Ye_Shenxi.

Ye_Shenxi.
Identification and Feedforward
Control of a Drop-on-demand
Inkjet Printhead
Master of Science Thesis
Shenxi Ye
Delft Center for Systems and Control
Identification and Feedforward Control
of a Drop-on-demand Inkjet Printhead
Master of Science Thesis
For the degree of Master of Science in Mechanical Engineering at Delft
University of Technology
Shenxi Ye
October 17, 2011
Faculty of DCSC/Mechanical, Maritime and Materials Engineering (3mE) · Delft University
of Technology
The work in this thesis was supported by Océ. Their cooperation is hereby gratefully acknowledged.
c Delft Center for Systems and Control (DCSC)
Copyright All rights reserved.
Summary
Inkjet technology is a very important key-technology from an industrial point of view. The
variety of its applications is very wide, with the document printing the most common one.
Usually, applications of inkjet technology are accompanied with some performance criteria
concerned with several drop properties such as the drop-velocity and the drop-volume, as
well as the consistency of these drop properties. In addition, requirements with respect to
the productivity and stability of the jetting process itself are also frequently imposed. These
performance criteria are expected to become more tighter for future applications.
For a typical piezoelectric inkjet printhead, a standard actuation pulse is applied so as to
meet the drop-on-demand requirement. However, two performance limitations are generally
encountered for such an approach: residual pressure oscillations and cross-talk. The former
one relates to the fact that the ink inside the channel is not at rest immediately after a drop
ejection, while the later one refers to the fact that the drop properties of the jetting channel
are affected if its neighboring channels are simultaneously actuated. Both phenomena would
limit the drop consistency and the productivity considerably.
In this thesis, a systems and control method is proposed to improve the printing performance.
First, the modeling of an inkjet printhead is taken to provide good insights for the system
dynamics. Given the complicated jetting conditions, in this thesis, system identification
method is adopted to estimate the models. Based on these models, an off-line optimizationbased method is used in a SISO case to design an optimal input actuation pulse for the piezo
actuator. Then the damping of residual pressure oscillations inside one ink channel can be
improved. In the MIMO case for reducing the cross-talk, the method used is just a more
complex vision, except that an additional input delay between neighboring channels is also
optimized for.
Simulation results found with the identified models show good applicability of the proposed
method. In addition, based on a real printhead setup, the experimental results also demonstrate a significant improvement of the printing performance with respect to the drop-velocity
consistency, e.g., the maximal velocity variation in DoD curve can be reduced from 6 m/s to
Master of Science Thesis
Shenxi Ye
ii
2 m/s in direct-channel, the velocity caused by cross-talk can also be eliminated in multiple
channels.
Shenxi Ye
Master of Science Thesis
Table of Contents
1 Introduction
1
2 Printhead description
3
2-1 Ink channel description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2-2 Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2-3 Operational limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2-3-1
Residual pressure oscillations . . . . . . . . . . . . . . . . . . . . . . . .
7
2-3-2 Cross-talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-4 Experimental setup description . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
11
3 Identification of ink channels
3-1 Identification procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
16
3-2 Multi-channel system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3-3 Direct-channel Hd identification . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3-4 Cross-talk channel identification
3-4-1 Hc1 identification . . . .
3-4-2 Hc2 identification . . . .
3-5 Concluding remark . . . . . . . .
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28
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33
37
4 Feedforward control
4-1 Feedforward control design . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
39
4-2 SISO control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2-1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2-2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
44
45
4-3 MIMO control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3-1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3-2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
52
55
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Shenxi Ye
iv
Table of Contents
5 Conclusions and recommendations
57
Bibliography
59
Glossary
61
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shenxi Ye
61
Master of Science Thesis
Chapter 1
Introduction
The ability of inkjet technology to deposit materials, with diverse chemical and physical
properties, onto a substrate has made it an important subject for both industry and academic research. In last fifties, a rapid development of inkjet technology started off and since
then, the inkjet technology has been a very versatile technique with a wide range of applications due to the low operational costs. Apart from the conventional document printing, the
inkjet technology has also been applied in the production of solar panels [1][2], LED and LCD
fabrication [3][4], organic tissue and organ printing [5], as well as 3D rapid prototyping [6], etc.
In this thesis, a piezoelectric inkjet printhead with two arrays of ink channels is addressed.
Each channel is equipped with one piezo actuator. When one droplet is needed, a specified
actuation pulse is applied to the piezo actuator and then a single ink drop can be ejected from
the nozzle [7]. After the ejection, the ink channel is not immediately at rest and one should
wait until the residual pressure oscillations are damped out before another actuation pulse
can be applied. Otherwise, the properties of the subsequent droplets cannot be guaranteed.
For this reason, the attainable jetting frequency is limited and that would then degrade the
printhead performance. Another limiting factor is often encountered during the jetting of
several channels. Given the so-called cross-talk, the drop properties through an ink channel
are affected when its neighboring channels are simultaneously actuated. Other undesired
phenomena include the formation of satellite droplet, variations in ink properties and disturbances in nozzle [8].
However, in nowadays inkjet printheads, a fixed actuation pulse which does not take the
aforementioned problems into account is mostly used. Generally speaking, the actuation
pulse only consists of one positive trapezoidal pulse. This particular pulse is also called the
standard pulse. The main drawback of this standard pulse is that it would generate great
residual oscillations. In the literature, there are various methods for the design of an actuation pulse which help damp the residual oscillations [8][9][10][11]. While, in this thesis, the
method to be used is a further study of that used in reference [12]. That paper proposed an
optimization-based method to design the actuation pulse for the piezo actuator in order to
Master of Science Thesis
Shenxi Ye
2
Introduction
improve the damping of the residual oscillations in the meniscus velocity. Here, the meniscus is an interface between the ink and the air in a nozzle. However, it is thought to be
very difficult to experimentally measure the meniscus position and velocity during the jetting
process. Therefore, in this thesis, the integration of a piezo-unit sensored signal is used to
indicate the pressure signal inside an ink channel because of the self-sensing characteristic of
a piezo-unit [13]. Briefly speaking, a piezo-unit can be simultaneously used as an actuator
and a sensor. When used as a sensor, its sensored signal is proportional to the derivative of
the ink channel pressure.
Thus, a similar systematic model-based approach as the reference [12] is applied in this
research. First, we require a model of the system that we want to control. Several models
such as the ‘two-port’ model [11] and the ‘narrow-gap’ model [14] have been used in other
research. However, the accuracies of these models are to some extent limited due to the
complicated physical relationships in the printhead system. Therefore, in order to improve
the model accuracy, a system identification method is chosen to describe the printhead dynamics. A model can be estimated by relating the actuation pulse voltage (i.e, the input) to
the piezo-unit sensored signal (i.e, the output). Then, using this model, we are able to obtain
the optimal actuation pulse with an optimization-based feedforward control method. A timedelay between neighboring channels is also optimized to reduce the cross-talk in multi-channel
system [15]. In conclusion,
Estimate an accurate experimental model to describe the relevant dynamics of the inkjet printhead and then use this model for the implementation of an optimization-based feedforward
control. Finally investigate the effects of this approach to reduce the two performance limitations: residual pressure oscillations and cross-talk.
The problem statement above can be clarified with the following research objectives:
• Identify the model by making use of the piezo-unit sensored signal to describe the system
dynamics.
• Implement an optimization-based feedforward control method, which is based on the integrated sensor signal, in a single channel so as to damp the residual pressure oscillations
in practice. This is referred to as the SISO case.
• Apply an input time-delay between neighboring channels so as to reduce the effects of
the cross-talk in multiple channels system. This is referred to as the MIMO case.
This thesis report is organized as follows. In chapter 2, structure of a typical printhead and its
working principle are introduced at first and then, two performance limitations are described
in details. Moreover, the experimental setup and its sensor functionalities are discussed.
In chapter 3, the modeling of the printhead is treated with system identification based on
piezo-sensor functionality. Both the models for direct-channel and cross-talk channels are
estimated. In next chapter, an optimization-based feedforward control method is introduced
in direct-channel system, as well as an additional time-delay implementation in multi-channel
system. At the ends of these two cases, the experiment results are shown to verify the
improvements. Finally, chapter 5 presents the conclusions and gives some recommendations
for future research.
