ILC Report · Title: ILC_Report.dvi Created Date: Thu Jun 30 11:25:46 2005

Iterative Learning Control

on an Inkjet Printhead

Niels Johannes Maria Bosch

Report No. DCT 2005.80

Research Report

Supervisors: M.B. Groot Wassink TU DelftO.H. Bosgra TU/e, TU DelftS.H. Koekebakker Oce

Eindhoven University of Technology

Department of Mechanical Engineering

Control Systems Technology Group

Oce-Technologies B.V.

R&D Research Report

Venlo, June 2005

Abstract

Inkjet is a technology with an increasing field of interest, making it an important subject forthe industry as well as for academic research. In this study, a piezoelectric inkjet printhead isconsidered, which is typically actuated by fixed actuation pulses. This form of passive controldoes not take into account several operational issues, like residual vibrations and cross-talk,which can be overcome by a switch from passive to active control. For this purpose, theprinthead dynamics are identified and modelled and an Iterative Learning Controller (ILC)is designed in the Lifted Setting. It is illustrated by means of simulation and experimentalresults that ILC can effectively deal with the operational issues.

Preface

This report is written for the Master’s Thesis assignment ‘Iterative Learning Control on anInkjet Printhead’ of the Eindhoven University of Technology, department of Mechanical Engi-neering, Control Systems Technology Group. The research is conducted at Oce-TechnologiesB.V. in Venlo.

First of all, I would like to thank my supervisors Sjirk Koekebakker, professor Okko Bosgraand, especially, Matthijs Groot Wassink for their support and advise during my research.Furthermore, I would like to thank all my colleagues at Oce for their help and the usefuldiscussions. Special thanks goes to Jan Simons and Marc van den Berg, who have been agreat help during the experiments.

De schrijver werd door Oce-Technologies B.V. in staat gesteld onderzoek te verrichten, datmede aan dit rapport ten grondslag ligt. Oce-Technologies B.V. aanvaardt geen verantwoorde-lijkheid voor de juistheid van de in dit rapport vermelde gegevens, beschouwingen en conclusies,die geheel voor rekening van de schrijver komen.

Nomenclature

General

A0 Nominal cross-sectional area of the ink channel [m2]Aeff Effective piezo surface [m2]An Cross-sectional area of the nozzle end [m2]∆A Change in cross-sectional area [m2]B Function, comprising viscosity effects and form of the cross-

section[-]

C Electrical capacity [F]c0,c Adiabatic and effective speed of sound in ink, respectively [m/s]c1,c2 Constants [Pa]~D Electric displacement vector [C/m2]Dr Remanent polarization [C/m2]Ds Saturation polarization [C/m2]d Matrix with piezoelectric charge constants [C/N]deff Effective piezoelectric constant [C/N]~E Electrical field vector [V/m]F Piezo force [N]f Frequency [Hz]h Height of the meniscus with respect to the nozzle end [m]i Electrical current [A]keff Effective piezo stiffness [N/m]L Length of the ink channel [m]l Length of a channel segment [m]p Pressure of the ink [Pa]Q Volume flow [m3/s]q Electrical charge [C]R Electrical resistance [Ω]Rn Radius of the nozzle (end) [m]r Cylindrical coordinate in radial direction [-]~S Strain vector [-]sE Matrix with mechanical compliances [m2/N]T Kinetic energy of the ink in the control volume [J]~T Stress vector [Pa]Td Kinetic energy of the droplet [J]

vi

t Time [s]U Input signal to the piezo actuator [V]Ubias Constant bias voltage applied to the piezo [V]Upiezo Piezo sensor signal [V]u Displacement of the piezo unit [m]un Velocity of the ink in the nozzle end [m/s]V Volume of the ink outside the nozzle [m3]Vd Volume of the droplet [m3]vd Speed of the droplet [m/s]v∗d Uncorrected speed of the droplet [m/s]∆vd Correction for the speed of the droplet [m/s]W Womersley number [-]x,y,z Cartesian coordinates in length, width and height direction

of the channel, respectively[-]

Greek Letters

α Matrix with influence parameters for change in cross-sectional area due to an input voltage

[1/V]

β Matrix with influence parameters for change in cross-sectional area due to pressure

[1/Pa]

εT Matrix with permittivity constants [C/Vm]ϑ Enlargement of the free surface [m2]κ Droplet volume factor [-]µ Dynamic viscosity of the ink [Pa.s]ρ0 Average density of the ink [kg/m3]σ Surface tension of the ink [N/m]ω Angular frequency [rad/s]

Systems and Signals

Ac,Bc,Cc,Dc State-space matrices of the continuous-time LTI systemAd,Bd,Cd,Dd State-space matrices of the discrete-time LTI systemC Discrete feedback controllerC Controllability matrix of the ILC systemC(s) Continuous feedback controllerCr Quadratic cost criteriond Disturbancee Tracking errorH Impulse response matrixH Actual impulse response matrixH∗ Non-square impulse response matrix∆H Model uncertainty

vii

Hdroplet Dynamics from the piezo actuator to the dropletHpiezo Dynamics from the piezo actuator to the piezo sensorHvmen Dynamics from the piezo actuator to the meniscus velocityh Impulse response consisting of Markov parametersI Identity matrixL Learning matrixL∗ Non-square learning matrixL(s) Continuous learning filterl Number of system outputsm Number of system inputsN Trial lengthN∗ Length of the actuation windowO Observability matrix of the LTI systemO Observability matrix of the ILC systemP Discrete plantP (s) Continuous plantPS Process sensitivity functionQ(s) Continuous robustness filterS Sensitivity functions Complex frequency∆t Sample timeU ,U∗ Matrices with output singular vectorsui i-th output singular vectoru System inputufb Feedback input signaluk Feedforward input signal for iteration k∆u Update of the system inputV ,V ∗ Matrices with input singular vectorsvi i-th input singular vectorWa,Wq,Wr Weighting matricesw New input signalX,X∗ Stabilizing solution of the DARExi i-th element of stabilizing solution of the DAREx System statey System outputyd Desired output or reference trajectoryz Discrete delay operatorα,β Tuning parameters for designing the learning matrixγ Scalar learning gainλ Closed-loop polesΣ Matrix with singular valuesσi i-th singular value

viii

Subscripts

i, j Element numberk Iteration numberm Segment numberr Spatial index

Operators

j =√−1 Imaginary unit

Z−1 One trial delay operatorλ(•) Eigenvalues| • | Absolute value• Time derivative• Mean~• Vector∫

Integration

Abbreviations

ASIC Application Specific Integrated CircuitCAD Computer Aided DesignCCD Charge-Couple Device cameraDARE Discrete-time Algebraic Riccati EquationDC Direct CurrentDOD Drop-On-DemandDSM Drop-Size ModulationFEM Finite Element MethodFRF Frequency Response FunctionIAE Integrated Absolute ErrorILC Iterative Learning Control(ler)LED Light Emitting DiodeLP Low-Pass filterLQ Linear Quadratic optimal controlLTI Linear Time-Invariant systemMIMO Multi-Input-Multi-OutputNUM/DEN Numerator-Denominator system representationPID Proportional, Integral en Differentiation feedback controllerSISO Single-Input-Single-OutputSVD Singular Value DecompositionZOH Zero-Order-HoldZPETC Zero-Phase-Error-Tracking-ControllerZPK Zero-Pole-Gain system representation

Summary

The inkjet technology is capable of depositing small droplets of various materials in certainpatterns on a substrate. These droplets may be in the order of picoliters and can be placedwith an accuracy of several micrometers. This makes inkjet an important technique for theindustry as well as for academic research.

In this study, a piezoelectric inkjet printhead is considered. This printhead comprises2 × 128 nearly identical ink channels with a high integration density, which can be actuatedby separate piezo actuators. The head works according to the Droplet-On-Demand (DOD)principle. When a droplet of ink is desired, a fixed actuation pulse is sent to one of the piezoactuators (passive control), resulting in a single droplet.

After droplet ejection, the channel acoustics are not immediately at rest and one shouldwait until the residual vibrations are damped out before a new droplet can be fired. Anotherproblem concerns the interaction between the different channels, which is called cross-talk. Asa result of cross-talk, the droplet properties may vary, what is an undesired effect. Moreover,passive control does not take into account production tolerances and changing dynamics ofthe printhead.

The mentioned operational issues can be overcome by a switch from passive to activecontrol. Residual vibrations can be actively damped and the effects of cross-talk can bereduced by active decoupling of the channels. Also the effects of changing dynamics andproduction tolerances can be reduced by an active controller. Furthermore, the applicationof active control makes it rather easy to produce droplets of different sizes. This technique iscalled Drop-Size Modulation (DSM).

Taking into account the highly repetitive character of the jet process, the technique Itera-tive Learning Control (ILC) has been applied. ILC requires a model of the system dynamics.Therefore, an experimental model of the printhead dynamics has been created. As sensorfunctionality, the piezo actuator is simultaneously applied as measuring device. It is demon-strated on experimental level that a Single-Input-Single-Output (SISO) ILC, which makes useof piezo sensor measurements, is capable of damping the residual acoustic vibrations, enablinghigher jet frequencies.

A second experimental model has been created to obtain knowledge about the dynamicsbetween the piezo sensor and the velocity of the ink-air interface, called meniscus. By makinguse of this second model, it is possible to construct a ‘smart reference’ signal for the ILC whichis based on the piezo sensor signal. It is shown by means of experiments that this methodresults in a comparable performance as ILC based on a ‘simple reference’ signal. Possibleexplanations for the fact that the smart reference does not lead to a better performance arethe rather strong nonlinearities of the meniscus dynamics and that the two-sided couplingbetween piezo and meniscus is not taken into account. It is demonstrated on simulation levelwhat meniscus movement is required to produce a droplet with arbitrary speed and volume

x

and DSM simulations are performed.To deal with the problems concerning cross-talk, a control concept is introduced which

makes use of Multi-Input-Multi-Output (MIMO) ILCs. Such a MIMO controller is capableof applying active decoupling of the ink channels, as is illustrated on simulation level. Toachieve this decoupling, also the neighboring, non-jetting, channels are actively controlled inorder to stay at rest. Two different implementations of ILC are considered, MIMO ILC andMultiloop SISO ILC, which both successfully reduce the effects of cross-talk.

Samenvatting

De inkjet technologie is in staat om kleine druppels van verschillende materialen in bepaaldepatronen aan de brengen op een substraat. Deze druppels kunnen in de orde van enkelepicoliters zijn en worden geplaatst met een nauwkeurigheid van enkele micrometers. Dit maaktinkjet een belangrijke technologie voor zowel de industrie als voor academische onderzoek.

In deze studie wordt een piezoelektrische inkjet printkop beschouwd. Deze printkopbestaat uit 2 × 128 vrijwel identieke kanalen met een hoge integratiedichtheid, die kunnenworden aangestuurd middels aparte piezo actuators. De kop werkt volgens het Droplet-On-Demand (DOD) principe. Wanneer een inktdruppel is gewenst, wordt er een vaste actu-atiepuls gestuurd naar een van de piezo actuators (passieve aansturing), dat resulteert in eenenkele druppel.

Na het vuren van een druppel is de kanaalakoestiek niet meteen in rust en men dient tewachten totdat de resttrillingen zijn uitgedempt voordat er een nieuwe druppel kan wordengejet. Een ander probleem betreft de interactie tussen de verschillende kanalen, overspraakgenoemd. Als een gevolg van overspraak kunnen de druppeleigenschappen varieren. Dit iseen ongewenst effect. Bovendien houdt de passieve aansturing geen rekening met produc-tietoleranties en veranderende dynamica van de printkop.

De genoemde nadelen kunnen worden overwonnen door een omschakeling van passievenaar actieve aansturing. Resttrillingen kunnen actief worden gedempt en de effecten vanoverspraak kunnen worden gereduceerd door middel van actieve ontkoppeling van de kanalen.Er kan ook gecompenseerd worden voor veranderende dynamica en productietoleranties mid-dels een actieve regelaar. Verder maakt de implementatie van een actieve regelaar het relatiefeenvoudig om druppels van verschillend volume te produceren. Deze techniek wordt Drop-Size-Modulation (DSM) genoemd.

Rekening houdende met het sterk repeterende karakter van het jet process is de techniekIterative Learning Control (ILC) toegepast. ILC vereist een model van de systeem dyna-mica. Daarom is er een experimenteel model van de printkop dynamica gecreeerd. Als sensorfunctionaliteit wordt de piezo actuator tegelijkertijd gebruikt als meetinstrument. Het is opexperimenteel niveau gedemonstreerd dat een Single-Input-Single-Output (SISO) ILC, diegebruik maakt van piezo sensor metingen, in staat is om de resttrillingen te dempen, om zohogere jet frequenties mogelijk te maken.

Er is een tweede experimenteel model gemaakt om kennis op te doen met betrekking tot dedynamica tussen de piezo sensor en de snelheid van de inkt/lucht interface, meniscus genoemd.Door gebruik te maken van dit tweede model is het mogelijk om een ‘slim’ referentie signaal teontwerpen voor de ILC die op het piezo sensor signaal is gebaseerd. Het is aangetoond middelseen experiment dat deze methode resulteert in vergelijkbare prestaties als ILC gebaseerd opeen ‘simpel’ referentie signaal. Mogelijke verklaringen voor het feit dat de slimme referentieniet leidt tot een betere prestatie zijn de nogal sterke niet-lineariteiten van de meniscus

xii

dynamica en dat er geen rekening is gehouden met de twee-zijdige koppeling tussen de piezoen de meniscus. Verder is op simulatie niveau gedemonstreerd welke meniscusbeweging nodigis voor het produceren van een druppel met willekeurige snelheid en volume en zijn er DSMsimulaties uitgevoerd.

Om de problemen betreffende overspraak aan te pakken is er een regelconcept geıntro-duceerd dat gebruik maakt van Multi-Input-Multi-Output (MIMO) ILCs. Een dergelijkeMIMO regelaar is in staat om de inktkanalen actief te ontkoppelen, zoals op simulatie niveauis geıllustreerd. Om deze actieve ontkoppeling te realiseren, worden ook de niet-jettendebuurkanalen actief aangestuurd om in rust te blijven. Twee verschillende implementaties vanILC zijn beschouwd, MIMO ILC en Multiloop SISO ILC, die beide leiden tot succesvolleonderdrukking van de overspraak.

Contents

Abstract i

Preface iii

Nomenclature v

Summary ix

Samenvatting xi

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Inkjet Printhead 5

2.1 Basic Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Principle of Piezo-Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Droplet Formation Process . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Residual Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Channel Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Nozzle Refill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Cross-Talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Types of Cross-Talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.2 Effects of Cross-Talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Overview of the Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.2 Sensor Functionalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 The Matlab Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Channel Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.2 The Dijksman Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.3 Piezo Sensor Functionality . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6.1 Frequency Response Measurements . . . . . . . . . . . . . . . . . . . . 27

2.6.2 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

xiv CONTENTS

2.6.4 Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Iterative Learning Control 33

3.1 Basic Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Lifted ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1 Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Loss in Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.4 The Learning Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.5 Convergence and Robustness . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.6 Additional Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3 MIMO Lifted ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Implementation of SISO ILC 49

4.1 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.1 Controller Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1.2 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 SISO ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.1 Derivation of the Learning Algorithm . . . . . . . . . . . . . . . . . . 51

4.2.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Integrated SISO ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59



4.3.3 DOD Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 DSM and the ‘Smart Reference’ 71

5.1 Additional System Identification . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Simulation Results based on the Meniscus Velocity . . . . . . . . . . . . . . . 73

5.2.1 DSM based on Amplitude Scaling . . . . . . . . . . . . . . . . . . . . 74

5.2.2 DSM based on Channel Resonance Modes . . . . . . . . . . . . . . . . 76

5.3 SISO ILC with a Smart Reference . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.1 Dynamics between Piezo Sensor and Meniscus . . . . . . . . . . . . . 81

5.3.2 Derivation of a Smart Reference . . . . . . . . . . . . . . . . . . . . . 84


5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 MIMO ILC Simulations 89

6.1 Generic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.1 Control Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.2 Generic MIMO ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 MIMO Printhead Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3 MIMO ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


6.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

CONTENTS xv

6.3.3 Droplet Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.4 Multiloop SISO ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4.1 Derivation of the Learning Algorithm . . . . . . . . . . . . . . . . . . 986.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.4.3 Droplet Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7 Conclusions and Recommendations 103

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

A Standard ILC 105

A.1 Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2 Application of Standard ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

A.2.1 Derivation of the Learning Algorithm . . . . . . . . . . . . . . . . . . 107A.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

B Lifted ILC without Actuation Window 113

C Reproducibility and Sensitivity of ILC 117

C.1 Reproducibility of ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117C.2 Sensitivity of ILC for Production Tolerances . . . . . . . . . . . . . . . . . . . 117

D Simplification of the Learned Input Signal 119

D.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119D.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

E Reference Signal for Multiple Droplets 123

Bibliography 125

Chapter 1

Introduction

1.1 Introduction

The inkjet technology is an important subject for the industry as well as for academic research.Inkjet can deposit droplets of various materials in patterns on a substrate. These dropletsmay be very small, namely in the order of picoliters, and they can be placed with an accuracyof several micrometers. This makes inkjet a very versatile technique with a wide field ofapplications.

A well-known application of inkjet is in the graphical industry. Here, inkjet is used fordocument printing, the production of posters and CAD drawings, for example. However,inkjet printing is also applied in the electronics industry for the production of polyLEDdisplays [Phi], [Dut] and the production of biochips for medical research [Sci]. Anotherapplication of inkjet is that of 3D printing for rapid prototyping [Rap], [Dut]. In thesemarkets, inkjet has not yet reached the same stage as in the graphical market, due to themuch tighter demands.

All different inkjet applications demand for their own performance requirements and,in general, these are quite high. For printing applications at Oce, an accuracy in terms ofmicroseconds, micrometers and picoliters is desired. It is expected that for future applicationsthese criteria become even more tight.

Oce is specialized in the market of printing applications for professional users. One ofthese applications is that of wide format color printing, for which the inkjet technology isapplied. Here, main targets are a high resolution, a constant quality and a high print speed.These demands can directly be translated into small droplets with constant properties andhigh jet frequencies of the printhead, respectively.

1.2 Problem Statement

In this report, a piezoelectric inkjet printhead is addressed that comprises two arrays of inkchannels with a high integration density, see Figure 1.1. Each channel is equipped with itsown piezo actuator and the printhead works according to the Droplet-On-Demand (DOD)principle. When a droplet of ink is desired, a voltage pulse is sent to one of the piezoelectricactuators and a single droplet is ejected from the nozzle. After ejection, the channel is notimmediately at rest and one should wait until the residual vibrations are damped out beforea new actuation pulse can be applied. Otherwise, the properties of the resulting droplets

2 Introduction

cannot be guaranteed. In the worst case the jet process can even become unstable. For thisreason the maximum jetting frequency is limited. A second phenomenon which is encounteredduring jetting is interaction between different channels, called cross-talk. Due to cross-talk,the droplet properties may vary, what is a highly undesired effect. Moreover, as a result ofproduction tolerances, the channels of a printhead are never exactly identical. This may resultin differences in droplet properties between the channels. Other undesired effects concernthe formation of satellite droplets and changing dynamics as a results of aging of the piezoactuator, variations in ink properties and disturbances in the nozzle.

Figure 1.1: Picture of a piezoelectric inkjet printhead.

In current inkjet printers mostly a fixed actuation pulse is used, which does not takeinto account the mentioned problems. This way of actuation is referred to as passive con-trol. A switch from passive control to active control can overcome the aforementioned pro-blems [Inta], [Bos04]. The residual vibrations can be actively damped and the channels can beactively decoupled. Also the effects of changing dynamics and production tolerances can bereduced by an active controller. Furthermore, by applying active control, it is also rather easyto produce droplets of different sizes. This printing technique is called Drop-Size Modulation(DSM).

The very small time scales involved make the implementation of an on-line digital feedbackcontroller infeasible, because of the limited time available for calculation. Since the genera-tion of droplets is a highly repetitive process, feedforward control by an Iterative LearningController (ILC) seems a proper choice. An important advantage of ILC is that calculationscan be performed off-line.

Active control demands for proper actuator and sensor functionality. The piezo unitis chosen as actuator, however, the choice of the sensor is less trivial. It is possible tosimultaneously use the piezo actuator as sensor by measuring the force which is actuatedon it by the ink, but this gives no direct information about the generated droplets [Intb].Other possible sensor functionalities are the observation of droplets with a Charge-CoupleDevice (CCD) camera, or measurement of the velocity of the ink/air interface (meniscus)using Laser-Doppler interferometry.

The goal of this study is to implement an Iterative Learning Controller on an inkjetprinthead. ILC is a model-based technique which requires a model of the system dynamics.

1.3 Objectives 3

Therefore, the printhead dynamics have to be identified first. Then, it is desired to verify onexperimental level whether the expectations of active control are feasible.

The problem statement of this thesis is:

Obtain an accurate experimental model which describes the relevant dynamics of the inkjetprinthead. Use this model for the implementation of Iterative Learning Control and investigateinto how active control can resolve the operational issues.

1.3 Objectives

The problem statement can be translated to the following objectives:

• Explore the working principle of a piezoelectric inkjet printhead and the operationalissues more extensively (see also [Bos04]).

• Identify and model the printhead dynamics by making use of the piezo sensor.

• Examine Lifted Iterative Learning Control in more detail (see also [Bos04]), aim thecontrol technique more at the operational issues of the inkjet printhead and investigatecontrollability, observability and robustness issues.

• Implement ILC based on the piezo sensor signal on a single channel, in order to activelydamp the residual vibrations in practice. This implementation is referred to as theSingle-Input-Single-Output (SISO) case.

• To effectively reduce the effects of cross-talk on simulation level, actively decouple thechannels by designing a Multi-Input-Multi-Output (MIMO) ILC for the actuation of anarray of channels.

1.4 Methodology

In Chapter 2, the working principle of a piezoelectric inkjet printhead and the droplet for-mation process are addressed. Moreover, the cause and effects of residual vibrations andcross-talk are discussed and the experimental setup with the different sensor functionalitiesis treated. An existing theoretical inkjet model is introduced and extended with the piezosensor functionality. The rest of this chapter concerns experimental system identification,modeling and model validation. Next, Chapter 3 handles the subject Iterative Learning Con-trol. Several simulation and experimental results of ILC implemented on a single channelare discussed in Chapter 4. Here, two different SISO ILC implementations are consideredwhich both make use of the piezo sensor functionality. The main control goal is here reduc-tion of the residual vibrations. In Chapter 5, it is studied whether it is possible to design asmarter reference signal for SISO ILC. For this purpose, additional identification is applied,based on the meniscus velocity sensor. The results are illustrated by means of simulationsand experiments. Chapter 6 addresses the subject MIMO ILC to deal with the problemsconcerning cross-talk. The general idea for the actuation of a complete array is illustratedand simulation results are reported for two different ILC implementations. The last chapter,Chapter 7, handles the conclusions and the recommendations for future research.

Chapter 2

The Inkjet Printhead

The control technique Lifted ILC demands for a model of the system dynamics. For this pur-pose, the path of experimental modeling is chosen, to obtain an analytical model. Before theidentification of the printhead dynamics is discussed, the working principle of a piezoelectricinkjet printhead is shortly resumed (see also [Bos04]). Moreover, two of the operational is-sues, residual vibrations and cross-talk, are discussed in more detail. To validate the availabletheoretical inkjet model, the theoretical and experimental results are compared.

The basic working principle of inkjet is addressed in Section 2.1. Here, the principle ofpiezo-jet and the droplet formation process are shortly discussed. Next, in Section 2.2, theresidual acoustic vibrations are treated and cross-talk is subject of Section 2.3. An overviewof the experimental setup and its sensor functionalities is provided in Section 2.4. Section 2.5concerns the existing theoretical model of the printhead and an extension to it. The subjectof system identification is treated in Section 2.6 and, finally, in Section 2.7, the conclusionswhich are made in this chapter are shortly recapitulated.

2.1 Basic Working Principle

In this study, an inkjet printhead is considered which is based on the piezo-jet principle.Basically, the piezoelectric printhead under investigation comprises two arrays of 128 nearlyidentical ink channels. All these channels are equipped with separate piezo actuators. Such apiezo actuator forms one side of the channel wall, where a foil is used to avoid direct contactbetween the actuator and the ink. A piezo actuator is made of piezo material which has theproperty to deform when it becomes electrically polarized. When a voltage is applied over thepiezo, it slightly deforms the wall of the channel. By fast deformation of the channel, acousticpressure waves are generated in the ink channel. When these waves are concentrated in thenozzle at the end of the channel and several conditions are met, this results in a droplet.

The ink in the channel is supplied from the reservoir which is located above the channels.Filters ensure that no dirt particles enter the ink channels. The nozzle plate which containsthe nozzles is mounted to the bottom of the head. The part of the head between a channeland a nozzle is called the connection. In Figure 2.1 [MGWK04], an exploded view of thepiezoelectric inkjet printhead is shown, together with a schematic representation of a singlechannel.

6 The Inkjet Printhead

Figure 2.1: Exploded view of a piezoelectric inkjet printhead (left) and schematic representa-tion of a single channel (right).

2.1.1 Principle of Piezo-Jet

In the current situation, the piezoelectric inkjet printhead is actuated by fixed trapezium-shaped pulses (passive control). In a simplified representation, the jet process can be dividedin several stages. The difference in cross-sectional area of the connection is not taken intoaccount in this simplified representation. Moreover, the channel geometry is assumed to be1D. The different stages can then be described as follows, see also Figure 2.2:

1. At time t = t0, a voltage is applied over the piezo actuator, resulting in an electricalcharge. Due to this charge, the piezo material deforms, enlarging the volume of the inkchannel. This causes a negative pressure wave with amplitude −p.

