Ω = −ΩTP Ω ≤ 0.

Example 3.8 (the parameters of the simulation model) The parameters used for the sim-ulations of the rolling mill are given as follows. Mill stand model, cf. Figure 3.2 (left):m1 = 105kg, m0 = 23×103kg, cg = 5.4×109N/m, d1 = 107Ns/m, d0 = 1.14×107Ns/m;hydraulic actuator, cf. Figure 3.3: A1 = 0.6567m2, A2 = 0.2082m2, Eoil = 1.6×109N/m2,p2 = 80 × 105N/m2, V0 = 0.02m3, αpcm = 50. Main mill drive: ΘR = 4723kgm2,dR = 100Nms/rad; Bridle roll systems (identical configuration of the entry and the exitbridle device): Rbr = 0.3m, Θbr,1 = 235kgm2, Θbr,2 = 100kgm2, dbr,1 = 10Nms/rad,dbr,2 = 4.4Nms/rad; Distances between the entry and exit bridle roll centers and theroll gap: Len = 2.2m, Lex = 2.4m; Strip parameters: ES = 2.1 × 1011N/m2, ρS =7.7 × 103kg/m3; strip width: BS = 0.5m. The roll gap parameters are due to Exam-ple 3.6.

Remark 3.21 (on the computational effort) Due to the “stiffness” of the mill dynam-ics (3.99) using the parameters as given in Example 3.8, which will be illustrated in thesubsequent Section 3.6.1, the numerical integration of (3.99) requires considerable com-putation time, typically about 250s per second “real-time”. This timing result is due tousing Matlab/Simulink [MS] with the ODE solver ode45 (“Dormand-Prince”; settings:max. time step: 10−3, tolerance bounds set to “auto”). See also Remark 3.18 addressingthe computational cost per evaluation of the roll gap model (including the calculation ofthe Jacobian, and the solution of the linear system related to one iteration step of theNewton method).

3. A Rolling Mill Model. . . 3.6.1. Analysis of the Linearized Mill Dynamics 88

3.6.1 Analysis of the Linearized Mill Dynamics

As a prerequisite for the control design task it is illustrative to investigate the locationof the eigenvalues of the linearized mill dynamics for different mill speeds vR. In thelight of a model reduction, the inspection of the root loci will motivate to discard thosesystem dynamics located beyond the frequency range accessible by the actuators. Beforeproceeding to the discussion of the root loci, we will first give the linearization of theimplicit roll gap model (3.95).

Remark 3.22 (linearization of the roll gap model (3.95)) Let yT(

λ, ξ)

= [Fr,Mr, ςen, ςex].From dξΩ =

(

∂λΩ) (

∂ξλ)

+ ∂ξΩ = 0 we have

∂ξλ = −(

∂λΩ)−1

∂ξΩ, with ∂ξΩ =[

∂h0Ω ν∗Sen+1 ν∗Sen+3

]

,

n = 2 + nu + nv, and ek, eik = δik, as the elements of the canonical basis of R

n+4. Then,the sensitivities of y w.r.t. ξ, are obtained as

dξy = − (∂λy)(

∂λΩ)−1

∂ξΩ + ∂ξy. (3.100)

Example 3.9 (root loci, calculated at different mill speeds) Consider the parameters asgiven in Example 3.8 and calculate the eigenvalues of the linearized mill dynamics, with themill speed vR taking values from 0.5m/s up to 20.5m/s. The Figures 3.26 and 3.27 showthe corresponding root loci, with the eigenvalues associated with vR = 0.5m/s indicated inred color.

The eigenfrequencies related to the mill stand oszillations are clearly beyond the dy-namic range of the hydraulic gap control under consideration, and, thus, they will bediscarded in view of the controller design. The motions of the bridle roll devices mightbe decomposed into a high-frequency pulley/pulley counter-phase oszillation mode and alower-frequency oszillation of both pulleys (thought of as notionally glued together). Asthe higher-frequency mode of oszillation is again beyond the dynamic range of the actu-ators, the pulley/pulley ozillations will also be discarded from the detailed bridle devicemodel. Figure 3.27 depicts a detailed view of Figure 3.26, containing the dynamics beingrelevant for the control design.

3. A Rolling Mill Model. . . 3.6.1. Analysis of the Linearized Mill Dynamics 89

−2500

−2000

−1500

−1000

−500

0

500

1000

1500

2000

2500

-300 -250 -200 -150 -100 -50 0

Im

Re

mill stand

mill stand

pulley/pulley osz.

bridle osz.

Figure 3.26: (cf. Example 3.9) Root loci of the linearized mill dynamics for different millspeeds (vR = 0.5m/s . . . 20.5m/s).

−150

−100

−50

0

50

100

150

−60 −50 −40 −30 −20 −10 0

Im

Re

bridle osz.

HGC MMD

Figure 3.27: Detail of Figure 3.26 (and comparison with the reduced-order model(black/magenta) to be introduced in the subsequent chapter).

chapter

FOUR

Flatness-based Rolling Mill Control

From the control point of view, particularly the handling of the acceleration and de-celeration procedures of the mill, which are intended to be kept as short as possible

for efficiency reasons by simultaneously meeting the tight tolerances for the elongationcoefficient, is known as a key objective in the industrial practice. Additionally, the controlsystem has to handle operator commands concerning the desired strip tensions σen andσex (see Figure 3.1), again by keeping the elongation coefficient within the given guaranteevalues. Usually, control engineers face these problems by adjusting additional feed-forwardschemes during the implementation and startup phase of the mill which frequently pointsout to be a very time-consuming task. Thus, besides the disturbance rejection perfor-mance of the control system, particular attention is to be paid to the trajectory planningand the trajectory tracking problem.

4.1 A Reduced-Order Model of the Rolling Mill

As a prerequisite for proposing a flatness-based approach to rolling mill control, we willdiscuss a reduced-order model, evolving from the investigations of Section 3.6.1, built inview of the control design.

4.1.1 A Quasi-static Mill Stand Model

Following the observations of Section 3.6.1, illustrated in the Figures 3.26 and 3.27, it isseen that the eigenfrequencies related to the mill stand oszillations are clearly beyond thedynamic range of the hydraulic actuator. Hence, in view of the control design it is suitableto discard those dynamics, which amounts to considering the mechanical sub-system ofthe mill stand as quasi-static. To this end, by setting d0 = d1 = 0, we have

Fh − cg (x1 − l1) −m1g = 0, Fr(

λ, h0

)

− Fh −m0g = 0, (4.1)

90

4. Rolling Mill Control. . . 4.1.1. A Quasi-static Mill Stand Model 91

see (3.1). From (4.1) and xp = x1 − h0 − l2, by arranging a change of coordinates fromFh to z, see (3.5), we obtain x1 = χ

(

λ, h0, z)

,

x1 = h0 + l2 +V0

A1

(

exp

(

z − Fr(

λ, h0

)

+m0g

EoilA1

)

− 1

)

, (4.2)

andFr(

λ, h0

)

− cg(

χ(

λ, h0, z)

− l1)

− (m0 +m1) g = 0. (4.3)

By assembling the dynamics (3.9) of the HGC-PCM, outlined in terms of z, and thequasi-static mill stand model, we find

z = αpcm

(

−z + EoilA1 ln

(

1 +A1

V0

xdp

)

+ Fr(

λ, h0

)

−m0g

)

0 =

[

Ω(

λ, h0, σen, σex, ζ)

Fr(

λ, h0

)

− cg(

χ(

λ, h0, z)

− l1)

− (m0 +m1) g

]

.

(4.4)

This consideration entails the formulation of a third setting of the implicit algebraic rollgap model besides (3.85) and (3.95), namely, the composition of the algebraic equationsof (4.4),

Ω(λ, ξ, ζ) = 0, ξT =[

z σen σex]

, (4.5)

with λT =[

λT , h0

]

from (3.81).

Remark 4.1 (HGC-FCM) With the HGC operating in force control mode, cf. Sec-tion 3.3.2, we obtain

z = αfcmEoil(

F dh − Fr (λ) +m0g

)

0 = Ω(λ, ξ, ζ)(4.6)

for the assembly of the actuator dynamics and the quasi-static mill stand model.

Again, with Ω of (4.5) meeting the conditions of the implicit function theorem, thereexists a C1 function ω, λ = ω(ξ, ζ), defined on a neighborhood of (λ0, ξ0, ζ0), withΩ(λ0, ξ0, ζ0) = 0 and ∂λΩ(λ0, ξ0, ζ0) regular. Thus, the DAE system of the reduced orderdynamics has again index 1.

Remark 4.2 (linearization of Ω) From dξΩ = (∂λΩ)(∂ξλ) + ∂ξΩ = 0, we have ∂ξλ =

−(∂λΩ)−1∂ξΩ. Due to the specific structure of (4.5), namely that Ω does not explicitly

depend on z and the roll force Fr is a function of λ only, the partial derivative ∂ξΩ yieldsthe particular simple representation

∂ξΩ =

0 ∂σenΩ ∂σex

Ω

−cg V0

EoilA21exp

(

z−Fr(λ)+m0gEoilA1

)

0 0

.

Let y (λ, σen, σex) = [Fr,Mr, ςen, ςex], cf. also Remark 3.22, then

dξy = (∂λy) (∂ξλ) + ∂ξy, ∂ξy =[

0 ∂σeny ∂σex

y]

is obtained.

4. Rolling Mill Control. . . 4.1.2. A Reduced-Order Model of the Bridle Rolls 92

4.1.2 A Reduced-Order Model of the Bridle Roll Dynamics

As for the bridle rolls the dynamics related to the pulley/pulley counter-phase oscillationsare also beyond the frequency domain accessible by the actuators, cf. Section 3.6.1, areduced order model obtained by thinking of the pulleys as rigidly glued together is used.The subsequent discussions exemplary refer to the subsystem “entry bridle device plusentry strip element”, i.e., (3.94) and (3.97), σen = ES/Len (ven − ω1R), see also Figure 3.1,regarding the roll gap entry velocity ven (t) as an input. Let Θ2/Θ1 = d2/d1 = γ, andarrange the notation Θ1 = Θ, Θ2 = γΘ, d1 = d, d2 = γd and κ = (1 + γ) /γ. Then, thedynamics of the considered subsystem read

Θω1 = M1 + (Fen − F )R− dω1

γΘω2 = M2 + (F − Fext,2)R− γdω2

F = cR (ω1 − ω2)

Fen = cen (ven − ω1R) ,

(4.7)

with the entry strip force Fen = σenBShnomen and the spring coefficient cen = ESBSh

nomen /Len.