Shenxi Ye
Master of Science Thesis
Chapter 2
Printhead description
The basis of a drop-on-demand (DoD) inkjet printhead is the use of a piezoelectric crystal to
convert an electric pulse into mechanical pressure. This pressure then is able to overcome the
surface tension forces holding the ink in a nozzle [14]. The name drop-on-demand comes from
the fact that one droplet is only jetted when an actuation pulse is provided. The varieties
encountered in piezoelectric inkjet printhead designs are enormous, however, there are three
common fundamentals.
• Ink channel design
Four basic components like the channel, nozzle, ink supply and piezo-unit are required
for an ink drop jetting.
• Working principle
The operation of such a printhead is very complicated while the acoustics inside the ink
channel play the most important role.
• Operational limitations
The printing performances are affected mainly by two factors that are associated with
the design and operation of a printhead: residual pressure oscillations and cross-talk.
Although the research presented in this thesis is elaborated on one particular printhead setup
at Océ Technologies, Netherlands, the results throughout are still generally applicable in
other piezoelectric inkjet printheads. In this chapter, the above mentioned three aspects are
discussed explicitly.
Master of Science Thesis
Shenxi Ye
4
Printhead description
2-1
Ink channel description
Several processes take place before an ink drop can be jetted [14]. In Fig.2-1, the prototype
drawing for a typical DoD inkjet printhead is depicted. In the jetting case, it starts with
melting the ink in the melting unit (a). A good heat transfer and draining of the melted ink
at a given temperature is necessary. Then the ink is filtered with unit (b). The next unit is
the reservoir (c) with enough volume to keep the total printhead dimensions within certain
limits. For the hose (d), it is used for closing static pressure in the reservoir. The lower part
of the printhead is the central part (e) where drop formation happens. More details are given
later. The required electric voltage for jetting is supplied by the electronic flex (f).
Figure 2-1: 3D CAD drawing of a printhead prototype showing (a) the melting unit, (b) the
filtering unit, (c) the reservior, (d) the static pressure hose, (e) the central part and (f) the
electronic driving supply.
The printhead setup used in our experiments consists of two parallel arrays of 128 ink channels each. All these ink channels are assumed to be identical. A cross-section view of such an
ink channel is shown in Fig.2-2.
Shenxi Ye
Master of Science Thesis
2-1 Ink channel description
5
Figure 2-2: A cross-section view of an inkjet channel.
As we can see, an ink channel is carved in channel plate. On the top of plate, a filter is placed
between the reservoir and channel to remove impurities from the melted ink. A metallic plate
with drilled holes is attached at the bottom of the channel plate. Every hole acts as a nozzle
for jetting. One wall of the ink channel is formed by a flexible foil with a piezo-unit attached.
On the application of a voltage, the piezo-unit, as an actuator here, would deform the flexible
foil and then this formation would generate pressure waves in the channel. When specific
conditions are met, an ink drop is jetted [12].
Figure 2-3: A schematic cross-section view of a DoD printhead.
In Fig.2-3, a cross-section view of a typical printhead is schematically depicted. It is clear
to see that all these piezo-units are connected to the same substrate and actuate their corresponding ink channels.
Master of Science Thesis
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6
2-2
Printhead description
Working principle
In this section, a description of the working principle of the the used piezoelectric inkjet printhead is given. A schematic side view of an ink channel and its working principle is illustrated
in Fig.2-4.
Figure 2-4: A schematic side view of an ink channel and its working principle.
In order to jet a droplet, a positive trapezoidal pulse is provided to the piezo actuator and
then, the following five steps occur inside the ink channel [11].
To start with, a negative pressure wave is generated by enlarging the volume in the channel
with the actuation voltage increasing.
Then, this pressure wave splits up and propagates in both directions.
Since the reservoir and the nozzle are different boundaries, these pressure waves are reflected
in different ways when they reach the ends.
Next, by decreasing the actuation voltage, a positive pressure wave is superimposed on the
reflected waves exactly when they get to the middle of the channel.
Consequently, the wave traveling towards the reservoir is canceled whereas the pressure wave
traveling towards the nozzle is amplified to be large enough to jet a droplet.
Shenxi Ye
Master of Science Thesis
2-3 Operational limitations
2-3
7
Operational limitations
For a piezoelectric inkjet printhead, some important drop properties, such as drop velocity,
drop volume and the consistency for them, are related to certain printing performances, for example, to avoid irregularities on printed images. In practical applications, these performance
requirements are severely affected by the following two operational limitations [14].
2-3-1
Residual pressure oscillations
Once an ink drop is jetted, due to the presence of the traveling pressure waves, the fluidmechanics within an ink channel are not at rest immediately. Generally, a fixed actuation
pulse is designed under the assumption that an ink channel is in steady state. If the subsequent ink drop is jetted before the settling of these residual vibrations, the resulting drop
properties will be different from the previous one. So one has to wait for these residual pressure oscillations to be sufficiently damped out to guarantee consistent drop properties. Since
it takes several microseconds for the pressure oscillations inside the channel to decay, the maximally attainable jetting frequency will be limited. If the residual pressure oscillations are
ignored and the jetting frequency increases nonetheless, the resulted drop properties would
start varying.
Therefore, an important characteristic named DoD curve, which represents the ink drop
velocity as a function of the jetting frequency, is proposed to show the effect of residual vibrations on drop velocity consistency. Ideally, as we can imagine, such a DoD curve must be
flat. However, due to the aforementioned reason, the curve is far from flat in practice.
Here we take Fig.2-5 below as an example, in plot (a), when a standard actuation pulse
is applied, obviously, the pressure waves inside the channel take about 30 − 40 microseconds
to damp out after the drop ejection. Therefore, the attainable jetting frequency would be
limited. If we generally increase the jetting frequency, the residual pressure oscillations cannot
be damped out sufficiently. Furthermore, the resulted drop properties cannot be guaranteed.
As shown in plot (b), considerable fluctuation results in DoD curve. The maximum speed
variation, as we can see, is as large as 6 m/s.
Master of Science Thesis
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8
Printhead description
Integrated sensor signal (Vs)
(a)
2
pressure response
standard pulse (scaled)
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
(b)
Drop velocity (m/s)
10
8
6
4
2
10
15
20
25
30
35
DoD jet ing frequency (kHz)
40
45
50
Figure 2-5: Residual vibrations (a) and its effect on DoD curve (b).
2-3-2
Cross-talk
The phenomenon that one ink channel cannot be actuated without affecting the fluid-mechanics
in its neighboring channels is called cross-talk. It mainly occurs in two ways. First, the acoustic cross-talk, which occurs via the ink reservoir, i.e., the pressure waves within one channel
Shenxi Ye
Master of Science Thesis
2-3 Operational limitations
9
influence other channels. Generally, its overall influence can be considered small and here we
mainly introduce the second one, the structural cross-talk, which can again occur in two ways.
For example, since all piezo-units are connected to a substrate, as shown in Fig.2-3, when a
piezo-unit is actuated, the reaction force of the substrate will be guided to its neighboring
non-actuated piezo-units. And then the channels which are not actuated will deform, too.
As shown in Fig.2-6 (a), the resulting deformation of the neighboring channels is opposite to
the deformation which is necessary for jetting a drop. Therefore, when a neighboring channel
is actuated together with the direct-channel, the drop velocity of the direct-channel will be
lower than that of only direct-channel is actuated.
Figure 2-6: Front view of the channel structures. (a) The reaction force of the substrate is
guided to the neighboring channels and this results in an opposite deformation of the neighboring
channels (b) The deformation of one channel results in an enlargement of its neighboring channels.
Another path is via the deformation of a channel itself since a positive pressure inside the
actuated channel would result a deformation of its neighboring channels. As shown in Fig.2-6
(b), the deformation of one channel results an enlargement of its neighboring channels and
diminution of the channel pressure, thus the drop velocity will also get lower [14].
In Fig.2-7, with actuation in the neighboring two channels, the effect of cross-talk on the
direct-channel pressure and the DoD curves are depicted. As we can see in plot (a), although
the pressure waves of two neighboring channels (red and green) are much smaller than that
of the direct channel (blue), this part can not be easily ignored. In plot (b), the resulting
drop speed of direct-channel (blue) is depicted when in turn the first neighboring channel
is actuated (black), as well as the first and second neighboring channels are simultaneously
Master of Science Thesis
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10
Printhead description
actuated (green), then we can see obvious drop-velocity decreases among these conditions.