2. Subsequently, the negative pressure wave splits up in two pressure waves with ampli-tude −p

2, travelling with the effective speed of sound in ink c in positive and negative

longitudinal direction of the channel.

3. The pressure wave that is travelling to the left hits the channel-reservoir interface. Thecross-section of the reservoir is much larger than that of the channel and the negativepressure wave is reflected with changed sign. The wave that is travelling to the right hitsthe channel-nozzle interface. The cross-section of the nozzle is much smaller than thatof the channel. As a result, the reflected pressure wave does not change sign. However,a negative pressure is induced inside the nozzle, causing the meniscus to retract.

4. At time t = t0 + Lc, with L the total length of the channel (length of the connection

included), the two reflected waves meet each other in the middle of the channel. Exactlyat this moment, the electrical charge is released. Due to the sudden decrease in channel

2.1 Basic Working Principle 7

volume, a positive pressure wave arises. Again, this wave splits up in a wave travellingto the left and one travelling to the right.

5. Now, a positive and a negative pressure wave are travelling to the left and they will(approximately) cancel each other out. Two positive pressure waves are travelling tothe right, leading to an amplification of the wave. When this doubled pressure wavehits the channel-nozzle interface, the pressure amplitude is large enough to change thedirection of the flow inside the nozzle and to overcome the impedance of the nozzle tocreate a droplet.

p i e z o a c t u a t o r / s e n s o r

I n kx = 0 x = L

U

t

- p

- p / 2 - p / 2

- p / 2+ p / 2

+ p

+ p

n o z z l er e s e r v o i r

x

( o p e n e n d ) ( c l o s e d e n d )

Figure 2.2: Simplified representation of the pressure waves in the ink channel and nozzleduring actuation with a trapezium-shaped pulse.

2.1.2 Droplet Formation Process

The droplet formation is a very complex process. For accurate simulation results of the dropletformation, the nonlinear Navier-Stokes equations have to be solved numerically. Mostly, thesemodels are 3D or 2D axisymmetric and numerically quite intensive. An alternative 1D modelis proposed by Dijksman [Dij84]. This model predicts whether a droplet is created or not andprovides reasonably good estimates of the droplet size and speed by considering an energybalance. An important advantage of the Dijksman model is that it is analytical and, therefore,very fast. The Dijksman model is discussed in more detail in Section 2.5.2.

For a typical printhead, the droplet formation process can be divided in four main stages.A simulation example of the meniscus velocity, which is calculated with the theoretical Mat-

lab model (see Section 2.5), is shown in Figure 2.3. Furthermore, an illustration of thedroplet formation, calculated with the fluid mechanics package Flow3D, is provided in Fi-gure 2.4 [MGWK04]. In both these figures, the different stages are indicated. It is assumedthat in the beginning the ink in the channel and nozzle is at rest.


0 5 10 15 20 25 30 35 40 45 50−3

−2

−1

0

1

2

3

4

1

2

3

4

actuation pulse (scaled)meniscus velocityinstant of droplet ejection

PSfrag replacements

Time [µs]

Men

iscu

sve

loci

ty[m

/s]

Figure 2.3: Simulation result of the meniscus velocity for a fixed actuation pulse; the numbersindicate the stages of the droplet formation.

1 2 3 4Figure 2.4: Illustration of the droplet formation process calculated with Flow3D; the numbersindicate the stages of the droplet formation.

1. In the first stage, the meniscus retracts due to the negative pressure inside the nozzle.This negative pressure is caused by the negative pressure wave that hits the channel-nozzle interface, as explained in the previous section.

2. During the second stage, the meniscus velocity starts to increase with positive accele-ration, due to the positive pressure wave that hits the channel-nozzle interface. First,the meniscus moves outwards without deformation. Shortly afterwards, the meniscussurface starts deforming in outward direction against the surface tension. The deformedarea grows in both radial and axial direction. After a while, the velocity of the inkreaches its maximum with positive acceleration.

2.2 Residual Vibrations 9

3. Then, in the third stage, the pressure becomes negative again and the meniscus velocitystarts to decrease, causing a decreasing flow of mass and kinetic energy in outwarddirection. Due to the winning surface tension, necking of the droplet’s tail takes placeat the tip of the nozzle.

4. In the last stage, the velocity of the ink becomes negative and, finally, the stretching tailbreaks and the droplet is released. During this process the tail might break up, formingsatellite droplets. These satellites may or may not catch up with the main droplet andmerge. Satellite droplets affect the printing quality negatively. Therefore, the combina-tion of ink properties (viscosity and surface tension), nozzle design and actuation pulsehave to be tuned in order to create well-defined droplets, without depositing satelliteson the substrate.

2.2 Residual Vibrations

As explained in Section 2.1.1, channel actuation results in excitation of the channel acoustics.These acoustics transport the actuation energy via pressure waves through the channel tothe nozzle where, when certain conditions are met, a droplet is created. In this Section, itis explained how the channel acoustics behave after droplet ejection. The subjects channeldamping and nozzle refill are both shortly addressed.

2.2.1 Channel Damping

In the simplified representation of the jet process (Section 2.1.1), all reflection coefficientsare unity, what indicates that no damping is included. In practice, however, the reflectioncoefficients at the different interfaces are smaller than unity. The resulting pressure wavesdamp out and, eventually, the channel is at rest again. The channel should only be actuatedwhen it is at rest, because for different initial conditions the droplet properties and even thestability of the jet process cannot be guaranteed. Unfortunately, the time needed for theresidual pressure waves to damp out is substantially larger than the short actuation time.Therefore, residual vibrations limit the maximum attainable jet frequency.

As illustration, a simulated time response of the meniscus velocity on a fixed trapezium-shaped pulse is depicted in Figure 2.5. Typically, a droplet is ejected after approximately 20µs. However, the meniscus velocity takes more than 200 µs to become at rest.

2.2.2 Nozzle Refill

During the time in which the channel acoustics damp out, the nozzle is filled with new ink.This process is called refill of the nozzle. Refill is mainly caused by the channel acoustics,according to the so-called nonlinear mass effect. The second positive pressure wave (first po-sitive wave after ejection) which hits the channel-nozzle interface feels only a small mass, sincethe nozzle is nearly empty, and the meniscus is accelerated in positive (outward) direction.Subsequently, the next negative pressure wave that hits the interface feels a larger mass andthe meniscus is decelerated less than it was accelerated. A sequence of several positive andnegative pressure waves leads to refill of the nozzle1. Another phenomenon which contributes

1Whether the nonlinear mass effect leads to successful refill of the nozzle in fact depends on both the nozzlegeometry and the ink properties.


0 20 40 60 80 100 120 140 160 180 200−3

−2

−1

0

1

2

3

4actuation pulse (scaled)meniscus velocityinstant of droplet ejection

PSfrag replacements

Time [µs]

Men

iscu

sve

loci

ty[m

/s]

Figure 2.5: Simulation result of the meniscus velocity for a fixed actuation pulse; illustrationof the residual vibrations.

to the refill process is capillarity, due to adhesive forces between the ink and the nozzle wall.However, the time scale of this process is much larger. For more information about the refillmechanisms, the reader is referred to [Intc].

2.3 Cross-Talk

As already mentioned, the printhead under investigation contains two arrays with ink channelswith a high integration density. These two arrays do not interact with each other. Unfor-tunately, this does not hold true for the channels in one and the same array. As a result ofthe compact design of the printhead, the actuation of one channel also affects neighboringchannels. This phenomenon is called cross-talk.

2.3.1 Types of Cross-Talk

Basically, three forms of cross-talk can be distinguished, which are:

• Electrical cross-talk

• Acoustic cross-talk

• Structural cross-talk

– Direct cross-talk

– Indirect cross-talk

Due to electrical cross-talk, a piezo is influenced when its direct neighbor is actuated. Theeffect of electrical cross-talk during normal operation of the printhead is very limited. With

2.3 Cross-Talk 11

acoustic cross-talk, the pressure waves in the actuated channel influence other channels viathe main ink reservoir. It is assumed that also this effect is not very large in current printheaddesigns. With structural cross-talk, the deformation of the actuated channel results in defor-mation of the printhead structure and, thus, a volume change of neighboring channels. Due tothe volume change, undesired pressure waves arise in these neighboring channels. Structuralcross-talk can be the result of the actuation of a neighboring channel or pressure waves ina neighboring channel. These two types of structural cross-talk are called direct or voltagecross-talk and indirect or pressure cross-talk, respectively.

2.3.2 Effects of Cross-Talk

The effects of cross-talk depend both on time and place. When a single channel is actuated,also pressure waves are induced in neighboring channels. Due to these pressure waves, themeniscus position of the neighboring channels may change. When one of the neighboringchannels is then actuated, it is possible that, as a result of the changed meniscus position,a droplet is obtained with deviating size and speed, for example. This is an example of theeffect of cross-talk in time. The effects of cross-talk in place are more trivial. When a certainchannel is actuated it has much more influence on its direct neighbors than on a channelwhich is located several channels further away.

A theoretical example of cross-talk in place is depicted in Figure 2.6 (upper plot). Theinfluence on the droplet speed of the center channel of an array of 21 channels when neigh-boring channels are active is shown in this figure. When the direct neighbor (channel 1 or-1) becomes active, a decrease in droplet speed of channel 0 is obtained. It can be observedthat for channels which are located further away, the influence of cross-talk decreases. Whena channel is located more than ten channels from the observed channel, cross-talk has barelyany effect.

In the bottom figure, a simulation result is shown which illustrates the effect of cross-talkin time. In this figure, the influence on the droplet speed of the center channel is depictedwhen different neighbors are active as well. There exists a certain time delay between theactuation of the measured channel and its neighbors. This delay is varied from -20 µs till 80µs. A delay of t µs means that the measured channel is fired t µs after its neighbors. Clearly,the effect of cross-talk depends on the time delay. When a smart value for the delay time ischosen, for instance -5 µs, the deviations in droplet speed are limited. Note that for a timedelay of 0 µs the same information is obtained as is shown in the upper plot.

Due to cross-talk, the actuation of one channel influences the acoustics in other channelsand thus affects the droplet formation process in these other channels. Therefore, cross-talkis a highly undesired phenomenon. As already illustrated in Section 2.1, the properties of thedroplets and even overall stability of the jet process cannot be guaranteed in case of non-zeroinitial conditions. Cross-talk may result in the following undesired effects [Intd]:

• Differences in droplet speed

• Differences in droplet volume

• Deflection of droplet’s angle

• Unstable jet behavior

• Channels stop functioning


−10 −8 −6 −4 −2 0 2 4 6 8 104.5

5

5.5

6

−20 −10 0 10 20 30 40 50 60 70 804.5

5

5.5

6

6.5

712345678910

PSfrag replacements

Active neighboring channel

Time delay [µs]

v drop,c

h0

[m/s

]v d

rop,c

h0

[m/s

]

Figure 2.6: Influence on the droplet speed of channel 0 (center channel) by actuation ofdifferent neighboring channels at the same time instant (upper plot) and with a varying timedelay between the channels (bottom plot).

2.4 Experimental Setup

To perform experiments with inkjet, several test setups are available at Oce. On these set-ups, experiments can be conducted with different piezoelectric printheads to examine the jetbehavior and the droplet formation process. Firstly, a schematic overview of such a setup isgiven. Here, the available equipment is shortly addressed and it is explained how the differentdevices are connected. Secondly, the subject of a sensor functionality is discussed and differentpossibilities of sensing are considered.

2.4.1 Overview of the Setup

With the setup it is possible to apply arbitrary input signals to the piezo actuators and monitorthe droplet formation and the printhead itself. The experimental setup of the printhead andits devices is depicted schematically in Figure 2.7.

For the printhead under investigation a hotmelt type of ink is used, which requires heatingof the head. A certain reference temperature is reached by PID controllers, which measurethe printhead temperature with thermocouples and control the input voltages of the heatingelements. To monitor the ink level inside the reservoir, a level sensor is put in this part ofthe head.

In the setup, a printhead is mounted in vertical direction with the nozzles faced down.To avoid that the ink simply flows out of the nozzles under the influence of gravity, an air

2.4 Experimental Setup 13

W a v e f o r m g e n e r a t o r A m p l i f i e r S w i t c h - b o a r d

I n k l e v e li n d i c a t o r

T e m p e r a t u r e c o n t r o l u n i t sP r i n t h e a d

P e r s o n a lc o m p u t e r

C C D c a m e r a +m i c r o s c o p e

S t r o b el i g h t

S c o p e

A i r p r e s s u r eu n i t

a c t u a t i o n s i g n a l

p i e z o s e n s o r s i g n a l

L a s e r V i b r o m e t e r + d e t e c t o r m i r r o r( 4 5 d e g . )

m e n i s c u sv e l o c i t y

i m a g e

Figure 2.7: Schematic overview of the printhead setup and its devices.

pressure unit makes sure that the pressure in the ink reservoir remains below the ambientpressure.

The printhead setup is connected to a personal computer. The software package which isof main interest for the setup is Labview. On the computer, the desired actuation signals canbe programmed and relevant date can be stored and processed. After defining the actuationsignal, it is sent to an arbitrary waveform generator. The waveform generator sends thesignal to an amplifier unit, which has a certain gain. From the amplifier the signal is fed toa so-called switch-board. The personal computer communicates with the switch-board andthe channels which have to jet can be selected. Finally, the switch-board is connected tothe individual piezo units of the printhead. For the capturing of signals an oscilloscope isavailable. This oscilloscope is connected to the computer and it is possible to save data to afile.

Measurements can be performed by a CCD camera and Laser-Doppler interferometry.Moreover, the piezo units itself can be applied as sensor. The different sensor functionalitiesand their working principle are further discussed in the next section.

2.4.2 Sensor Functionalities

For the application of ILC a proper sensor functionality is required. The signal obtainedfrom this sensor, the system output, is needed to perform adjustments to the system input.Obtaining relevant data from the jet process and the droplet formation is rather complicated.Ideally, one would like to have a single sensor signal per channel which provides all relevantinformation about the jet behavior and droplet formation and which is continuously available.Unfortunately, no such sensor signal exists. At the moment, three sensor functionalities canbe used which all have their advantages and disadvantages. The three sensor functionalities


are schematically illustrated in Figure 2.8, together with the relevant dynamics Hpiezo, Hvmen

and Hdroplet. In the following subsections, these different techniques to obtain informationabout the jet process are discussed.

p i e z o

c h a n n e ln o z z l e

r e s e r v o i r d r o p l e tm e n i s c u sPSfrag replacements

Hpiezo

Hvmen

Hdroplet

Figure 2.8: Schematic overview of a single channel with its sensor functionalities and relevantdynamics.

CCD Camera

The generated droplets can be observed by a Charge-Couple Device (CCD) camera, equippedwith a microscope. A stroboscope provides a short light flash at a defined instant after thedroplet is ejected and an image is obtained on which the droplet seems to be fixed in theair. Both the time duration and the distance that the droplet has travelled are known. Byusing this information, an estimate of the droplet speed can easily be obtained. Moreover, itis possible to estimate the volume of the droplet, because the droplet diameter can be deter-mined. Other information which can be obtained concern the droplet’s angle, the formationof satellites and the stability of the jet process. A great advantage of the CCD camera is thatdirect information about a droplet is obtained. But, unfortunately, this information is onlyavailable at discrete time instants.

Laser-Doppler Interferometry

The principle of Laser-Doppler interferometry consists of the splitting of a laser beam in twodifferent paths and, finally, combine the beams again. One beam travels over a fixed path andthe path of the other beam is varied. In case a beam is reflected against a moving object, aDoppler shift takes place. When the object is moving towards the beam, the frequency of thesignal increases and when the object is moving away from the beam, the frequency decreases.In this way, the combined signal contains information about the phase difference and thefrequency shift between the two signals. This information is measured by a detector. With aLaser-Doppler interferometer or laser-vibrometer it is possible to measure the velocity of themeniscus inside a nozzle. Here, the meniscus surface is the moving object which reflects thebeam. Unfortunately, this type of measurement can only be applied to a small range of thedynamics. It is namely not possible to jet during this measurement, without taking specialmeasures.

2.5 The Matlab Model 15

Piezo Sensor

It is well known that a piezoelectric transducer can be used as actuator and as sensor,see [Waa]. In the actuator functionality, a voltage over the piezo results in deformationof the piezo material (indirect piezoelectric effect). This function of the piezo is used duringthe jet process to generate pressure waves which result in droplets. Moreover, the pressureinside the ink channel acts on the piezo surface, resulting in a force which causes a smalldeformation of the piezo material, resulting in an electrical current, according to the sensorfunctionality (direct piezoelectric effect). This current, due to pressure waves inside the chan-nel, is called the piezo signal. The piezo signal is a measure for the time-derivative of thepressure in the channel, integrated over the surface of the piezo unit. An advantage of thepiezo signal is that the signal can be measured continuously, also during jetting. However,the signal does not provide direct information about the droplet formation. In this study,main attention is paid to the piezo sensor signal as system output.

2.5 The Matlab Model

Within Oce, several theoretical models of the piezoelectric printhead are available. One ofthese models is known as the Matlab model or low-reduced frequency model [Bel94], [Bel98]and comprises a linear 1D model of an array of channels. The Matlab model is based on the1D wave equation for channels with flexible walls, which is solved analytically by Beltman.Because this equation is considered in the frequency domain, time-varying effects are nottaken into account. Viscosity effects of the ink, however, are taken into account. As a resultof friction between the ink and the channel wall, the wavefront will generally not be straight.Therefore, the model can in fact be seen as a ‘1.5D’ model. The droplet formation process isnot implemented in the Matlab model. However, by using the Dijksman model it is possibleto calculate the size and speed of the resulting droplet. There is no interaction between thetwo models and the calculation of the droplet properties is typically a post-processing step. Agreat advantage of the model is that structural cross-talk is accounted for. The quality of theresults is reasonable and the model is, due to the analytical character, very fast. Importantshortcomings are that the piezo unit is not implemented and that the acoustics are alsoassumed to be valid inside the nozzle. Moreover, the change in mass inside the nozzle whena droplet is jetted is not taken into account.

2.5.1 Channel Acoustics

In the Matlab model, a special form of the wave equation is used to describe the acousticsinside the ink channel. This equation is named the narrow-channel equation and it is based onthe narrow-gap equation, which is derived in [Bel94]. To obtain the narrow-channel equation,use is made of the Navier-Stokes equation, the equation of continuity, several thermodynamicrelationships and a function which takes into account the flexibility of the channel wall. Byintroducing several assumptions and simplifications it is possible to actually solve the waveequation for an ink channel in one dimension, leading to an analytical solution. The resultswill shortly be addressed here, but for a detailed derivation the reader is referred to [Intd],[Bel98], or [Intc].


Assumptions

In order to solve the narrow-channel equation analytically, several assumptions are madewhich are summarized here. The assumptions which hold are indicated by a ‘+’ as bullet andthose which do not by a ‘−’. The last two assumptions are fulfilled in the channel but not inthe nozzle and are, therefore, indicated by a ‘−’.

+ Body forces are negligible.

+ The ink is homogeneous and isotropic with constant viscosity.

+ There is no flow of heat or generation of heat (by viscous dissipation).

+ Rotation and dilatation terms can be neglected.

+ Velocities perpendicular to the direction of the length of the channel are negligibly small.

+ The pressure in the nozzle transients linearly to zero.

− The fluid is an ideal gas.

− The nozzle is completely filled all the time.

− All perturbations are small, there is no main flow.

− The dimensions of the characteristic cross-section are much smaller than the dimensionsof the characteristic length.

Regarding these assumptions, it can be concluded that most of them are satisfied for the inkchannel. The major drawback of the model is that several important assumptions do not holdfor the nozzle.

Narrow-Channel Theory

According to the narrow-channel theory, the pressure inside an array of ink channels can bedescribed by the second-order partial differential equation

B∂2p

∂x2+

ω2

c20

p = −ρ0ω2 ∆A

A0

, (2.1)

which is expressed in the frequency domain. Here, p represents a column with complexamplitudes of the pressure disturbance in the different channels, x is the coordinate in thelength direction of the channels and ω is the angular frequency. The adiabatic speed of soundin ink is denoted by c0, ρo is the average density of ink, A0 the nominal cross-sectional areaof the channels, ∆A the change in cross-sectional areas and B is a function which describesviscosity effects and the form of the cross-section. The right-hand term in (2.1) is the actuationterm, which can be subdivided in two parts. Changes in the cross-sectional area can namely bethe result of elastic deformation caused by the ink pressure in the channels and of deformationof the piezoelectric elements when a voltage is applied. The actuation term equals

∆A

A0

= αU + βp. (2.2)


In (2.2), U is a column with applied voltages to the piezo actuators of the array and α andβ are matrices with influence parameters which depend on the elasticity of the printheadstructure and the properties of the piezo units. Note that the direct and indirect structuralcross-talk are captured by α and β, respectively. The parameters in these matrices arecalculated by a Finite Element Method (FEM) model of the printhead structure, by makinguse of a 2D static calculation. For this purpose use is made of the software package Ansys.An example of structural deformation due to actuation and due to pressure inside a channelis given in Figure 2.9. In this figure, a cross-section of an array of channels and piezo unitsis shown, of which the most left channel is responsible for the structural deformation. Fromthese deformations, the α and β parameters can be extracted.

u n d e f o r m e d

d e f o r m a t i o n b yp i e z o v o l t a g e

d e f o r m a t i o n b yc h a n n e l p r e s s u r e

Figure 2.9: Structural deformation of the printhead, calculated with Ansys.

Combining (2.1) and (2.2) then results in

B∂2p

∂x2+

(ω2

c20

+ ρ0ω2β

)

p = −ρ0ω2αU. (2.3)

The analytical solution for the pressure p equals the sum of the homogenous and the particularsolution of (2.3), leading to

p = c1ekx + c2e

−kx +ρ0ω

2αU

Bk2, (2.4)

where

k2 = −ω2

c20

+ ρ0ω2β

B. (2.5)

The two constants c1 and c2 can be derived by using proper boundary conditions.


Boundary Conditions

In the Matlab model, the channel, the connection as well as the nozzle are taken into account.These parts are divided in segments, which all have their own cross-section. The flexibilityof the channel wall of these segments and the change in cross-section due to actuation orpressure may differ per segment. The narrow-channel equation (2.1) is solved per segment,for each frequency. For the boundary conditions of the complete channel geometry, a zeropressure at the entrance of the channel and a zero pressure at the nozzle end are assumed.These two conditions, obviously, hold for the beginning of the first segment and the end ofthe last one. The coupling of the segments is described by the equations

pm(lm) = pm+1(0),

Qm(lm) = Qm+1(0),(2.6)

where pm and Qm represent the pressure in and the volume flow through the m-th segment,respectively. The length of segment m is denoted by lm. In fact, equations (2.6) simplyindicate that the pressure at the end of segment m equals that at the beginning of segmentm + 1 and that there is a preservation of flow.

Pressure Profile

In Figure 2.10, the amplification of the Frequency Response Function (FRF) from actuationvoltage to pressure is shown per segment, resulting in the theoretical pressure profile insidean ink channel. Note that in the 1D Matlab model only deviations in longitudinal directionare taken into account, therefore, this direction is shown as position coordinate. The firstresonance modes and the higher-order resonance modes can clearly be recognized in thefigure. Obviously, the first mode has a single maximum at the center of the channel andthe higher-order modes show multiple maxima. These extrema equal the antinodes of thepressure waves.

2.5.2 The Dijksman Model

The droplet formation process depends on the viscosity, the surface tension and the velocitydistribution of the ink inside the nozzle. A model which computes approximates of the dropletspeed and size is derived by Dijksman [Dij84]. Here, the droplet formation process itself is notsolved. The model is based on an energy balance and predicts whether a droplet is createdand, if one is created, it provides estimates of the droplet speed and size. Despite of itssimplifying character, reasonably good results are obtained.

Assumptions

The Dijksman model is based on several assumptions. Again, a ‘+’ indicates that the as-sumptions holds and a ‘−’ that it does not.

+ The droplet formation depends on the ratio of kinetic energy transported outwards tothe energy needed to form the surface of the droplet. If this number is larger than unity,then in principle a droplet can be formed. The speed of the resulting droplet dependson the amount by which this ratio exceeds unity. The viscosity does not directly play arole here.


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PSfrag replacements

Frequency [kHz]

x-position [m]

Mag

nit

ude

[-]

Figure 2.10: Surface plot of the theoretical pressure profile in an ink channel, as a functionof the longitudinal channel position and the frequency.

+ The influence of ambient air can be neglected.

+ No long jets or tails are formed.

− The droplet formation is completely situated outside the nozzle.

Derivation

At time t = t1, the velocity of the ink in the nozzle end has changed sign from negative topositive (in outward direction). At this time instant, a control volume is defined just in frontof the meniscus, see Figure 2.11. The volume of ink which is transported through the controlsurface is

V (t) = An

∫ t

t1

undt, (2.7)

where An is de cross-sectional area of the nozzle end and un is the mean velocity of the ink inthe nozzle end. Assuming a viscid Poiseuille flow, the amount of transported energy is twicethat of an inviscid flow when both flows have the same mean velocity. The amount of kineticenergy transported through the control surface in case of a viscid flow then equals

T (t) = ρ0V (t)

∫ t

t1

u2ndt =

1

2ρ0V (t)

∫ t

t1

unun|r=0dt. (2.8)

In (2.8), ρ0 is the average density of the ink and un|r=0 is the maximum velocity of the ink,equal to the velocity at radius r = 0. For a Poiseuille flow, the maximum velocity equalstwice the mean velocity, see Figure 2.12. By writing the expression for the kinetic energy in


c h a n n e l

n o z z l e

m e n i s c u sc o n t r o l s u r f a c e

c o n t r o l v o l u m e

Figure 2.11: Schematic representation of the nozzle and the control volume.

terms of both the maximum and the mean velocity, the similarity with the expression for thekinetic energy of a point mass (T = 1

2mv2) becomes clear.