Remark 4.3 Notice that the general case (with Θ2/Θ1 = γ, d2/d1 = η 6= γ) can be tracedback to the dynamics of the “symmetric configuration” given above by setting Θ1 = Θ,Θ2 = γΘ, d1 = γ

ηd, d2 = γd, and introducing M1 = M

′

1 + (γ/η − 1) dω1 with the new

input M′

1.

Now, apply the input transformation[

M1

M2

]

= R

[

0Fext,2

]

+

[

α Rγα −R

] [

uw

]

, (4.8)

see also [FSK03], [FGSK03]. The reason for this particular choice will become apparentin a moment. For discarding the dynamics related to the pulley/pulley-oscillations, takethe limit c → ∞, i.e., limε→0 εF = R (ω1 − ω2), ε = 1/c, rendered from the singularperturbation point of view. Hence, we have ω1 = ω2 = ω, and, thus, the reduced-ordermodel representing the motion of (4.7) confined to the slow manifold reads

ω =1

(1 + γ) Θ((1 + γ)αu+RFen − (1 + γ) dω)

Fen = cen (ven − ωR)

(4.9)

and

F =1

κFen + w. (4.10)

Especially, the choice α = 1/ (1 + γ) for the input transformation (4.8) yields a particularappealing representation of the reduced order dynamics,

Θredω = u+RFen − dredω

Fen = cen (ven − ωR) ,(4.11)

with Θred = Θ1 + Θ2 = (1 + γ) Θ and dred = d1 + d2 = (1 + γ) d.

4. Rolling Mill Control. . . 4.2. Rolling Mill Control Design 93

4.2 Rolling Mill Control Design

In order to summarize and combine the results obtained from the previous model reductionprocedure, the final setting of the entire reduced-order model of the rolling mill is stated inthe following. With (4.4), (4.5), (3.96), (3.97) and the reduced-order bridle roll dynamicsof the preceding section, the basis for the control design is given as

z = αpcm

(

−z + EoilA1 ln

(

1 +A1

V0

xdp

)

+ Fr (λ) −m0g

)

ωR =1

ΘR

(MR −Mr (λ) − dRωR)

σen =ESLen

(ςen (λ, σen, σex)ωRR− ωebrRebr)

σex =ESLex

(ωxbrRxbr − ςex (λ, σen, σex)ωRR)

ωebr =1

Θred,ebr

(

uebr + Rebrσen − dred,ebrωebr)

ωxbr =1

Θred,xbr

(

uxbr − Rxbrσex − dred,xbrωxbr)

0 = Ω(

λ, z, σen, σex, ζ)

(4.12)

with the abbreviations Rebr = Bhnomen Rebr, Rxbr = Bhnomex Rxbr. The control inputs u of(4.12) are

uT =[

xdp MR uebr uxbr]

. (4.13)

Let us consider the decomposition w = fw(λ, ξ, w, u),˙ξ = f ξ(λ, ξ, w, u), 0 = Ω(λ, ξ, ζ)

of (4.12) with wT = [ωR, ωebr, ωxbr], and ξ as defined in (4.5). Then, the correspondingexplicit representation suitable for the simulation purpose, see also Remark 3.20, reads

w = fw(λ, ξ, w, u)

˙ξ = f ξ(λ, ξ, w, u)

λ = −(∂λΩ)−1(

(∂ξΩ)f ξ − P Ω)

,

with P > 0 introduced to asymptotically stabilize the system’s motion with respect tothe manifold defined by Ω.

Example 4.1 To conclude on the validity of the model reduction procedure leading to(4.12), the root loci of (4.12) are illustrated in Figure 3.27 (black/magenta) in comparisonwith those of the detailed mill dynamics, cf. Example 3.9.

The control concept for the rolling mill is based on the flatness property of the reduced-order dynamics (4.12), which is stated in the following proposition.


Proposition 4.1 The dynamics (4.12) are exact input/state linearizable via static statefeedback, and, thus, differentially flat. A (practically suitable) representative of the flatoutput is given e.g. as

yT =

[

ωxbrRxbr

ωebrRebr

− 1 ωR σen σex

]

, (4.14)

entailing the relative degree r = (1, 1, 2, 2).

Proof. The static state feedback equivalence is easily seen e.g. by invoking the implicitfunction theorem on Ω, which (locally) gives λ = ω(z, σen, σex, ζ), see Section 4.1.1. Theverification of the relative degree r = (1, 1, 2, 2) for (4.14) is straight-forward.

Remark 4.4 The function y1 of (4.14) is the so-called elongation coefficient, which isthe crucial control variable in temper rolling, y2 is the angular velocity of the main milldrive, and y3, y4 are given as the strip tensions. The representative (4.14) of the flatoutput is particularly useful as it coincides with the set of control variables of this typcialmill configuration.

Given a flat output, the theory of flatness based control, cf., e.g., [FLMR95], [Rot97],provides a systematic way for the control system synthesis. The application of the non-linear static state feedback entailing the exact linearization, y1 = v1, y2 = v2, y3 = v3,y4 = v4, with the new input v, yields a linear and time-invariant dynamics of the trackingerror e = y−yd. Here, yd denotes the desired trajectory of the flat output. These trackingerror dynamics are easily shaped by means of linear techniques.

Example 4.2 Givem the parameters of Example 3.8, Figure 4.1 finally shows simulationresults of the flatness based tracking control. The quantization of the position signal ofxp is chosen as 2µm, and the amplitude of the noise added to the strip tension signals is0.4N/mm2.


0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

elon

g.co

eff.y

1(%

)

-10

-5

0

5

10

15

20

xp−xs p

(µm

)

xpxdp

45

50

55

60

65

70

75

ωR

(rad

/s)

-200

20406080

100120140

MR−M

s R(k

Nm

)

60

65

70

75

80

85

0 0.5 1 1.5 2 2.5 3 3.5 4

σen,σex

(N/m

m2)

t (s)

σen

σex

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3 3.5 4

Mebr,M

xbr

(kN

m)

t (s)

Mebr

Mxbr

Figure 4.1: Simulation results of the flatness-based rolling mill control, applied to thedetailed dynamics of Section 3.6. The figures on the left side represent the trajectories y,yd of the flat output, and the figures on the right depict the associated control inputs. Fora better view, selected quantities are given as deviations from the selected steady state ofthe mill (indicated by the superscript s).

“Viele Systeme sind flach.”

J. Rudolph

chapter

FIVE

Non-linear Vehicle Dynamics Control – A Flatness-based

Approach

This chapter proposes a novel approach for the non-linear vehicle dynamics controlwhich is essentially based on the observation that the dynamics of the planar holo-

nomic bicycle model depicted in Figure 5.1 are differentially flat. The contact betweenthe tires and the road is modelled in terms of contact forces, which implies that the tiresare enabled to slip and slide on the road. The steering angle and the longitudinal tireforces are regarded as control inputs.

PSfrag replacements

Flv

Fsv (v, β, r, δ)

Flh

Fsh (v, β, r)

δ

vβ

ψ

r,Md

αh

αv

Ξ

x

y

C

X

Y

Figure 5.1: The planar (holonomic) bicycle model.

While the flatness property of the non-holonomic kinematic vehicle, which is basedon the arrangement of ideal rolling contact of the wheels, is well-known in the literatureand numerously exploited for tracking applications of vehicles (with trailers), cf. e.g.[FLMR95], the differential flatness of the (holonomic) bicycle model of Figure 5.1 has notbeen reported yet.

96

5. Non-linear Vehicle Dynamics Control 5.1. The Bicycle Model 97

The bicycle model as introduced in [RS40] emerges from the four-wheel vehicle bygluing together the front and the rear wheels to a single (mass-less) front and rear wheel,respectively, located at the longitudinal axis of the car. This planar model, known asa well-established basis for the design of vehicle dynamics control systems, see, e.g.,[ABO99], [Bun98], [Rit04], is capable of rendering the longitudinal, lateral and yaw dy-namics of the vehicle. The pitch and roll dynamics of a vehicle are clearly not involvedin the scope of this model.

This chapter is organized as follows: After a brief revisit of the bicycle dynamics inSection 5.1, the discussion is concerned with the system analysis in paragraph 5.2 yieldingthe flatness property, and, in particular, a physically relevant representative for the flatoutput of the bicycle model as the main result. This representative of the flat output doesnot depend on the particular choice of the vehicle’s actuation, i.e., it holds for the rear-,front- and all-wheel driven car equivalently. Up to a certain family of functions whichhave to be excluded, the system analysis does not refer to particular representatives forthe functions describing the lateral tire forces.

The task of (real-time) trajectory planning amounts to mapping the inputs suppliedby the driver, i.e., the current position of the throttle/brake pedal and the angle of thesteering wheel, to suitable trajectories for the flat output. The physical meaning of theflat output is regarded to facilitate this objective significantly. Informations gathered byscanning the environment, e.g., regarding the conditions of the road, or the position of adetected obstacle, might also be incorporated for the real-time shaping of the trajectories.The on-line trajectory shaping task (to also give a pleasant feeling for the driver) is anopen problem and not addressed in this thesis. Finally, to illustrate the proposed controlapproach, simulation results given in Section 5.5 conclude this chapter.

5.1 The Bicycle Model

To start with, let us first revisit the modelling assumptions and the dynamics of thebicycle model depicted in Figure 5.1. A detailed discussion of these issues can be founde.g. in [RS40] or [Bun98]. The global position (X,Y ) of the center of mass C and theorientation ψ of the longitudinal axis represent the degrees of freedom of the bicycle. Theparameters are given as follows: m denotes the vehicle mass, J is the moment of inertiawith respect ot the yaw axis fixed at C, and lv, lh denote the distances between C and thefront and the rear wheel, respectively. The steering angle δ as well as the longitudinaltire forces Flv and Flh, which are due to the motor (or the braking) torque, are regardedas control inputs. The torque Md denotes a disturbance acting w.r.t. the yaw axis. Theinputs are collected to the vector uT = [δ, Flv, Flh,Md].