This in turn shows that the effect of cross-talk on the drop-speed is particularly substantial
when multiple channels are simultaneously actuated.
Integrated sensor signal (Vs)
(a)
1.5
direct channel
first neighboring channel
second neighboring channel
1
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (μs)
(b)
30
35
40
45
50
drop velocity (m/s)
8
Direct channel jet ing
Direct and first neighboring channels jet ing
Deriect, first and second neighboring channels jet ing
7
6
5
4
20
22
24
26
DoD jet ing frequency (kHz)
28
30
32
Figure 2-7: Cross-talk (left) and its effect on the drop-speed (right).
Shenxi Ye
Master of Science Thesis
2-4 Experimental setup description
2-4
11
Experimental setup description
In references [12] and [15], a discrete-time model relating the piezo input voltage to the meniscus velocity was used. Namely, the input is the piezo actuation pulse and the output is the
meniscus velocity. The meniscus is the interface between ink and air in a nozzle. However, it
is very difficult to experimentally measure the meniscus position and velocity while jetting an
ink drop [16]. Thus in this thesis, we adopt to use the piezo-unit as both an actuator and a
sensor simultaneously because of its so-called ‘self-sensing’ capability. A schematic overview
of the experimental setup is depicted in Fig.2-8.
Figure 2-8: A schematic overview of the experimental setup.
With the computer, the actuation signals can be programmed and processed, then they are
sent to a waveform generator. Through an amplifier, the actuation signal is fed to switch
board, which is used to determine which channel is provided with the appropriate actuation
pulse. An oscilloscope is used for the tracing of both the actuation and sensor signals. The
CCD camera equipped with a microscope is used to observe the generated droplets. Since
both the time duration and the distance a droplet has traveled are known, an estimate of the
drop velocity can easily be obtained. Unfortunately, all these measurements only give details
on the ink flow outside the printhead, while for the phenomena inside the ink channels need
to be investigated with the piezo-unit sensored signal. Some fundamental explanations for
piezo-unit’s self-sensing characteristic are introduced as follows.
Master of Science Thesis
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12
Printhead description
As generally known, a piezo-unit can be used as an actuator or a sensor [17]. For the actuator
effect, if an electrical potential V is applied to the piezo-unit, a deformation u of the piezounit is generated. While for the sensor effect, if a force F is applied to the piezo’s surface, an
electric charge q generates. Together, this behavior can be described as in equation (2-1).
"
u
q
#
"
=
d 1/k
C
d
#"
V
F
#
(2-1)
with C the piezo’s capacity, d the piezoelectric charge constant, k the piezo’s stiffness coefficient.
When a single piezo-unit is used both as an actuator and a sensor simultaneously, it possesses several advantages over the use of two separate units acting as individual actuator and
sensor, e.g. much lighter and costlier. Furthermore, the collocation of actuating and sensing
allows the control signal to be applied at the point of measured response, thereby eliminating
the capacitive coupling between individual actuator and sensor [13]. In practical applications,
specially designed electric circuit as introduced in reference [14], the so-called bridge circuit,
is used to realize the self-sensing concept.
For a piezo-unit sensor, its measured signal is an electric charge q. In our experiments,
the measured signal q is made up of two contributions, as shown in Fig.2-9. One is generated
from the applied actuation voltage V via the piezo’s capacity C, which is also referred to
as the direct-path. The other one originates from the force F , exerted by the ink inside the
channel, via the piezoelectric charge constant d and is also called as the indirect-path.
Figure 2-9: Division into a piezo- and ink-block diagram.
As the indirect-path contribution is related to the jetting process inside the ink channel, it
is the required sensor signal and needs to be extracted from the measured output signal q.
However, since the contribution of the direct-path is considerably larger than that of the
indirect-path, it is quite difficult to acquire this part while using the piezo-unit as an actuator. In this thesis, we use the off-line compensation approach proposed in [18] to reconstruct
the sensor signal.
Shenxi Ye
Master of Science Thesis
2-4 Experimental setup description
13
As schematically depicted in Fig.2-10, two experiments are carried out with applying the
same input. The first one is carried out with the ink inside the channel and the measured
piezo output signal q_fill is stored. During the second experiment, the channel is kept empty
by removing the ink from the printhead, then the measured output signal q_empty is stored.
By a subtraction of the signal q_empty from the signal q_fill , the required sensor signal q is
obtained. Here during the experiments in two situations, piezo capacity differences caused by
temperature difference and deformation of substrate are neglected.
Figure 2-10: The basic principle to obtain the actuation and sensor signal simultaneously as
used in the piezo-sensing device.
Having discussed the technical implementation of using piezo-unit as an actuator and a sensor, as well as the practical approach to obtain the required sensor signal, the next question
is to know the representation of this sensor signal. Physically speaking, a piezo-unit senses
the force which results from the pressure distribution in the channel. This force creates a
charge on the piezo-unit, as shown in equation (2-1). Therefore, we have a voltage as our
final sensor signal. This voltage signal is thought to be proportional to the derivative of the
pressure inside the ink channel because the force on the piezo-unit is induced from the channel
pressure [8]. As a result, the obtained sensor signal can be regarded as a representation for the
jetting process and plays an important role in investigating a systems and control approach
to improve printing performance. This part will be introduced in next chapter.
Master of Science Thesis
Shenxi Ye
14
Shenxi Ye
Printhead description
Master of Science Thesis
Chapter 3
Identification of ink channels
Generally, the piezo actuation pulse only consists of one positive trapezoidal pulse, namely
the standard pulse as shown in Fig.2-5 (a). It is determined mainly by manual tuning on
a printhead setup. The main drawback of this standard pulse is that it generates residual
pressure oscillations, as discussed in section 2-3-1. Thus the printing quality and the jetting
frequency would be limited. Therefore, in this thesis, a systems and control approach is used
to design the actuation pulse for the DoD inkjet printhead so as to improve the printing
performance.
For this purpose, we require a model of the system that we want to control, at first. Basically, there are two different ways for modeling [19]
• physical modeling
A model is derived by formulating the physical laws that the system obeys, such as
mass-balance, energy-balance and so on.
For ink channel systems, two such models can be found in the literature. One is the
‘two-port’ model, in the reference [8], the ink channel is divided into several functional
subsystems. Each of them then is modeled as a two-port system using first principle
modeling only. At last, a modeling approach based on the notion of bilaterally coupled
systems is proposed. The other model is called the ‘narrow-gap’ model [14], which is
an analytical model. It describes the system dynamics mainly based on viscous and
thermal wave propagation principles.
• system identification
A model is constructed by fitting a parametric model to the measured input and output (I/O) data, without concerning the physical interpretation of the model parameters.
As in many engineering problems, system identification has been successfully applied
and sometimes, it is the only method that can be applied to do the modeling because
Master of Science Thesis
Shenxi Ye
16
Identification of ink channels
of high system complexity or insufficient knowledge about the physical behaviors. However, due to the desired simplicity, a model estimated from system identification can
not perfectly describe a complex system.
Although some analytic and numerical models are available, they are more appropriate for
printhead system design and not sufficiently accurate for the purpose of control design. For
high performance control problem, it requires accurate models. So in this thesis, system
identification method is applied as the best modeling method to get an accurate model for
further control design. At the very beginning, a brief introduction about system identification
procedures is given.
3-1
Identification procedure
As shown in Fig.3-1, the system identification procedure has a natural logical flow: first is
collecting data, then choosing a model set and picking the ‘best’ model from this set. It is
quite likely that the model first obtained will not pass the model validation tests. Various
steps of the procedure thus need revision [20].
Figure 3-1: A flow diagram of system identification.
Generally, the I/O data Z N = {y(t) , u(t) | t = 1, ..., N } is generated according to a true
system S.
S : y(t) = G0 (q)u(t) + H0 (q)e(t)
The choice of input signals is of very substantial influence since the input signals determine
the operating point of the system and which modes of the system are excited during the experiments [21]. For the identification of a linear system, there are some basic facts governing
Shenxi Ye
Master of Science Thesis
3-1 Identification procedure
17
the choices of the input signal. First, from practical considerations, the input signal must
have limited amplitudes. Second, it should be persistently exciting of a certain order, which
at least should be equal to the number of parameters to be estimated in the plant G(θ). For
example, if the numerator and denominator of a transfer-function model have a same number
of parameters n, then the input signal should be persistently exciting of order 2n. This means
that the spectrum of the input Φu (ω) should be nonzero at 2n points.