PSfrag replacements

r

un un|r=0

un(r)

Figure 2.12: Illustration of the Poiseuille velocity profile.

By making use of (2.7), the kinetic energy can be reformulated as

T (t) = ρ0An

∫ t

t1

u3ndt. (2.9)

Basically, the droplet formation consists of a conversion of kinetic energy into surface energy.To obtain an expression for the surface energy, the enlargement of the free surface has to bedetermined. The height of the free surface relative to the nozzle end is

h(r, t) =

∫ t

t1

un(r, t)dt. (2.10)

This equation expresses that the emerging fluid from the nozzle is modelled as flowing indiscrete concentric cylindrical shells, like an expanding telescope. The emerging fluid fromthe nozzle is depicted schematically in Figure 2.13. Application of Pythagoras’ theorem leadsto an expression for the enlargement of the free surface (see [MGWK04]), which equals

ϑ(t) = 2π

∫ Rn

0

√

1 +(∂h(r, t)

∂t

)2

rdr − An, (2.11)

where Rn is the radius of the nozzle end. Suppose that at time instant t = t2, the situationis reached in which the difference between the kinetic energy transported through the controlsurface and the energy needed to enlarge the free surface is just equal to the instantaneouskinetic energy of the fluid inside the control volume. This situation is described by

T (t2) − σϑ(t2) =1

2ρ0V (t2)un(t2)un(t2)|r=0. (2.12)


c h a n n e l

n o z z l e

e x p a n d i n g t e l e s c o p e

Figure 2.13: Schematic representation of the nozzle and the emerging fluid.

In (2.12), σ is the surface tension of the ink and V (t2) is the volume of ink outside the nozzleat t = t2. This is the situation when a droplet can be created. The volume of the droplet isthen Vd = κV (t2), where κ is the droplet volume factor. Calculations of the droplet formationwith Flow3D show that not the complete volume of the fluid outside the nozzle results in adroplet [MGWK04]. This effect can be captured by choosing 0 < κ < 1. In the rest of thisstudy, however, it is assumed that κ = 1, for simplicity. Reformulation of (2.12) yields

∫ t2

t1

[

u2n(t) − u2

n(t2)]

dt =2σϑ(t2)

ρ0V (t2)> 0. (2.13)

The right-hand term of (2.13) is always positive, thus the left-hand side must also be positive.From this it follows that equality (2.12) can only be satisfied when the velocity at t = t2 hasgone through a maximum. This provides a condition whether a droplet can be created or notin a certain situation. An example of the different energies involved in the droplet formationis given in Figure 2.14. Here, the kinetic energy transported into the control volume, theenergy needed to enlarge the free surface, the net available energy and the instantaneouskinetic energy in the control volume are shown as a function of the time. At the time instantat which the net available energy equals the instantaneous kinetic energy, a droplet is formed.Obviously, the droplet is formed after the instantaneous kinetic energy (and thus the velocity)has gone through a maximum.

In (2.12), the last term represents the net available energy which equals the kinetic energyof the droplet which is still connected to the main fluid by a stretching fluid thread. Forseparation of the droplet it is assumed that an extra free surface of 1

2An has to be created in

case of a Poiseuille flow. The kinetic energy of the resulting droplet then equals

Td =1

2ρ0Vd(v

∗

d)2 =

1

2ρ0V (t2)un(t2)un(t2)|r=0 −

1

2σAn. (2.14)

Now, the speed of the resulting droplet, v∗d, can be calculated by

v∗d =

√

un(t2)un(t2)|r=0 −σAn

ρ0Vd. (2.15)

When the argument of the square root is positive, (2.15) provides in the droplet speed.However, when the argument is negative, the surface tension was too large compared to thekinetic energy transported to the outside and no droplet is created. In Figure 2.15, an exampleof the energies is given, which does not result in a droplet. In this situation there there is no


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1

1.5

2

2.5

x 10−10

Ttransported

Tsurface

Tnet

Tinstantaneous

equilibrium

PSfrag replacements

Time [µs]

Ener

gy[J

]

Figure 2.14: Example of the energies involved in the droplet formation model of Dijksman,resulting in a droplet.

14 15 16 17 18 19 20 21 22 230

1

2

3

4

x 10−11

Ttransported

Tsurface

Tnet

Tinstantaneous

PSfrag replacements

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Ener

gy[J

]

Figure 2.15: Example of the energies involved in the droplet formation model of Dijksman,not resulting in a droplet.


intersection point between the curve of the instantaneous kinetic energy and the net availableenergy, and thus no droplet can be created.

During stretching of the tail, the droplet is slowed down by viscous dissipation or elonga-tional viscosity. Therefore, a correction has to be applied to (2.15) to obtain the final dropletspeed. This correction factor equals

∆vd =3µ

2ρ0

1∫ t2t1

un(t)dt. (2.16)

One is referred to [Dij84] for a detailed derivation of this expression. Taking the correctionfactor for the velocity into account, the final droplet speed is given by

vd = v∗d − ∆vd =

√

un(t2)un(t2)|r=0 −σAn

ρ0Vd− 3µ

2ρ0

1∫ t2t1

un(t)dt. (2.17)

In the Dijksman model, it is assumed that the droplet formation starts when the meniscusis outside the nozzle. However, in [MGWK04] it is shown that for the nozzle geometry ofthe printhead type under investigation the droplet formation starts when the meniscus is stillretracted. Despite of this difference, the Dijksman model provides reasonably good results.For this reason, it is assumed in the rest of this study that the droplet formation starts outsidethe nozzle.

2.5.3 Piezo Sensor Functionality

For validation purpose and to be able to perform simulations based on the piezo sensor signal,the Matlab model is extended with the piezo sensor functionality. The measured piezo signalis a voltage which is a measure for the time-derivative of the force on the piezo. Firstly, asimplified piezo model is introduced, after which the equations are given to transform thepressure of the ink inside the channel to the voltage signal which is available for measurement.Secondly, the theoretical Frequency Response Function (FRF) from piezo actuator to piezosignal is shown and discussed. A numerical improvement which has been made in the accuracyof the time responses of the theoretical model is shortly addressed, thirdly.

Two-port Piezo Model

According to the piezoelectric effect a piezo material deforms under the influence of an elec-trical field [Waa]. The relation between the electrical deformation D, or polarization, and theapplied electrical field E can be illustrated by the dielectric hysteresis curve. An illustrationof such a curve is shown in Figure 2.16.

Obviously, the relation between D and E is nonlinear. The maximum displacement whichcan be obtained is the saturation polarization Ds. Moreover, as the name of the curvealready indicates, there is a hysteresis effect. When an applied electrical field is removed,the displacement does not become zero. The value of the electrical displacement when noelectrical field is applied is called the remanent polarization Dr.

When only small fluctuations are considered, the mechanical-electrical behavior of thepiezo can be linearized. The linearized piezo dielectric curve around E = Ubias is also indicatedin Figure 2.16. The relations between the applied electrical field ~E and stress ~T on the one


l i n e a r i s a t i o n

b i a sv o l t a g e

p u l s es i n es w e e p

PSfrag replacementsE [V/m]

D [C/m2] Ds

Dr

−Ds

−Dr

0

Figure 2.16: Illustration of the piezo dielectric hysteresis curve and the bias voltage which isapplied to the piezo actuators.

hand and the electrical displacement ~D and the strain ~S on the other are then given by[

~S~D

]

=

[sE dT

d εT

] [~T~E

]

. (2.18)

Here, sE is a 6 × 6 matrix with the mechanical compliances in case of a constant electricalfield, d is a 3× 6 matrix with the piezoelectric charge constants and εT is a 3× 3 matrix withthe permittivity constants in case of constant stress. All these parameters hold for the bulkpiezo material.

For the piezo actuators in the inkjet printhead several assumptions can be made whichlead to a simplification of (2.18). For the printhead which is considered in this study, it isassumed that there is no shear stress and shear deformation and that the piezo only operatesin the polarization direction, the so-called d33-operation. Furthermore, a typical printheadpiezo actuator may comprise several layers of piezo material. The behavior of the piezoactuator can then be described by

[uq

]

=

[1/keff deff

deff C

] [FU

]

, (2.19)

where the inputs F and U are the force on the piezo and the voltage over the piezo, res-pectively. The outputs are the piezo displacement u and the electrical charge q. The piezobehavior itself can be described by the three functional parameters keff , deff and C whichare respectively the effective stiffness, the effective piezoelectric constant and the electricalcapacity. For the derivation of these functional parameters, the reader is referred to [Inte].The piezo model (2.19) consists of a mechanical and an electrical part, namely the first andsecond equation, respectively. The effective piezoelectric constant deff is responsible for thecoupling between the mechanical and electrical domain. It is obvious that there is a strongrelation between (2.18) and (2.19).


Moreover, there is a certain relation between the effective piezo parameters deff and keff

on the one side and α and β in (2.2) on the other. The parameter β is the mechanicalcompliance, thus β ∼ 1/keff . However, β 6= 1/keff , which is caused by a difference in theexact physical meaning of the two parameters. The parameter keff concerns the mechanicalstiffness of only the actuator unit and β concerns the mechanical compliance of the completeprinthead structure. The parameters α and deff are not identical either. deff relates theactuation voltage to the deflection of the piezo and α describes the relation between theactuation voltage and the change in cross-sectional area of the channel.

According to the direct piezoelectric effect, deformation of the piezo results in an electricalfield. When the printhead piezo is used as sensor it produces an electrical charge due to theforce which acts on the piezo. This force is a result of the pressure of the ink inside thechannel which presses against the piezo surface, according to

F =

∫

Aeff

pdA, (2.20)

where p is the pressure of the ink in the channel and Aeff the effective piezo surface. Accordingto the piezoelectric effect, this force results in an electrical charge q which equals

q = deffF. (2.21)

Differentiating q in time leads to the electrical current

i =dq

dt. (2.22)

Finally, the piezo signal which is measured equals the voltage drop over a resistance R,according to

Upiezo = Ri. (2.23)

Here, Upiezo is the obtained piezo sensor signal which is a measure for the time-derivative ofthe force on the piezo.

Piezo Frequency Response Function

With the extension of the piezo signal incorporated in the Matlab model, the theoreticalFrequency Response Function from piezo input to piezo output can be determined. Thistheoretical FRF is depicted in Figure 2.17. In this figure, several resonance frequencies canbe observed. These resonance frequencies correspond to the acoustic modes, see also thepressure profile in Figure 2.10. The first acoustic mode, for instance, is found at 48 kHz. Thehigher-order modes are also clearly present, but they are more averaged out due to the filteringeffect of (2.20). As a result, not all higher order modes can be observed equally well by thepiezo sensor. From the figure it also follows that the global slope of the FRF after the firstmode equals +1. This rather unusual slope in the high-frequency range is caused by the time-derivative in (2.22). Due to this time-derivative, higher order resonance modes have a largeramplification that the first resonance mode, which is dominant for the jet operation. A highgain for high frequencies leads to the undesired amplification of high-frequency measurementnoise, resulting in a rather poor signal-to-noise ratio.

At first sight it may be striking that in the high-frequency range the amplitude has a +1slope and the phase is -90 instead of +90 degrees. The explanation for this is as follows. Apositive actuation signal leads to an increase in channel volume and thus to negative pressurewaves and a negative force. This causes a shift in phase of 180 degrees in de FRF.


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Frequency [kHz]

Mag

nit

ude

[dB

]P

has

e[d

eg.]

Figure 2.17: Theoretical Frequency Response Function from the piezo input to the piezo output.

Numerical Improvement

In the theoretical Matlab model, the available Frequency Response Functions are trans-formed to impulse response functions for simulation purposes, by making use of the fft rou-tine. Simulations have shown that this transformation is rather sensitive for the time andfrequency settings. Numerical improvement can be obtained by fitting good models on theFRFs and use these models instead of the impulse responses for simulation purposes. Notethat the impulse responses can still be obtained by simulation of the fitted models.

2.6 System Identification

For the application of the control strategy Lifted ILC, a model of the system dynamicsis required. The theoretical Matlab model is due to its simplified character not accurateenough for this purpose. There exist various other theoretical inkjet models (see [Bos04]), but,unfortunately, an increase in accuracy comes together with an increase in computation time,which is not desired. In order to obtain a fast model, which describes the system dynamicsaccurate enough, the route of experimental modeling is chosen. Moreover, by experimentallyidentifying the printhead dynamics, it is possible to validate the theoretical models. For now,it is assumed that the actual printhead system behaves linear enough to be able to applylinear identification techniques, which is favorable for experimental modeling. Frequencyresponse measurements are performed via two different methods and a linear empirical modelis obtained by fitting the frequency response data. Next, the obtained empirical model isvalidated and it is shown what the effect is of the nonlinearities which are present in practice.In the last section, the subject model uncertainty is shortly addressed.

2.6 System Identification 27

2.6.1 Frequency Response Measurements

The Frequency Response Function from the actuation signal to the piezo sensor signal ismeasured in two different ways. The first method which is followed is a pseudo sine sweep.For each desired frequency point a clean harmonic signal (with conformable frequency) isoffered and the system time response is stored. By making use of least-square techniques, theFourier coefficients and trend components per frequency point are determined. This makesthe pseudo sine sweep a rather accurate estimation technique, also when the measurementsare disturbed by slowly varying components known as trends. For further information onthe pseudo sine sweep, the reader is referred to [PGP96]. Measurement of the FRF via thesine sweep automatically results in correction for mean values. In this way the offset which ispresent in the sensor signal is reasonably compensated for. However, the mean value does notnecessarily equal the offset exactly. Furthermore, the sine sweep can directly be performedfor the desired frequency range. A drawback of this measurement method is that it takesrelatively much time, what increases the possibility of piezo drift. Piezo drift has two majorcauses. In the first place, a bias voltage which is applied to the piezo results in drift. This biasmakes sure that the total voltage over the piezo has always the same sign, such that the piezomaterial cannot be depolarized. In the second place, drift can be the result of temperaturefluctuations of the piezo, according to the pyroelectric effect [Waa]. The application of a sinesignal to the piezo, superposed on the bias voltage, is schematically illustrated in Figure 2.16.

The second method which is followed is the step response. By applying a fast step inputsignal to the system, practically all printhead dynamics are excited at the same time. Whenthe Fourier transformed of the step response is differentiated in time, the result equals thedesired FRF. An advantage of the step response is that this measurement can be performedvery fast. There is only actuation at time t = 0, so undesired piezo effects, like drift, playno role. However, the method via the step response does not automatically correct for offsetand the frequency vector can not be chosen arbitrarily, but results from the time settings.

It is known that the piezo eigenmodes are in the order of several MHz, therefore, the samplefrequency is chosen equal to 10 MHz for experiments. This is assumed to be a good value toavoid aliasing effects. The two measured FRFs together with the theoretical result are shownin Figure 2.18. From the figure it can be concluded that the FRFs match reasonably well.The most important resonance modes can clearly be observed in all three results. However,there are several differences between the FRFs.

Firstly, the measured FRFs contain an amplifier which introduces substantial phase lagdue to its limited bandwidth. For frequencies far below this bandwidth, the phase lag canbe approximated by a linear phase delay of 0.08 deg./kHz, resulting in 80 degrees delay at 1MHz. The signal sampling in combination with a Zero-Order-Hold (ZOH) also introduces acertain delay. For a sample frequency of 10 MHz this results in 18 degrees delay at 1 MHz.Another issue concerns a possible delay between the internal clock of te waveform generatorand that of the scope. Both clocks are sampling at 10 MHz, but during most measurementsthese clocks were not coupled. In the worst case this results in a delay of 0.1 µs or a phasedelay of 36 degrees at 1 MHz. The reason for not coupling the two clocks during measurementsis that it restricts the time settings of the scope too much. In total, the phase delay at 1MHz equals 134 degrees in the worst case situation. This does still not fully explain therather large difference in phase between theory and measurement for high frequencies. Themismatch in phase makes it assumable that there exist additional delays in the hardwareloop. One of these is probably the phase delay introduced by the electric circuitry which is


used to amplify the piezo sensor signal. Unfortunately, the phase delay of this component isnot exactly known.

Secondly, there is a certain mismatch in amplitude in the low-frequency range betweentheory and practice. In theory, the magnitude of FRF goes to −∞ for low frequencies andin practice it goes to a certain small, but constant, value. This indicates that in theorythe system has no steady-state response. Regarding the working principle of the piezo-jetprinthead this is what one expects. When a step input is applied to the system, acousticwaves arise and they damp out eventually. This supports the assumption that the mismatchis caused by electrical effects which occur during the measurement and that the printheaddynamics itself have no steady-state response.

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200

measured via sine sweepmeasured via step responsetheoretical

PSfrag replacements

Frequency [kHz]

Mag

nit

ude

[dB

]P

has

e[d

eg.]

Figure 2.18: Measured and theoretical Frequency Response Functions from the piezo input tothe piezo output.

2.6.2 Model Fitting

Experiments proved that the FRF measured with the sine sweep is better reproducible. More-over, it is more smooth in the high-frequency range than the FRF resulting from the stepresponse. For these reasons it is chosen to use the FRF which is obtained with the sine sweepfor experimental modeling. In the considered frequency range the global slope of the FRF is+1. This makes the dynamics improper, which is not desired for model fitting. To obtaina proper FRF and to suppress the high-frequency piezo behavior, a 4th-order ButterworthLow-Pass (LP) filter is added with a cut-off frequency of 500 kHz.

As already illustrated, the sample delay has a relatively small influence in the consideredfrequency range. Therefore, it is chosen to directly fit a continuous state-space model on themeasurement data, instead of a discrete-time model. For the fit procedure use is made of the


Matlab command frsfit, which is a least-squares fit routine. This leads to a state-spacemodel of the form

x = Ac x + Bc u

y = Cc x + Dc u,(2.24)

where x is the system state, u is the input, y is the output and Ac, Bc, Cc, Dc are the state-space matrices. The resulting 16th-order fit together with the measured FRF with LP filter areshown in Figure 2.19. From this figure it can be observed that the fitted model captures theimportant resonance modes. Note that the mentioned hardware delays are not compensatedfor during the fitting procedure, because these delays will also be present when ILC is appliedto the printhead.

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200

measurement16th−order fit

PSfrag replacements

Frequency [kHz]

Mag

nit

ude

[dB

]P

has

e[d

eg.]

Figure 2.19: Measured Frequency Response Function and continuous fit from the piezo inputto the piezo output with Low-Pass filter.

The fit routine which is applied produces a state-space model in the control canonicalform. A drawback of this canonical form is the numerical fragility. This, in combination withthe +1 slope and the very high resonance frequencies of the printhead, results in a very largecondition number of the system matrix. Problems which are encountered due to the badnumerical condition of the model are further treated in Chapter 4 and Appendix A.

2.6.3 Model Validation

In order to test the accuracy of the obtained linear model several experiments are performed.In Figure 2.20, the simulated as well as the measured system time responses are shown for anon-jetting and a jetting situation, respectively. In case of the non-jetting situation, relativelysmall signals are used and the simulation matches very well with the measurement. For the


jetting situation with larger signals there are some inaccuracies, but, generally, the responsesstill match reasonably well.

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zosi

gnal

[V]

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1measurementsimulationactuation pulse (scaled)

PSfrag replacements

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Pie

zosi

gnal

[V]

Figure 2.20: Measured and simulated piezo response in case of a non-jetting situation (left)and in case of a jetting situation (right).

The mentioned inaccuracies are mainly caused by problems concerning uncertainty andnonlinearities of the printhead system. This latter cause can be illustrated relatively simpleby superposition tests. According to the superposition principle

y(u1) + y(u2) = y(u1 + u2), (2.25)

with ui the i-th input and y the output, must hold for any linear system. In Figure 2.21, themeasured system responses are shown just below and just above the required jet voltage. Inthe left plot, u1 = u2 is a 5/5/3 trapezium pulse with an amplitude of 11.5 V and in the rightplot u1 = u2 is a 12.5 V 5/5/3 trapezium pulse. From the figure it follows that in case ofthe non-jetting situation (2.25) does not exactly hold, thus the system does not behave fullylinear. Here, no droplets are jetted, therefore, the nonlinearities have to be due to nonlinearpiezo effects and nonlinear structural effects of the printhead. In case of the jetting situationthe inaccuracies in (2.25) become larger. Obviously, the discontinuity in mass when a dropletis jetted plays a role. In Figure 2.21, it can be observed that the sudden decrease in massinside the nozzle results in a small increase in frequency of the first and dominant resonancemode and in a slightly larger amplitude. During the refill of the nozzle the old situation isgradually restored.

Figures 2.20 and 2.21 clearly illustrate that the actual printhead is not completely linear,as expected. However, the nonlinear dynamics can be described reasonably good by the linearempirical model. Fortunately, ILC does not require a perfect model of the system dynamics.A model which describes the most important resonances is sufficient. Moreover, ILC is wellcapable of dealing with small non-linearities, as is illustrated in Chapter 4.

2.6.4 Model Uncertainty

The obtained linear model is denoted by H (or Hpiezo) and can be used for simulation oftime responses and for the derivation of the learning algorithm, see Chapter 4. However, themodel is not an exact representation of the actual nonlinear dynamics of the printhead H as


0 10 20 30 40 50 60 70 80 90 100−0.8

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0.8y(u

1)+y(u

1)

y(u1+u

2)

actuation pulse (scaled)

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Time [µs]

Pie

zosi

gnal

[V]

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0.8y(u

1)+y(u

1)

y(u1+u

2)

actuation pulse (scaled)

PSfrag replacements

Time [µs]

Pie

zosi

gnal

[V]

Figure 2.21: Superposition test for a non-jetting situation (left) and a jetting situation (right).

has already been illustrated with the time responses in the previous section. The differencebetween the actual nonlinear dynamics and the linear model equals the model uncertainty∆H, according to

H = H + ∆H. (2.26)

Roughly speaking, the model uncertainty is introduced by three major causes:

• The assumption that the printhead dynamics are linear.

• The uncertainty in the measured Frequency Response Function.

• Modeling errors which are introduced during the fit procedure.

The first cause of model uncertainty is rather trivial. By working with linear identificationtechniques, only a linear description of the system dynamics can be obtained. In fact, thislinear description equals the actual system dynamics, linearized around a certain operatingpoint. In case of the inkjet printhead, this operating point is equal to U = Ubias. For smalldeviations with respect to the chosen operating point, the linearized dynamics are an accuraterepresentation of the actual dynamics. However, when the deviations from the operating pointincrease, the linearized representation may become inaccurate.

Moreover, there exists a rather large uncertainty in the measurement of the FrequencyResponse Function itself. This is illustrated in Figure 2.22, where three measured FRFs aredepicted for the same ink channel and with the same amplitude of the input signals. All threeFRFs are measured with the pseudo sine sweep identification technique. It can be seen thatthe FRFs do not exactly match.

Finally, model uncertainty is introduced during the fit procedure. The actual dynamicshave an infinitely high order and can, therefore, never be exactly described by a model offinite order, see also Figure 2.19.


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measurement 1measurement 2measurement 3

PSfrag replacements

Frequency [kHz]

Mag

nit

ude

[dB

]P

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e[d

eg.]

Figure 2.22: Measured Frequency Response Functions from the piezo input to the piezo output;illustration of the measurement uncertainty.

2.7 Conclusions

• The main drawbacks which occur during operation of the inkjet printhead are residualacoustic vibrations in the ink channels and cross-talk between the channels. These ef-fects restrict the maximum attainable jet frequency and endanger the droplet properties(print quality) and even overall jet stability.

• The theoretical inkjet model is extended with the piezo sensor functionality and the the-oretical Frequency Response Function from the piezo input to the piezo output matchesreasonably well with the measured FRFs.

• It is illustrated that the dynamics from piezo input to piezo output behave rather linearand a linear fit is made on the measured FRF in order to obtain an empirical model.By means of time responses it is shown that the empirical model provides reasonablygood results.

Chapter 3

Iterative Learning Control

Passive control of the inkjet printhead does not take into account several serious drawbackswhich occur during jetting, as explained in Chapter 2. These drawbacks can be overcomeby a switch from passive to active control. The very high sample frequency (in the order ofseveral MHz) in combination with the highly repetitive character of the jet process make thecontrol strategy Iterative Learning Control very suited for this purpose, because calculationscan be performed off-line. However, this is not the only possibility for the application ofactive control, see [Bos04].

The basic working principle of ILC is explained in Section 3.1 and the mathematicalbackground of the control strategy Lifted ILC is addressed in Section 3.2. In Section 3.3,the extensions of ILC to the MIMO case are shortly sketched and in Section 3.4, severalconcluding remarks are made.

3.1 Basic Working Principle

The technique Iterative Learning Control has been developed several decades ago from apractical environment. The first English publication about ILC was presented in 1984 byArimoto [AKM84]. Because a sound theoretical foundation was not yet available, manydifferent ILC design techniques were published since. It was not before the year 2000 that aunified theory on ILC was presented by Phan, Longman and Moore [PLM00]. Good overviewsof Iterative Learning Controllers are provided in [Moo99] and [Lon00].

The basic working principle of ILC can be illustrated with help of Figure 3.1. In this figure,a feedback loop is shown with plant P and feedback controller C. The feedback controllermay be a PID controller which stabilizes the plant. This part of the scheme operates on-line.The feedforward signal is generated by the learning controller ILC which operates off-line.In words, the ILC algorithm works as follows. The time period in which the system has toperform its (repetitive) task is called a trial and each task is called an iteration. Duringiteration k, the sum of feedback and feedforward input ufb + uk = u is applied to the plant,resulting in output y. The error signal e = yd − y, which equals the difference between thedesired output and the actual output of the plant, is determined and both the error and theinput are stored. Then, using this information, the new feedforward uk+1 for iteration k + 1is computed off-line, according to a certain update law of the learning algorithm. Moreover,a disturbance d may be acting on the system.