Let v represent the (magnitude of the) velocity of C, and let β denote the angle betweenthe longitudinal axis and the velocity vector X∂X + Y ∂Y at C, i.e.,

v =√

X2 + Y 2, β = arctan(

Y /X)

− ψ. (5.1)

5. Non-linear Vehicle Dynamics Control 5.1. The Bicycle Model 98

Thus, we have

X = v cos (β + ψ) , Y = v sin (β + ψ) , ψ = r, (5.2)

with r denoting the yaw rate. Notice that in the scope of vehicle dynamics control (toassist the driver in emergency situations), only the case v > 0 is considered.

In order to describe the tire/road contact via suitable mathematical models for thelateral tire forces Fsv and Fsh, the literature offers a wide variety of modelling assumptions.A very common and well-established assumption, cf. e.g. [BPL89], is that the lateral tireforces, which allow for changes of the vehicle’s orientation, are regarded as functions ofthe respective side-slip angles of the wheels. The side-slip angles αh and αv represent theangles between the velocity vector at the rear/front wheel and the associated tire plane,see Figure 5.1,

αh (v, β, r) = − arctan

(

v sin β − lhr

v cos β

)

,

αv (v, β, r, δ) = δ − arctan

(

v sin β + lvr

v cos β

)

.

(5.3)

However, in the scope of the system analysis to be given in Section 5.2, we do nota-priori select a particular representative for the functions of the lateral tire forces, i.e.,we will regard the functions Fsh (v, β, r), Fsv (v, β, r, δ) as arbitrary smooth functions1.The dynamics of the bicycle model read

˙x = f (x, u) , xT =[

v β r]

(5.4)

with the vector field f given as

f =

1

m(Fsv (v, β, r, δ) sin (β − δ) + Flv cos (β − δ) + Fsh (v, β, r) sin β + Flh cos β)

−r +Fsv (v, β, r, δ) cos (β − δ) − Flv sin (β − δ) + Fsh (v, β, r) cos β − Flh sin β

mv1

J(lv (Fsv (v, β, r, δ) cos δ + Flv sin δ) − lhFsh (v, β, r) +Md)

.

(5.5)The reason for selecting the representation (5.4), (5.5) is twofold: First, this particularchoice for the coordinates x implies that the functions fα do not depend on the vehi-cle’s orientation ψ (and the global position X, Y as well). Thus, these coordinates areoften referred to as vehicle coordinates. Secondly, as the vehicle dynamics control underconsideration is not concerned with position control of the car, but with control objec-tives depending on x only, the subsystem (5.2) has been dropped from the entire bicycledynamics.

1This C∞ assumption is arranged to avoid mathematical subtleties. Thus, it is a sufficient conditionin the scope of the following discussions.

5. Non-linear Vehicle Dynamics Control 5.2. The Flatness Property. . . 99

5.2 The Flatness Property of the Bicycle Dynamics

Let Md = 0 in the scope of the following system analysis. We will commence by in-vestigating the general case of the so-called all-wheel driven vehicle, which means thatthe motor (or braking) torque can be supplied to the front and the rear wheel with agiven transmission ratio. To this end, let us introduce the distribution of the aggregatelongitudinal force Fl to the front and the rear wheel as

Flh = γFl , Flv = (1 − γ)Fl, (5.6)

with γ ∈ [0, 1] referred to as the transmission ratio. This ratio γ is regarded either asa function of the time, γ (t) : R 7→ [0, 1], or as a function of the vehicle’s state x. Thefirst point of view amounts to regarding γ as an input given by the driver, the secondviewpoint, however, is understood as an action of a vehicle control system.

The central result for this type of vehicle is given in the following proposition.

Proposition 5.1 Let the lateral tire forces Fsh (v, β, r), Fsv (v, β, r, δ) be arbitrary (smooth)functions up to the requirements

Fsh (v, β, r) 6= (mlv)2

J (lv + lh)v2 sin β cos β + κh

(

v cos β, v sin β − J

mlvr

)

(5.7)

and

Fsv 6=κv (v, β, r) − (1 − γ) (γ sin δ + (1 − γ) δ)Fl

1 − γ (1 − cos δ), (5.8)

with κh, κv as arbitrary (smooth) functions. Then, the bicycle dynamics (5.4)-(5.6),uT = [δ, Fl], Md = 0, with γ ∈ [0, 1] regarded either as a function of the time or the state x,are differentially flat (for v 6= 0). Moreover, this system is exact input/state linearizablevia static state feedback. An output y = (y1, y2) entailing the exact linearization withrelative degree (1, 2), i.e., a flat output, is given as

y1 = c1 (x) = v cos β (5.9)

and

y2 = c2 (x) = v sin β − J

mlvr. (5.10)

Proof. The exact linearizability is straight-forwardly verified with (5.9), (5.10). Forthe (local) coordinates transformation z = ϕ (x),

z1

z2

z3

=

c1 (x)c2 (x)

Lfc2 (x)

=

v cos βv sin β − Jr/ (mlv)

(lv + lh) (mlv)−1 Fsh (v, β, r) − vr cos β

, (5.11)

to qualify as a diffeomorphism, the requirement

Jv (∂vFsh) sin β + J (∂βFsh) cos β +mlvv (∂rFsh)

m2l2v− v2 cos β

lv + lh6= 0 (5.12)

5. Non-linear Vehicle Dynamics Control 5.2.1. Some Key Observations 100

has to be fulfilled due to the implicit function theorem. The requirement (5.12) amountsto excluding the family (5.7) of functions for the lateral rear tire force Fsh. Remark 5.5provides a comment on the condition (5.7) to show that it does not imply a practicallyrelevant restriction on the choice of Fsh.

The static state feedback entailing the linear and time-invariant dynamics outlined inthe coordinates z, with the new input wT = [w1, w2], is obtained (locally) as the solutionof

(

Lfc1)

(x, u) = w1,(

L2fc

2)

(x, u) = w2 (5.13)

with respect to uT = [δ, Fl]. Following the implicit function theorem, (local) solvability isguaranteed iff the condition

γFsv sin δ − (1 − γ (1 − cos δ)) (∂δFsv + (1 − γ)Fl) 6= 0, (5.14)

as well as (5.12) is met. Thus, additionally to (5.7), we have to impose the requirement(5.8). Again, Remark 5.5 provides a comment on this restriction.

5.2.1 Some Key Observations Associated with the Flatness Property ofthe Bicycle Dynamics

The following remarks provide some vital issues and consequences related to Proposi-tion 5.1.

Remark 5.1 Note that, except for the restrictions (5.7) and (5.8), the property of exactlinearizability for (5.9), (5.10) does not depend on the particular choice of the functionsFsv (v, β, r, δ) and Fsh (v, β, r). Additionally, note that the restrictions (5.7), (5.8) areassociated with the particular choice for c1 (x), c2 (x) as given in (5.9), (5.10).

Remark 5.2 The function v sin β−J/ (mlv) r occurring in (5.10) depicts the y-componentvΞy of the velocity of the point Ξ located on the vehicle axis with the (vehicle) coordinates

(x, y) = (−J/ (mlv) , 0), see Figure 5.1. Thus, the output c2 of (5.10) is attached with aclear physical meaning.

Remark 5.3 The function c1 (x) of (5.9) depicts the x-component of the velocity of pointslocated on the vehicle’s longitudinal axis.

Remark 5.4 The location of the point Ξ as introduced above only depends on the para-meters J , m and lv which are known (rather) accurately in applications.

Remark 5.5 Typically, as sketched in Section 5.1, the lateral tire forces Fsh and Fsv areconsidered as functions of the respective side-slip angles αh, αv, cf. e.g. [BPL89] fora very well-established approach. In view of this, the conditions (5.7) and (5.8) do notimpose practically relevant restrictions. Particularly, to this end, note that the (arbitrary,smooth) function κv of the condition (5.8) on Fsv must not depend on the steering angleδ. Additionally, note that the function κh involved in the restriction (5.7) for Fsh is afunction of the velocity components vΞ

x , vΞy of the point Ξ.

5. Non-linear Vehicle Dynamics Control 5.2.1. Some Key Observations 101

Remark 5.6 Note that the transmission ratio γ, regarded as γ (t) or γ (v, β, r), does notexplicitely appear in y2 = z3 = Lfc2, see (5.11).

The point Ξ is interesting also from another point of view. To this end, let us calculatethe (x, y)-decomposition of the acceleration of a point Q located at the x-axis at thedistance x = ζ from the center of mass C. Let R = SO (2), i.e., the group of rotations inthe plane,

R (ψ) =

[

cosψ sinψ− sinψ cosψ

]

, 0 ≤ ψ < 2π

denote the mapping from the inertial frame (X,Y ) to the vehicle coordinates (x, y). Vari-ables augmented with the subscripts x, y are related to the vehicle coordinate system,whereas the subscripts X, Y indicate the respective inertial frame representation. Thevelocity of Q is given as

[

vQxvQy

]

=

[

v cos βv sin β + ζr

]

,

[

vQXvQY

]

= R−1

[

vQxvQy

]

,

and the acceleration reads[

aQxaQy

]

= Rd

dt

[

vQXvQY

]

=

[

vQxvQy

]

+

[

−vQyvQx

]

ψ.

Thus, we have

aQx =1

m(Flv cos δ + Flh − Fsv (v, β, r, δ) sin δ) − ζr2 (5.15)

for the longitudinal component, and

aQy =1

m

((

1 − ζmlhJ

)

Fsh (v, β, r) +

(

1 + ζmlvJ

)

(Flv sin δ + Fsv (·) cos δ)

)

+ζ

JMd

(5.16)for the lateral acceleration, with (·) = (v, β, r, δ).

Remark 5.7 The lateral acceleration aΞy at the point Ξ, ζ = −J/ (mlv),

aΞy =

1

m

((

1 +lhlv

)

Fsh (v, β, r) − 1

lvMd

)

, (5.17)

does not explicitly depend on the contact forces of the front wheel, i.e, Fsv (v, β, r, δ) andFlv.