Next to the choice of the input signal, the selection of the sampling interval is also of importance. Generally, the information loss caused by sampling is best described in frequency
domain. To avoid aliasing, the signal being sampled should not contain frequencies beyond
the Nyquist-frequency fN , which is defined as half of the sampling frequency fS [22].
The choice of model structure is quite subjective. Generally, we denote the model structures by M, while a particular model corresponding to the parameter value θ is described as
M(θ). Such a parametrization is instrumental in searching the ‘best’ models. For example,
given a model structure below
M : {G(q, θ) , H(q, θ) | θ ∈ DM }
Then one aspect of the choice of a good model structure is to select M so that S ∈ M holds
for the given description of S, namely, S = M(θ0 ) for some values θ0 [20]. Normally, S is
unknown and this will typically involve trials of several different structures M.
The choice of an appropriate model structure is based on the insights about the system
to be identified. It is conceivable that various nonparametric techniques could be helpful for
finding suitable model structures and estimating the order of a linear system. Usually, the
empirical transfer function estimate (ETFE) is applied to give valuable information about
the resonance peaks and the high frequency roll-off. All these would give a hint as to what
model orders would be required to give an adequate description of the interested dynamics.
Based on the data set Z N over the interval 1 ≤ t ≤ N , the estimate of the frequency response
of the true plant transfer function G0 (eiω ) can be described as
ĜN (eiω ) =
YN (ω)
UN (ω)
(3-1)
with two Fourier transforms
N
1 X
YN (ω) = √
y(t)e−iωt
N t=1
,
N
1 X
UN (ω) = √
u(t)e−iωt
N t=1
In practice, several model sets are commonly used, such as ARX, OE, BJ, etc. Here, as an
example, we consider the model structure M corresponding to a generalized transfer-function
structure
C(q)
B(q)
y(t) =
u(t) +
e(t)
(3-2)
F (q)
D(q)
Master of Science Thesis
Shenxi Ye
18
Identification of ink channels
with the parameter vector
θ = [ b1 , · · · , bnb , f1 , · · · , fnf , c1 , · · · , cnc , d1 , · · · , dnd ]
Here θ includes all the coefficients of the polynomials involved. The orders of the polynomials
are nb , nf and so on. By tuning the orders, a corresponding true system S can be uniquely
represented such that S = M(θ0 ) holds.
As we know, the selection of a parameterized model structure is vital for identification problem. This is the link between system identification and parameter estimation techniques.
Basically, there are two methods, prediction-error identification method (PEM) and subspace
model identification (SMI).
In the PEM framework, the prediction error {(t, θ)} is used to create a cost-function which
is then optimized with respect to the parameters of the model. The prediction error is
(t, θ) = y(t) − ŷ(t, θ)
with the predicted output given by
ŷ(t, θ) = H(q, θ)G(q, θ)u(t) + (1 − H(q, θ)−1 )y(t)
The cost-function is usually defined as the average of two-norm of the prediction error
VN (θ) =
N
1 X
ky(t) − ŷ(t, θ)k22
N t=1
(3-3)
The resulting estimation of θ̂N can be obtained by the minimization of equation (3-3)
θ̂N = arg min VN (θ)
θ
Usually, the PEM is considered to be the basic approach for system identification because
of three important advantages. First, it is applicable to general model structures. Second,
it has an optimal asymptotic accuracy when the true system can be represented within the
model structure (S ∈ M). Third, it has reasonable properties when the true system can not
be represented within the model structure (S ∈
/ M).
For a linear system with several outputs, which requires a model structure with many parameters, the SMI methods form a valuable alternative. They have the advantage of allowing an
estimate by using efficient and numerically robust calculations without iterative search. For
detailed information about SMI, we can refer to the reference [23].
Shenxi Ye
Master of Science Thesis
3-1 Identification procedure
19
With the parameter estimation procedure, a ‘best’ model within the chosen model structure
can be obtained yet. Then, model validation is the next crucial step to check if the model
performs sufficiently well. There exist several methods with different characters. At first, a
particular useful technique, the so-called residuals analysis is introduced.
The residuals here mean the differences between the model predicted output and the measured output. Thus, the residuals represent the portion of the data which are not explained
by the model.
(t) = (t, θ̂N ) = y(t) − ŷ(t, θ̂N )
Generally, residuals analysis consists of two tests: the whiteness test R̂ and the independence
test R̂u , given as following two equations.
R̂N (τ ) =
N
1 X
(t)(t − τ )
N t=1
N
1 X
N
(t)u(t − τ )
R̂u
(τ ) =
N t=1
(3-4)
The numbers R̂N (τ ) carry the information about whether the residuals can be regarded as
white. If they are not small for τ 6= 0, then part of (t) could have been predicted from
past data. This means that y(t) could have been predicted from past data, which is a sign
N (τ ) express the covariance between residuals and
of deficiency of the model. The number R̂u
past inputs. If they are not small, then there are traces of past inputs in the residuals. This
means there is a part of y(t) that originates from the past inputs has not been properly picked
up by the model. Hence, the model can be improved.
Another commonly used validation method is to compare the fit between the model simulated output and the measured output:
Fit = (1 −
|y(t) − ŷ(t, θ̂N )|
) × 100%
|y(t)|
(3-5)
If a model can pass the validation tests, it is regarded to be sufficient for intended application.
Otherwise, the identification procedure should go back to previous steps, such as choosing
new input signals or model structures, and find a better model.
Master of Science Thesis
Shenxi Ye
20
3-2
Identification of ink channels
Multi-channel system
As we previously discussed in Chapter 2, this thesis mainly focuses on improving the printing
performance of a printhead by restraining two operational factors, residual pressure oscillations in a single ink channel and cross-talk among several ink channels, respectively. The
following figure shows a diagram for a multi-channel system. Here, only five channels are
considered.
−2
−1
( −2)( −2)
+1
( +1)( +1)
( −1)( −1)
( −1)
( −2)
+2
( +2)( +2)
( +2)
( +1)
( +1)
( −1)
( −2)
( +2)
+
Figure 3-2: A block diagram of the multi-channel system.
As depicted in Fig.3-2, the models have two sub-index. Each of them has an input and output
in the channel denoted by the second and first index, respectively. For example, an input for
Shenxi Ye
Master of Science Thesis
3-2 Multi-channel system
21
channel n + 1, denoted as un+1 , goes through model Hn(n+1) to have an effect on the output
of channel n, which is represented by yn . Moreover, yn is the sum of the outputs from five
models, Hn(n−2) , Hn(n−2) , Hnn , Hn(n+1) and Hn(n+2) :
yn = yn(n−2) + yn(n−2) + ynn + yn(n+1) + yn(n+2)
In our case, two assumptions are made: all channels in the printhead are identical and bilateral
symmetrically distributed. Thus, with the same input for every channel, the output effect
on the middle channel would be identical, no matter from the left neighboring channel or
the right one. As a result, we can simplify such a multi-channel system into three models,
denoted as follows
Hnn = Hd
Hn(n−1) = Hn(n+1) = Hc1
(3-6)
Hn(n−2) = Hn(n+2) = Hc2
Here, Hd means the model for the direct-channel dynamics. Hc1 and Hc2 represent the models for the cross coupled dynamics from the first and second neighboring channel, respectively.
Now, the system identification problem for a printhead can be solved by estimating three
SISO models. In subsequent sections, these models are discussed individually.
Master of Science Thesis
Shenxi Ye
22
Identification of ink channels
3-3
Direct-channel Hd identification
In this section, we will introduce the practical system identification procedures for directchannel. A model for Hd , as shown in Fig.3-2 is given in the end.
To collect data for further analysis, a few decisions have to be made. During the experiments, we selected a sampling interval of 0.1 µs. The input was chosen to be a periodic signal
since the drop jetting is an iterative process. Due to the practical limitations on the waveform shape of the actuation pulse, the input signal we used should be in trapezoidal form.
As shown in Fig.3-3, a standard pulse was tried at first. However, this pulse would cause
residual pressure oscillations, as discussed in section 2-3-1, the identification results were not
good especially when we considered different jetting frequency conditions.