For the design of the learning algorithm, often use is made of model information. This

34 Iterative Learning Control

P+ _

I L Co f f - l i n eo n - l i n e

C ++

++

PSfrag replacements

yd e

uk

uufb yd

Figure 3.1: Feedback loop and ILC feedforward loop in the time domain.

model may be both theoretical or empirical. A learning controller which makes use of infor-mation about the system dynamics is referred to as model-based ILC.

When applying ILC, it is assumed that every iteration starts with the same zero initialconditions. A properly tuned ILC may then be able to learn the complete systematic partof the error, resulting in a large decrease in tracking error. It is important to realize thatILC is only useful when the deterministic part of the tracking error is much larger than thestochastic part. The deterministic part is, namely, reproducible and only that part can belearned.

An Iterative Learning Controller iteratively determines the input needed to optimallytrack a given reference trajectory. In case of a printhead, this reference may be the desiredmovement of the meniscus, for example. When the reference is a feasible trajectory, theresulting response can converge to this trajectory within several iterations.

3.2 Lifted ILC

Two important types of model-based learning algorithms are Standard ILC [SM00], [B+] andLifted ILC [TMB01], [BS03], [BS03], [Dij04]. Standard ILC is a technique which is definedin the frequency domain and makes use of frequency response data. Lifted ILC is definedin the trial domain and requires impulse response data of the system. In this study, mainattention is paid to Lifted ILC, because this strategy has several advantages over standardILC. First of all, the stability criterion of Lifted ILC, defined in a finite time setting, is lessconservative than that of Standard ILC, which is defined in an infinite time setting. Thisgreatly increases the flexibility of controller design. Furthermore, Lifted ILC can be applied toSISO as well as MIMO systems and it is rather easy to apply weightings and time windows.A drawback of Lifted ILC is that for long trials the procedure becomes numerically veryintensive. To illustrate the differences between the two learning strategies, a simulation resultwith Standard ILC applied to the printhead system is provided in Appendix A.

As stated before, the model-based technique Lifted ILC is defined in the trial-domain.In the lifted representation, the input and output of the system are discrete finite vectors oflength N . The length of the vectors equals the trial length. The continuous dynamics of theplant are ‘lifted’ to a static, discrete-time mapping function and the ILC feedforward becomesa feedback action in the trial domain. Stabilization of this feedback system is equivalent toconvergence of the ILC in the time domain. Lifted ILC can be used as an add-on feedforwardcontroller which improves the tracking performance of a system that performs repetitive tasks.

3.2 Lifted ILC 35

This system usually is a closed-loop system with a stabilizing feedback controller, but mayalso be a stable plant. In the lifted setting it is possible to make use of optimal controltechniques and to tune the convergence behavior with a weighting parameter, resulting in atrade-off between input effort/noise amplification and convergence/stability.

In this section, firstly the working principle of Lifted ILC is addressed and loss in rankis considered, secondly. Next, the Singular Value Decomposition is treated and two methodsfor deriving the learning matrix are explained. Finally, the convergence properties of LiftedILC are considered and several techniques for additional tuning of the ILC are provided.

3.2.1 Working Principle

Suppose that the linear time-invariant system (Ad, Bd, Cd, Dd) is the discrete-time state-spacerepresentation of the process sensitivity (PS), according to

x(t + ∆t) = Ad x(t) + Bd u(t)

y(t) = Cd x(t) + Dd u(t),(3.1)

where x is the system state. In the trial domain, for iteration k, the closed-loop dynamics (3.1)can then be defined by the convolution

yk(j) =

j∑

i=0

h(i)uk(j − i), (3.2)

for j = 0, 1, . . . , N − 1. Here, h is the impulse response vector and the argument (•) indicatesthe time instant within the trial. Expression (3.2) is equivalent to the static mapping function

yk(0)yk(1)yk(2)

...yk(N − 1)

=

h(0) 0 0 . . . 0h(1) h(0) 0 . . . 0

h(2) h(1) h(0). . .

......

. . .. . .

. . . 0h(N − 1) · · · h(2) h(1) h(0)

uk(0)uk(1)uk(2)

...uk(N − 1)

, (3.3)

or yk = Huk, with H the impulse response matrix. Time invariance implies that each diagonalcontains the same elements, matrix H is Toeplitz. Moreover, H is lower-triangular becausethe system is causal. In (3.3), h(i) are called the Markov parameters of the impulse responsewhich equal

h(0)h(1)h(2)

...h(N − 1)

=

Dd

CdBd

CdAdBd

...

CdAN−2

d Bd

. (3.4)

In the trial domain, both the reference trajectory yd and the deterministic part of the dis-turbance d are constant. According to the Internal Model Principle, a model of the distur-bance should be present in the ILC feedback controller to be able to suppress this distur-bance [FW76]. Asymptotic rejection of a constant disturbance can only be obtained when


each controller disturbance mode can propagate through the plant, thus rank H = N musthold. Then, the ILC system is fully observable.

The Lifted ILC loop is depicted in Figure 3.2. In the figure, the mapping function H is theimpulse response matrix of the closed-loop system PS and the mapping function S representsthe sensitivity. Note that in case of open-loop dynamics (as is the case for the printhead),there is no feedback controller. PS then becomes the plant P and S the identity I. Thelearning matrix is represented by L, which may be non-causal and time-varying. Z−1 = z−1Iis a one trial delay operator and can be seen as a memory block. The one trial delay operatorwith the positive feedback loop forms a bank of integrators which is a copy of the disturbancemodel. As long as the integrators are uncompromised and the ILC feedback loop is stable,asymptotic convergence is obtained.

The signals in Figure 3.2 all have length N and are defined as follows. uk is a vectorwhich represents the system input and yk the system output. yd is the desired output, d isa disturbance and ek is the error signal. The update of the system input is ∆uk and uk+1 isthe input signal for iteration k + 1.

L

HZ - 1

S++

+_PSfrag replacements

∆uk

uk+1 uk yk

yd − d

ek

yd

Figure 3.2: Lifted ILC loop in the trial domain.

The new system input is calculated by adding the previous error multiplied by the learningmatrix L to the input of the previous iteration. The expressions of the lifted ILC loop arethen

ek = −yk + yd = −Huk + yd,

uk+1 = uk + ∆uk,

∆uk = Lek,

u0 = 0.

(3.5)

In (3.5), yd = S(yd−d) and it is assumed that every iteration starts with zero initial conditions.In the trial domain, the closed-loop expression equals

uk+1 = (I − LH)uk + Lyd. (3.6)

3.2 Lifted ILC 37

The evolution of the error signal ek = yd − yk during learning can be expressed by

ek+1 = yd − yk+1

= yd − Huk+1

= yd − H(uk + Lek)

= ek − HLek

= (I − HL)ek.

(3.7)

The necessary and sufficient condition for stability, resulting in asymptotic convergence, is

|λ(I − HL)| < 1, (3.8)

where λ(I − HL) are the eigenvalues of (I − HL). Stated otherwise, the eigenvalues of theclosed-loop system (3.6) have to lie inside the unit disk to obtain a contraction mapping.

In fact, the lifted ILC loop can be seen as an N × N Multi-Input-Multi-Output closed-loop system with unity input matrix and unity system matrix, H as output matrix and nothroughput matrix. The states of the ILC system are then denoted by uk, the iterationnumber k refers to the time and L is a feedback matrix. The controllability matrix C of theILC system then equals

C =[

IN ININ · · · IN−1

N

](3.9)

and the observability matrix O is given by

O =

HHIN

...

HIN−1

N

, (3.10)

where IN is the identity of size N ×N . Regarding the rank of both matrices, it easily followsthat the ILC feedback system is always fully controllable and only fully observable when Hhas full rank.

3.2.2 Loss in Rank

All possible system outputs yk lie in the image of matrix H, yk ∈ im H. When H is singular,rank H < N , the feedback loop cannot affect all poles and asymptotic convergence is notpossible. There are two reasons for loss in rank [Bos03]. For example, when a delay of nsamples is present in the system, the first n Markov parameters of the impulse response areequal to zero and, as a result, n modes are unobservable. Loss of rank can also be caused bytransmission zeros outside the unit circle, which are called nonminimum phase zeros. This isnow explained in more detail. Suppose that a system has one nonminimum phase zero. Bydefinition of zeros there always exist an initial input u0 and an exponential input signal uz

containing this zero as exponential factor such that

Huz + Ou0 = 0. (3.11)


In (3.11), O is the observability matrix of the underlying LTI system, which equals

O =

Cd

CdAd

CdA2d

...

CdAN−1

d

. (3.12)

Then, the upper bound on the smallest non-zero singular value σ of H is given by

σ(H) <‖Ou0‖‖uz‖

. (3.13)

For a nonminimum phase zero, ‖uz‖ will grow exponentially for larger dimensions N , forcingσ(H) to be small. Thus, for increasing dimension, additional loss in rank occurs when non-minimum phase zeros are present.

3.2.3 Singular Value Decomposition

When H does not have full rank, the Singular Value Decomposition (SVD) provides the rank.The SVD of H is given by

H = UΣV T =[

U1 U2

][

Σ1 00 Σ2

] [V T

1

V T2

]

. (3.14)

Here, V is a matrix with singular vectors which represent input trajectories and U is a matrixwith singular vectors of the output trajectories. Both V and U are orthogonal matrices,UT U = V T V = I. Σ is a diagonal matrix containing the singular values σi, ordered fromlarge to small. Thus, trajectories in the direction vi are amplified with gain σi in the directionui. Σ2 contains unobservable part of the system and is (nearly) equal to zero. The rank ofH equals the number of singular values in Σ1.

For an infinitely large impulse response matrix (N → ∞) of an LTI system, the SVD infact provides the same information as the well-known Frequency Response Function in thefrequency domain, as is illustrated in [Dij04]. The frequency components of a certain singularvector have identical system gain, thus same magnitude in a Bode plot, which is equal tothe corresponding singular value, see for example Figure 4.3. Phase shift is given by thedifference between the input and the output singular vectors, as is illustrated in Figure 4.4.To circumvent further problems in case H is rank-deficient, a new input w can be defined by

Hw = Hw1 + Hw2, (3.15)

where w1 ∈ im V1 and w2 ∈ im V2. Thus

Hw ≈ U1Σ1VT1 w1, (3.16)

as V T1 w2 = V T

2 w1 = 0. Now, w1 is chosen as

w1 = V1u. (3.17)

For the closed-loop system the expressions

uk+1 = (I − L∗HV1)uk + L∗yd,

ek = −HV1uk + yd

(3.18)

are obtained. By omitting Σ2, model reduction is applied. As a result, the learning matrix isnon-square (indicated by the ‘∗’) and the dimension of vectors uk and ∆uk is reduced.

3.2 Lifted ILC 39

3.2.4 The Learning Matrix

The fact that the Lifted ILC loop in the trial domain can be regarded as a feedback loop makesit possible to apply general linear feedback techniques for designing the learning matrix L. Inthis section, two of these methods are treated, namely pole assignment and Linear Quadratic(LQ) optimal control.

Pole Assignment

In case of pole assignment the desired location of the poles is chosen and the resulting learningmatrix can be determined. Let

L∗HV1 = L∗U1Σ1VT1 V1 = αI, (3.19)

then the learning matrix equals

L∗ = αΣ−11

UT1 . (3.20)

For α = 1, dead-beat control is obtained. All poles of the closed-loop system equal zero. Incase of dead-beat control, convergence takes place in just one iteration. However, the systemoutput yk only follows the reference yd in subspace im H = im U1. A drawback of dead-beatcontrol is the large sensitivity for model uncertainties. The parameter α can be used to weighrobustness against convergence speed.

LQ-Optimal Control

A method which works more subtle is LQ-optimal control. Here, a quadratic criterion isminimized by solving a Discrete-time Algebraic Riccati Equation (DARE), resulting in anoptimal solution for the learning matrix. The LQ-criterion, with WQ a matrix for outputweighting and WR a matrix for input weighting, equals

Cr =∞∑

k=1

(yTk WQyk + ∆uT

k WR∆uk)

=∞∑

k=1

(uTk V T

1 HT WQHV1uk + ∆uTk WR∆uk).

(3.21)

Note that WQ must be positive definite and WR positive semi-definite. A useful choice forthe weighting matrices is WQ = I and WR = βI. The criterion then becomes

Cr =∞∑

k=1

(uTk Σ2

1uk + β∆uTk ∆uk). (3.22)

In this way there is one parameter, β, left for tuning of the learning algorithm. The solutionto this LQ-optimal control problem is

∆uk = −(βI + X)−1Xuk, (3.23)

with X the stabilizing solution of the DARE

X = X + V T1 HT HV1 − X(βI + X)−1X, (3.24)

which can be simplified to

0 = Σ21 − X(βI + X)−1X. (3.25)


Approximated Solution From (3.25) it becomes clear that X is a diagonal matrix. Theentries of X on its diagonal are xi and those of Σ1 equal σi. The expression for xi is given by

xi =1

2σ2

i

(

1 +

√

1 +4β

σ2i

)

≈ σ2i + β. (3.26)

This approximation holds if 4β

σ2

i

1 and results in

X = Σ21 + βI. (3.27)

The approximated solution is in fact the exact solution to an LQ-optimal control problemwith a slightly different weighting WQ [DB02]. Substitution of (3.27) in (3.23) yields

∆uk = −(2βI + Σ21)

−1(Σ21 + βI)uk

= −L∗HV1uk

(3.28)

for the ILC feedback law. In the second equation of (3.28) only the loop behavior is considered.Solving the feedback law (3.28) for L∗ results in an expression for the learning matrix

L∗ = (2βI + Σ21)

−1(Σ1 + βΣ−11

)UT1 . (3.29)

The learning matrix can be seen as an adjusted inverse of the plant. The adjustment is causedby the introduced weighting β. Generally, the learning matrix is non-causal and time-varying.By using (3.29), the closed-loop matrix becomes

I − L∗HV1 = I − (2βI + Σ21)

−1(βI + Σ21) = βI(2βI + Σ2

1)−1, (3.30)

with closed-loop poles

λi =β

2β + σ2i

. (3.31)

Exact Solution Without approximating the solution of the DARE, an exact expression forthe learning matrix can be obtained. The learning matrix is then given by

L∗ = (βI + X)−1XΣ−11

UT1 (3.32)

and the closed-loop matrix equals

I − L∗HV1 = I − (βI + X)−1X = βI(βI + X)−1. (3.33)

The closed-loop poles of the exact solution are

λi =β

β + xi

=β

β + 1

2σ2

i

(

1 +√

1 + 4β

σ2

i

) .(3.34)

3.2 Lifted ILC 41

3.2.5 Convergence and Robustness

A model of a system never exactly captures all system dynamics. There always exists modeluncertainty, see also Section 2.6.4. How large this uncertainty is depends both on the sys-tem and the quality of the model. Suppose that the uncertainty in system H equals ∆H.Convergence is then obtained when for all closed-loop trial poles

|λ(I − L(H + ∆H))| < 1 (3.35)

holds. The feedback matrix can be written as

L = V1

l1 0 · · · 0

0 l2. . .

......

. . .. . . 0

0 · · · 0 lN∗

UT1 , (3.36)

with the elements given by

li =2

1 +√

1 + 4β

σ2

i

σ−1

i . (3.37)

Regarding (3.37), two extreme situations can be distinguished. When β is chosen much largerthan a singular value, σ2

i β, then

li ≈ 0 (3.38)

holds. Thus, for these singular values ILC is inactive and robust stability is achieved. Usually,the model uncertainty is the largest for small singular values and the model uncertainty ∆Hhas practically no influence on them. For the closed-loop trial poles it then follows that

λi ≈ 1. (3.39)

When β is much smaller than a singular value, σ2i β, the ILC behaves like an inverse of

the system, according to

li ≈ σ−1

i . (3.40)

This results in dead-beat control. The closed-loop trial poles are now located at

λi = λi(−viσ−1

i uTi ∆H), (3.41)

where vi and ui are singular vectors of V and U , respectively. Obviously, in case there existsno model uncertainty, the trial poles are located at 0. When an uncertainty is present and βis at least chosen in the same order of magnitude as ∆H, then the trial pole will be smallerthan 1 and robust convergence is obtained. This implies that the tuning parameter β can beseen as as upper limit for model uncertainty.

Summarizing, for large singular values σi the closed-loop poles lie near the origin and LQ-optimal control approximately provides dead-beat performance. For small singular values theclosed-loop poles lie close to the unit circle. Now, the gain in the feedback loop is almostzero, resulting in very slow convergence. The choice for the tuning parameter β determines for


which singular values the system is inverted and thus which singular values can be effectivelylearned. Roughly speaking, only the singular values with σi >

√β contribute to the feedback

loop. A small β results in high performance (large error reduction) and fast convergence.For small β, large input signals are allowed, see the cost criterion (3.22). Applying largeinput signals does not result in a robust solution. Moreover, the system model is usually notexactly known for high frequencies, where the singular values are normally small. Therefore,to obtain a robust solution, β must not be chosen too small and the choice should dependon the quality of the system model. Obviously, there is a trade-off between performance andconvergence speed on the one side and robustness on the other.

A properly tuned lifted ILC is capable of strongly suppressing all disturbances which areconstant or vary a little from trial to trial. However, this is at the cost of amplification ofworst case trial-varying noise which is acting on the system, according to Bode’s sensitivityintegral in the trial domain. The effect of ILC on trial-varying disturbances can be illustratedby a trial domain sensitivity function, as is shown in [Dij04].

3.2.6 Additional Tuning

It may be desirable to add additional possibilities for tuning of the learning controller. Inthis section, the introduction of a scalar learning gain, an additional input weighting and anactuation window are discussed.

Scalar Learning Gain

A first, relative simple, addition is the introduction of a scalar learning gain γ. With thisparameter it can be chosen to what extent the calculated update signal ∆uk is actually usedto update the input signal. In case a scalar learning gain is present, the expression for theclosed-loop poles becomes

λi =β + (1 − γ)xi

β + xi

=β + 1

2(1 − γ)σ2

i

(

1 +√

1 + 4β

σ2

i

)

β + 1

2σ2

i

(

1 +√

1 + 4β

σ2

i

) .

(3.42)

Obviously, for γ = 1, (3.42) results in (3.34). By comparing these expressions it follows thatin case 0 < γ < 1 the pole locations are shifted towards 1, where model uncertainties haveless influence. Thus, by choosing 0 < γ < 1 the lifted ILC can be made more robust.

Note that there is a fundamental difference between the tuning parameters β and γ. Thisis graphically illustrated in Figure 3.3. β determines which singular values have effect in thefeedback loop and γ determines how large the actual update step is in the calculated direction∆uk. Thus, γ influences the convergence speed and the robustness, but it does not influencethe final tracking error, provided that that ILC is stable. The tuning parameter β, however,does affect the final tracking error.

Additional Input Weighting

When it is desired that the input is only active in the first part of the trial, this can simplybe imposed by an input weighting matrix Wa which cuts off all inputs after a defined sample.

3.2 Lifted ILC 43

PSfrag replacements

0 element number i

1

λi

1

γ

1

β

Figure 3.3: Graphical illustration of the influence of γ and β on the closed-loop trial poles.

This matrix may be diagonal with entries equal to unity for the first N ∗ elements and thefollowing N − N∗ entries equal to zero. When desired, it is also possible to define a moresmooth transition from unity to zero. The reduced ILC loop with additional learning gain γand weighting filter Wa incorporated is depicted in Figure 3.4.

L

Z - 1

S++

+_H V 1W a

*

PSfrag replacements

∆uk

uk+1 uk yk

yd − d

ek

yd

Figure 3.4: Reduced Lifted ILC loop in the trial domain, with scalar learning gain and addi-tional input weighting filter.

Actuation Window

It is possible to let actuation and/or observation take place only in specific time ranges duringa trial, by the introduction of time windows. Firstly, it is explained how time windows workand what time windows can mean for control of the printhead. Secondly, the influence oftime windows on the observability is treated.

Working Principle For a pick-and-place robot, for example, the tracking error during themotion itself is not so important, but the final position is, see [DB03], [Oos03] and [Dij04].Here, actuation only takes place during the motion and observation during stand-still, wherea small error is desired.


In case of an inkjet printhead, the tracking behavior in the complete trial is important.However, it is desirable to restrict the actuation to the first part of the trial. This avoidsthat the duration of the actuation signal becomes too long, which is undesired for high jetfrequencies. In Figure 3.5, the principle of an actuation window is schematically illustrated.The trial length is denoted by N and the length of the actuation window by N ∗.

a c t u a t i o n

o b s e r v a t i o nPSfrag replacements

0 N element number [-]

1

0 N∗ element number [-]

1

Figure 3.5: Illustration of an actuation window.

Consider once again the system matrix H and let N ∗ be the sample where the actuationmust stop. The system mapping is then defined by

yk(0)...

yk(N∗)

yk(N∗ + 1)...

yk(N − 1)

=

[H11 0H21 H22

]

︸︷︷︸

H

uk(0)...

uk(N∗)

uk(N∗ + 1)...

uk(N − 1)

. (3.43)

Since only actuation is applied until sample N ∗, the relevant part of the mapping matrix Hequals

yk(0)...

yk(N∗)

yk(N∗ + 1)...

yk(N − 1)

=

[H11

H21

]

︸︷︷︸

H∗

uk(0)...

uk(N∗)

. (3.44)

The Riccati equation for the relevant part of the system matrix becomes

0 = (H∗)T H∗ − X∗(βI + X∗)−1X∗, (3.45)

with X∗ the solution to the Riccati equation. For the feedback interconnection it then follows

L∗H∗ = (βI + X∗)−1X∗

= (X∗)−1(H∗)T H∗

L∗ = (X∗)−1(H∗)T .

(3.46)

3.2 Lifted ILC 45

Just like the addition of an input weighting matrix, the actuation windows results in a zeroinput after a predefined sample. However, the strategy with the actuation window is moresubtle, because it takes the zero input in the last part of the trial into account during thedesign process of the learning algorithm and not just cuts off the input after sample N ∗.

The effect of an actuation window can in principle also be reached by proper tuning ofweighting filters WQ and WR. Advantages of the actuation window above tuning the weightingmatrices are that it is more easy to implement and that a reduction of the impulse responsematrix is possible, what has a positive effect on the computation time.

Influence on Observability Having explained the motivation for an actuation windowis illustrated and the working principle, the question may arise what the influence is on theobservability of the ILC. The Singular Value Decomposition of the reduced impulse responsematrix H∗ of size N × N∗ (with N∗ < N) is as follows

H∗ = U∗Σ∗(V ∗)T . (3.47)

In (3.47), U∗ is an N ×N matrix of which the N columns are the output singular vectors and(V ∗)T is an N∗ ×N∗ matrix of which the N ∗ rows are the input singular vectors. Matrix Σ∗

is of size N × N∗ and contains the N ∗ singular values, according to

Σ∗ =

σ∗

1 0 · · · 0

0 σ∗

2

. . ....

.... . .

. . . 00 · · · 0 σ∗

N∗

0 0 · · · 0

0 0. . .

......

. . .. . . 0

0 · · · 0 0

. (3.48)

Matrix U∗ obviously has the same column space as U , im U ∗ = im U . This could lead tothe false assumption that also in the reduced setting with the actuation window all arbitraryoutput signals can be realized, but this is not true. The explanation for this is as follows.Because every input signal consists of N ∗ elements, it is only possible to influence maximalN∗ elements of the output signal independently. The remaining N − N ∗ elements of theoutput signal are linearly dependent on the first N ∗ elements. This explanation supportsthe intuitive observation that when the size of the actuation window is decreased it becomesharder to reach a certain control goal.

Alternatively, above statement can be illustrated by means of the image of the impulseresponse matrix. For simplicity, it is assumed that the impulse response matrix H has fullrank. Without actuation window, the image of H then equals

im H = im IN , (3.49)

where IN is the identity of size N ×N . In case an actuation window is applied, the image ofH∗ becomes

im H∗ = im

[IN∗

F

]

. (3.50)


In (3.50), IN∗ is the identity of size N ∗ × N∗ and F is a full matrix of size (N − N ∗) × N∗.The image of H∗ clearly illustrates that it is only possible to influence N ∗ elements of theoutput signal independently.

The application of an actuation window thus implies a certain loss in observability and,therefore, a decrease in performance. It is no longer possible to influence all elements ofthe output signal independently. However, from simulations and experiments it follows that,when the actuation window is not chosen too small, the decrease in performance is acceptable.The fact that the length of the actuation signal can be reduced is far more important.

3.3 MIMO Lifted ILC

As stated before, the printhead under investigation has a high integration density of inkchannels. Due to this high integration density the channels interact with each other, which iscalled cross-talk. In case of active control by ILC, it should be possible to learn the systematiceffects of these interactions and to reduce the cross-talk. For this purpose, a MIMO learningcontroller can be applied, for the actuation of an array of channels. Although Lifted ILC inSection 3.2 is explained for the SISO case, the derivation of the algorithm also holds for theMIMO case. The framework of the model-based technique Lifting ILC is in fact designed forthe MIMO case.

In Figure 3.6, two neighboring channels are considered, denoted by A and B. The inputsof the channels are uA and uB and the outputs equal yA and yB. The transfer function frominput ux to output yx is given by Hx and cross-talk between the channels is captured by HAB

and HBA. In case both channels are exactly identical (no production tolerances), HA = HB

and HAB = HBA hold.

A B

ink ch

annelPSfrag replacements

uA uB

yA yB

HA HB

HAB

HBA

Figure 3.6: Graphical representation of two neighboring ink channels and the involved transferfunctions.