Remark 5.8 For ζ = J/ (mlh), the lateral acceleration does not explicitly depend on thelateral force Fsh (v, β, r) of the rear wheel,

aPy =1

m

((

1 +lvlh

)

(Flv sin δ + Fsv (v, β, r, δ) cos δ) +1

lhMd

)

. (5.18)

This point P, with coordinates (x, y) = (J/ (mlh) , 0), occurs in the analysis of Ackermannleading to the “robustly decoupling control law of DLR” [ABO99], [Bun98].

5. Non-linear Vehicle Dynamics Control 5.2.2. Front- and Rear-Wheel Drive 102

5.2.2 The Front- and the Rear-Wheel Driven Bicycle as Special Cases

The flatness property of the rear-wheel driven vehicle (γ = 1) and the front-wheel drivenvehicle (γ = 0) follow as special cases from Proposition 5.1. Again, let us emphasizethat the representative (5.9), (5.10) of the flat output does not depend on the vehicle’sactuation. For the rear-wheel driven car, the restriction (5.14) reads Fsv tan δ−∂δFsv 6= 0,and, thus, the requirement

Fsv (v, β, r, δ) 6= κv (v, β, r) cos−1 δ, (5.19)

is to be imposed, see (5.8). In the case of the front-wheel driven bicycle, the restriction

Fsv (v, β, r, δ) 6= κv (v, β, r) − Flδ (5.20)

has to be obeyed, see (5.8) again.

Chronologically, the investigation of the rear-wheel driven vehicle (γ = 1) regardingpossibly useful structural properties was the author’s first attack to the vehicle dynamicscontrol problem, which finally led to the revelation of the flat output as given in Propo-sition 5.1. Not only from this chronological point of view, but rather because of theobservation that the construction of the flat output (5.9), (5.10) is particularly illustra-tive for the rear-wheel driven case, we will sketch this original approach in the followingdiscussion.

First, note that (with γ ∈ [0, 1]) the PDEs

L[∂Fl,f]c

2 = 0, L[∂δ,f ]c2 = 0, (5.21)

reflecting the requirements on the output c2 (x) to entailing relative degree 2 are integrableby virtue of Frobenius’s theorem, as the distribution

∆ = span [∂Fl, f ] , [∂δ, f ]

is regular (for (5.14)) and involutive.For the rear wheel driven vehicle, γ = 1, the PDEs (5.21) read

v cos β∂vc2 − sin β∂βc

2 = 0 (5.22)

and

(∂δFsv (v, β, r, δ) sin (δ − β) + Fsv (v, β, r, δ) cos (δ − β)) ∂vc2+

+1

v(−∂δFsv (v, β, r, δ) cos (δ − β) + Fsv (v, β, r, δ) sin (δ − β)) ∂βc

2+

+mlvJ

(−∂δFsv (v, β, r, δ) cos δ + Fsv (v, β, r, δ) sin δ) ∂rc2 = 0. (5.23)

A solution to (5.22), (5.23) can be constructed by first observing that (5.22) admits thesolution

c2 (v, β, r) = λ (v sin β, r) = λ (ξ, r)

5. Non-linear Vehicle Dynamics Control 5.3. On Configuration Flatness 103

with an arbitrary (smooth) function λ. This solution is substituted into the PDE (5.23)yielding

A (∂δFsv (v, β, r, δ) cos δ − Fsv (v, β, r, δ) sin δ) ∂ξλ (v sin β, r) +

+ (∂δFsv (v, β, r, δ) cos δ − Fsv (v, β, r, δ) sin δ) ∂rλ (v sin β, r) = 0

and, thus,A∂ξλ (ξ, r) + ∂rλ (ξ, r) = 0, (5.24)

with the abbreviation A = J/ (mlv). The solution to (5.24) is

λ (ξ, r) = ϑ (ξ − Ar) ,

and, therefore, finally, the solution to the problem (5.22), (5.23) reads

c2 (v, β, r) = ϑ

(

v sin β − J

mlvr

)

, (5.25)

with an arbitrary (smooth) function ϑ.

Before proceeding to the discussion of a flatness-based approach to the non-linearvehicle dynamics control based on Proposition 5.1, we will discuss certain issues relatedconfiguration flatness (Section 5.3) and to the non-holonomic vehicle (Section 5.4).

5.3 A Connection to Configuration Flatness

There is an interesting relation of the bicycle’s flatness property to prior work on flatness,namely, on configuration flatness of Lagrangian systems underactuated by one control[RM98]. The terminus “configuration flatness” indicates the dependence of the flat out-put on the Lagrangian system’s configuration only. E.g., the planar rigid body actuatedby two body-fixed forces allows for a configuration-flat output (Huygens center of oscilla-tion), see Example 5.1, with the PVTOL being a well-known representative of this class.However, in contrast to the flatness based PVTOL (position) control, the proposed ve-hicle dynamics control involves static state feedback only, which is a consequence of thecontrol objective involving functions of the “velocity coordinates” instead of configurationcoordinates. To reveal these issues, we will commence by revisiting the geometric the-ory introduced by Muruhan Rathinam and Richard M. Murray [RM98] on configurationflatness of Lagrangian systems with n degrees of freedom and n − 1 controls, with therange of control forces depending on the configuration only, and the Lagrangian havingthe structure of kinetic energy minus potential, L = T − V . Additionally to providingnecessary and sufficient conditions for concluding on this structural property, this theory,relating configuration flatness to Riemannian geometry, gives a constructive method fordetermining all possible configuration flat outputs.

Let us consider a Lagrangian system evolving on an n-dimensional smooth configurationmanifold Q, with Lagrangian L : T Q → R, and the range of control forces depending on


the configuration only, i.e., P ⊂ T ∗Q, with m = dimP as the number of independentcontrols. Referring to local coordinates q = (q1, . . . , qn) on Q, all feasible motions arerequired to meet the following underdetermined system of equations,

aik

(

d

dt

(

∂L

∂qi

)

− ∂L

∂qi

)

= 0, k = 1, . . . , n−m,

with the vector fields aik∂i spanning the annihilator of P , i.e., P⊥ = span aik∂i.The subsequent discussions will exclusively deal with the case m = dimP = n − 1,

i.e., underactuation degree 1, with the Lagrangian given as

L (v) =1

2g (v, v) − V τQ (v) ,

with v as a section of the tangent bundle T Q, i.e., v ∈ Γ (T Q), and g as the (non-degenerate) Riemannian metric according to the kinetic energy. Here, V : Q → R is thepotential energy function, and τQ denotes the projection T Q → Q.

As a prerequisite for stating the proposition of [RM98] on configuration flatness of theconsidered class of Lagrangian dynamics, chose any ξ ∈ Γ (T Q) such that P⊥ = span ξ,and define the following distribution,

D = span ξ,∇Zξ : Z ∈ Γ (T Q) , (5.26)

with ∇ denoting the covariant derivative given by the Levi-Civita connection. The fol-lowing result is formulated in intrinsic geometric terms (with Ty : T Q → R

n−1 denotingthe tangent map associated to y : U ⊂ Q → R

n−1).

Proposition 5.2 [RM98] Let q be a point on Q, and U be an open neighborhood of q,and suppose y : U ⊂ Q → R

n−1 is a submersion. If (y1, . . . , yn−1) is a configuration flatoutput, then

g (kerTy,D) = 0. (5.27)

Conversely, if g (kerTy,D) = 0 and if the regularity condition (stated below) holds at q,then (y1, . . . , yn−1) is a configuration flat output at q. The regularity condition is that theratios of functions in the following set should not all be the same at q,

∇η (g (ξ, Z)) : g (ξ, Z) , ∇η (g (∇Z1Z2, ξ)) : g (∇Z1Z2, ξ) , ∇η (ξ (V )) : ξ (V )

,(5.28)

where Z, Z1, Z2 are arbitrary vector fields around q that are y-related to some vectorfield on R

n−1 and ξ, η are non-vanishing vector fields such that P⊥ = span ξ andkerTy = span η.

The proof can be found in [RM98]. The application of this proposition to check onconfiguration flatness and to compute all possible configuration flat outputs might proceedas follows. Given the control forces, and, hence, P , the distribution D is calculated via


(5.26). In local coordinates, the covariant derivative ∇ZX of the vector field X = Xk∂kalong the integral curves of Z = Zj∂j reads

∇ZX =(

ZjXkΓijk + Zj∂jXi)

∂i,

with the Christoffel symbols Γijk = Γikj given as

Γijk =1

2(∂jglk + ∂kglj − ∂lgjk) g

li, i, j, k = 1, . . . , n,

gikgkj = δij. Hence, we have

D = span ξ,∇∂iξ : i = 1, . . . , n (5.29)

by linearity of ∇.If D = T Q, then the system is clearly not configuration flat: Given any y, one can

always find a vector field Z ∈ D = T Q such that g (kerTy, Z) 6= 0. If dimD < n,then chose a one-dimensional distribution (spanned by a vector field η) that is orthogonalto D, i.e., g (η, Z) = 0, Z ∈ D. This one-dimensional distribution, with its annihilatorspanned by the 1-forms dy1, . . . , dyn−1, induces a foliation of the configuration manifold Q.The functions (y1, . . . , yn−1) qualify as a configuration flat output provided the regularityconditions (5.28) hold.

In local coordinates, the regularity conditions take a particular appealing representa-tion when carrying out the calculations in the coordinates q associated with the foliation,i.e., q = (y1, . . . , yn−1, z). Let ∂k = ∂/∂qk. Then, (5.28) amounts to calculating thefollowing ratios of functions:

∂z (g (ξ, ∂ı)) : g (ξ, ∂ı) , i = 1, . . . , n− 1

∂z(

g(

∇∂k∂ı, ξ

))

: g(

∇∂k∂ı, ξ

)

, i, k = 1, . . . , n− 1

∂z (ξ (V )) : ξ (V ) .

(5.30)

In case of identical equality of all these ratios (in a local neighborhood), the functionsy1, . . . , yn−1 are differentially dependent, thus violating the requirements imposed on aflat output. Below, we will follow this procedure for investigating configuration flatnessfor three examples.