Actuation input signals
30
Test pulse
Standard pulse
25
20
Voltage (V)
15
10
5
0
−5
−10
−15
0
50
100
150
Time (μs)
200
250
300
Figure 3-3: Test pulse (blue) used as input signal and a standard pulse (red).
In reference [12], an additional pulse is applied after the standard pulse in order to damp the
residual oscillations in the meniscus velocity, namely the test pulse as shown in Fig.3-3. And
we decided to choose it as our input signal for the identification problem. The output signal
is the piezo-unit sensored signal as discussed in section 2-4, which is a subtraction of the piezo
outputs from the ink filled-channel and empty-channel.
Shenxi Ye
Master of Science Thesis
3-3 Direct-channel Hd identification
23
A record of 60, 000 samples was collected and a portion of the data is shown in Fig.3-4
below. The data set was then split into two halves, to be used for estimation and validation,
respectively.
Output signal
Sensored signal (V)
0.1
0.05
0
−0.05
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−4
x 10
Input signal
Actuation pulse (V)
30
20
10
0
−10
−20
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 3-4: Time plot for direct-channel.
For the input signal, we also checked its spectrum. For example, as shown in Fig.3-5, there
were a large number (16) of nonzero values in different frequencies up to 330 kHz. Thus we
thought this input signal was persistently exciting enough for our model estimation.
Periodogram
0
10
−2
10
−4
Amplitude
10
−6
10
−8
10
−10
10
−12
10
2
10
3
10
4
10
Frequency (Hz)
5
10
Figure 3-5: Input spectrum.
Master of Science Thesis
Shenxi Ye
24
Identification of ink channels
With a high sampling frequency of 10 MHz, if we applied the ETFE to have a first insight
about the system, we can find the data contains high-frequency noise outside the frequency
range of the system dynamics, as shown in Fig.3-6.
Frequency response
0
Amplitude
10
−5
10
2
3
10
10
4
5
10
10
6
10
7
10
4
x 10
Phase (deg)
0
−1
−2
−3
2
10
3
10
4
5
10
10
6
10
7
10
Frequency (Hz)
Figure 3-6: ETFE for direct-channel I/O data.
Therefore, a resampling of the data signals without information loss is necessary and helpful.
In our experiments, a resampling factor 15 was selected. As shown in Fig.3-7, from the ETFE
of the resampled data, we can clearly see two resonance peaks in a frequency interval from
20 kHz to 200 kHz. Thus, at least a fourth-order model is required.
Since our modeling focuses on the dynamics rather than the disturbance properties, a outputerror (OE) model would be sufficient. Its structure can be represented by
y(t) =
Shenxi Ye
B(q)
u(t) + e(t)
F (q)
Master of Science Thesis
3-3 Direct-channel Hd identification
25
Frequency response
−2
Amplitude
10
−3
10
ETFE of the data
FR of the model OE880
5
10
Phase (deg)
−100
−200
−300
−400
−500
5
10
Frequency (Hz)
Figure 3-7: ETFE and FR of an estimated model for direct-channel.
After several trials, an eighth-order OE model was chosen, with its frequency response shown
in Fig.3-7. Obviously, it has a good fit within the frequency range from 20 kHz to 200 kHz.
Master of Science Thesis
Shenxi Ye
26
Identification of ink channels
For model validation, here we use two approaches. Fig.3-8 shows that this chosen OE model
has a Best Fit of 94.76 to the validation data. That means the model can reproduce the data
set quite well.
Best Fit for model OE880
0.06
Measured output
Model predicted output
0.04
Sensored signal (V)
0.02
0
−0.02
−0.04
−0.06
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 3-8: Best Fit between measured and simulated outputs.
Fig.3-9 shows the residual analysis results.
Autocorrelation of residuals for output y1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−100
−80
−60
−40
−20
0
20
40
60
80
100
40
60
80
100
Cross corr for input u1 and output y1 resids
0.2
0.1
0
−0.1
−0.2
−100
−80
−60
−40
−20
0
Samples
20
Figure 3-9: Residual tests for direct-channel.
Shenxi Ye
Master of Science Thesis
3-3 Direct-channel Hd identification
27
As we just mentioned, for a output error model, the modeling focus is on the dynamics G(q, θ).
So the model could just show independence test of the residual (t) and the input u(t), while
pay less attention to the whiteness test of (t). Generally, for a good model, the required
correlation function should not go significantly outside a confidence region, as we introduced
in section 3-1. The confidence region corresponds to the range of residual values with a specific probability of being statistically insignificant fro the system. The system identification
toolbox in Matlab uses the estimated uncertainty in the model structures to calculate this
confidence intervals. Here in Fig.3-9, we use a 95% confidence region, which means that the
region around zero represents the range of residual values that have a 95% probability of
being statistically insignificant.
As shown in Fig.3-9, the cross-correlation R̂u passes the residuals test. This allows to determine a suitable model set M. Besides the residuals analysis, we can also compare the
frequency response between the model G(eiω , θ̂N ) and the standard deviation of its variance
q
cov(G(eiω , θ̂N )). For example, if we want to use the model for control, the modeling error
q
cov(G(eiω , θ̂N )) ) has to be small up to the bandwidth. A rule of thumb is
(measured by
q
cov(G(eiω , θ̂N )) < 0.1kG(eiω , θ̂N )k
(3-7)
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
5
6
10
10
Frequency (rad/s)
Figure 3-10: Comparison between G(eiω , θ̂N ) (red) and
Master of Science Thesis
q
cov(G(eiω , θ̂N )) (blue).
Shenxi Ye
28
Identification of ink channels
As shown in Fig.3-10, the equation (3-7) holds, which means the uncertainty is small and the
model identification is good enough. Thus it seems reasonable to pick this eighth-order OE
model as the final choice for our direct-channel model Hd .
Its numerical value is
Hd :
y(t) = [B(q)/F (q)] + e(t)
B(q) = 0.0001165 − 0.001259q −1 + 0.001601q −2 + 0.002929q −3
− 0.009304q −4 + 0.01049q −5 − 0.006042q −6 + 0.001469q −7
F (q) = 1 − 3.477q −1 + 5.893q −2 − 6.732q −3
+ 5.592q −4 − 3.296q −5 + 1.236q −6 − 0.2782q −7 + 0.06331q −8
3-4
Cross-talk channel identification
A quite similar procedure with previous section is used for the cross-talk channel estimation.
The models for Hc1 and Hc2 , mentioned in section 3-2, are given in the end of the following
two subsections, respectively.
3-4-1
Hc1 identification
To estimate a ‘best’ model for the first cross-talk channel, we proceeded as follows. A record
of 60, 000 samples was collected during the experiments. The input signal is the test pulse
applied in the first neighboring channel (n + 1), while the output signal is the piezo-unit
sensored signal of channel (n). Due to the setup limitations, some high-frequency disturbances
intervened. As shown in Fig.3-11, with a fast Fourier transform of the output data, we can
clearly see some disturbances around 1 MHz caused by piezo actuator dynamics, which are
beyond the frequency range of interest for the system dynamics.
Shenxi Ye
Master of Science Thesis
3-4 Cross-talk channel identification
29
−3
Single−Sided Amplitude Spectrum of y(t)
x 10
6
5
|Y(f)|
4
3
2
1
0
0
5
10
15
Frequency (Hz)
5
x 10
Figure 3-11: Fast Fourier transform for output data.
Since the input-output relation for a linear system will not be changed by filtering the input
and output data through a same filter [20], a digital low-pass filter of 500 kHz was applied to
preprocessing the data. A portion of the filtered data is shown in Fig.3-12. Then the filtered
data was split into an estimation set, consisting of the first 30, 000 samples, and a validation
set, consisting of the remaining 30, 000 samples.
−3
Output signal
x 10
Sensored signal (V)
5
0
−5
−10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−4
x 10
Actuation pulse (V)
Input signal
20
10
0
−10
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 3-12: Time plot for the first cross-talk channel.
Master of Science Thesis
Shenxi Ye
30
Identification of ink channels
Then a factor of 15 was selected to resample the estimation data, the ETFE of this data set
is depicted in Fig.3-13. As we can see, there are three resonance peaks, thus, a sixth-order
model is needed, at least. Due to the reasons we discussed in direct-channel identification
problem, a OE model structure was also chosen for the estimation problem here. After several
trials, a ninth-order OE model was obtained, with its frequency response depicted in Fig.3-13.