3.3 MIMO Lifted ILC 47

Just like in the SISO case, the system dynamics can be transformed into a static mappingfunction in the trial domain. For two channels this results in

yA(0)yB(0)yA(1)yB(1)

...yA(N − 1)yB(N − 1)

=

hA(0) hBA(0) 0 0 . . . 0 0hAB(0) hB(0) 0 0 . . . 0 0

hA(1) hBA(1) hA(0) hBA(0). . .

......

hAB(1) hB(1) hAB(0) hB(0). . . 0 0

......

. . .. . .

. . . 0 0hA(N − 1) hBA(N − 1) · · · hA(1) hBA(1) hA(0) hBA(0)hAB(N − 1) hB(N − 1) · · · hAB(1) hB(1) hAB(0) hB(0)

uA(0)uB(0)uA(1)uB(1)

...uA(N − 1)uB(N − 1)

.

(3.51)

In short, this is again denoted by yk = Huk and the learning algorithm can be designed in asimilar way as for the SISO case. Extension of (3.51) for more channels is straightforward.

A MIMO lifted ILC is fully observable when m ≥ l, with m the number of system inputsand l the number of system outputs, and the impulse response matrix has full rank, rankH = Nl. Stated otherwise, all columns of H should be linearly independent. Disturbancerejection is obtained if m ≤ l. Thus, both conditions can only be fulfilled when the systemis square. When these requirements are not satisfied, not all modes are asymptotically stableand the disturbance may not be fully rejected. A serious drawback of MIMO Lifted ILC isthat the involved impulse response matrix and the learning matrix become even larger asin the SISO case. Therefore, MIMO Lifted ILC demands for more computer memory andcomputation power. For systems with several inputs and outputs the ILC may even becometoo large for a computer to handle. To significantly reduce the required computer memory andcomputation power, one can make use of a Hamiltonian simulation to calculate the update ofthe input rather than using to full matrix expressions. For more information on this solution,the reader is referred to [Bos04].

Because of the strong similarity between SISO and MIMO lifted ILC, MIMO ILC is notfurther discussed here. For additional information on ILC for MIMO systems, the reader isreferred to [RB00a], [RB00b], [BS03] and [Kaa04].


3.4 Conclusions

• The robustness of the Iterative Learning Controller can be adjusted with the controlparameters β and γ, which both influence the closed-loop trial poles in a different way.

• By introducing an actuation window, the length of the actuation signal can easily berestricted. However, working with an actuation window means a loss in observability,which may lead to a decrease in performance.

• It is illustrated how the Lifted SISO ILC implementation can be extended for the ap-plication of MIMO ILC on an array of ink channels.

Chapter 4

Implementation of SISO ILC

In this chapter, the implementation of Iterative Learning Control on an actual inkjet printheadis considered. For this purpose, a SISO ILC is derived which is applied to a single channel.Use is made of the piezo sensor signal as controlled output. The control goal in this chapteris active damping of the residual vibrations in order to be able to increase the jet frequencywithout affecting the droplet properties negatively. A high jet frequency is favorable forachieving higher printing speeds. To achieve this goal, two different implementations of SISOILC are discussed.

First, in Section 4.1, it is explained how a controller can be implemented on the inkjetprinthead. In Section 4.2, the derivation and results of an ILC are treated which makes directuse of the measured piezo sensor signal. An alternative implementation, which is based on thetime-integrated sensor signal, leads to better results and is considered in Section 4.3. Finally,Section 4.4 summarizes the conclusions which are made in this chapter.

4.1 Implementation Issues

Before the experimental results of ILC are given, it is shortly described in this section howa controller can be implemented on the inkjet printhead. Firstly, the implementation of thecontroller itself is considered and, secondly, several practical considerations are addressed.

4.1.1 Controller Implementation

In Figure 2.7, it has already been illustrated which devices are relevant in the setup and howthey are connected to each other and to a personal computer. However, it has not yet beenexplained what happens on software level. The relevant software and the communicationlines are schematically depicted in Figure 4.1. The software package which is of main interestis Labview. With an interface card this program is coupled to the hardware. Labview iscapable of reading data from the different sensors and sending control signals to the relevantdevises. On software level Labview communicates with Matlab, in which the controlleris implemented. Moreover, Matlab is used for complex calculations. For data storage anddata reading, use is made of Excel. In Excel, all relevant time signals are stored.

50 Implementation of SISO ILC

L a b v i e w

E x c e l M a t l a b

H a r d w a r e

d a t as t o r a g e

d a t ar e a d i n g

c o n t r o ls i g n a l s

s e n s o rs i g n a l s

c a l c u l a t i o nr e q u e s t s

c a l c u l a t i o nr e s u l t s I n t e r f a c e

Figure 4.1: Schematic illustration of the communication between the software and the hard-ware.

4.1.2 Practical Considerations

Next, several practical issues have to be considered in order to implement the ILC successfullyon the printhead. These considerations are summarized below. The first three issues also holdfor the identification experiments.

Offset Correction

The measured piezo signal comprises a certain DC-offset which is induced by several electricalcomponents. This offset is not a fixed value, but drifts in time. Because the exact value ofthe offset is unknown and not constant, it is chosen to subtract the mean from the measuredpiezo signal. Note that the mean value is not necessarily equal to the offset.

Signal Averaging

The measurement of the piezo signal is rather sensitive for disturbances and contains high-frequency noise. Therefore, it is chosen to always use an averaged signal instead of a singlemeasurement. For all experiments the piezo signal is averaged 150 times before it is used forprocessing.

Temperature Isolation

The piezo sensor signal is also rather sensitive for piezo drift. One cause of piezo drift is tem-perature fluctuation. When the temperature of the piezo changes, this causes an electricalcharge according to the pyroelectric effect. In order to minimize temperature fluctuations,the printhead is isolated with a metal cover which is provided with heating elements. Duringexperiments, the cover is then heated till it reaches a temperature just below the tempera-ture of the printhead itself. In this way, temperature fluctuations of the piezo actuator areminimized, without affecting the temperature control of the head.

4.2 SISO ILC 51

Low-Pass Filtering

As already mentioned in Section 2.6.2, a Low-Pass filter is added to the printhead dynamicsto suppress the high-frequency piezo dynamics and in order to obtain a proper fit. This filterconcerns a 4th-order Butterworth filter with a cut-off frequency of 500 kHz. Also during ILCexperiments, use is made of this filter.

Initial Conditions

One property of ILC is that every iteration requires the same zero initial conditions. Statedotherwise, the system has to be at rest before the next iteration can be performed. This,however, concerns exactly one of the operational issues of the inkjet printhead. Due to thelow passive damping of the channel acoustics, the maximum jet frequency is limited. In orderto obtain zero initial conditions, which are needed for ILC, the jet frequency is set at a lowvalue of 1 or 2 kHz during learning. In case of passive control, it takes in the order of 200µs for the system to become at rest. Thus for the chosen jet frequency the channel is by allmeans at rest. Reducing the jet frequency may seem conflicting, because here the goal ofILC is to increase the jet frequency. However, when the ILC is converged and the residualvibrations are actively damped, then the jet frequency can be increased.

4.2 SISO ILC

In this section, active damping of the channel acoustics after droplet ejection stands central.For this purpose an ILC is applied which makes use of direct measurement of the piezo sensorsignal. The learning algorithm is derived and some numerical issues are treated. Next, severalexperimental results of ILC are given and discussed.

4.2.1 Derivation of the Learning Algorithm

By making use of the 16th-order model which has been fitted on the measured FRF frompiezo input to piezo output, the impulse response h can be determined. A trial length of100 µs seems a proper value, see Figure 4.2 (left). For this choice, all relevant dynamics arecaptured without letting the trial length become too large. For a sample frequency of 10MHz, a trial length of 100 µs leads to vectors of length N = 1000. The impulse response canbe used to construct the impulse response matrix H. For the chosen trial length this resultsin a H matrix of size 1000 × 1000. The impulse response function and the singular values ofthe impulse response matrix are depicted in Figure 4.2. Due to the +1 slope of the FRF, the(for the jet process) important first resonance mode can hardly be observed in the impulseresponse, which is dominated by the higher harmonics. The rank of the impulse responsematrix equals 996, what corresponds with a loss in rank of 4. This loss in rank can also beobserved in the figure with the singular values, where the last four are significantly smallerthan the others. In this case, the loss in rank is caused by four nonminimum phase zeros ofthe underlying LTI model. Possibly, these nonminimum phase zeros are introduced duringthe fit procedure.

In Chapter 3, it is explained that the Singular Value Decomposition in fact containsthe same information as the familiar Frequency Response Function. This is illustrated inFigure 4.3, where it can be observed which singular values correspond to which frequencies.


0 10 20 30 40 50 60 70 80 90 100−0.015

−0.01

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Am

plitu

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[V/V

]

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10−4

10−2

100

σi

sqrt(β)PSfrag replacements

Element number i

Sin

gula

rva

lue

σi

Figure 4.2: Impulse response from the piezo input to the piezo sensor signal (left) and singularvalues of the impulse response matrix (right).

From the figure it can also be seen that the first resonance mode does not correspond to thelargest singular values, as is usually the case. The reason for this is the unusual +1 slopeof the FRF which results in large singular values for the high instead of the low frequencycontent. For control purposes and robustness this effect is not favorable, because now therelevant system dynamics are not captured by the largest singular values. Moreover, the highfrequency dynamics, where the model uncertainty is typically the largest, are now capturedby the large singular values.

To be able to attain high jet frequencies it is undesired to work with long actuationsignals. When the full model is used this would lead to a signal length of 100 µs which isvery long compared to the fixed pulse of 13 µs. For this reason an actuation window of 50µs is introduced. Experiments have shown that with a smaller window ILC is not capableanymore of controlling the system properly. With the introduction of the actuation windowN∗ becomes 500 and the new non-square impulse response matrix H∗ is now of size 1000×500.The impulse response matrix now has full rank, rank H∗ = 500, and is used for the derivationof the learning matrix.

4.2 SISO ILC 53

200 400 600 800 1000−60

−55

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−10

PSfrag replacements

Element number i

20lo

g10(σ

i)[d

B]

Frequency [kHz]M

agnit

ude

[dB

]

Figure 4.3: Logarithm of the singular values (left) and amplitude plot of the Frequency Res-ponse Function from the piezo input to the piezo sensor signal (right).

The 1st and 71st input and output singular vector of the reduced impulse response matrixH∗ are depicted in Figure 4.4. Singular vectors v1 and u1 are the vectors with the largestamplification σ1 and singular vectors v71 and u71 correspond to the first resonance mode ofthe channel acoustics. When the input singular vectors are compared to the output singularvectors, two things can be observed. In the first place, the input singular vectors have half thelength of the output singular vectors. The applied actuation window is obviously responsiblefor this. In the second place, it can be seen that an input singular vector approximatelyequals the first 500 elements of its output singular vector. The only difference is the delaywhich is present between the signals. This delay is caused by the phase delay of the FrequencyResponse Function, as is explained in Section 3.2.3.

In Figure 4.5, the frequency spectrum of both the 1st and the 71st output singular vectoris shown. The dominating frequency component of u1 equals 400 kHz. This is indeed inagreement with Figure 4.3, where it can be seen from the FRF that the largest amplificationtakes place at 400 kHz. The dominating frequencies of u71 are 50 kHz and 750 kHz, accordingto Figure 4.5. The fact that the first resonance mode, which is dominant for the jet process,is not captured before singular value 71, again illustrates that the first resonance mode is notwell observable in the current situation.


100 200 300 400 500 600 700 800 900 1000

−0.1

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0

0.05

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0.151st output sing. vect.71st output sing. vect.

100 200 300 400 500 600 700 800 900 1000

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−0.05

0

0.05

0.1

0.151st input sing. vect.71st input sing. vect.

PSfrag replacements

Element number i

Val

ue

Val

ue

Figure 4.4: 1st and 71st input and output singular vectors of the reduced impulse responsematrix H∗.

100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5x 10−6

1st output sing. vect.71st output sing. vect.

PSfrag replacements

Frequency [kHz]

Mag

nit

ude

[-]

Figure 4.5: Frequency spectrum of the 1st and 71st output singular vector of the reducedimpulse response matrix H∗.

4.2 SISO ILC 55

For tuning of the ILC, use is made of the tuning parameters β and γ. These parametersare chosen equal to β = 2 · 10−3 and γ = 0.5, which is a rather robust choice. The reasons forthis robust choice are the rather large model uncertainty and the bad numerical condition ofthe model. In Figure 4.2(right),

√

(β) is plotted together with the singular values. Roughlyspeaking only the singular values which are larger than

√

(β) do significantly contribute tothe ILC feedback loop1.

There are several ways to solve the linear quadratic criterion and obtain the learningmatrix L. These methods are the following:

• Exact solution of the Discrete-time Algebraic Riccati Equation, solved in matrix formwith the Matlab routine dare.

• Exact solution of the DARE, solved element-wise.

• Simulation of the update of the input, by making use of a Hamiltonian simulation.

• Approximated solution of the DARE.

The first method can in principle be applied, but, when the impulse response matrix becomestoo large, the routine fails due to numerical problems. Alternatively, the exact solution canbe determined element-wise. However, during experiments, it appeared that the ILC basedon the exact solution result in rather large input signals. These large input signals are notfavorable with respect to robustness and the limited actuation range of the hardware. Insteadof calculating the learning matrix, the update of the input signal can also be determined bymaking use of a Hamiltonian simulation, see [Bos04]. The Hamiltonian simulation is basedon the dynamics of underlying LTI system and produces nearly identical results as the exactsolution. Unfortunately, due to the bad numerical condition of the experimental printheadmodel, this method cannot be applied. The forth method, the approximated solution, is infact the exact solution to a DARE with slightly different output weighting. The approximatedsolution results in satisfying convergence behavior, without producing too large input signals.Therefore, the approximated solution is used for the ILC experiments. A drawback of workingwith the approximated solution is that it is not possible to extract certain singular values fromthe feedback loop. The extraction of singular values from the solution is in fact possible whenthe exact solution is applied. A learning matrix which is based only on certain singular values,which are capable of describing the relevant dynamics, can be favorable to avoid instabilityof high-frequency components, where the model uncertainty is typically large.

The resulting learning matrix, obtained with the approximated solution, is depicted inFigure 4.6. Obviously, the size of the learning matrix is 500 × 1000, as it determines thenew 50 µs input signal based on a 100 µs measurement. It can be seen that the elements onthe ‘diagonal’ of the matrix are not constant, but vary in time. Moreover, the elements ofan arbitrary column are unequal to zero above the diagonal, what is due to the anti-causalbehavior. This illustrates that the optimal feedback interconnection of the ILC system is amixed causal/anti-causal, time-varying operation, which is typical for ILC.

1The SVD of H∗ results in only 500 singular values instead of 1000. Now, the same system information has

to be spread over only 500 singular values, therefore, these 500 singular values are not equal to the first 500singular values in Figure 4.2(right).


Figure 4.6: SISO ILC; surface plot of the learning matrix.

4.2.2 Experimental Results

In case of passive control, the inkjet printhead is actuated with a fixed trapezium-shapedpulse. This pulse has a rise time of 5 µs, then stays at a constant level for another 5 µs,followed by a drop time of 3 µs. In future, this pulse will simply be referred to as a 5/5/3pulse. In case of active control, it is chosen to jet the same droplets as are produced by a5/5/3 pulse with an amplitude of 30 V , but now active damping of the residual vibrationsis applied. A trivial choice for the initial input u0 is then the 30 V 5/5/3 pulse itself. Theresponse of the piezo signal on the fixed pulse is shown in Figure 4.7. This experiment isdenoted by iteration 0.

The desired control goal, active damping of the residual acoustic vibrations, can be at-tained by choosing a proper reference trajectory. To obtain the same droplets as in the passivecase, the desired response equals the measured response for the first 25 µs. The droplet isejected between 20 and 25 µs, thus it is expected that this choice does not disturb the dropletproperties. After 25 µs the desired piezo signal is damped during a single period of thedominating harmonic. Experiments have shown that it is not possible to damp the channelmuch faster, without affecting the first part of the response where the droplet is generated.Moreover, refill of the nozzle has to be ensured and high voltage input signals have to beavoided.

4.2 SISO ILC 57

0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0

0.5

1measurementreference

0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

30

40

PSfrag replacements

Time [µs]

Pie

zosi

gnal

[V]

Input

[V]

Figure 4.7: SISO ILC; measured piezo signal, desired piezo signal and input signal for iteration0.

In Figure 4.8, the same signals are shown, but now for iteration 25. It can clearly be seenthat ILC is capable of actively damping the residual channel acoustics. To achieve this, ILCadjusted the first 50 µs of the input signal. Roughly speaking, the input signal is shapedopposite to the dominating harmonic in order to actively damp it. Moreover, the figureshows that dynamics with a high frequency component are present in the input as well asthe output signal. Apparently, ILC is unstable for this high frequency, which corresponds toone of the resonance modes of the piezo unit itself. This may be due to a mismatch betweenthe empirical model and the measured FRF in the high-frequency range (see Figure 2.19),which results in instability, despite of the applied LP filter. When the model uncertainty ∆Hbecomes larger than β, several closed-loop trial poles may shift outside the unit circle. Itis possible that the observed high-frequency component is only an electrical effect and doesnot affect the channel acoustics and the droplet formation, but divergence of the ILC is stillundesired.

As error criterion the Integrated Absolute Error (IAE) is chosen. The IAE is given by

IAEk =N∑

i=1

|ek(i)|, (4.1)

which equals the sum of the absolute error. The IAE as a function of the iteration number isshown in Figure 4.9. From this figure it can be seen that ILC has converged in approximately20 iterations and the IAE is reduced by a factor 4. After iteration 20, the IAE slightly startsincreasing. This is due to the high frequency instability which causes these components tostart dominating the signals.


0 10 20 30 40 50 60 70 80 90 100−1

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1measurementreference

0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

30

40

PSfrag replacements

Time [µs]

Pie

zosi

gnal

[V]

Input

[V]

Figure 4.8: SISO ILC; measured piezo signal, desired piezo signal and input signal for iteration25.

5 10 15 20 250

10

20

30

40

50

60

70

PSfrag replacements

Iteration number [-]

IAE

[V]

Figure 4.9: SISO ILC; Integrated Absolute Error as a function of the iteration number.

The spectrum of the tracking error ek is depicted in Figure 4.10 for iteration 0 and 25.From this figure it becomes obvious that the first resonance mode at 50 kHz, which dominates

4.3 Integrated SISO ILC 59

the channel acoustics, is strongly suppressed. However, for the higher frequency componentsthe ILC is unstable, especially in the range 700 - 900 kHz, just like was observed in the timesignals.

101 102 103−180

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iteration 0iteration 25

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Mag

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[-]

Figure 4.10: SISO ILC; error spectrum for iteration 0 and 25.

4.3 Integrated SISO ILC

The results of the ILC which is directly based on the measured piezo signal are not satisfac-tory. Therefore, another implementation is considered in this section. Here, it is chosen towork with the time-integrated piezo signal, see also [GWBBK05]. This choice offers severaladvantages over the former implementation. Firstly, it is physically more logical to work withthe integrated piezo signal, because this is a measure for the pressure inside the channel2.Secondly, this choice is numerically more attractive as will be illustrated in the section wherethe ILC algorithm which is based on the time-integrated piezo sensor signal is derived. Ex-perimental results are provided and discussed and the positive effect of active damping onthe jet frequency is illustrated by means of a so-called DOD curve.


The measured Frequency Response Function from the piezo input to the piezo signal dividedby jω is depicted in Figure 4.11. Moreover, the figure shows the new 17th-order model whichis obtained by adding a pure integrator to the 16th-order model. The result of this time-integration is a change in slope of the amplitude of −1 and a change in phase of -90 degreeswith respect to the original FRF.

2Remember that the piezo signal itself is a measure for the time-derivative of the pressure.


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Mag

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[dB

]P

has

e[d

eg.]

Figure 4.11: Measured Frequency Response Function and fitted model from the piezo input tothe piezo output, integrated in time.

The new ILC feedback loop in the trial domain is illustrated in Figure 4.12. Clearly, thenew system can be seen as the old system together with the integrator block.

L

Z - 1++

+_1 / sHPSfrag replacements

∆uk

uk+1 uk yk ek

yd

Figure 4.12: Lifted ILC loop in the trial domain, based on the integrated piezo signal.

Again, the same time settings are chosen as for the previous experiments and the newimpulse response h is shown in Figure 4.13. The impulse response illustrates that now thefirst resonance mode is dominant. By making use of the impulse response, the new impulseresponse matrix can be obtained. The singular values of the impulse response matrix can alsobe found in Figure 4.13. The rank of H now equals 995 what corresponds to a loss in rank offive. This additional loss in rank with respect to the previous section is due to the time delaywhich is introduced by the added integrator.


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σi

sqrt(β)PSfrag replacements

Element number i

Sin

gula

rva

lue

σi

Figure 4.13: Impulse response from the piezo input to the integrated piezo signal (left) andsingular values of the new impulse response matrix (right).

To compare the singular values with the integrated FRF, the results are shown togetherin Figure 4.14. Now, the important dynamics correspond to the large singular values, whichis favorable for control design.

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i)[d

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[dB

]

Figure 4.14: Logarithm of the singular values (left) and amplitude plot of the FrequencyResponse Function from the piezo input to the integrated piezo sensor signal (right).

To avoid long actuation pulses, an actuation window of 60 µs is introduced. Experimentsproved that in case of integrated ILC a slightly larger actuation window is needed to obtainproper convergence. In order to illustrate the effect of the loss in observability which is


introduced by the window, the results of an ILC experiment without actuation window areshown in Appendix B.

In Figure 4.15, the input and output singular vectors are depicted for singular value σ1

and σ2, which are the two dominating singular values. Here, the length of the input singularvectors equals 600 elements. Again, it can be observed that the output singular vectors havea certain delay with respect to the input singular values. The frequency spectra of the firsttwo output singular vectors are shown in Figure 4.16. This picture makes obvious that bothsingular values correspond to the first acoustic mode at 50 kHz. The fact that the greatestsingular values correspond to the most important resonance mode for the jet process directlyillustrates the improvement in observability of the first resonance mode as a result of theadded integrator block.

100 200 300 400 500 600 700 800 900 1000−0.1

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PSfrag replacements

Element number i

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ue

Val

ue

Figure 4.15: 1st and 2nd input and output singular vectors of the new reduced impulse responsematrix H∗.

The tuning parameters are now chosen as β = 1 · 10−15 (see Figure 4.13) and γ = 0.25,which are again rather robust settings. The learning matrix, calculated by making use ofthe approximated solution of the DARE, is shown in Figure 4.17. The non-square learningmatrix is now of size 600× 1000. Again, the mixed causal/anti-causal time-varying characterof ILC can be observed from the figure.


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Mag

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[-]

Figure 4.16: Frequency spectrum of the 1st and 2nd output singular vector of the new reducedimpulse response matrix H∗.

Figure 4.17: Integrated SISO ILC; surface plot of the new learning matrix.


For the experiments the same practical considerations hold as discussed earlier, but the time-integrated piezo signal is required. To obtain this signal, a relative simple cumulative numeri-


cal integration routine is applied which is based on the Simpson rule. The Simpson rule makesuse of a weighted sum of both the midpoint method and the trapezium method. The midpointmethod and the trapezium method are both first-order accurate, but, by proper choice of theweights, the Simpson rule is third-order accurate. Now, the offset-correction becomes extraimportant, because when it is omitted this results in drifting of the time-integrated signal.

The same control goal is applied as for the ILC experiment which is based directly on themeasured piezo signal, namely active damping of the residual vibrations. Again, the fixed5/5/3 pulse can be used as initial input. The initial system response and the initial input areshown in Figure 4.18. The desired response follows the initial response until 30 µs and thendamps the signal during half the period of the dominating harmonic.

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pie

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[V]

Figure 4.18: Integrated SISO ILC; integrated piezo signal, desired piezo signal and input signalfor iteration 0.

The results for iteration 25 are depicted in Figure 4.19. It can be seen that the ILCsuccessfully damps the residual channel acoustics with an actuation signal of 60 µs. At theend of the actuation window the input signal shows a few peaks by which ILC tries to dampseveral resonance modes just before the input becomes inactive. Moreover, it is striking thatnow there is no high frequency component present in the signals. Note that here a differentreference is chosen than during the previous experiment. Due to this, the fact that the lengthof the actuation window is changed and other ILC settings are chosen, the learned inputsignal is not the same is for the previous experiment. Further information on the practicalimplementation of learned ILC actuation signals is handled in Appendix D.

In Figure 4.20, the IAE is given as a function of the iteration number. The ILC is convergedin approximately 15 iterations and the error criterion is by a factor 5. After convergence, theerror remains approximately at a constant level. This error after convergence is called theresidual error. There are several possible causes for the residual error to be unequal to zero.


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Figure 4.19: Integrated SISO ILC; integrated piezo signal, desired piezo signal and input signalfor iteration 25.

Firstly, a system is never exactly trial-invariant and the implementation of Lifted ILC is onlycapable of full elimination of repetitive disturbances. Secondly, the desired system outputmay be an infeasible output for the actual system dynamics. When the system is not fullyobservable, not any arbitrary reference can be followed. Loss in observability is here mainlycaused by the introduction of the actuation window.

The spectrum of the tracking error is depicted in Figure 4.21. This figure illustrates thatthe error is now reduced over a much larger frequency range. Unfortunately, ILC is not stablefor all frequencies, but the unstable components are far less amplified as in the previous case.As long as ILC is turned off after convergence, these instabilities do not deteriorate the resulttoo much. When it is required that ILC is absolutely stable for all frequencies, one couldconsider even more robust settings of the tuning parameters. Other possible improvementsare a better measurement of the Frequency Response Function and making a better fit on thisdata. For experimental results with respect to the reproducibility of ILC and the sensitivityfor production tolerances, the reader is referred to Appendix C.