Example 5.1 (The planar rigid body actuated by two body-fixed forces). Consider therigid body (mass m, moment of inertia J) depicted in Figure 5.2 (left), which is movingin the (vertical) plane under the action of (gravity and) two control forces F1, F2 beingbody-fixed regarding the points of application S1,S2 and the lines of action as well. Thelines of action intersect in point A, and a is the length of CA, with C as the center ofmass. The (clearly non-flat, as not controllable) case S1 = S2 = C is excluded. Thesystem is known to be configuration flat, see, e.g., [MRS95], [MMR03], with the PVTOLbeing a well-known representative. The reason for discussing this example, in particularvia application of Proposition 5.2, is to reveal relations to the planar holonomic bicycle


PSfrag replacements

Fsh (v, β, r)

F3

α1

α2

δ

βv

ψ

ψ

xy

S1

S2 C

C

F1

F1

F2

F2

A

A

Ξ (config. flat output)

Ξ (config. flat output)

X

Y

Figure 5.2: (cf. Example 5.1 and Remark 5.12) The planar rigid body actuated by twobody-fixed forces (left), and a link to the bicycle model (right).

dynamics, and the distinctions as well. Clearly, the system evolves on the configurationmanifold Q = SE (2) = R

2 × S1. In coordinates q = (X,Y, ψ), with (X,Y ) localizing C,we have T = m

2(X2 + Y 2) + 1

2Jψ2, and constant metric [gij] = diag (m,m, J). The range

of control forces span the codistriubtion P,

P = span cos (ψ − α1) dX + sin (ψ − α1) dY − a sin (α1) dψ,cos (ψ + α2) dX + sin (ψ + α2) dY + a sin (α2) dψ ,

whose annihilator P⊥ is spanned by

ξ = a sin (ψ) ∂X − a cos (ψ) ∂Y + ∂ψ.

Notice that ξ does not depend on the angles α1, α2, hence, the same will hold for the flatoutput. The distribution D of (5.26), (5.29),

D = span ξ, a cos (ψ) ∂X + a sin (ψ) ∂Y ,

has rank n− 1 = 2. The distribution orthogonal to D is spanned by the vector field

η = − J

masin (ψ) ∂X +

J

macos (ψ) ∂Y + ∂ψ,

which implies the requested foliation of Q by solving for the solutions y1, y2 : Q → R ofthe PDE Lηy = 0,

y1 = X − J

macos (ψ) , y2 = Y − J

masin (ψ) . (5.31)

Notice that up to now the potential V has not been involved. Indeed, V enters via the reg-ularity conditions only, thus, interestingly, configuration flatness is primarily determinedby the metric g (and the way of which the control forces are applied, i.e., P, of course).


To proceed with the check of (5.28), let q = (y1, y2, z) = Φ (q), z = ψ. The push-forwardof ξ by Φ is

ξ = Φ∗ξ =

(

a+J

ma

)

sin (z) ∂y1 −(

a+J

ma

)

cos (z) ∂y2 + ∂z

and the components of the metric tensor, referring to the coordinates q, are

[gij] =[

∂ı(

Φ−1)k∂j(

Φ−1)lgkl

]

=

m 0 −Ja

sin (z)0 m J

acos (z)

−Ja

sin (z) Ja

cos (z) J(

1 + Jma2

)

.

The first set of regularity conditions of (5.30) reads

∂

∂z

(

g

(

ξ,∂

∂y1

))

: g

(

ξ,∂

∂y1

)

= cos (z) : sin (z) ,

∂

∂z

(

g

(

ξ,∂

∂y2

))

: g

(

ξ,∂

∂y2

)

= − sin (z) : cos (z) ,

thus implying the qualification of (5.31), indicated as point Ξ in Figure 5.2 (left), asconfiguration flat output by virtue of Proposition 5.2, regardless of the potential V .

Remark 5.9 (Huygens “center of oscillation”) The distinguished point Ξ, revealed as theflat output of the planar rigid body, is historically known as “center of oscillation” fromplanar pendulum dynamics (Christiaan Huygens, 1629–1695). Think of the planar rigidbody of Figure 5.2 (left) fixed at “pivot” A and left oscillating as a pendulum under theaction of gravity g, then the equation of motion reads

(

J +ma2)

ψ = −mga cos (ψ) .

On the other hand, consider a (mathematical) pendulum, with pivot A, and the mass mlocated at point Ξ, i.e.,

m

(

a+J

ma

)2

ψ = −mg(

a+J

ma

)

cos (ψ) ,

then the dynamics are identical for both systems.

Remark 5.10 (Exact linearization; Example 5.1 cont’d) It is easy to check that the planarrigid body dynamics of Example 5.1, i.e.,

mX = F1 cos (ψ − α1) + F2 cos (ψ + α2)

mY = F1 sin (ψ − α1) + F2 sin (ψ + α2)

Jψ = −aF1 sin (α1) + aF2 sin (α2) .

(5.32)

are not exact input/state-linearizable by static state feedback, but require for dynamic(quasi-static) feedback.


Remark 5.11 (Differential parametrization of the rigid body dynamics (5.32) via the flatoutput (5.31); Example 5.1 cont’d) From

y1 =1

m

(

F1 cos (α1) + F2 cos (α2) +J

aψ2

)

cos (ψ) ,

y2 =1

m

(

F1 cos (α1) + F2 cos (α2) +J

aψ2

)

sin (ψ) ,

we findtanψ = y2/y1. (5.33)

Hence, with the identity cos−2 ψ = (1 + tan2 ψ) and (5.31), we obtain the parametrizationof the center of mass C,

X = y1 +J

ma

y1

√

(y1)2 + (y2)2, Y = y2 +

J

ma

y2

√

(y1)2 + (y2)2. (5.34)

Finally, the control inputs are obtained from

F1 cos (α1) + F2 cos (α2) =my1

cos (ψ)− J

aψ2

F1 sin (α1) − F2 sin (α2) = −Jaψ

as[

F1

F2

]

=

[

sin(α2)sin(α1+α2)

cos(α2)sin(α1+α2)

sin(α1)sin(α1+α2)

− cos(α1)sin(α1+α2)

][

m√

(y1)2 + (y2)2 − Jaψ2

−Jaψ

]

, (5.35)

with ψ and ψ given in terms of y and its derivatives up to order 4, see (5.33).

Now, we will turn towards the bicycle dynamics by revealing certain links to theplanar rigid body example, and the distinctive features as well. In order to draw theselinks even more illustrative, the nomenclature F1, F2, F3 for the tire forces (instead of Flv,Fsv and Flh) has been introduced in Figure 5.2 (right). First, we will state the followingobservation, though evolving from an unrealistic point of view (Fsh ≡ 0).

Remark 5.12 In a very first view, let Fsh ≡ 0 and regard the forces Fi, i = 1, 2, 3, ascontrol inputs. Obviously, F3 ∈ P, with P as the codistribution spanned by the fronttire forces F1, F2. (This is also reflected in (5.6) by using this dependency of the controlforces to distribute the aggregate engine/braking torque as longitudinal tire forces to thefront and the rear tire via the transmission ratio γ, i.e., F3 = γFl and F1 = (1 − γ)Fl,γ ∈ [0, 1]). Hence, with F1, F2 as control inputs, this (simplified) bicycle model isimmediately identified as a special case of the rigid body example of Figure 5.2 (left).

5. Non-linear Vehicle Dynamics Control 5.4. On the Non-holonomic Vehicle 109

Clearly, regarding the lateral tire forces Fsh (v, β, r) and F2 = Fsv (v, β, r, δ) as func-tions of v, β, r (and δ) does not directly fit into the framework of configuration flatness ofLagrangian systems as discussed above, as the force Fsh (v, β, r) is not required to evolvefrom a potential, and F2 does not depend on the configuration only.

The steering angle δ and the aggregate longitudinal tire force Fl (see Remark 5.12)represent the control inputs, u = (δ, Fl). Again, let Fsh ≡ 0. Then, (5.33) and (5.34) holdfor the differential parametrization of ψ and X, Y . To solve for the contol inputs (δ, Fl),consider the representation of the resulting tire force in the so-called vehicle coordinateframe (x, y), i.e., F = Fxdx+ Fydy,

F = (Fl (γ + (1 − γ) cos δ) − Fsv (v, β, r, δ) sin δ) dx+

+ (Fl (1 − γ) sin δ + Fsv (v, β, r, δ) cos δ) dy.

The independence of Fx and Fy requires for

det

[

∂δFx ∂FlFx

∂δFy ∂FlFy

]

= γFsv sin δ − (1 − γ (1 − cos δ)) (∂δFsv + (1 − γ)Fl) 6= 0.

Notice that this condition appears as (5.14).

The check of qualification of (5.31) as the configuration flat output for the (exclusivelyrelevant) case of non-vanishing lateral rear tire force Fsh (v, β, r) is somewhat more tedious(as (5.33) and (5.34) do not hold in this case) and amounts to invoking the implicit functiontheorem to validate the (local) existence of the differential parametrization of the bicycledynamics.

Remark 5.13 The vehicle dynamics control approach to be discussed in Section 5.5 restsupon the qualification of the velocity components of Ξ as the flat output (of the dynamics(5.4), (5.5) obtained by dropping the subsystem (5.2) related to the global position andorientation, which is not involved in the control objective), cf. Proposition 5.1. However,also the flat ouput (5.31) of (5.2), (5.4), (5.5), i.e., the global position of Ξ, might bevery valuable for applications: One might think, e.g., of the control objective of tracking avehicle (equipped with a global-positioning system) on a pre-specified path on a test track.