Frequency response
−3
Amplitude
10
−4
10
ETFE of the data
FR of the model OE990
4
5
10
10
500
Phase (deg)
0
−500
−1000
−1500
−2000
4
10
5
10
Frequency (Hz)
Figure 3-13: ETFE and FR of an estimated model for first cross-talk channel.
Shenxi Ye
Master of Science Thesis
3-4 Cross-talk channel identification
31
Next task is the model validation.
−3
Best Fit for model OE990
x 10
Model predicted output
Measured output
2
Sensored signal (V)
0
−2
−4
−6
−8
3
3.1
3.2
3.3
3.4
3.5
Time (s)
3.6
3.7
3.8
3.9
4
−4
x 10
Figure 3-14: Best Fit between measured and simulated outputs.
Autocorrelation of residuals for output y1
1
0.5
0
−0.5
−1
−100
−80
−60
−40
−20
0
20
40
60
80
100
40
60
80
100
Cross corr for input u1 and output y1 resids
0.2
0.1
0
−0.1
−0.2
−100
−80
−60
−40
−20
0
Samples
20
Figure 3-15: Residual tests for first cross-talk channel.
According to Fig.3-14, we see a best fit of 70.09 between the measured and predicted data.
The residuals analysis is given in Fig.3-15. The values of cross-correlation R̂u are inside a
95% confidence region, which means the model passes the residuals test.
Master of Science Thesis
Shenxi Ye
32
Identification of ink channels
Furthermore, we take a look at the frequencyqresponse comparison between the model G(eiω , θ̂N )
and the standard deviation of its variance
cov(G(eiω , θ̂N )).
−2
10
−3
10
−4
10
−5
10
−6
10
−7
10
−8
10
5
6
10
10
Frequency (rad/s)
Figure 3-16: Comparison between G(eiω , θ̂N ) (red) and
q
cov(G(eiω , θ̂N )) (blue).
As shown in Fig.3-16, the equation (3-7) holds, which means the uncertainty is small and
the model identification is good enough. So it seems quite reasonable that we did a good
modeling job with a fairly simple model OE990 for our first cross-talk channel model Hc1 .
Its numerical value is
Hc1 :
y(t) = [B(q)/F (q)] + e(t)
B(q) = 0.0002768 − 0.001107q −1 + 0.001826q −2 − 0.001446q −3
− 0.0001079q −4 + 0.001686q −5 − 0.002164q −6 + 0.001432q −7 − 0.0004426q −8
F (q) = 1 − 1.894q −1 + 1.215q −2 + 0.4509q −3
− 1.56q −4 + 1.525q −5 − 0.7671q −6 − 0.07883q −7 + 0.2018q −8 − 0.02906q −9
Shenxi Ye
Master of Science Thesis
3-4 Cross-talk channel identification
3-4-2
33
Hc2 identification
The identification procedure for the second cross-talk channel is quite the same with that for
the first one. So here only a brief introduction is given. After the I/O data was collected, a
low-pass filter was also used to remove the disturbances of high frequencies that we do not
want to be included in the modeling process. Then the filtered data was split into two parts
for estimation and validation, respectively. A portion of that data is given in Fig.3-17 below.
−3
Sensored signal (v)
4
Output signal
x 10
2
0
−2
−4
−6
−8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−4
x 10
Input siganl
Actuation pulse (V)
30
20
10
0
−10
−20
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 3-17: Time plot for the first cross-talk channel.
In Fig.3-18, we can see the ETFE of the resampled data with a resampling factor of 15. There
are three resonance peaks in the frequency range from 10 kHz to 200 kHz. So the order of a
sufficient model should be at least six. For the model structure, we still chose the output-error
form. After several trials, a ninth-order OE model was adopted. Its frequency response is
also displayed in Fig.3-18.
Master of Science Thesis
Shenxi Ye
34
Identification of ink channels
Frequency response
−2
10
ETFE of the data
FR from the model OE990
−3
Amplitude
10
−4
10
−5
10
4
5
10
10
500
Phase (deg)
0
−500
−1000
−1500
4
10
5
10
Frequency (Hz)
Figure 3-18: ETFE and FR of an estimated model for second cross-talk channel.
Shenxi Ye
Master of Science Thesis
3-4 Cross-talk channel identification
35
With the following two figures, model validation is discussed.
−3
Best Fit for model OE990
x 10
3
Model predicted output
Measured output
2
Sensored signal (V)
1
0
−1
−2
−3
−4
−5
−6
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
0.9
1
−4
x 10
Figure 3-19: Best Fit between measured and simulated outputs.
Autocorrelation of residuals for output y1
0.5
0
−0.5
−100
−80
−60
−40
−20
0
20
40
60
80
100
40
60
80
100
Cross corr for input u1 and output y1 resids
0.1
0.05
0
−0.05
−0.1
−100
−80
−60
−40
−20
0
Samples
20
Figure 3-20: Residual tests for second cross-talk channel.
A best fit of 79.02 between the measured and predicted data, as shown in Fig.3-19, together
with the residuals analysis in Fig.3-20 illustrate the model identification is good enough.
Master of Science Thesis
Shenxi Ye
36
Identification of ink channels
Furthermore, let us have a look at the frequency response
comparison between the model
q
G(eiω , θ̂N ) and the standard deviation of its variance
cov(G(eiω , θ̂N )).
−3
10
−4
10
−5
10
−6
10
−7
10
−8
10
5
6
10
10
Frequency (rad/s)
Figure 3-21: Comparison between G(eiω , θ̂N ) (red) and
q
cov(G(eiω , θ̂N )) (blue).
As shown in Fig.3-21, the uncertainty of the model is small and such a OE990 model is a
reasonable choice for the second cross-talk channel model Hc2 . Its numerical value is
Hc2 :
y(t) = [B(q)/F (q)] + e(t)
B(q) = − 0.0002019 + 0.0003613q −1 − 0.00001565q −2 − 0.000714q −3
+ 0.0009255q −4 − 0.0003435q −5 − 0.0004342q −6 + 0.0005931q −7 − 0.0002703q −8
F (q) = 1 − 1.376q −1 + 0.5288q −2 + 1.317q −3
− 2.199q −4 + 1.515q −5 − 0.0376q −6 − q −7 + 0.8364q −8 − 0.4067q −9
Shenxi Ye
Master of Science Thesis
3-5 Concluding remark
3-5
37
Concluding remark
Till now, we have got three models for the direct-channel and two cross-talk channels, namely
Hd , Hc1 , Hc2 shown in Fig.3-2. The frequency responses with their corresponding model
uncertainties of these three models are depicted in following Fig.3-22.
Frequency response
−2
10
−3
Amplitude
10
−4
10
Hc2
Hc1
−5
Hd
10
5
10
Frequency (Hz)
Figure 3-22: Frequency responses with model uncertainties for Hd , Hc1 , Hc2 .
As we can see, the amplitude of direct-channel Hd is much larger than the other two models.
Now we briefly discuss about the model uncertainty. When we estimate the model parameters
from the data, we can obtain their nominal values which are accurate within a confidence region. For the size of this region, it is determined by the values of the parameter uncertainties
during estimation process. The magnitudes of the uncertainty provide us a measure of the
model reliability. For Hd , the plot shows that the uncertainty is very low within 20 − 200 kHz
frequency range. That indicates the estimated model Hd is quite reliable in this frequency
interval. While for Hc1 and Hc2 , large uncertainties happen at some frequency resonances.
In next chapter, these three models for direct-channel and two cross-talk channels are used
for the control application.
Master of Science Thesis
Shenxi Ye
38
Shenxi Ye
Identification of ink channels
Master of Science Thesis
Chapter 4
Feedforward control
In previous section 2-3, the performance requirements of a printhead as well as the corresponding limitations are discussed in detail. For our thesis, the control objective is focused
on improving the printing performance with respect to the following two requirements, dropconsistency and productivity. Furthermore, the residual pressure oscillations and cross-talk
are the major performance limiting phenomena when considering the above two requirements.
Therefore, the control objective can be shifted to minimize the residual pressure oscillations
and cross-talk. In this chapter, the corresponding research will be discussed.
4-1
Feedforward control design
For the printhead setup under investigation, feedback control is not applicable due to the
following limitations
• No sensor is available for real-time measurement of the ink channel pressure.