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Figure 4.20: Integrated SISO ILC; Integrated Absolute Error as a function of the iterationnumber.

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Figure 4.21: Integrated SISO ILC; error spectrum for iteration 0 and 25.

4.3.3 DOD Curve

To illustrate the positive effect of active damping of the residual channel acoustics on the jetfrequency, the so-called DOD curve is considered in this section. When the droplet speed is


depicted as a function of the jet frequency, the DOD speed curve is obtained. Such a DODcurve clearly illustrates the variations in droplet speed when the jet frequency changes.

The instant when a droplet is fired depends on the position of the carriage on which theprintheads are mounted. The velocity of the carriage is not exactly constant, thus the timeperiod between two droplets can vary. Moreover, it depends on the information that has tobe printed whether a droplet is required or not at a certain position. Due to these effects, thetime period between two following droplets, and thus the jet frequency, varies. It is highlyundesired that variations in jet frequency have any effect on the print quality. Therefore, thedroplet properties have to be as constant as possible when the jet frequency changes.

The physical background of the dependency of the droplet speed on the jet frequency liesin the residual channel vibrations, which lead to oscillations of the meniscus. When the timeperiod between droplets is relatively long (a low jet frequency), then the meniscus is at restwhen the next droplet is fired. In this way, identical droplets are produced. However, whenthe time periode between droplets is shorter than the time that the system takes to become atrest, the meniscus is still moving when the next droplet is fired. When the meniscus velocityis positive (in outer direction) when the next droplet is fired, the available kinetic energy islarger than in the nominal situation and the droplet speed will be higher. Contrarily, whenthe meniscus velocity is negative there is less kinetic energy available and the resulting dropletwill go slower. These two situations are called overfill and underfill of the nozzle, respectively.It is obvious that these changes in droplet speed are undesired, because a change is dropletspeed has as result that the droplet is placed at the wrong position.

Both the DOD curve for the fixed 30 V 5/5/3 pulse and the learned ILC wave based onthe integrated piezo signal are measured. For this experiment the jet frequency is varied from2 kHz till 40 kHz. This means that for the highest jet frequency which is applied a timeperiod of 25 µs is available. In case of the fixed pulse, which has a duration of 13 µs, this isno problem, but for the ILC wave with a duration of 60 µs it is. To solve this problem, use ismade of the knowledge that the dynamics from piezo input the piezo signal behave reasonablylinear and that for a linear system the superposition principle holds. Therefore, it may beallowed to sum overlapping ILC waves. The end of the previous wave than makes sure thatthe channel acoustics become at rest and, in the same time interval, the beginning of the newwave initiates the new acoustic waves in order to jet a droplet. The summed ILC actuationsignal at a jet frequency of 25 kHz together with its individual ILC waveforms is shown inFigure 4.22. In this figure, it can clearly be observed how the continuous ILC actuation signalis obtained from the separate waveforms.

The two DOD curves are shown in Figure 4.23. In case of the fixed pulse the mentionedvariations in jet frequency can clearly be observed3. In [Bos04] it was explained that the firstacoustic mode corresponds to the 1

3-lambda resonator (see also Figure 2.10), according to

f1 = c3L

≈ 50 kHz. Taking into account that variations in droplet speed are mainly causedby this first mode, it follows that the distance between two maxima equals 1

f1= 3L

c≈ 20µs.

The maxima are thus located at the jet frequencies: 50 kHz (not shown), 50

2kHz, 50

3kHz,

etcetera. Furthermore, it can be seen that the curve has a positive trend. When the jetfrequency increases, better refill of the nozzle is obtained, resulting in higher droplet speeds.For high jet frequencies, 30 and 34 kHz, erroneous measurements occur and extreme valuesof the droplet speed are measured. The cause of these peaks are eigenmodes of the printhead

3The jet frequency is not actually a frequency, but the reciproque of the time period between the firing ofdroplets and, therefore, the DOD curve cannot be seen as a frequency spectrum.


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Figure 4.22: Summed ILC actuation signal and individual ILC waveforms for a jet frequencyof 25 kHz.

structure itself which lie in this frequency range. Especially for low droplet speeds erroneousmeasurements are obtained in this range.

In Figure 4.23, also the DOD curve for the learned ILC wave is depicted. For very lowjet frequencies, below 4 kHz, there is not much difference between both curves. Here, thereis a relatively long time period between the droplets, which is long enough for the meniscusto become passively at rest (see Figure 2.5). Over the full frequency range, ILC is capableof decreasing the variations in droplet speed considerably with respect to the fixed pulse,what illustrates that ILC indeed enables higher jet frequencies. However, the variations indroplet speed are not fully eliminated, for which there are several explanations. Firstly, bytracking the chosen reference signal which damps the integrated piezo signal after jetting, themeniscus movement is not necessarily damped, because there are dynamics present betweenthe piezo sensor and the meniscus. Secondly, the integrated piezo signal is actively dampedin approximately 40 µs. This means that, even with active control, the the channel is notat rest for jet frequencies higher than 25 kHz. Thirdly, the summation of overlapping ILCwaves, which is required for jet frequencies higher than 1

60µs= 16.7 kHz, is an approximation,

because the system is not completely linear. This approximation becomes more inaccuratefor high jet frequencies when several waves are summed.

In Appendix E, an alternative method with long references for multiple droplets is consi-dered, where the dynamics are not necessarily damped after each droplet. An advantage ofthis method is that summation of overlapping waveforms, which introduces inaccuracies incase of nonlinear dynamics, is avoided. Possibly, this alternative method is able to furthereliminate the variations in droplet speed. Another possibility is to develop a new sensor, whichpreferably leads to linear, continuous dynamics and provides good information about the

4.4 Conclusions 69

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Figure 4.23: DOD speed curves; droplet speed as a function of the jet frequency for both thefixed trapezium pulse and the learned input wave.

droplet formation, for future experiments. A third route is to identify the dynamics betweenthe piezo sensor and the meniscus movement and taking this additional model knowledge intoaccount during ILC experiments which are based on the piezo sensor signal. This is exactlythe route which is chosen in Chapter 5.

4.4 Conclusions

• With the implementation of SISO ILC it is possible to actively damp the residualvibrations of the piezo sensor signal after droplet ejection.

• Making use of an ILC implementation which is based on the time-integrated piezo signaloffers several advantages over the implementation which makes direct use of the piezosensor signal.

• It is shown by means of a DOD curve that the changes in droplet speed for increasing jetfrequency are significantly reduced by ILC, what indicates that the residual meniscusvibrations have been actively reduced. However, variations in droplet speeds are notcompletely eliminated by active damping of the piezo sensor signal.

• Possibly, further improvement can be realized by making use of long reference signalsfor multiple droplets, a new sensor, or additional model knowledge.

Chapter 5

DSM and the ‘Smart Reference’

The DOD curve which is obtained with the ILC input signal illustrates that variations indroplet speed are not fully eliminated when the (integrated) piezo signal is actively damped.One of the reasons for this is that there are dynamics present between the piezo signal andthe meniscus movement. This illustrates that the piezo sensor functionality may not be thebest choice for active control of the inkjet printhead. To have a better influence on the dropletformation, one would ideally like to control the droplet formation process itself. However,direct control of the droplet formation requires a hybrid type of controller, because the ejectionof a droplet is a discrete event, see [Bos04] for further information. It is chosen to refer tothe meniscus velocity instead. The meniscus velocity is continuously available and closelyrelated to the droplet formation. By means of simulations, the meniscus movement can bedetermined which results in the desired droplet properties. Control of the meniscus velocityrequires a different sensor functionality, namely Laser-Doppler interferometry. The subject ofthis study, however, is the implementation of ILC, based on the piezo sensor signal. Therefore,it is chosen to create an experimental model which describes the dynamics between the piezosignal and the meniscus movement. For control purpose it may be sufficient to control themeniscus, based on piezo sensor measurements, by making use of this model. In fact, themodel is used to derive a so-called smart reference signal for the piezo sensor signal. In theblock scheme of Figure 5.1 it is schematically illustrated how, via a smart reference for thepiezo sensor signal, the input signal can be determined which produces the desired droplets.

In Section 5.1, the dynamics from the piezo input to the meniscus velocity are identified,by making use of Laser-Doppler interferometry. Next, in Section 5.2, it is illustrated by meansof a DSM simulation with the Matlab model what the possibilities are of controlling themeniscus. Furthermore, it is discussed how the reference signal for the meniscus velocity canbe chosen to result in a droplet with arbitrary properties, which is again illustrated by a DSMsimulation. In Section 5.3, it is shown how the knowledge of the dynamics between the piezosensor and the meniscus velocity can be used to derive a smart reference for the piezo sensorsignal. Experimental results are reported for an ILC which is based on the integrated piezosensor signal and makes use of this smart reference to indirectly control the meniscus. Finally,Section 5.4 resumes the conclusions which are made in this chapter.

72 DSM and the ‘Smart Reference’

H d r o p l e t

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?

s i m u l a t i o n sI L C s i m .I L C e x p .

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U droplet

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Figure 5.1: Schematic illustration of the determination of the input signal which results inthe desired droplet.

5.1 Additional System Identification

To be able to control the meniscus, additional system identification is required. Again, linearidentification techniques are applied for this purpose. The Frequency Response Function fromthe actuation signal to the meniscus velocity can be measured by making use of Laser-Dopplerinterferometry, as is shortly explained in Section 2.4.2.

In practice, the meniscus velocity profile has the shape of a so-called Womersley profile,see [MGWK04]. A Womersley profile describes unsteady flows, such as arise in the nozzledue to the pulsating acoustic pressure waves. The inertial and viscous forces in an unsteadyflow are related by the Womersley number, which equals

W = Rn

√ωρ

µ, (5.1)

where Rn is the radius of the nozzle. When W < 1, the flow can be considered as quasi-steady,but when W increases, the inertia forces become more important and result in a delay of thebulk flow. This expresses itself in a velocity profile which becomes more flat in the center.As a result of the Womersley profile, the meniscus dynamics do not behave fully linear. Inthe linear Matlab model, the velocity profile is approximated by the much simpler Poiseuilleprofile. When the measured and theoretical FRFs are compared, one has to keep in mind thisdifference. The velocity which is measured by the laser is the maximum meniscus velocityun|r=0 and not the mean velocity. Therefore, the maximum meniscus velocity is used duringexperiments.

In Figure 5.2, the measured FRF from the piezo input to the maximum meniscus velocityis depicted, together with the theoretical FRF which is obtained with the Matlab model.From the amplitude plot it can be observed that till 300 kHz a reasonable match is obtainedbetween theory and practice. The resonances in this frequency range are predicted ratheraccurate by the model. In the frequency range 300 - 500 kHz the measured FRF behavesrather strange and the match between the two FRFs is poor. The cause of this behavior is notclear, but fortunately the most important (low-frequency) modes are successfully captured

5.2 Simulation Results based on the Meniscus Velocity 73

by the theoretical model. When the phase plots are compared, it can be seen that thecurves match reasonably well. However, it has to be mentioned that the measured FRF iscompensated for phase delay. This delay is mainly caused by the laser decoder.

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Figure 5.2: Measured and theoretical Frequency Response Function from the piezo input tothe maximum meniscus velocity.

The limited bandwidth of the amplifier causes a phase lag, which, for frequencies far belowthis bandwidth, can be approximated by a linear phase delay of 0.04 deg./kHz. At 1 MHz,this corresponds to a delay of 40 degrees. The phase delay of the laser is twofold. Firstly,the laser system suffers from a time delay of 1.05 µs. This time delay can be translated to aphase delay of 0.38 deg./kHz, resulting in a delay of 380 degrees at 1 MHz. Secondly, the laserdecoder contains a 3th-order Bessel Low-Pass filter with a cut-off frequency of 1.5 MHz. Forfrequencies far below the cut-off frequency, the phase lag of this filter can be approximatedby a phase delay of 0.07 deg./kHz, what corresponds to a delay of 70 degrees at 1 MHz.Summation of all delays leads to a total delay of 487 degrees at 1 MHz. When these hardwaredelays are compensated for, a reasonably good match in phase is obtained between theoryand practice. Note that sample delay is not taken into account here. The reason for this isthat the FRF from the piezo input to the meniscus velocity is measured at a very high samplerate of 40 MHz.

5.2 Simulation Results based on the Meniscus Velocity

In this section, it is illustrated on simulation level what the possibilities are of controlling themeniscus velocity. When the meniscus velocity is controlled, one should be able to directlyeliminate residual vibrations of the meniscus, which prevents variations in droplet properties.


Moreover, it is assumed that when the meniscus always makes a well-defined movement, thechance that a dirt particle or air enters the nozzle decreases. It is shown by two simulationexamples, including Drop-Size Modulation, how a reference signal for the meniscus velocitycan be designed, resulting in a certain desired droplet. In the first simulation, DSM is realizedby amplitude scaling of the input signal and in the second simulation, DSM is realized bymaking explicitly use of the channel resonance modes. It is chosen to work with the meniscusvelocity instead of the meniscus position, because the latter is discontinuous in time whena droplet is fired. It is assumed that the time-derivative of the position, the velocity, isa continuous signal, also during jetting. Continuity is a great advantage for identificationand control, because a discontinuity implies a hard nonlinearity and for systems with hardnonlinearities, linear identification techniques as a sine sweep cannot be applied. Besides,by fitting the frequency response data only linear models can be obtained. The reason foronly considering simulation results is that in this study the accent lies on the implementationof ILC based on the piezo signal. The experimental implementation of ILC based on themeniscus velocity is subject of another, but related, study. Moreover, the test setups forpiezo sensing and Laser-Doppler interferometry are at different locations and can, therefore,not be applied at the same time.

5.2.1 DSM based on Amplitude Scaling

For the simulation, use is made of the acoustic Matlab model which provides the theoreticalFrequency Response Function from the piezo actuator to the meniscus velocity, see Figure 5.2.This FRF cannot directly be used for simulations. Firstly, a model has to be fitted on it.Therefore, a continuous 18th-order state-space model is created which provides a very goodfit on the theoretical FRF. Subsequently, an ILC is designed in a similar way as in Chapter 4.The tuning parameters are chosen equal to β = 5 · 10−3 and γ = 1 and an actuation windowof 50 µs is applied, thus N ∗ = 500. Here, the learning matrix is calculated by making use ofthe exact solution of the DARE.

It is chosen to include Drop-Size Modulation in the example. For this purpose two actu-ation pulses are applied, which only differ in amplitude, resulting in two different drop-sizes.The first pulse is chosen such that it produces 30 pl droplets and the second one results in 20pl droplets. The control goal is defined as follows. Jet a sequence of droplets with a volumeof 20, 30, 20, 20, 30, 30 and 20 pl at a constant jet frequency of 40 kHz, where active dampingof the residual vibrations is applied between the droplets. This simulation is performed forthe converged ILC waves as well as for fixed trapezium-shaped pulses.

The simulation results are depicted in Figure 5.3. In the bottom plot the input signalsare shown. For a jet frequency of 40 kHz, the time duration between the pulses is only 25 µs.Therefore, again summation of overlapping input waves is applied. It can be seen that thepeak value of the meniscus velocity in case of fixed pulses varies per droplet. To investigatethe effect of this on the droplet formation, the droplet properties are considered.

The droplet properties are predicted by the Dijksman model, by making use of the menis-cus velocity signal. In Figure 5.4, the droplet volume and the droplet speed are shown forboth the fixed pulses and the ILC signal. This figure illustrates that a deviating meniscus ve-locity directly results in different droplet properties and in case of the fixed pulses the desireddroplets are not obtained. In the ILC case, however, the correct droplets are obtained as aresult of active damping of the residual meniscus vibrations between the firing of droplets.


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Figure 5.3: DSM based on amplitude scaling; simulation results of the meniscus velocities andinput signals for the fixed pulses and the ILC signal.

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Figure 5.4: DSM based on amplitude scaling; simulation results of the droplet volume anddroplet speed as a function of the droplet number for the fixed pulses and the ILC signal.


5.2.2 DSM based on Channel Resonance Modes

For printing jobs where DSM is applied it is not desired that droplets of different sizes havea different speed, because then the droplets cannot be places accurately1. In the consideredexample there is a fixed relation between the droplet volume and the droplet speed. This isas a result from the fact that only one parameter, the input amplitude, is changed to obtaindroplets of different volume. When not only the amplitude but also the form of the inputsignal is varied it could be possible to produce droplets with different sizes which still havethe same speed.

In Section 2.5.2, it is explained that, according to the Dijksman model, only the part ofthe meniscus response where the velocity is positive (and large enough to result in a droplet)determines the properties of the resulting droplet. Here, the Dijksman model is applied toderive a smart reference signal for the meniscus velocity. To restrict all possible forms of thepositive part of the meniscus velocity, it is chosen to consider only harmonic signals. Thus,the positive part of the meniscus velocity has the shape of half the period of a sine function,see Figure 5.5. As a result of this choice, there are two tuning parameters left, the periodtime and the amplitude of the sine. The Dijksman model is used to calculate the dropletproperties for a 2D grid of these two parameters.

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Figure 5.5: Illustration of the chosen form of the meniscus velocity.

A surface plot of the droplet volume and the droplet speed as a function of half the periodtime of the sine function, tsine, and the sine amplitude vsine are shown in Figure 5.6. Bothsurface plots indicate that a certain minimum amplitude and time duration of the positivepart of the meniscus velocity are required to produce a droplet. Moreover, one can observethat there exists a minimum droplet volume. The left figure shows that the droplet volumeheavily dependents on both the amplitude and the period time of the sine signal. Thisis exactly what one expects, because the droplet volume equals the the time-integral of temeniscus velocity times the nozzle area, see (2.7), and thus depends both on the amplitudeand the time duration. It is rather striking, however, that the droplet speed mainly dependson the velocity amplitude for a large range of the grid. The explanation for this is as follows.Examination of the equation for the droplet speed (2.15) shows that the expression containsa term with the square of the meniscus velocity and for most conditions this term dominatesthe other terms. The fact that the droplet speed is rather insensitive for changes in the periodtime of the sine can be exploited for the purpose of DSM. For example, when tsine decreases

1In theory, the problem concerning differences in droplet speed can also be solved by tuning the timingof ejection for the different drop sizes, provided that the droplet speeds and distance from the nozzle to thesubstrate is exactly known.


the droplet volume also decreases, but the droplet speed remains approximately constant.

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Figure 5.6: Droplet volume (left) and droplet speed (right) as a function of tsine and vsine.

To minimize the position error of droplets, high and constant droplet speeds are desired.In this simulation example it is chosen to work with a droplet speed of 7 m/s. The question isnow how the two tuning parameters have to be chosen. This is done graphically, by makinguse of the data of the two surface plots. In Figure 5.7, the contour plot of the droplet volumeas a function of tsine and vsine is depicted. Moreover, the curve with a constant dropletspeed of 7 m/s is indicated by a dashed line. To obtain a droplet with a speed of 7 m/sand an arbitrary volume, the intersection point between the dashed curve and that of thedesired droplet volume has to be found. The required parameter settings can then simplybe obtained, as they equal the coordinates of the intersection point. In theory, this methodworks fine, but it has not been taken into account that the desired meniscus movement cannotbe prescribed directly. The meniscus movement is namely the dynamic response on a certainpiezo actuation signal. Especially changes in period time of the meniscus response may behard to realize. The printhead is namely designed for operation mainly in the first acousticresonance mode. When there is a mismatch between the desired frequency of the meniscusand the first resonance mode of the channel it may require considerably more control effortto achieve the desired response. Therefore, it is chosen to use resonance mode I and II forobtaining two different drop sizes, to make optimally use of the channel dynamics. Note thatthe amplitude of mode II is much smaller than that of mode I. It may still require muchcontrol effort to jet a droplet in mode II.

Half the period time of the sine which corresponds to mode I and mode II is also indicatedin Figure 5.7. The time duration tsine is fixed and the droplet volume cannot freely be chosenanymore. The droplet volumes which can be produced in mode I and II now follow fromthe intersection points with the dashed curve. The parameter settings and the correspondingdroplet volume are given in Table 5.1 for mode I and II, respectively. Note that the velocityamplitude of the meniscus is hardly changed, just as expected.

Having determined the required parameter settings, ILC can be applied. For the simu-lations, again use is made of the theoretical Matlab model. Simulations have shown thatit is not possible to produce a large positive meniscus velocity directly from a rest situation.For this reason, half a sine period with a negative sign is added to the reference signal, justbefore and after the positive part. The period time is chosen equal to that of the positive partand the amplitude equals −3 m/s. Moreover, the reference of the small droplet is delayed


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Figure 5.7: Contour plot of the droplet volume in [pl] as a function of tsine and vsine. Thedashed line represents the contour of 7 m/s droplet speed and the solid lines indicate half theperiod time tsine of the first and second resonance mode of the channel.

Table 5.1: Parameter values for jetting droplets in mode I and mode II.

Mode

number

tsine [µs] vsine [m/s] Vdrop [pl] vdrop [m/s]

1 10.73 6.52 35.7 7.002 5.70 6.79 19.6 7.00

to come to a match in firing instant between the two drop sizes. The resulting referencesare both depicted in Figure 5.8. During the learning process it proved to be much harder tolearn the small droplet, which corresponds to mode II. For this reason it was necessary tochange the ILC tuning parameters, resulting in the following settings: β = 1 ·10−4, γ = 1 andN∗ = 800. With these settings a reasonably good convergence is obtained. Unfortunately,this is at the cost of an increase of the size of the actuation window and a rather high controleffort, especially in case of the small droplet. Moreover, with this choice of the meniscusreference signals, refill of the nozzle is not taken into account.

The two learned ILC input waves are again used for jetting a sequence of droplets, namely19.6, 35.7, 19.6, 19.6, 35.7, 35.7 and 19.6 pl at a constant jet frequency of 40 kHz. Thissimulation is performed for the ILC signal, where overlapping waves are summed, as well asfor fixed 5/5/3 pulses. In case of the fixed pulses only the amplitude is adjusted to produce the


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Figure 5.8: DSM based on channel resonance modes; references for the application of DSMon simulation level.

desired droplet volumes. The resulting meniscus responses and actuation signals are shownin Figure 5.9. The bottom figure clearly illustrates the large increase in control effort in caseof ILC.

The properties of the obtained droplets are again calculated with the Dijksman modeland these are depicted in Figure 5.10. In case of the fixed pulses rather large deviations inthe desired droplet volumes are obtained and the droplet speeds are far from constant, justlike in Figure 5.4. In case of ILC, however, approximately the correct droplet volumes areobtained, all with a speed of 7 m/s. The small deviations which can be observed for theactively controlled case are due to the fact that it was not possible to immediately bring themeniscus to a rest after jetting.

It is not likely that such large actuation signals, as are obtained in the simulation, canbe applied in practice. However, the simulation does illustrate how DSM can be applied intheory. It is illustrated that the droplet speed mainly depends on the velocity amplitude ofthe meniscus and it suggests that a high amplification of both the first and second resonancemode of the printhead is favorable for DSM. Besides, the illustrated simulation example isonly restricted to 1D sine-form meniscus velocities. More complex flows of the meniscuscould lead to comparable results with less control effort. Further research is desired in orderto realize different droplet volumes with a constant speed in practice.

An alternative solution for DSM is to let the ILC learn long reference signals containingmultiple droplets at a certain jet frequency. An advantage of the long reference is that thesummation of overlapping input waves, which is an approximation for not completely linearsystems, can be avoided. For more information on this subject, the reader is referred toAppendix E.


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Figure 5.9: DSM based on channel resonance modes; simulation results of the meniscus ve-locities and input signals for the fixed pulses and the ILC signal.

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Figure 5.10: DSM based on channel resonance modes; simulation results of the droplet volumeand droplet speed as a function of the droplet number for the fixed pulses and the ILC signal.

5.3 SISO ILC with a Smart Reference 81

5.3 SISO ILC with a Smart Reference

The ILC experiments in Chapter 4 resulted in active damping of the residual channel acoustics,though, the dynamics of the meniscus movement were not yet considered explicitly. Thechoice of bringing the (integrated) piezo signal to zero may not be optimal with respect tothe damping of the meniscus. In this section, it is illustrated how a smart reference for thepiezo signal can be determined, by making use of the knowledge of the meniscus dynamics.

5.3.1 Dynamics between Piezo Sensor and Meniscus

The printhead dynamics which are known are the dynamics from the piezo input to the piezosensor signal, Hpiezo, and those from the piezo input to the meniscus velocity, Hvmen . Ofcourse, one could design an ILC which is directly based on the measurement of the meniscusvelocity. This is exactly the subject of a related study about active control on an inkjetprinthead. Here, however, the accent lies on the piezo signal and only this sensor signal isused for ILC experiments. To be able to obtain information about the meniscus velocity whenonly the piezo signal is measured, a model is desired which describes the dynamics betweenthe piezo signal and the velocity of the meniscus.

Assumptions

Several assumptions are made in order to come to an experimental model which approximatesthe dynamics from piezo sensor to meniscus velocity.

− All considered dynamics are linear.

− There is no interaction between the different parts of the jet process (no closed-loopdynamics) and the two measured transfers concern open-loop dynamics. Interactionbetween the parts is referred to as ‘two-sided coupling’, see [Bos04] or [Intb] for furtherinformation on this subject.

− Division of two FRFs is allowed and this does not introduce problems with respect tothe system’s order and causality.

All three assumptions are indicated by a ‘−’, what means that none of them is satisfied.However, the model which is derived in the following section is still the best possible approxi-mation of the actual dynamics between piezo sensor and meniscus velocity, which is availableat this moment.