5.4 A Connection to the Non-holonomic Vehicle

Typically, the lateral tire forces Fsh, Fsv are considered as functions of the side-slip anglesαh, αv, see Remark 5.5. In order to illustrate a connection between the (holonomic)bicycle model introduced in the preceding sections and the non-holonomic vehicle relatedto the assumption of rolling contact without slipping, let us commence by considering thelimit as the “tire stiffness” (to be defined) tends to infinity. To this end, let the smoothfunctions Fsh (αh) : S1 ⊃ V → R and Fsv (αv) : S1 ⊃ V → R denote the lateral tire forceswith the associated Taylor expansions

Fsh (αh) = chαh + Oh (2) , Fsv (αv) = cvαv + Ov (2) (5.36)


evolved at the points αh = αv = 0. Without loss of generality we will assume that thetire stiffnesses ch and cv are related by cv = κch with 0 < κ < ∞. With ε = 1/ch, thesubstitution of (5.36) into the bicycle dynamics (5.2), (5.4)-(5.6) yields

X

Y

ψ

=

v cos (β + ψ)

v sin (β + ψ)

r

(5.37)

and

ε

v

β

r

=

(καv (·, δ) sin (β − δ) + αh (·) sin β + εb1 (·, δ)) /m(καv (·, δ) cos (β − δ) + αh (·) cos β + εb2 (·, δ)) / (mv)

(lvκαv (·, δ) cos δ − lhαh (·) + εb3 (·, δ)) /J

(5.38)

with the abbreviations (·) = (v, β, r) and

b1 (·, δ) = Ov sin (β − δ) + Oh sin β + Fl ((1 − γ) cos (β − δ) + γ cos β)

b2 (·, δ) = Ov cos (β − δ) + Oh cos β − Fl ((1 − γ) sin (β − δ) + γ sin β) −mvr

b3 (·, δ) = lv (Ov cos δ + (1 − γ)Fl sin δ) − lhOh.

The representation (5.37), (5.38) already indicates a decomposition appropriate for thesingular perturbation point of view with ε serving as the perturbation parameter. Theframework of the singular perturbation theory (see, e.g., [Kha96]) emanates from themodel

ξαξ = fαξ (t, ξ, η, ε) , αξ = 1, . . . , nξ (5.39a)

εηαη = gαη (t, ξ, η, ε) , αη = 1, . . . , nη (5.39b)

with the C1–functions fαξ , gαη and the so-called (small) perturbation parameter ε. Forε = 0, the differential equations (5.39b) degenerate to the set of algebraic equations

0 = gαη (t, ξ, η, 0) . (5.40)

Following [Kha96], the model (5.39) is referred to as the singular perturbation standardform iff (5.40) exhibits k ≥ 1 isolated real roots

ηαη = hαη

i (t, ξ) , i = 1, . . . , k

for each (t, ξ) ∈ [0, t1] × D, with D ⊆ Rnξ denoting some open connected set. Then, the

i-th root of (5.40) is associated with a well–defined reduced order model

ξαξ = fαξ (t, ξ, hi (t, ξ) , 0) (5.41)

evolving on a nξ–dimensional manifold.


Clearly, the bicycle model (5.37), (5.38) exhibits the structure (5.39), or, more pre-cisely,

ξαξ = fαξ (ξ, η) ξ = (X,Y, ψ)

εηαη = gαη (η, δ, Fl, ε) η = (v, β, r) ,

however, it is not in standard form as the algebraic equations gαη (η, δ, 0) = 0,

καv (v, β, r, δ) sin (β − δ) + αh (v, β, r) sin β

καv (v, β, r, δ) cos (β − δ) + αh (v, β, r) cos β

lvκαv (v, β, r, δ) cos δ − lhαh (v, β, r)

=

0

0

0

, (5.42)

with the slip angles αv, αh due to (5.3), are functionally dependent, which is seen bycalculating the associated 1–forms dgαη = ∂βη

gαηdηβη yielding dg1 ∧ dg2 ∧ dg3 = 0.In order to reveal the relations between v, β, r and δ imposed by (5.42), note that the

equations

g1 (v, β, r, δ) cos β − g2 (v, β, r, δ) sin β = −καv (v, β, r, δ) sin δ = 0

g1 (v, β, r, δ) cos (β − δ) − g2 (v, β, r, δ) sin (β − δ) = αh (v, β, r) sin δ = 0

imply

αv (v, β, r, δ) = δ − arctan

(

v sin β + lvr

v cos β

)

= 0 (5.43)

and

αh (v, β, r) = arctan

(

v sin β − lhr

v cos β

)

= 0, (5.44)

which are clearly attached with an obvious physical interpretation, namely the non-holonomic constraint of rolling contact without slipping.

With the velocity v at the center of mass serving as a free coordinate, the slip angleβ and the yaw rate r are obtained from (5.43) and (5.44) as

β = arctan

(

lhl

tan δ

)

, r =v

lhsin

(

arctan

(

lhl

tan δ

))

, (5.45)

arranging the abbreviation l = lh+ lv. Thus, by substituting (5.45) into (5.37) the bicyclemodel takes the form

X

Y

ψ

=

v cos

(

arctan

(

lhl

tan δ

)

+ ψ

)

v sin

(

arctan

(

lhl

tan δ

)

+ ψ

)

v

lhsin

(

arctan

(

lhl

tan δ

))

, (5.46)


PSfrag replacements

vlatv = 0

vh

vlath = 0

δδ

vβ

ψ

r = ψ

r = ψ

(Xh, Yh)

x

y

C

M

X

Y

Z

Figure 5.3: The non–holonomic bicycle, with M denoting the instantaneous center ofrotation.

representing the kinematic behavior of the system, with the variables v and δ regardedas inputs. A more convenient representation of the system (5.46) is attained by applyingthe diffeomorphic coordinates transformation

ξαξ = φαξ (ξ) , uαu = φαu (u) ,

namely,

Xh

Yhψ

=

X − lh cosψY − lh sinψ

ψ

,

[

vh

δ

]

=

v cos

(

arctan

(

lhl

tan δ

))

δ

, (5.47)

see also Figure 5.3. The application of the transformation (5.47) finally reveals the familiarrepresentation of the kinematic vehicle,

Xh = vh cosψ, Yh = vh sinψ, ψ =vhl

tan δ (5.48)

with the inputs u = (vh, δ). The following corollary summarizes some results on thestructure of the kinematic vehicle well-known in the literature.

Corollary 5.3 The kinematic vehicle (5.48) is differentially flat with the position y =(Xh, Yh) qualifying as a flat output. However, it is not static state feedback equivalent toa linear time–invariant system.

Proof. The property of differential flatness is easily shown by explicitly calculatingthe system variables (Xh, Yh, ψ, vh, δ) in terms of the flat output y = (Xh, Yh) and its timederivatives, yielding

Xh = y1, Yh = y2, ψ = arctan

(

y2

y1

)


and

vh = ±√

(y1)2 + (y2)2, δ = ± arctan

(

ly1y2 − y1y2

(

(y1)2 + (y2)2)3/2

)

.

The second statement of the corollary concerning the exact input/state linearizability viastatic state feedback is obtained by the observation that the distribution

∆ = span [∂vh, f ] , [∂δ, f ] =

= span

cosψ∂X + sinψ∂Y +tan δ

l∂ψ,

vhl

(

1 + tan2 δ)

∂ψ

related to the existence of an output function c (X,Y, ψ) with relative degree 2 is notinvolutive.

Another connection between the bicycle model of Figure 5.1 admitting side-slip of thewheels and the non-holonomic vehicle is attained by invoking the Lagrange formalism withthe non-holonomic constraints of ideal rolling contact incorporated by means of Lagrangemultipliers.

To this end, let the system’s motion be modelled on the bundle (E , pr1,B) with localcoordinates (t) for the base manifold B ⊆ R and (t, qα),

(qα) = (X,Y, ψ, δ) , (5.49)

for the total configuration manifold E = B × M, M = R2 × S1 × S, S ⊂ S1, and pr1

denoting the natural projection pr1 : E → B. The velocities qα are denoted as vα = qα,synonymously.

Due to the requirements imposed on the wheels not to slip in the lateral direction,i.e., vlath = vlatv = 0, see Figure 5.3, the system’s motion is restricted by means of theconstraints

g (q, q) =

[

−q1 sin q3 + q2 cos q3 − lhq3

−q1 sin (q3 + q4) + q2 cos (q3 + q4) + lv q3 cos q4

]

= 0. (5.50)

However, more geometrical insight into the problem is achieved by considering the con-straints (5.50) via the representation

ω1 = − sin q3dq1 + cos q3dq2 − lhdq3

ω2 = − sin(

q3 + q4)

dq1 + cos(

q3 + q4)

dq2 + lv cos q4dq3

with the 1-forms ω1, ω2 ∈ Γ (T ∗E) implying the restrictions imposed on the sections v ofthe tangent bundle T E by means of the requirement vcωj = 0, j = 1, 2. By virtue ofFrobenius’ Theorem it is easily seen that the regular codistribution ∆∗ = span ω1, ω2 isnon-integrable as

dω1 ∧ ω1 ∧ ω2 = 0

dω2 ∧ ω1 ∧ ω2 = (lh + lv) dq1 ∧ dq2 ∧ dq3 ∧ dq4 6= 0,


i.e, there does not exist a regular submanifold of the configuration manifold E the system’smotion is restricted to.

By introducing the first jet manifold J 1 (E) with local coordinates (t, qα, qα1 ), the kineticenergy of the vehicle reads

T (q1) =1

2m(

(

q11

)2+(

q21

)2)

+1

2J(

q31

)2+

1

2Js(

q41

)2, (5.51)

with Js denoting the moment of inertia of the front wheel, related to the Z–axis. The equa-tions of motion of the constrained system are obtained by invoking the Euler–Lagrangeformalism

d1

(

∂1αL)

− ∂αL = Qα, α = 1, . . . , 4

with the augmented Lagrangian L,

L (q, q, λ) = L0 (q, q) + λigi (q, q) = T (q) + λigi (q, q) , (5.52)

incorporating the non-holonomic constraints (5.50) by means of the Lagrange multipliersλi, i = 1, 2, see, e.g., [CH93]. This results in the differential equations

d1

(

∂1αT)

+ λi1∂1αgi + λid1

(

∂1αgi)

− λi∂αgi = Qα, (5.53)

which have to be met additionally to the restrictions (5.50). The variables Qα denote thegeneralized forces, related to the coordinates qα, i.e.,

Q1 = Flv cos (q3 + q4) + Flh cos q3, Q3 = Flvlv sin q4 +Md

Q2 = Flv sin (q3 + q4) + Flh sin q3, Q4 = Ms

(5.54)

with the steering torque Ms. With (5.50) and (5.51), the equations (5.53) finally give

mq12 − λ1

1 sin q3 − λ21 sin (q3 + q4) − λ1 cos q3q3

1 − λ2 cos (q3 + q4) (q31 + q4

1) = Q1

mq22 + λ1

1 cos q3 + λ21 cos (q3 + q4) − λ1 sin q3q3

1 − λ2 sin (q3 + q4) (q31 + q4

1) = Q2

Jq32 − λ1

1lh + λ21lv cos q4 − λ2lv sin q4q4

1+

+λ1 (q11 cos q3 + q2

1 sin q3) + λ2 (q11 cos (q3 + q4) + q2

1 sin (q3 + q4)) = Q3

Jsq42 + λ2 (q1

1 cos (q3 + q4) + q21 sin (q3 + q4) + lvq

31 sin q4) = Q4.