• The driving electronics limit the waveform of the actuation pulse. Only trapezoidal
shape can be used in practice.
• Sampling time would be very short for the control computation due to a high jetting
frequency.
Thus, a feedforward strategy, as discussed in reference [12], is applied in this thesis. The
ultimate goal is to generate a trapezoidal actuation pulse for the piezo actuator to meet the
control objective. In general, a single positive trapezoidal pulse, namely the standard pulse,
is applied to jet a single droplet with specified properties. The parameters describing such a
pulse can be found by extensive examination of an experimental setup. However, this pulse
cannot damp the residual pressure oscillations inside the ink channel after jetting an ink
drop. With the difference from the reference [12], in which the meniscus velocity inside the
nozzle is used as an image of the channel pressure, the integrated piezo-unit sensored signal
Master of Science Thesis
Shenxi Ye
40
Feedforward control
is adopted in this thesis. As discussed in section 2-4, the piezo-unit can be also used as a
sensor. Its sensored signal represents the derivative of the pressure inside the ink channel.
Numerically, damping this derivative to zero does not imply the channel is at rest. Therefore,
the sensored signal needs to be first integrated so as to focus on the channel pressure itself.
Here, in Fig.4-1, a plot of the integrated sensor signal resulting from a standard pulse with
the direct-channel model Hd (q) is given to show the pressure oscillations, as well as the time
instant of drop-ejection is indicated, around 9 µs.
2
Integrated sensor signal
Standard pulse (scaled)
Integrated sensor signal (Vs)
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
Figure 4-1: Pressure oscillations of direct channel model due to a standard actuation pulse input.
As can be seen in Fig.4-1, it takes about 30 to 40 µs for the oscillations to be sufficiently
damped out. Therefore, a negative trapezoidal pulse is usually added to the standard pulse
so as to damp the residual pressure oscillations, as depicted in Fig.4-2.
Shenxi Ye
Master of Science Thesis
4-1 Feedforward control design
41
Figure 4-2: Proposed piezo actuation pulse.
As we can see, this actuation signal then consists of two parts. A positive trapezoidal pulse,
also called resonating pulse, is responsible for jetting an ink drop. The following negative
trapezoidal pulse, which is named the quenching pulse, is responsible for damping the oscillations. This actuation pulse can be characterized by the rise time (tr ), dwell time (tw ), fall
time (tf ) and the amplitude (VR and VQ ) for both the resonating pulse and the quenching
pulse, besides a time interval tdQ between them. Thus the input signal u(k, θ) can be defined
by using the parameter vector θ, with the time units given in microseconds (µs).
θ = [trR , twR , tf R , VR , tdQ , trQ , twQ , tf Q , VQ ]T
As opposed to the physical approaches [9], in this thesis, an optimal parameter vector of the
actuation pulse is determined by using a systematic optimization-based approach. In order
to define an optimization problem and then find the optimal parameter vector θopt . At first,
a reference trajectory yref as the desired ink channel pressure needs to be defined.
As shown in Fig.4-3, the integrated sensor signal, which is proportional to the channel pressure, presents a response from a standard pulse in two parts.
Master of Science Thesis
Shenxi Ye
42
Feedforward control
Part A is the response to the resonating pulse which allows the ink drop to be jetted and
part B contains the unwanted residual pressure oscillations which would influence the correct
jetting of subsequent droplet. Therefore, a desired reference signal yref is to keep part A
unchanged and brings part B to zero gradually.
2
actual response
reference trajectory
Integrated sensor signal (Vs)
1.5
1
Part A
Part B
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
Figure 4-3: Reference trajectory for the integrated sensor signal.
Generally, there are two important constraints for the reference trajectory construction. First,
the pressure oscillations are not brought to a rest immediately after the ejection so as to ensure the refill of the nozzle. Second, a gradually damping can avoid a high actuation voltage.
If an optimal actuation pulse can found in such a way that the actual channel pressure
y(k) follows the reference trajectory yref (k), the channel will come to a rest state very quickly
after the drop ejection. As a result, the residual oscillations can be reduced and then the attainable jetting frequency can be increased. The procedure to find such an optimal actuation
pulse is introduced in the next section.
Shenxi Ye
Master of Science Thesis
4-2 SISO control
4-2
43
SISO control
In this section, we examine a single ink channel and apply the optimization-based approach
to improve its performance with respect to reducing the residual pressure oscillations.
At first, we can formulate the optimization problem. The objective function can be defined
as the following sum of square errors between the reference signal yref (k) and the integrated
sensor signal y(k).
2
N 1 X
J (θ) =
yref (k) − y(k, u(k, θ))
N k=0
2
N k
X
1 X
=
yref (k) −
S(p)
N k=0
p=0
(4-1)
S(p) = Hd (q)u(p, θ)
where N = TTs with Ts the sampling time, T the chosen response time. Hd (q) is the directchannel model we estimated in section 3-3. q is the forward shift operator and u(p, θ) is the
proposed actuation pulse parameterized by θ. A summation of Hd (q)u(p, θ) expresses the
integration calculation.
Thus, the optimal actuation pulse θopt is the parameter vector θ by solving the optimization problem
min J (θ),
θ
subject to
θLB ≤ θ ≤ θU B
with θLB and θU B the vectors containing the lower and upper bounds on each element of the
parameter vector θ. Generally, the parameter bounds are determined by physical insights of
the printhead.
This problem is a constrained nonlinear optimization problem and can be solved off-line
with standard algorithms. Recall that part A of the reference signal is kept the same as a
standard pulse response, so we can also adopt such standard pulse as our resonating pulse.
The Optimization toolbox in Matlab is used to solve this optimization problem with an initial
value chosen as
θinit = [1.5 3 1.5 25 0 0 0 0 0]T
An optimal parameter vector θopt for the optimal actuation pulse can be obtained
θopt = [1.5 3 1.5 25 6.5 3.5 1.0 1.1 − 7.11]T
Master of Science Thesis
Shenxi Ye
44
4-2-1
Feedforward control
Simulation results
Now, we compare the standard pulse and optimal actuation pulse which is parameterized by
θopt in Fig.4-4(b).
(a)
Integrated sensor signal (Vs)
2
reference signal
standard response
optimal response
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
30
35
40
45
50
(b)
30
standard pulse
optimal pulse
Voltage (V)
20
10
0
−10
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
Figure 4-4: Channel pressure response to standard and optimal actuation pulses.
As expected, the optimal input pulse contains two components, the resonating pulse and
the quenching pulse. The quenching pulse plays an important part in damping the residual
oscillation. This also enables the channel pressure signal to track the reference trajectory
very closely, as shown in Fig.4-4(a). The sum of square errors with optimal pulse response
is 78.11% smaller than that of the standard pulse response (from 5.799 to 1.269). Since the
ink channel is brought to rest very quickly, the ink drops can thus be jetted with higher
frequencies using this optimal actuation pulse.
Shenxi Ye
Master of Science Thesis
4-2 SISO control
4-2-2
45
Experimental results
The simulation results show that significant improvements can be achieved by using the optimal input u(k, θopt ). In this subsection, we present the experimental results to validate this
claim.
First, a standard actuation pulse was applied for the direct-channel while the jetting frequency was set as 20 kHz. As introduced in section 2-4, the experiment consists of two
conditions: the ink-filled channel and the empty channel. In these two conditions, the corresponding output signals were stored. Then, the sensored signal for standard pulse can be
obtained by a subtraction of these two outputs. The same experiment was also implemented
for the optimal actuation pulse and another sensored signal for optimal pulse can be obtained.
The sensored signals were then numerically integrated to express the pressure situation inside
the direct-channel during jetting process. In Fig.4-5, the integrated sensor signals from a
standard pulse and an optimal pulse are compared.
Experiment result comparison
1.5
standard actuation pulse
optimal actuation pulse
Integrated sensor signal (Vs)
1
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
Figure 4-5: Experiment results comparison between standard and optimal actuation pulses.
As can be seen from Fig.4-5, with the standard actuation pulse, it takes about 30 µs to gradually damp out the residual pressure oscillations. While with the optimal pulse, the integrated
Master of Science Thesis
Shenxi Ye
46
Feedforward control
sensor signal is brought to zero much better.