Derivation

The Womersley profile of the meniscus velocity is not taken into account and it is assumedthat the meniscus dynamics behave fully linear. Furthermore, by neglecting the two-sidedcoupling, Hvmen can be considered as the series connection of Hpiezo and Hpiezo→vmen , justlike is illustrated in Figure 5.1. The latter transfer refers to the FRF from piezo sensor tomaximum meniscus velocity. In Figure 5.11, this idea is graphically illustrated in a blockscheme, where both the system input and output, which are to be used for ILC experiments,are indicated. In fact, the colored block concerns a transfer function between two sensors. The


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Figure 5.11: Block scheme with the relevant transfer functions for the design of a smartreference for the piezo sensor signal.

FRF from the piezo signal to the meniscus velocity is obtained by dividing the two measuredFRFs on each other, according to

Hpiezo→vmen =Hvmen

Hpiezo. (5.2)

Before Hvmen can be used for calculating Hpiezo→vmen , this FRF has to be compensated forphase delay. Otherwise, Hpiezo→vmen would incorporate a delay between the piezo signal andthe meniscus movement which is physically not present. The reason for compensating onlyHvmen is that the transfer is desired between the measured piezo signal (including delay) tothe actual meniscus velocity (without delay).

In order to obtain a proper FRF, which can be fitted with a model, it is chosen to determinethe FRF from the unfiltered, non-integrated piezo signal to the meniscus velocity. This FRFis depicted in Figure 5.12, together with a 24th-order fit2. From the figure it can be observedthat the dynamics between the piezo signal and the meniscus velocity have a global slopeof -1, so the dynamics roughly behave like a 1st-order system. As a result, low-frequencysignals are amplified and high-frequency signals are suppressed by the FRF. This correspondswith the general observation that the meniscus velocity has a much more smooth characterthan the piezo signal, which is dominated by high-frequency components. Moreover, severalfrequency components are amplified or weakened, illustrating that not all resonance modesare equally important for the piezo signal and the meniscus velocity. From the phase plot itcan be observed that for frequencies below the 80 kHz a mismatch between FRF and modelis obtained. This mismatch may be caused by problems concerning causality. The calculatedFRF from piezo sensor to meniscus velocity is not causal per definition, however, the fitprocedure can only produce causal models. It should be mentioned that, because of a lightdamage of the ink channel which is used in Chapter 4, in this section another channel of thesame printhead is considered. Due to production tolerances the dynamics of this channel mayslightly differ.

By making use of the model Hpiezo→vmen , it is now possible to simulate the fellow meniscusvelocity response of a measured piezo signal. An example of the simulated meniscus responsefor the first and last iteration of an experiment with integrated ILC based on the piezo sensormeasurement, as performed in Chapter 4, is shown in Figure 5.13. From this figure it canbe observed that damping of the integrated piezo signal results in strong damping of themeniscus. However, the meniscus is not completely at a rest due to the dynamics which arepresent between these two sensors. This observation motivates for the derivation of a smart

2Note that fitting a model on the FRF which results from the division of the two measured FRFs may leadto a mismatch in the order of the models. Stated otherwise, the sum of the orders of the models Hpiezo andHpiezo→vmen

does not necessarily correspond to that of Hvmen.


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reference signal for the integrated piezo sensor signal, where the dynamics of the meniscusare taken into account.

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Figure 5.13: Simulation of the meniscus velocity responses, based on the measured piezo sensorsignals which are obtained during an experiment with integrated SISO ILC.


5.3.2 Derivation of a Smart Reference

When a desired meniscus velocity is chosen, the fellow piezo signal has to be determined toobtain the smart reference. For this purpose, actually the FRF from meniscus velocity to thepiezo signal is required. In theory, this FRF can be obtained by inverting (5.2). Unfortunately,this results in an improper FRF which cannot be fitted with a linear model. The problemof inverting a system can be avoided by again making use of Iterative Learning Control.The piezo sensor signal which corresponds to the desired meniscus velocity can be learnediteratively on simulation level.

The Matlab model is not capable of describing the actual nonlinear dynamics accurateenough to apply the meniscus velocity reference derived in Section 5.2 as a reference signalfor ILC experiments. Therefore, it is chosen to measure the unfiltered piezo sensor responseon a fixed pulse and determine the meniscus response with the model. Next, this signal isused to construct the reference, just like is done in Chapter 4. During ILC experiments basedon the smart reference it appears that the droplets are slowed down by the damping of themeniscus. Therefore is is necessary to work with an initial pulse of 35 V and apply ratherslow damping of the meniscus velocity.

Because the simulation does not suffer from system disturbances and measurement noise,high performance settings can be chosen for the ILC. The tuning parameters are chosen equalto β = 0.01, γ = 1 and an actuation window of 50 µs is applied. In Figure 5.14, the initialzero piezo sensor input, the initial meniscus response and the desired response are depicted.The ILC converges rather fast and the results four iterations later are shown in Figure 5.15.Subsequently, this learned input signal, after Low-Pass filtering and numerical integration,can be used as a reference signal for SISO ILC experiments which are based on the integratedpiezo signal.

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Figure 5.14: SISO ILC simulation; piezo sensor input, meniscus output and constructedmeniscus reference signal for iteration 0.


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Figure 5.15: SISO ILC simulation; piezo sensor input, meniscus output and constructedmeniscus reference signal for iteration 4.


Now the smart reference signal for the piezo sensor signal is determined, the ILC experimentcan finally be performed. For this experiment, the same controller settings are chosen as forthe integrated ILC experiment in Section 4.3. In Figure 5.16, the results are depicted foriteration 0. It can be seen that the reference signal suffers from a small trend between the 50and 100 µs. This trend results from offset problems during the numerical integration, whichis applied to derive the smart reference.

The ILC converges fast and the results of iteration 8 are shown in Figure 5.17. This figureillustrates that the reference signal can be tracked reasonably well by the system output.Subsequently, the newly learned ILC actuation signal can be applied to obtain a DOD curve.

In Figure 5.18, the DOD curves are depicted for a fixed 35 V 5/5/3 pulse, an ILC wavewith a ‘simple reference’ signal (just like in Section 4.3) and an ILC wave with a ‘smartreference’ signal. Again, summation of overlapping actuation signals is applied for higher jetfrequencies.


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Figure 5.16: SISO ILC experiment; integrated piezo signal, smart reference signal and piezoinput for iteration 0.

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Figure 5.17: SISO ILC experiment; integrated piezo signal, smart reference signal and piezoinput for iteration 8.


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Figure 5.18: DOD speed curves; droplet speed as a function of the jet frequency for fixedtrapezium pulses, learned waves based on a ‘simple reference’ and learned waves based on a‘smart reference’.

It can be observed that both ILC implementations are capable of reducing the variationsin droplet speed considerably. The ILC input waves both have a time duration of 60 µs.From this it follows that for jet frequencies higher than 16.7 kHz the waves overlap. Since thesystems dynamics are not fully linear, an error is introduced by summation of overlappingwaves. This can clearly be observed in the figure, by a small increase in droplet variations.Due to the damping of the meniscus, the droplets obtained by ILC with a smart reference areslightly slowed down, what causes the offset in the figure. Despite of this, both ILC imple-mentations lead to a comparable performance. The fact that ILC with the smart referencedoes not explicitly perform better is possibly caused by the rather strong nonlinearities ofthe meniscus dynamics and by neglecting the two-sided coupling between piezo and menis-cus. This results in a poor quality of the linear model Hpiezo→vmen , which is used to derivethe smart piezo sensor reference. Moreover, effects like refill, the discontinuity in meniscusposition and wetting are not taken into account during the experiment.


5.4 Conclusions

• The dynamics from the piezo actuator to the maximum meniscus velocity are identifiedexperimentally and provides a reasonable match with the theoretical Matlab model.

• It is illustrated on simulation level how a reference signal for the meniscus velocity canbe designed, which results in a certain desired droplet. From simulations it results thatthe droplet speed mainly depends on the velocity amplitude of meniscus and is ratherinsensitive for changes in the time duration of the positive part of the meniscus velocity.

• By considering both the dynamics from the piezo input to the piezo sensor and to themeniscus velocity, a model is created which approximates the dynamics from the piezosensor signal to the meniscus velocity.

• The model between the two sensors is applied to derive a ‘smart reference’ signal forILC based on the integrated piezo sensor signal. Experimental results show that withthis method a comparable performance is obtained as with a ‘simple reference’. Possibleexplanations for the fact that the smart reference does not lead to better performanceare the rather strong nonlinearities of the meniscus dynamics and that the two-sidedcoupling is not taken into account.

Chapter 6

MIMO ILC Simulations

In Chapter 4 and 5, it is shown how the variations in droplet speed have been reduced by activedamping of the residual vibrations with SISO ILC. As stated before, the droplet properties arealso affected by cross-talk between the different channels. To decrease the undesired effectsof cross-talk, MIMO ILC can be applied for the actuation of an array of channels.

In Section 6.1, the general idea of controlling a complete array of channels is introduced.Three classes of control architectures for systems which comprises many identical actua-tor/sensor units are discussed and it is illustrated in what way MIMO ILC can be appliedfor this purpose. The theoretical MIMO printhead model, which is used for simulations, isshortly introduced in Section 6.2 and simulation results of MIMO ILC are reported in Sec-tion 6.3. Moreover, an alternative strategy, which is called Multiloop SISO ILC, is consideredin Section 6.4. Finally, in Section 6.5, some concluding remarks are made.

6.1 Generic Control

Ultimately, it is desired to obtain a generic control algorithm for the actuation of the com-plete array of channels. This algorithm should incorporate interactions due to cross-talk andpossible boundary effects at both ends of the printhead structure. A generic control algo-rithm can be obtained by using several control techniques. Here, a generic control algorithmis proposed which makes use of MIMO ILC controllers. This idea exploits the fact that theinkjet printhead comprises many nearly identical actuator/sensor units. Systems with manyidentical subsystems, such as the printhead, which can be actuated and sensed individuallyare called spatially-invariant. Other typical examples of spatially-invariant systems are vehi-cle platoons, arrays of micro-cantilevers, and flows with boundary control. Furthermore, it isassumed that the printhead dynamics behave linear. For the dynamics from piezo input topiezo sensor signal this assumption is approximately fulfilled.

6.1.1 Control Architectures

Control architectures for the control of spatially-invariant systems can be divided into threeclasses: centralized, localized (or semi-decentralized) and fully decentralized control, see[Bos04]. These three classes are depicted in Figure 6.1. In this figure, Pr and Cr representthe r-th subsystem and the r-th controller, respectively, with r = −∞, . . . ,−1, 0, 1, . . . ,∞.

90 MIMO ILC Simulations

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C e n t r a l i z e d : L o c a l i z e d : F u l l y D e c e n t r a l i z e d :

Figure 6.1: Control architectures for spatially-invariant systems; centralized, localized (heredepicted with only nearest neighbor interactions) and fully decentralized control, respectively.

The centralized controller comprises a single complex controller which actuates and sensesall subsystems. Centralized control leads to the best possible performance, because it hasaccess to information of all subsystems. Unfortunately, centralized control is numericallyvery demanding which results in problems with the practical implementation of the controller,since communication between all single units is required.

In the localized control architecture, each unit has its own relatively simple controller.Because the single systems themselves are spatially-invariant, the controllers can also be.Thus, all controllers are identical. Here, the controllers are allowed to use only informationof neighboring systems (not necessarily only the direct neighbors). As a result of the factthat a controller does not receive information from all other controllers, there is a decrease inperformance. However, for a properly designed localized controller, the reduction in compu-tational effort and ease of implementation may be far more important than the decrease inperformance.

The third class is that of fully decentralized controllers. Here, rather simple and identicalcontrol algorithms are applied to all single systems and there is no exchange of informationbetween neighboring controllers. This implies a further reduction in computational effort, butalso in performance. Moreover, fully decentralized controllers are rather easy to implement,because a controller does not need information from its neighbors.

6.1.2 Generic MIMO ILC

The general idea of using MIMO ILC to obtain a generic control algorithm is as follows.Assume that in total an array of K channels is affected by the actuation of a single channelin the center of this array, due to cross-talk. Subsequently, a MIMO learning algorithm canbe implemented for this array of K channels. This MIMO ILC has to be designed in such away that the channel in the center generates prescribed droplets and all K − 1 neighboringchannels stay at rest. To accomplish this, for the center channel a reference trajectory ischosen which results in a droplet of desired volume and with a desired speed, after whichthe residual vibrations are actively damped. The neighboring channels have to follow a zeroreference trajectory. Note that the neighboring channels are actively controlled in order tostay at rest.

It is a assumed that the printhead dynamics are approximately linear and for a linearsystem the superposition principle (2.25) holds. Taking into account the spatial invariance ofthe printhead, the designed MIMO ILC controller can be seen as a localized controller of thesingle channel in the center of the array. By equipping all channels with this localized ILCcontroller, it may be possible to eliminate the undesired effects of cross-talk. It should be

6.2 MIMO Printhead Model 91

taken into account that the printhead comprises a finite number of ink channels and boundaryeffects have to be treated carefully.

6.2 MIMO Printhead Model

The working principle of MIMO ILC is illustrated by simulations with the theoretical Matlab

model. For the simulations an array of K = 15 neighboring channels is considered. RegardingFigure 2.6, this is a reasonable choice. As sensor output use is made of the meniscus velocity.The theoretical model is fully linear, therefore, no problems are expected with this sensorchoice. The calculated Frequency Response Functions from the center channel to itself andto its neighbors are depicted in Figure 6.2. Only the neighboring channels on one side ofthe center channel are considered, because of symmetry. Considering the amplitude plot itcan be seen that the magnitude of the cross-transfers is approximately one decade smallerthan that of the center channel. Moreover, the same resonances and anti-resonances can beobserved in all transfers. When considering the phase plot, it can be concluded that below100 kHz the phase of the cross-transfers differs 180 degrees from that of the center channel.The explanation for this is as follows. When a positive actuation signal is provided to thecenter channel, the channel volume increases which results in a retraction of the meniscus.However, this increase in volume of the center channel results in a decrease in volume of theneighboring channels due to deformation of the printhead structure, see also Figure 2.9.

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Figure 6.2: Theoretical Frequency Response Function from the actuation signal to the menis-cus velocity of the actuated channel (0) and its neighbors (1-7).


With a sample frequency of 10 MHz and a trial length of 100 µs the MIMO impulseresponse matrix would become of size KN × KN = 15000 × 15000. In Matlab, the storageof this matrix at double precision requires nearly 2 GB. For a standard personal computerthis is computationally too intensive and it is chosen to work with a sample frequency ofapproximately 1 MHz instead, resulting in a 1545 × 1545 impulse response matrix. Becausethe high-frequency piezo dynamics are not present in the theoretical model, this reduction insample frequency is allowed.

In the following two sections, two different strategies of ILC are discussed. The firststrategy makes use of a MIMO learning controller and the second one of 15 SISO learningcontrollers. This latter controller is referred to as Multiloop SISO ILC and can be seen as analternative for MIMO ILC.

6.3 MIMO ILC

In this section, the strategy MIMO learning controller is considered. For the consideredarray of 15 channels, the MIMO ILC can be seen as a centralized controller. Firstly, it isshortly described how the learning algorithm is derived and the simulation results of thelearning procedure are treated, secondly. Next, the effectiveness of MIMO ILC, with respectto dealing with the cross-talk, is verified by considering the droplet properties.


The MIMO ILC strategy makes use of full model knowledge, comprising cross-talk, for thederivation of the learning algorithm. This results in the optimal MIMO ILC for the desiredsettings of the tuning parameters. The MIMO learning matrix is derived by solving theDARE element-wise, leading to the exact solution. The tuning parameters for each channelare chosen as follows: β = 5 · 10−3 and γ = 1 and an actuation window of 50 µs is appliedto each input. Taking the same values for the tuning parameters for each channel seems alogical choice, since all subsystems are identical. In Figure 6.3, the remaining 765 singularvalues of H∗ (1545×765) are shown. The reduced impulse response matrix now has full rank.

The value of√

(β) is plotted in the same figure as the singular values and it can be seenthat only the largest singular values play an active role in the feedback loop. These largestsingular values correspond to the first resonance frequency of the channel acoustics, whichare by far the dominating dynamics for the meniscus movement. The closed-loop trial polesare also shown in Figure 6.3. All poles lie in between 0 and 1 and a stable ILC feedback loopis obtained1.

1Note that the LQ optimal solution is per definition a stable solution.

6.3 MIMO ILC 93

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Figure 6.3: MIMO ILC simulation based on the meniscus velocity; singular values of thereduced MIMO impulse response matrix (left) and closed-loop trial poles (right).

6.3.2 Simulation Results

The control goal for the 15 × 15 MIMO system is defined as follows. The center channel hasto jet a droplet, after which the residual meniscus vibrations are actively damped. All 14neighbors are actively controlled to stay at rest, in spite of the cross-talk. For calculationsthe complete 15 × 15 MIMO model is used, but for the presentation of the results only halfthe array is considered. The simulated time responses, the desired responses and the inputsignals of iteration 0 are shown in Figure 6.4. As initial input a 50 V 5/5/3 pulse is appliedto the center channel, which results in a certain droplet. Note that all neighboring channelsapproximately show the same response, but with a smaller amplitude. Moreover, they are inanti-phase with respect to the actuated channel.

The same signals are shown in Figure 6.5, but now for iteration 30. In this figure it canbe seen that good tracking behavior of all subsystems is obtained. The meniscus of the centerchannel moves smoothly to produce a droplet and all neighbors stay at rest. Regarding thelearned input signals it can be observed that the neighboring channels are indeed actuated inorder to stay at rest. Furthermore, it can be seen that the inputs of the neighboring channelsare similar to that of the center channel, but with smaller amplitude. Obviously, all inputsare in phase. This means that all channels are actuated in the same direction to come to asituation that the center channel is able to jet a droplet and all neighbors stay at rest.


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Figure 6.5: MIMO ILC simulation based on the meniscus velocity; simulated meniscus res-ponses, desired meniscus responses and input signal for iteration 30.

6.3 MIMO ILC 95

The MIMO system takes approximately ten iterations for all subsystems to converge, ascan be seen in Figure 6.6. In this figure, the IAE for each channel is depicted together withthe summed IAE. Clearly, all subsystems show a monotone convergence behavior. The IAEreduction rates after 30 iterations are given in Table 6.1 per channel. The second column inthe table shows that the error criterion for all channels is reduced.

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Figure 6.6: MIMO ILC simulation based on the meniscus velocity; Integrated Absolute Errorsas a function of the iteration number.

Table 6.1: Reduction rates of the IAE after 30 iterations for both the MIMO ILC and theMultiloop SISO ILC strategies.

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6.3.3 Droplet Properties

So far, the simulation result with MIMO ILC only illustrated that it is possible to producea droplet without affecting the meniscus of the neighboring channels too much. Interestingis now what the effect of this is on the droplet properties, because the droplet propertiesdetermine the final print quality. To be able to show the effect of MIMO ILC on the dropletproperties, four different jet situations are considered:

• Only the center channel jets a droplet, by actuation with a fixed pulse (‘fixed solo’).

• Both the center channel and its direct neighbor jet a droplet, by actuation with fixedpulses (‘fixed double’).

• Only the center channel jets a droplet and the array is actuated by the learned inputwaves (‘ILC solo’).

• Both the center channel and its direct neighbor jet a droplet, where both channels areactuated by the learned input waves (‘ILC double’).

In the last situation, use is made of the superposition principle. Here, channel 0 is actuated byinput signal 0 (see Figure 6.5) and its neighboring channels by input signals 1-7. Superposedon these signals, channel 1 is also actuated by input signal 0 and its neighbors by inputs 1-7,just like is explained in Section 6.1.2.

In Figure 6.7 and 6.8, the calculated droplet volumes and droplets speeds are shown,respectively, for the four different situations. The results are plotted as a function of theiteration number. Moreover, the difference between the two fixed situations and betweenthe two ILC situations is provided. It can be observed, that there is indeed a considerabledifference in droplet properties between ‘fixed solo’ and ‘fixed double’. Obviously, both thesetwo situations stay at a constant level for increasing iteration number. Regarding the curve‘ILC solo’, it can be seen that for iteration 0 the same droplet is obtained as for ‘fixed solo’,because as initial input the fixed pulse is applied. During learning, the curve ‘ILC solo’ tendsaway from the ‘fixed solo’ curve and finally converges to a certain constant value. Whenthe curve ‘ILC double’ is considered it can be seen that this curve starts at the same valueas the ‘fixed double’ situation. During learning also this curve tends away from the initialvalue and converges to approximately the same value as ‘ILC solo’. This means that almostidentical droplets are obtained with MIMO ILC whether one or two neighboring channels areactuated at the same time. In this way it is proven that the effects of cross-talk can activelybe suppressed on simulation level.

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Figure 6.8: MIMO ILC simulation based on the meniscus velocity; droplet speed for fixedpulses and learned input waves as a function of the iteration number.


6.4 Multiloop SISO ILC

All system knowledge is taken into account in the MIMO ILC strategy and, theoretically, thisleads to the best control result. Moreover, when all noise sources in the loop are uncorrelated,the MIMO controller has a reducing effect on the noise, because the noise is averaged out,see [Kaa04]. This are two advantages of MIMO ILC. A drawback of MIMO ILC is that all K2

transfer functions are required. In practice, this would require 15 × 15 = 225 identificationexperiments for an array of 15 channels. Because the number of transfer functions increasesquadratically with the number of subsystems, the problem becomes relatively fast unmanage-able in practice. Moreover, during preliminary experiments it appeared that the mentionedelectrical cross-talk (see Section 2.3), which has minor influence under normal printhead op-eration conditions, has a large influence on the piezo sensor signal when neighboring channelsare sensed. This makes it extra difficult to measure good cross-transfers.

Regarding the drawback of MIMO ILC, in this section an alternative strategy is consi-dered, named Multiloop SISO ILC. The Multiloop SISO strategy concerns a fully decentralizedcontrol architecture for the considered array of 15 channels. Firstly, it is shortly explainedhow the Multiloop SISO ILC works and the learning algorithm is derived. Secondly, thesimulation results are considered and, finally, the droplet properties are discussed.


In the special case of the inkjet printhead, cross-talk can be considered as a repetitive distur-bance. This would justify a so-called Multiloop SISO ILC which does not require knowledgeof the cross-transfers in the controller design. Here, cross-talk can be reduced by ILC, be-cause it behaves constant each trial. An advantage of this strategy is that no cross-transfersbetween the channels have to be identified. The ILC is now solely based on the 15 SISOtransfer functions of the subsystems. Note that in the simulation model all channels are ex-actly identical and ILC only requires a single SISO transfer function. For the simulation ofthe time responses, however, still the full MIMO impulse response matrix is needed.

It should be mentioned that the assumption that cross-talk behaves like a repetitive dis-turbance, and thus is constant each iteration, is not completely correct. When an array ofchannels is passively controlled (by fixed pulses), the assumption indeed holds. But, when ac-tive control in the form of ILC is applied, the actuation signals change each iteration and thecross-talk will change too. The effect of this should be considered carefully during learning.

The learning matrix is derived in a similar fashion as for MIMO ILC and the same valuesfor the tuning parameters are used. Moreover, an actuation window of 50 µs is applied. Thesingular values of the reduced 1545×765 impulse response matrix H∗ are shown in Figure 6.9(left). The reduced impulse response matrix has again full rank. For comparison with theimpulse response matrix which comprises cross-talk (see Section 6.3), also the singular valuesof this matrix are given in the figure. It can be seen that the singular values of both matricescorrespond reasonably well. However, the singular values of the matrix comprising cross-talkdrop off faster. This is due to the fact that the cross-talk transfers show relatively smallamplifications with respect to the direct transfer, which expresses itself in the singular values.

In Figure 6.9 (right), the closed-loop trial poles are depicted for the Multiloop SISOcontroller. Also the closed-loop poles of the MIMO ILC are shown in the figure. Again, allpoles are stable and it can be observed that the trial poles of both strategies are approximatelyidentical.

6.4 Multiloop SISO ILC 99

0 100 200 300 400 500 600 700 80010−3

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σi comprising cross−talk

σi not comprising cross−talk

sqrt(β)

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comprising cross−talknot comprising cross−talkPSfrag replacements

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Figure 6.9: Multiloop SISO ILC simulation based on the meniscus velocity; singular values ofthe reduced MIMO impulse response matrix with and without cross-talk (left) and closed-looptrial poles (right).

6.4.2 Simulation Results

The control goal has stayed the same for the Multiloop SISO controller, and the same referencetrajectories are applied. The trial results of iteration 0 are identical to those in Figure 6.4and are, therefore, omitted. In Figure 6.10, the signals of iteration 30 are shown. It can beobserved that also in case of the Multiloop SISO strategy reasonably good tracking behavioris obtained. However, after 40 µs the meniscus of the different channels is not as well at restas in case of MIMO ILC.

The IAE error criterion is shown in Figure 6.11. In this figure it can clearly be seen thatconvergence is obtained for all channels. However, the IAE does not converge monotonouslyand shows an oscillating behavior. This is probably caused by the fact that the ILC has noknowledge about the cross-talk, which does not behave perfectly trial-invariant. In spite ofthis, the simulation example illustrates that, as long as disturbances do not change too muchper iteration, ILC is still well capable of decreasing the tracking errors. The reduction in IAEwhich is obtained after 30 iterations is given in Table 6.1, in the third column. The resultshows that all errors are reduced. However, the error reduction is less than in case of MIMOILC, as was already observed from the trial signals in Figure 6.10.


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Figure 6.10: Multiloop SISO ILC simulation based on the meniscus velocity; simulated menis-cus responses, desired meniscus responses and input signal for iteration 30.

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[m/s

]

Figure 6.11: Multiloop SISO ILC simulation based on the meniscus velocity; Integrated Abso-lute Errors as a function of the iteration number.