(5.55)To further reveal the structure of the system (5.50), (5.55) it is illustrative to focus onthe state space representation

qα = vα,

[

H11 H12 (q)0 0

] [

v

λ

]

+

[

W (q, v, λ)g (q, v)

]

=

[

Q0

]

(5.56)

with the matrices

H11 =

[

H11 00 Js

]

, H11 = diag (m,m, J) , (5.57)


H12 (q) =

[

H12 (q)0

]

, H12 (q) =

− sin q3 − sin (q3 + q4)cos q3 cos (q3 + q4)−lh lv cos q4

, (5.58)

and the function W T (q, v, λ) =[

W T , U]

composed as

W =

−λ1 cos q3v3 − λ2 cos (q3 + q4) (v3 + v4)−λ1 sin q3v3 − λ2 sin (q3 + q4) (v3 + v4)λ1 (v1 cos q3 + v2 sin q3) + λ2 (v1 cos (q3 + q4) + v2 sin (q3 + q4) − lvv

4 sin q4)

U = λ2(

v1 cos(

q3 + q4)

+ v2 sin(

q3 + q4)

+ lvv3 sin q4

)

(5.59)

The differential–algebraic system (5.56)–(5.59) can easily be transformed to an explicitsystem by first taking the time derivative of the constraints g (q, v) = 0,

H21 (q) v + C (q, v) = 0,

with

H21 (q) =[

∂1αgi]

=[

H21 0]

, H21 (q) =

[

− sin q3 cos q3 −lh− sin (q3 + q4) cos (q3 + q4) lv cos q4

]

and

C (q, v) = [∂αgivα] =

[

−v1v3 cos q3 − v2v3 sin q3

− (v3 + v4) (v1 cos (q3 + q4) + v2 sin (q3 + q4)) − lvv3v4 sin q4

]

,

and concluding that the DAE system exhibits index 1, as the matrix

H (q) =

[

H11 H12 (q)H21 (q) 0

]

is regular for all q,

detH (q) = Js(

J +ml2h − (J −mlv (2lh + lv)) cos2 q4)

> 0. (5.60)

Thus, the associated explicit representation according to (5.56)–(5.59) reads

qα = vα,

[

v

λ

]

= H−1 (q)

[

−W (q, v, λ) +Q−C (q, v)

]

, (5.61)

where the initial conditions (q0, v0) have to comply with the constraints (5.50) definingthe manifold C ⊂ J1 (E) the system’s motion is restricted to. In order to render thisconstraint manifold attractive for the purpose of numerical integration, the system (5.61)is augmented as

qα = vα,

[

v

λ

]

= H−1 (q)

[

−W (q, v, λ) +Q−C (q, v) − Pg (q, v)

]

(5.62)

5. Non-linear Vehicle Dynamics Control 5.5. A Flatness-based Approach. . . 116

with the positive definite matrix P . The asymptotic stability of (5.62) with respect tothe constraint manifold can be seen with the Lyapunov function

V (q, v) =1

2gTg

yieldingV = gT g = gT (C (q, v) +H21 (q) v) = −gTPg,

by observing that H21 (q) v = −C (q, v) − Pg (q, v).

Remark 5.14 Typically, it is appropriate to consider the vehicle with a kinematic steer-ing device, i.e., with Js = 0, which amounts to taking the variable δ = v4 as an input. Byintroducing the notation v = [v1, v2, v3]

T, u = δ = v4, the associated differential–algebraic

model is obtained from (5.56)–(5.59) as

qα = vα, α = 1, 2, 3

q4 = u

[

H11 H12 (q)0 0

] [

˙v

λ

]

+

[

W (q, v, u, λ)g (q, v)

]

=

[

Q0

]

with Q = [Q1, Q2, Q3]T , see (5.54). With

H (q) =

[

H11 H12 (q)H21 (q) 0

]

, det H (q) = detH (q) /Js > 0,

see also (5.60), the associated explicit representation reads

[

˙v

λ

]

= H−1 (q)

[

−W (q, v, u, λ) + Q−C (q, v, u) − Pg (q, v)

]

, (5.63)

again augmented with −Pg (q, v), P > 0, in order to guarantee the asymptotic convergenceof the motion to the constraint manifold.

5.5 Flatness-based Vehicle Dynamics Control

Returning from the excursion on the non-holonomic vehicle, let us now focus on thenon-linear vehicle dynamics control, essentially based on Proposition 5.1, invoking theflatness-based methods.

Given the trajectory of the flat output y, the associated trajectories of the systemvariables xT = [v, β, r] and the control inputs uT = [δ, Fl] are obtained as the solution of

F = span c1 (x) = y1, c2 (x) = y2, Lfc2 (x) = y2,

Lfc1 (x, u, γ) = y1, L2fc

2 (x, u, γ) = y2 .(5.64)


PSfrag replacements

z = ϕ (x)

trackingcontroller

bicycledynamics

nonlinearstate feedback

(5.11)

(5.4)-(5.6)(5.13)(5.66),(5.67)

y

γ (t, v, β, r)

driver’s demandon the transmission ratio

calc. trans-mission ratio

γ

uw

x

x

yd, yd, .

zreal-timetrajectoryplanning

drivers’s demandon the long. and lat. dynamics(throttle/brake ped. pos., steering angle δ)

Figure 5.4: Scheme of a flatness based vehicle dynamics control approach.

By obeying the restrictions regarding the lateral tire forces Fsv, Fsh given in Proposi-tion 5.1, the implicit function theorem guarantees the local solvability of (5.64).

Figure 5.4 depicts the scheme of a flatness-based vehicle dynamics control approach.The controller structure for the asymptotic stabilization of the tracking error dynamics(exact linearization of the bicycle dynamics, asymptotic stabilization of the resulting LTIerror dynamics) follows the standard scheme of the flatness-based methods, cf. e.g.,[FLMR95], [Rot97], [Rud03].

Remark 5.15 Note that, as a vital benefit, the incorporation of the driver’s demand γ (t)on the transmission ratio or the action γ (v, β, r) of a traction/braking control system,respectively, qualifies to fit very seamlessly into this control approach, see Proposition 5.1and, in particular, Remark 5.6. This means that the proposed control approach onlyrequires a transmission control system of the type γ (t, v, β, r), see Figure 5.4.

Remark 5.16 Clearly, for an application, the shaping of the trajectories of the flat outputis to be done in real-time, based on the inputs supplied by the driver, see Figure 5.4. Theposition of the throttle/brake pedal is thought of as to reflect the driver’s demands onthe longitudinal dynamics. Accordingly, the angle of the steering wheel is regarded as thedriver’s demand on the lateral dynamics. This on-line trajectory planning task, which alsohas to give a pleasent feeling for the driver, is still an open problem.

The non-linear state feedback ui = χi (x, w, γ), i = 1, 2, derived from (5.13) entailsthe exact linearization of the bicycle dynamics, z1 = w1, z2 = z3, z3 = w2, see also (5.11).This implies the linearity and time-invariance of the tracking error dynamics, with thetracking error e = y − yd. Here, the subscript d is used to indicate the desired values.


The asymptotic stabilization of the tracking error dynamics

e1 = y1 − y1d = z1 − y1

d = w1 − y1d

e2 = y2 − y2d = z3 − y2

d = w2 − y2d

(5.65)

can be obtained by means of the control law

w1 = y1d − µe1 − µξ1 = y1

d − µ(

z1 − y1d

)

− µξ1

w2 = y2d − ν1e

2 − ν2e2 − νξ2 = y2

d − ν1

(

z2 − y2d

)

− ν2

(

z3 − y2d

)

− νξ2(5.66)

with the integral partξ1 = e1, ξ2 = e2 (5.67)

and µ > 0, µ > 0, and ν1, ν2, ν such that the characteristic polynomials are Hurwitz.Thus, the error dynamics read e1 = −µe1 − µξ1, e2 = −ν1e

2 − ν2e2 − νξ2 with (5.67).

In order to illustrate the proposed vehicle dynamics control approach, a close-to-realitymodelled sports car which is provided by the multi-body simulation program msc.adams

[MSC], is used as a testrig. This demo vehicle comprises fully-detailed models of thesuspension, the powertrain and the steering system, while the bodywork is consideredas a rigid body, spring-mounted on the chassis. The total number of degrees of freedomamounts to 96.

The parameters of the corresponding bicycle model, which have been extracted fromthe msc.adams vehicle, are given by m = 1529 kg, J = 1344 kgm2, lh = 1.08 m andlv = 1.481 m. The point Ξ, which is attached with particular meaning regarding theflatness property, see Remark 5.2, is located at (x, y) = (−0.594 m, 0).

The lateral forces Fsh, Fsv acting on the tires are modelled following the widely-usedmodel of Pacejka [BPL89],

Fsk (αk) = 2Ds sin (Cs arctan (Bsαk − Es (Bsαk − arctan (Bsαk)))) , (5.68)

k ∈ v, h, often referred to as “Pacejka’s magic formula” in the literature. Thus, thelateral forces are given as functions of the respective side-slip angles, see also Remark 5.5.For instance, the parameters of the rear tire model are chosen to be Bs = 13 rad−1,Cs = 1.65, Ds = 4789 N and Es = 0.68, see Figure 5.5.

The real-time trajectory planning as sketched in Remark 5.16 is an open problem.For the current sake of illustration, the desired trajectory yd (t) : [0, T ] → R

2 of the flatoutput is chosen as

y1d (t) = v0 +

3t2T − 2t3

T 3(vT − v0) (5.69)

for the longitudinal component of the velocity at Ξ, and

y2d (t) =

p (t− t1, a1, t2 − t1) , t ∈ [t1, t2)p (t− t2, a2, t3 − t2) , t ∈ [t2, t3)

0, else(5.70)

5. Non-linear Vehicle Dynamics Control 5.6. Conclusions. . . 119

−10

−8

−6

−4

−2

0

2

4

6

8

10

−20 −15 −10 −5 0 5 10 15 20

late

raltire

forc

eFs

[kN

]

side-slip angle α []

Figure 5.5: The lateral tire force as a function of the side-slip angle, see also Remark 5.5,following the model (5.68) of Pacejka [BPL89].

for the lateral component, with

p (t, a, τ) = −at3 (τ − t)3

τ 6.

Here, v0 and vT denote the x-component of the velocity at the time t = 0 and t = T ,respectively. Let t1 = 1.5 s, t2 = 2.5 s, t3 = 3.5 s, T = 5 s, v0 = 27.7 m/s, vT = 33.3 m/s,a1 = 50 m/s, a2 = −57 m/s, then this trajectory yd (t) of (5.69), (5.70) implies a singlelane change maneuver, associated with an acceleration of the vehicle, see Figure 5.6.Figure 5.7 finally depicts the simulation results of the proposed flatness-based control.The coefficients of the tracking controller (5.66) are chosen as µ = µ = 10, and ν1 = 1200,ν2 = 60, ν = 8000.

The following remarks provide comments on the simulation results given in Figure 5.7.

Remark 5.17 The difference between the actual rear tire longitudinal force Flh and thepredicted value (dashed line) is due to the fact that the msc.adams model involves aero-dynamic forces and the road resistance, which have to be overcome to follow the desiredtrajectory.

Remark 5.18 The deviation of the yaw rate and the steering input results from unequalloading of the inner and the outer track during cornering. This unequal loading clearlyaffects the lateral and longitudinal tire forces via the normal force.

5.6 Conclusions, Perspectives and Future Research

Individually controlled braking of all four wheels (ESP) to maintain the vehicle’s stabilityand steering response has proven as very valuable to increase safety. Comparably, making


0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100 120 140 160

Y[m

]

X [m]

Figure 5.6: Single lane change maneuver due to (5.69), (5.70): Trajectory of the vehicle’scenter of gravity C.

available the steering system to the vehicle dynamics control offers additional potential tosupport the driver in emergency situations. To this end, the differential flatness propertyof the bicycle model may offer new perspectives to cope with the vehicle dynamics controlproblem involving the longitudinal forces of the tires and the steering angle as control in-puts. The flat output revealed in this contribution could be identified as the longitudinaland the lateral component of the velocity of a certain point located on the vehicle’s longi-tudinal axis. The location of this distinguished point is determined in terms of the mass,the moment of inertia and the distance between the front wheel and the center of mass,which can be regarded as well-known parameters in practical applications. Additionally,this flat output does not depend on the particular actuation of the vehicle, i.e., it holdsfor the rear-, front- and all-wheel driven car equivalently.

Besides the objective of real-time trajectory shaping for the flat output, our futureresearch will be concerned with observer design and parameter identification of the tirecharacteristics. While the first issue deals with driver’s experience during demandingmaneuvers, the second point focuses on the fact that the automotive industry, in general,is strongly reluctant to implement side-slip angle sensors. Finally, vehicle dynamics controlis required to work seamlessly under various road conditions.


27.0

28.0

29.0

30.0

31.0

32.0

33.0

34.0

y1

[m/s

]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

β[

]

-0.8-0.6-0.4-0.20.00.20.40.60.81.0

y2

[m/s

]

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

r[rad

/s]

-2.0-1.5-1.0-0.50.00.51.01.52.0

αh

[]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

δ[

]

-8.0-6.0-4.0-2.00.02.04.06.08.0

0 1 2 3 4 5

aC y

[m/s

2]

t [s]

-0.50.00.51.01.52.02.53.03.54.0

0 1 2 3 4 5

Flh

[kN

]

t [s]

Figure 5.7: Simulation results of the proposed flatness based vehicle dynamics controlapproach, using a close-to-reality modelled sports car provided by the multi-body simu-lation program msc.adams [MSC]. The desired trajectories are indicated with dashedlines. See Remarks 5.17 and 5.18 for notes on these results.

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Curriculum Vitae

Personal data

name: Dipl.Ing. Stefan Fuchshumer

date/place of birth: January 2, 1972, Linz

citizenship: Austria

marital status: married with Mag. Susanne Fuchshumer

address: Graben 17, A–4722 Peuerbach

email: [email protected]

Education

09/1978–06/1982 elementary school, Linz

09/1982–06/1990 grammar school Linz–Auhof, passed with distinction

10/1990–01/1997 diploma studies in Mechatronics at the Johannes Kepler Univer-sity (JKU) of Linz, passed with distinction.

diploma thesis: Bestimmung der magnetischen Eigenschaften vonElektroblech mit dem Tafeljoch; prepared at the Department ofAutomatic Control and Control Systems Technology in coopera-tion with voestalpine Stahl Linz.

since 01/1999 doctoral studies on Technical Sciences at the Johannes Kepler Uni-versity of Linz.

Professional experience

02/1997–03/1997 contract for services at voestalpine Stahl Linz

04/1997–11/1997 military service

02/1998–10/1998 project engineer at the Department of Automatic Control andControl Systems Technology, JKU Linz. Kplus pilot project onMIMO temperature control of extruders, in coop. with Bernecker& Rainer Industrie–Elektronik, Eggelsberg, Austria

01/1999–12/2005 research assistant at the Christian Doppler Laboratory for Auto-matic Control of Mechatronic Systems in Steel Industries installedat the Institute of Automatic Control and Control Systems Tech-nology, JKU Linz, in cooperation with Voest-Alpine Industriean-lagenbau GmbH Linz (VAI)

since 1999 lecturer at the Institute of Automatic Control and Control Sys-tems Technology, JKU Linz.

since 2002 co-supervisor of diploma theses

Publications

Chapters of books:

• S. Fuchshumer, K. Schlacher, A. Kugi: Mathematical Modelling and Nonlinear Con-trol of a Temper Rolling Mill. Selected Topics in Structronics and Mechatronic Sys-tems. Eds: A.K. Belyaev and A. Guran. Series on Stability, Vibration and Controlof Systems, vol. 3, World Scientific, 2003, pp. 175–219.

• K. Schlacher, S. Fuchshumer, J. Holl: Some Applications of Differential Geometryin Control. Advanced Dynamics and Control of Structures and Machines. Eds: H.Irschik and K. Schlacher. CISM Courses and Lectures No. 444, Springer, 2004,pp. 249–260.

Articles:

• S. Fuchshumer, G. Grabmair, K. Schlacher, G. Keintzel: Automatisierungstechnikin der Mechatronik: Zwei Beispiele aus der Stahlindustrie. e&i Elektrotechnik undInformationstechnik, Heft 5, 2003, pp. 164–171.

• S. Fuchshumer, K. Schlacher, G. Grabmair, K. Straka: Flachheitsbasierte Fol-geregelung des Labormodells Ball on the Wheel. e&i Elektrotechnik und Informations-technik, Heft 9, 2004, pp. 301–306.

• S. Fuchshumer, K. Schlacher, T. Rittenschober: Ein Beitrag zur nichtlinearen Fahr-dynamikregelung: Die differentielle Flachheit des Einspurmodells. e&i Elektrotech-nik und Informationstechnik, Heft 9, 2005, pp. 319–324.

• K. Schlacher, J. Holl, S. Fuchshumer: Zur Modellierung und aktiven Schwingungs-unterdruckung in Stahlwalzanlagen. at – Automatisierungstechnik, Oldenbourg,Heft 3, 2005, pp. 114–124.

Conference papers:

• S. Fuchshumer, K. Schlacher, A. Kugi: Mathematical Modelling of a Temper RollingMill. In: Proc. Metal Forming 2000, Sept. 3–7, 2000, Krakow, Poland.

• S. Fuchshumer, K. Schlacher, A. Kugi: A Nonlinear Control Concept for a TemperRolling Mill. In: Proc. XXVIII Summer school on Actual Problems in Mechanics(APM 2000), June 1–10, 2000, St.Petersburg, Russia.

• S. Fuchshumer, G. Grabmair: A Nonlinear Control Concept for Rolling Mills.In: Proc. Annual Meeting of the “Gesellschaft fr angewandte Mathematik undMechanik” (GAMM), Feb. 12-15, 2001, Zrich, Switzerland.

• S. Fuchshumer, K. Schlacher, A. Kugi: Elongation and Tension Control for a Tem-per Rolling Mill. In. Proc. APM 2001, June 21–30, 2001, St.Petersburg, Russia.

• S. Fuchshumer, K. Schlacher, M. Polzer, G. Grabmair: Flatness Based Control ofthe System Ball on the Wheel. In: Proc. 6th IFAC Symposium on Nonlinear ControlSystems (NOLCOS 2004), Sept. 1–3, 2004, Stuttgart, Germany.

• S. Fuchshumer, K. Schlacher, G. Keintzel: A Novel Non-circular Arc Rollgap Model,Designed from the Control Point of View. In: Proc. 11th IFAC Symposium onAutomation in Mining, Mineral and Metal processing (MMM 2004), Sept. 8–10,2004, Nancy, France.

• K. Schlacher, S. Fuchshumer, G. Grabmair, J. Holl, G. Keintzel: Active VibrationRejection in Steel Rolling Mills. In: Proc. IFAC World Congress, July 4–8, 2005,Prague, Czech Republic.

• S. Fuchshumer, K. Schlacher, G. Keintzel: A Non-Circular Arc Roll Gap Model forControl Applications in Steel Rolling Mills. In: Proc. IEEE Conference on ControlApplications (CCA 2005), August 28–31, 2005, Toronto, Canada. (invited paper)

• S. Fuchshumer, K. Schlacher, T. Rittenschober: Nonlinear Vehicle Dynamics Con-trol – A Flatness based Approach. In: Proc. 44th IEEE Conference on Decision andControl, and European Control Conference ECC 2005, Dec. 12–15, 2005, Seville,Spain.

Eidesstattliche Erklarung

Ich erklare an Eides statt, dass ich die vorliegende Dissertation selbstandig und ohnefremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutztsowie die wortlich oder sinngemaß entnommenen Stellen als solche kenntlich gemachthabe.

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