6.4 Multiloop SISO ILC 101

6.4.3 Droplet Properties

To check how effective the Multiloop SISO controller has reduced the cross-talk, the dropletproperties are determined by making use of the Dijksman model. Again the same four jetsituations are considered as in the previous section. The droplet volume and droplet speed as afunction of the iteration number are depicted in Figure 6.12 and 6.13, respectively. Moreover,the (absolute) difference between the two fixed and between the two ILC situations are plotted.Also in case of the Multiloop SISO ILC a large reduction in droplet differences is obtained.However, after iteration 12 the difference in droplet volume tends to increase a little.

0 5 10 15 20 25 300

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fixed soloILC solofixed doubleILC doublefixed |difference|ILC |difference|

PSfrag replacements


Dro

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e[p

l]

Figure 6.12: Multiloop SISO ILC simulation based on the meniscus velocity; droplet volumefor fixed pulses and learned input waves as a function of the iteration number.

By comparison of the two ILC strategies it can be stated that MIMO ILC results in theoptimal learning controller for the chosen values of the parameters. Multiloop SISO ILCresults in sub-optimization of the different subsystems. This means that a certain channeldoes not take into account its neighbors, but only itself. In fact, the SISO controllers can beseen as fully decentralized controllers of the considered array of 15 channels.

With respect to the inkjet printhead it can be concluded that the MIMO controller out-performs the Multiloop SISO controller. However, the latter still performs reasonably welland it does not require identification of the cross-talk what is a great advantage, especiallyfor experiments.


0 5 10 15 20 25 300

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[m/s

]

Figure 6.13: Multiloop SISO ILC simulation based on the meniscus velocity; droplet speed forfixed pulses and learned input waves as a function of the iteration number.

6.5 Conclusions

• To suppress the effects of cross-talk, a MIMO Iterative Learning Controller can beapplied, which controls an array of channels. It is illustrated by simulations for bothMIMO ILC and Multiloop SISO ILC that differences in droplet properties due to cross-talk can be reduced considerably.

• A MIMO ILC, which makes use of full model knowledge, obviously results in the bestperformance. However, for an array of K channels, K2 transfer functions are required.For large arrays and during experiments this may become a serious problem.

• Full identification of all relevant transfer functions can be avoided by applying a Mul-tiloop SISO strategy instead of MIMO ILC. This alternative strategy is based on theassumption that cross-talk approximately behaves like a repetitive disturbance and theMultiloop SISO controller requires no knowledge of the cross-talk.

Chapter 7

Conclusions and Recommendations

In this final chapter, all important conclusions which have been made in the report aresummarized. Moreover, several recommendations for future research are given.

7.1 Conclusions

The main conclusion of this thesis is:

Iterative Learning Control is capable of actively suppressing the residual vibrations in practice,enabling higher jet frequencies. On simulation level, the effects of cross-talk can be eliminatedby a MIMO ILC which controls an array of channels.

The implementation of a Lifted Iterative Learning Controller requires a model of the sys-tem dynamics. It has been demonstrated that the dynamics from the piezo input to the piezosensor are rather linear and that the obtained experimental model provides reasonably goodresults.

A SISO ILC, which is based on the measured piezo sensor signal, is capable of applyingactive damping of the residual vibrations in practice, enabling higher jet frequencies. However,for high-frequency components the ILC is unstable. An alternative implementation of ILC,which is based on the time-integrated piezo signal, produces better results.

By means of a DOD speed curve it has been demonstrated that deviations in droplet speedare considerably reduced with ILC. To be able to attain high jet frequencies, summation ofoverlapping ILC input waves has been applied. This method introduces an error as a resultof the nonlinear printhead dynamics, but it still works reasonably well.

An experimental model, which approximates the dynamics between the piezo sensor andthe meniscus velocity, makes it possible to derive a ‘smart reference’ signal for the ILC whichis based on the piezo sensor. Unfortunately, ILC based on the smart reference signal doesnot perform explicitly better than the ILC with the ‘simple reference’ signal. Probably, thisis caused by the nonlinear behavior of the meniscus dynamics and the fact that the two-sidedcoupling is not taken into account.

It has been illustrated by simulations what meniscus velocity is required for producingdroplets with arbitrary speed and volume. By making use of this knowledge, Drop-SizeModulation, based on learned input waves, is capable of producing droplets with constantspeed and different sizes on simulation level.

104 Conclusions and Recommendations

A proposed generic control algorithm, which is based on MIMO Iterative Learning Con-trollers and exploits the spatial invariance of the printhead dynamics, can deal with theundesired effects of cross-talk. With simulation results it is demonstrated that MIMO ILCsuccessfully eliminates the cross-talk, by applying active decoupling of the channels.

Also a Multiloop ILC can be applied, assuming that the cross-talk behaves repetitive.This alternative strategy does not require full identification of the MIMO system like MIMOILC does. The MIMO controller outperforms the Multiloop SISO controller, however, alsothe Multiloop strategy is capable of reducing the effects of cross-talk on simulation level.

7.2 Recommendations

First of all, it is desired to conduct further investigation into the reproducibility, the stabilityand the robustness of ILC on an inkjet printhead. This is especially important because ofthe rather large model uncertainty which exists when use is made of the piezo sensor signal.Here, additional tuning of the available ILC parameters may be desirable, where also thelength of the actuation window and the settings of the Low-Pass filter can be regarded astuning parameters. Possibly, improvement of the results can be realized with a different sensorfunctionality for future experiments, which has a better signal-to-noise ratio and/or betterobserves the dynamics which are relevant for droplet formation.

It is desirable to investigate in more detail what meniscus movement is required to producea certain droplet. Here, effects like jet stability and refill should be carefully taken intoaccount. Additional knowledge is desired in how the meniscus movement should be chosento minimize the chance of capturing air or dirt in the nozzle. Moreover, for experiments athigh DOD frequencies, it may be desired to learn the input signal for a sequence of dropletsinstead of applying summation of overlapping input waves, which introduces an error in caseof nonlinear dynamics.

Furthermore, both MIMO ILC and Multiloop SISO ILC have to be implemented on theprinthead in practice and one has to check the convergence behavior of ILC and verify thedroplet properties. Next, one of these two implementations, can be extended to the proposedgeneric control algorithm. This may induce several practical problems, among others, withthe required computational power. For the application of ILC on a complete array of channels,one probably has to refer to the Hamiltonian simulation technique. Another path is to learnand store input waves which lead to a satisfying performance for all channels.

Appendix A

Standard ILC

There exist two important model-based ILC techniques, namely Standard and Lifted ILC.Lifted ILC is already discussed in detail in Chapter 3. Here, the subject of Standard ILCis addressed. In Section A.1, the working principle of Standard ILC is explained and inSection A.2 the results of the application of Standard ILC on simulation level are reported.

A.1 Working Principle

Standard ILC is a model-based technique for learning control in the frequency domain, see forexample [SM00] and [B+]. Usually, this technique is applied to a stable closed-loop system asdepicted in Figure A.1. Here, P (s) is a continuous plant and C(s) is a continuous feedbackcontroller which stabilizes the plant. This feedback controller may comprise a stabilizing leadfilter (PD), an integral action (I) and a LP filter to suppress high-frequency dynamics. Theoutput of the system for the k-th iteration is yk, the reference trajectory is yd, the trackingerror equals ek and d is a disturbance which may be acting on the system. The input signalu consists of the sum of two signals, the output signal of the feedback controller, ufb, and thefeedforward signal uk.

P ( s )C ( s )+ ++ +

+_

L ( s )Q ( s )+ +

PSfrag replacements

yd

ek

ufb

uk

u

uk+1

d

yk

Figure A.1: Standard ILC loop in the frequency domain.

The learning controller can be regarded as an add-on controller for the given closed-loopsystem, just like the Lifted ILC. The fact that the feedback controller and the ILC can be

106 Standard ILC

designed separately is a great advantage for practical implementation. In the figure, L(s)represents the learning filter and Q(s) the robustness filter. After each iteration the trackingerror ek is stored and then filtered with learning filter L. The learning filter determines whichpart of the error is relevant for learning. Subsequently, the sum of the previous input signaland the output of the learning filter is filtered with robustness filter Q. The result, whichequals the new input uk+1, is stored in a table and is used as feedforward signal for the nextiteration. The two filters L and Q determine the performance and have to assure convergence.Mathematically, the tracking error equals

ek = S(yd − d) − PSuk, (A.1)

with S = 1

1+PCand PS = P

1+PCthe sensitivity and the process sensitivity function, res-

pectively. For convergence analysis, a small-gain type of argument is used, for which it isassumed that yd = d = 0 and the expression for the tracking error reduces to

ek = −PSuk. (A.2)

The learning update rule is given by

uk+1 = Q(uk + Lek). (A.3)

By combining (A.2) and (A.3) and eliminating the input signal, expression

ek+1 = Q(1 − L PS)ek (A.4)

is obtained for the propagation of the error signal. Obviously, convergence is obtained whenthe criterion

|Q(1 − L PS)| < 1 (A.5)

is satisfied. A suitable choice for L would be L = PS−1, which leads to zero tracking errorjust in one trial. This is called dead-beat control. However, the process sensitivity is strictlyproper in most cases and cannot be inverted. Moreover, if the process sensitivity containsnonminimum phase zeros the inverse would be unstable. For this reason, one often uses anapproximated inverse of the process sensitivity. To determine an approximated stable inverseof the process sensitivity, the Zero-Phase-Error-Tracking-Controller (ZPETC) algorithm canbe applied [TTC88]. The discrete NUM/DEN representation of the process sensitivity equals

PS(z−1) = z−d B(z−1)

A(z−1), (A.6)

were B and A are the numerator and denumerator polynomial, respectively. Nonproperbehavior and delay is captured by d and z is the discrete delay operator. The learning filter,which is the stable inverse of (A.6), then equals

L(z−1) = zd A(z−1)B−(z)

βB+(z−1)= zd+p A(z−1)B∗(z−1)

βB+(z−1). (A.7)

Here, p is the number of nonminimum-phase zeros and β is a scaling parameter. The stablepart of B equals B+ and the anti-stable part is captured by B−, which equals B−(z) =zpB∗(z−1).

A.2 Application of Standard ILC 107

Due to the approximation, convergence is usually not obtained for all frequencies, espe-cially for the higher frequency content. To assure convergence for all frequencies, the robust-ness filter Q is applied. Usually, Q is a LP filter with a properly chosen cut-off frequency. Toavoid phase loss, anti-causal filtering can be applied. The signal is then filtered two times inamplitude, once forward in time and once backward in time, leading to zero phase distortion.Anti-causal filtering is possible, because the learning algorithm operates off-line.

The process sensitivity is effectively inverted for frequencies below the cut-off. As a result,only frequency components of the error in this range can be learned. Note that applyingthe robustness filter yields to compromised integration and zero tracking error is no longerguaranteed.

The presented ILC scheme typically needs three to ten iterations to decrease the trackingerror with a factor 10 − 100, depending on the quality of the model and the size of thestochastic part with respect to the deterministic part of the error. When the performance ofthe learning algorithm is satisfactory, the ILC may be turned off and the feedforward signalof the last iteration is memorized.

It should be mentioned that the control technique Standard ILC, which is explained forthe closed-loop case, does also hold for stable open-loop systems, such as the inkjet printhead.Here, there is no feedback controller, thus C = 0. This reduces the sensitivity to the identityand the process sensitivity to the plant. However, the derivation of the algorithm is still valid.

A.2 Application of Standard ILC

In this section, a Standard ILC is derived for the inkjet printhead. Next, simulation resultsare reported for a Standard ILC on simulation level.

A.2.1 Derivation of the Learning Algorithm

The Matlab routine frsfit, which is used for fitting frequency response data, produces acontinuous state-space model in the control canonical form. A drawback of this canonicalform is the limited numerical accuracy. This, in combination with the +1 slope and the veryhigh resonance frequencies of the printhead FRF from the piezo input to the piezo signal,results in a very large condition number of the system matrix Ac.

During the design process of Standard ILC several transformations to other system des-criptions have to be performed. Firstly, the continuous state-space model has to be dis-cretized and it has to be transformed to the ZPK-representation, secondly. Next, the ZPETCalgorithm is applied to calculate a stable inverse of the system. All these transformationsdeteriorate the already ill-conditioned model and the obtained learning filter L is useless dueto numerical errors.

Additional transformation techniques in order to improve the numerical condition of thecontinuous state-space representation did not lead to any significant improvement. Therefore,it is chosen to directly fit a discrete NUM/DEN model on the measured FRF with the routineinvfreqz. The obtained discrete fit can then directly be inverted by the ZPETC algorithm.

108 Standard ILC

The measured FRF together with the discrete fit are depicted in Figure A.21. The ob-tained discrete model provides a reasonably good fit, but, unfortunately, it does not captureall dynamics properly. Furthermore, there is a certain mismatch in amplitude in the low-frequency range.

100 101 102 103−60

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Mag

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Figure A.2: Measured Frequency Response Function from the piezo input to the piezo signaland 16th-order discrete fit.

The stable inverse of the FRF, the learning filter, is shown in Figure A.3 before andafter phase compensation. Moreover, the product of the plant model and the learning filterPL is shown, which ideally should be 1. The scaling parameter β is used to tune the firstresonance mode which is assumed to be the most important. Due to numerical issues ZPETCstill does not provide a very good inverse of the system dynamics. To assure stability, a2th-order robustness filter Q with a cut-off frequency of 70 kHz is chosen. This relatively lowcut-off frequency is needed because of the rather bad inverse of the plant. Furthermore, theconvergence criterion is slightly changed in order to obtain a stable learning algorithm. Itnow equals

|1 − QLP | < 1. (A.8)

Essentially, this criterion is the same as (A.5), but the robustness filter Q is shifted to anotherlocation in the Standard ILC loop. Note that, because the inkjet printhead concerns anopen-loop system, the plant P is used instead of the process sensitivity PS. The convergencecriterion is depicted in Figure A.4. From the figure it follows that the criterion is fulfilled.

1Here, the LP filter is omitted, because with the LP filter no good fit was obtained. The obtained fit onthe FRF without LP filter seems improper, but the amplitude drops for frequencies above the 1 MHz.


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Figure A.3: Standard ILC; discrete plant, uncorrected learning filter, phase-corrected learningfilter and product of the plant and the learning filter.

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Figure A.4: Standard ILC; convergence criterion for the designed learning filter.

110 Standard ILC

A.2.2 Simulation Results

The control goal is again active damping of the residual vibrations in a single channel. Thesimulation results of Standard ILC for iteration 0 and iteration 5 are depicted in Figure A.5and A.6, respectively. As initial input a fixed trapezium-shaped pulse is applied which pro-duces the desired droplet. For signal filtering it proved to be necessary to enlarge the triallength and to cut effects of signal leakage. It can be observed that the Standard ILC con-verges to the desired solution and Figure A.7, in which the Integrated Absolute Error isshown, supports this observation.

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Figure A.5: Standard ILC; simulated system response, desired response and system input foriteration 0.

Unfortunately, the application of Standard ILC to the inkjet printhead is not robust atall. The convergence criterion (A.8) is hardly fulfilled and only dynamics till approximately70 kHz can be learned effectively. This low cut-off frequency is also the reason for the shiftof the Q-filter in the ILC loop. The frequency content of the fixed trapezium pulse, whichis assumed to be the base of the ILC input signal, cannot be learned by the ILC algorithm,because of the low cut-off. A solution to this problem is to use the initial input unfiltered.The resulting ILC waveform then exist of the sum of a 30 V 5/5/3 pulse and adaptationsof the input signal, with a frequency content till 70 kHz, to realize active damping of theresidual channel vibrations.

This trick illustrates that it is possible to obtain convergence on simulation level. However,one can conclude that Standard ILC, based on the piezo sensor signal, is not suitable forcontrol of the actual inkjet printhead due to numerical problems. Therefore, experimentalresults with Standard ILC are omitted and the related learning technique Lifted ILC is appliedfor the implementation on the printhead.


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Figure A.6: Standard ILC; simulated system response, desired response and system input foriteration 5.

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Figure A.7: Standard ILC; Integrated Absolute Error as a function of the iteration number.

Appendix B

Lifted ILC without Actuation

Window

In Section 4.3, the implementation of a SISO Lifted ILC has been considered, which is basedon the integrated piezo sensor signal. To avoid that the duration of input wave becomes toolong, an actuation window has been applied. A drawback of this actuation window is a lossin observability, and thus in performance, as is illustrated in Section 3.2.6. Here, the resultsare discussed of an ILC with the same settings for the tuning parameters, but without anactuation window, thus N ∗ = N = 1000. The rank of the impulse response matrix H equals995, see also Figure 4.13. This indicates that the system is almost fully observable. Theoutput singular vectors of the five least significant singular values are shown in Figure B.1and represent the system trajectories which cannot be realized. It appears that all fiveoutput singular vectors are only active during the first 3 µs of the trial and are, therefore,not considered very important for the convergence behavior of the ILC.

5 10 15 20 25 30 35 40−0.8

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Figure B.1: Integrated SISO ILC without actuation window; output singular vectors of thefive least significant singular values.

114 Lifted ILC without Actuation Window

The relevant time signals for iteration 0 are depicted in Figure B.2. Again, the samecontrol goal is pursued, namely, active damping of the residual vibrations after jetting of adroplet. Therefore, the same reference signal is chosen as in Section 4.3.

0 10 20 30 40 50 60 70 80 90 100−2

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Figure B.2: Integrated SISO ILC without actuation window; integrated piezo signal, desiredpiezo signal and input signal for iteration 0.

For iteration 35, the time signals are shown in Figure B.3. From the upper plot it followsthat almost perfect tracking of the reference signal is obtained. Moreover, it can be seenthat the input remains active during the complete trial. However, during the last 40 µs theinput signal is very small. This illustrates that indeed the first part of the trial is the mostimportant for actuation, as is assumed for the implementation of ILC with actuation window.

The Integrated Absolute Error as a function of the iteration number is depicted in Fi-gure B.4. In this figure it can be seen that a large error reduction is obtained in the first fiveiterations and between iteration 5 and 25 the error signal decreases further. After iteration25, the IAE remains approximately at a constant level and the ILC is converged. The finalreduction of the IAE equals a factor 14 and the residual error signal now mainly consists oftrial-invariant disturbances and noise.

115

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Figure B.3: Integrated SISO ILC without actuation window; integrated piezo signal, desiredpiezo signal and input signal for iteration 35.

5 10 15 20 25 30 350

0.5

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IAE

[V]

Figure B.4: Integrated SISO ILC without actuation window; Integrated Absolute Error as afunction of the iteration number.

116 Lifted ILC without Actuation Window

In Figure B.5, the error spectrum is shown for iteration 0 and 35. From this figure it canbe seen that until 250 kHz a large reduction in error is obtained. The first acoustic resonancemode of the channel is almost completely suppressed.

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Figure B.5: Integrated SISO ILC without actuation window; error spectrum for iteration 0and 35.

Appendix C

Reproducibility and Sensitivity of

ILC

In this section, the reproducibility of ILC on one and the same ink channel is illustrated byexperimental results. Moreover, the sensitivity of ILC for production tolerances is tested bymeans of an experiment. For both experiments use is made of the integrated SISO ILC whichis discussed in Section 4.3.

C.1 Reproducibility of ILC

Several experiments have been performed to obtain a feeling for the reproducibility of the ILCresults. With these experiments it is tested how well the exact same result can be reproducedat another time instant. The IAE for three of these experiments is shown in Figure C.1. Thisfigure illustrates that the reproducibility is reasonably good. However, during experimentsthis was not always the case.

C.2 Sensitivity of ILC for Production Tolerances

To check the sensitivity of ILC for production tolerances between the different channels,the learned ILC input signal (of channel 50) is also applied to other channels of the sameprinthead. For a perfectly homogenous head, exactly the same responses are expected foreach channel. In practice, however, no two channels are exactly identical due to productiontolerances.

The time responses of the integrated piezo signal for several neighbors are shown in Fi-gure C.2. From this figure it can be seen that the best performance is obtained for channel50. This is a rather trivial result, because the ILC is designed for this channel. Moreover, thetime responses of neighboring channels 51, 55 and 60 are shown in the figure. These responsesindicate that there is a decrease in performance due to production tolerances. However, theactively damped case still provides an improvement with respect to the passively controlledcase, which takes substantially more time for the channel to become at rest.

This experiment illustrates that active damping of a channel can be obtained withoutidentification and application of ILC on each separate channel, however, this is at the cost ofthe performance.

118 Reproducibility and Sensitivity of ILC

0 5 10 150

0.5

1

1.5

2

2.5x 10−4

Experiment 1Experiment 2Experiment 3

PSfrag replacements


IAE

[V]

Figure C.1: Integrated SISO ILC; Integrated Absolute Error as a function of the iterationnumber as an illustration of the reproducibility of ILC.

0 10 20 30 40 50 60 70 80 90 100−2

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0

1

2

3

4x 10−6

channel 50channel 51channel 55channel 60channel 50 (before learning)

PSfrag replacements

Time [µs]

Int.

pie

zosi

gnal

[Vs]

Figure C.2: Integrated SISO ILC; integrated paint responses of neighboring channels on thelearned ILC input signal as an illustration of the sensitivity of ILC for production tolerances.

Appendix D

Simplification of the Learned Input

Signal

In normal operation of the printhead, the ink channels are actuated by trapezium-shapedinput pulses which are implemented on an Application Specific Integrated Circuit (ASIC).The learned actuation signals are, unfortunately, far too difficult to store on an ASIC. Onlyrelatively simple signals, consisting of a limited number of points connected by straight linesegments, can be implemented. Therefore, when it is desired to store a learned actuationsignal on an ASIC, the signal first has to be simplified. In Section D.1, an example of thesimplification of a learned actuation signal is given on simulation level and Section D.2 handlesseveral experimental results.

D.1 Simulation Results

In Figure D.1, the learned ILC actuation signal with an actuation window of 50 µs is shown,together with the response of the integrated piezo signal. This response is simulated bymaking use of the experimental printhead model. The learned input causes the output to beat a rest in approximately 42 µs. It should be mentioned that, because of a light damageof the ink channel which is used in Chapter 4, in this section another channel of the sameprinthead is considered.

Next, the learned actuation signal, consisting of 500 elements, is simplified by hand to asignal which consists only of 7 points, connected by straight line segments. It can be seenthat on simulation level almost an identical performance of the output signal is obtained, onlywith a rather crude approximation of the learned input signal.

120 Simplification of the Learned Input Signal

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Figure D.1: Integrated SISO ILC; learned input signal, simplified input signal and simulatedsystem responses.

D.2 Experimental Results

In case of experiments, the system is no longer fully linear, there are disturbances acting onthe system and the sensor measurements are corrupted with noise. The fixed input pulse andthe learned input signal with an actuation window of 60 µs are both depicted in Figure D.2and the system responses are shown in Figure D.3. It can be seen that the learned inputsignal is now more difficult to simplify.

With simplification A, the learned input signal, consisting of 600 elements, is approximatedby a signal comprising 8 points. Unfortunately, this crude approximation does not result inproper damping of the piezo sensor signal. To improve the performance of the simplifiedsignal, it is chosen to approach the ILC input signal better by simplification B. Here, 16points are used to approximate the 600 elements signal. From Figure D.3 it can be observedthat in this case a rather good performance is obtained.

From the experiments it can be concluded that it is indeed possible to simplify the actu-ation signal without give in too much in system performance, as long as the ILC input signalis not approximated too crudely.

D.2 Experimental Results 121

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Figure D.2: Integrated SISO ILC; fixed actuation pulse, learned actuation signal and simplifiedactuation signals.

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Figure D.3: Integrated SISO ILC; measured responses on fixed actuation pulse, learned actu-ation signal and simplified actuation signals.

Appendix E

Reference Signal for Multiple

Droplets

Until now, all ILC simulations and experiments are performed for a situation where a singledroplet is jetted. When it was desired to jet multiple droplets, the learned actuation signal wasapplied several times. For high jet frequencies, where the time duration between two followingdroplets is shorter than the length of the actuation signal, summation of overlapping inputsignal was applied. This method is exact for linear systems, however, for nonlinear systemsinaccuracies are introduced.

An alternative method is to design a long reference signal for ILC, which contains multipledroplets at a certain desired jet frequency. Here, the dynamics are not necessarily dampedafter each droplet. One could think of a sine-shaped reference signal, for example, where adroplet is produced each period. An advantage of this method is that summation of over-lapping input signals is no longer required. A drawback of this alternative is that a longerreference signal requires a longer trial length. Since the size of the impulse response ma-trix increases quadratically with the trial length, the computational effort may become toolarge when many droplets are jetted. Another drawback is that for every sequence of droplets(possibly of different volume) and for every jet frequency a new input signal has to be learned.

Simulations are performed with ILC, where the method of summation of overlappinginput signals is compared to the method with the long reference signal. The results of thesesimulations, making use of the experimental model from piezo input to piezo sensor, are shownin Figure E.1. Moreover, the result for fixed actuation pulses is depicted in the figure. In thisexample, four droplets are jetted at a jet frequency of 40 kHz and the integrated piezo sensorsignal is applied as system output. Both ILC methods approximately result in the samesystem response. In case of summation of overlapping input signals, however, the system isnot yet fully at rest when the next droplet has to be produced. This results in a small, butconstant, difference in peak value of the piezo sensor signal, compared to the response of ILCwith the long reference. Regarding the difference in response, this suggests that the ILC withthe long reference signal for multiple droplets results in a slightly better performance.

When the ILC with a reference for multiple droplets is applied which is based on themeniscus velocity or the non-integrated piezo signal, one should design this reference signalvery carefully. Otherwise, this could result in drifting of the input signal, because the meniscusvelocity, or derivative of the ink pressure is controlled and not the meniscus position or inkpressure, respectively.

124 Reference Signal for Multiple Droplets

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Figure E.1: Integrated SISO ILC; simulation results of jetting multiple droplets with summa-tion of overlapping input signals and with a long reference signal.

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