As previously discussed in section 2-3-1, the DoD curve is also an important performance
indicator for the minimization of the residual pressure oscillations. In the experiments, with
the jetting frequency creasing from 10 kHz to 52 kHz, a CCD camera was used to measure
the drop velocity. Here a DoD curve comparison with the standard pulse and the optimal
pulse is depicted in Fig.4-6.
DoD curves for direct channel
10
Standard pulse
Optimal pulse
9
8
Drop velocity (m/s)
7
6
5
4
3
2
1
0
10
15
20
25
30
35
DoD jetting frequency (kHz)
40
45
50
Figure 4-6: Comparison between standard and optimal actuation pulses with DoD curves.
From Fig.4-6, we can find that the speed fluctuation with standard pulse is quite large, especially within high frequencies. The maximum of the speed variation is approximate 5.8 m/s
(2.2 m/s-8.0 m/s). However, with the optimal actuation pulse, it is well substituted by a
rather flat curve, with a smaller velocity variation of 2 m/s (5.1 m/s-7.1 m/s). This also
demonstrates a reasonable good result in pressure oscillations reduction with the proposed
optimization-based method.
Shenxi Ye
Master of Science Thesis
4-3 MIMO control
4-3
47
MIMO control
In a real printhead, besides improving the performance of a single channel, the performances
when multiple ink channels are simultaneously actuated, should also be taken into account.
As shown in Fig.3-2, we know that an input for one channel will influence the output of its
neighboring channels, i.e. the cross-talk. In this section, possible method to compensate the
cross-talk is discussed.
Fig.4-7 shows a diagram in which cross-talk phenomena are displayed. In this thesis, the
situation that at most five channels are simultaneously actuated is considered, since the effects from the third or further neighboring channels are very small and can be negligible
during this research. All these channels are applied with the same input signal as we got from
the SISO case in previous section. Moreover, the output yn is the sum of the outputs from
the direct-channel model Hd , two cross-talk channel models Hc1 and Hc2 . The effects from
the left or right side are identical as we assumed.
𝑛−2
channel
𝑢
𝑦𝑛(𝑛−1)
𝑢
𝑢
𝐻𝑐1
𝐻𝑑
𝐻𝑐1
𝑛+2
𝑛+1
𝑢
𝑢
𝐻𝑐2
𝑛
𝑛−1
𝑦𝑛𝑛
𝐻𝑐2
𝑦𝑛(𝑛+1)
𝑦𝑛(𝑛−2)
𝑦𝑛(𝑛+2)
+
𝑦𝑛
Figure 4-7: Inputs of neighboring channels affect output through cross-talk.
Master of Science Thesis
Shenxi Ye
48
Feedforward control
With a consideration that an input pulse can either be applied or not, there are 16 permutations in which the four cross-talk channels (n − 2, n − 1, n + 1, n + 2) are jetting an
ink drop or not. And due to the fact that there exist different permutations, the objective
function to be finally optimized should be in the worst case.
Based on the reference [15], we know that in a three-channel system, the worst case occurs when the bilateral symmetry channels are actuated simultaneously. So in this thesis, for
a five channels system, we only need to consider three possible worst permutations: the first
neighboring channels (n − 1)&(n + 1), the second neighboring channels (n − 2)&(n + 2), as
well as all these channels (n − 2), (n − 1), (n + 1)&(n + 2) are simultaneously actuated with
direct-channel (n). The experiments were also separate into two conditions, ink-filled channel
and empty channel and then the sensored signals were obtained by a subtraction of those
two output signals. The corresponding integrated sensor signals for the direct-channel yn are
shown in Fig.4-8.
The largest difference, from the response of direct-channel actuated only, happens when we
apply the same input pule to all five channels. This could also shown with numerical values.
The objective function in these three permutations, which is defined in equation (4-1) as the
sum of square errors between the reference signal yref (k) and the integrated sensor signal
y(k), can be calculated and compared. The values are 6.404, 15.043 and 23.417 for above
mentioned three permutations. The largest one comes from the condition that five channels
are simultaneously actuated. So in the sequel, this situation is considered as the worst case
to be investigated.
Shenxi Ye
Master of Science Thesis
4-3 MIMO control
49
Integrated sensor signal (Vs)
(a)
2
without cross talk (n)
with cross talk (n−2,n+2))
reference trajectory
1
0
−1
0
5
10
15
20
25
(b)
30
35
40
45
50
Integrated sensor signal (Vs)
2
without cross talk (n)
with cross talk (n−1,n+1)
reference trajectory
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
(c)
30
35
40
45
50
Integrated sensor signal (Vs)
2
without cross talk (n)
with cross talk (n−2,n−1,n+1,n+2)
reference trajectory
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
Figure 4-8: Cross-talk responses for different permutations.
One manner with which the cross-talk can be reduced is to introduce a time-delay between
the neighboring channels. This is investigated by making a distinction between odd and even
channels. That is, all the odd channels (n − 1), (n + 1) will be actuated simultaneously, as will
the even channels (n − 2), (n), (n + 2), but the odd and even channels will be delayed relative
to each other. Fig.4-9 shows the block diagrams when the channel (n) is an odd channel and
Fig.4-10 the channel (n) is an even channel, with the time-delay unit denoted by q −d block.
Master of Science Thesis
Shenxi Ye
50
Feedforward control
𝑛−2
channel
𝑢
𝑛
𝑛−1
𝑢
𝑢
𝑢
𝑢
𝑞 −𝑑
𝐻𝑐2
𝑛+2
𝑛+1
𝑞 −𝑑
𝐻𝑐1
𝐻𝑑
𝐻𝑐1
𝑦𝑛(𝑛−1)
𝑦𝑛𝑛
𝑦𝑛(𝑛−2)
𝐻𝑐2
𝑦𝑛(𝑛+1)
𝑦𝑛(𝑛+2)
+
𝑦𝑛
Figure 4-9: Odd channel system: inputs of odd channels are delayed with respect to even
channels.
𝑛−2
channel
𝑢
𝑞 −𝑑
𝑢
𝑢
𝑞 −𝑑
𝑞 −𝑑
𝐻𝑐1
𝑦𝑛(𝑛−2)
𝐻𝑐1
𝐻𝑑
𝑦𝑛(𝑛−1)
𝑦𝑛𝑛
𝑛+2
𝑛+1
𝑢
𝑢
𝐻𝑐2
𝑛
𝑛−1
𝐻𝑐2
𝑦𝑛(𝑛+1)
𝑦𝑛(𝑛+2)
+
𝑦𝑛
Figure 4-10: Even channel system: inputs of even channels are delayed with respect to odd
channels.
Shenxi Ye
Master of Science Thesis
4-3 MIMO control
51
Then an optimization problem wherein the time-delay d will be optimized is proposed. The
corresponding objective function J (θopt , d) only has a variable d since we set the input pulse
as the optimal input parameter vector θopt which we got from the SISO case. Moreover,
because a differentiation is made between odd and even channel systems, the final choice
for the objective function should be the worst case. Such an optimization problem can be
described as follows
dopt = arg min J (θopt , d)
d
subject to
dLB ≤ d ≤ dU B
where
J (θopt , d) =
Jodd (θopt , d) =
max [ Jodd (θopt , d) , Jeven (θopt , d) ]
2
N k
X
1 X
yref (k) −
S1 (p)
N k=0
p=0
S1 (p) = Hd (q)u(p, θopt ) + 2Hc1 (q)u(p − d, θopt ) + 2Hc2 (q)u(p, θopt )
Jeven (θopt , d) =
2
N k
X
1 X
yref (k − d) −
S2 (p)
N k=0
p=0
S2 (p) = Hd (q)u(p − d, θopt ) + 2Hc1 (q)u(p, θopt ) + 2Hc2 (q)u(p − d, θopt )
with
dLB
Lower bound for the input delay
dU B
Upper bound for the input delay
N
Number of time samples
q
Discrete-time forward shift operator
Hd
Model for direct-channel
Hc1
Model for first cross-talk channel
Hc2
Model for second cross-talk channel
d
Delay of the samples
θopt
Optimal parameter vector for the input pulse and here equal to
[1.5 3 1.5 25 6.5 3.5 1.0 1.1 − 7.11]T
The values for the objective function J (θopt , d) can be calculated and the result is depicted
in Fig.4-11. An optimal value for the time-delay is equal to 4.4 µs.
Master of Science Thesis
Shenxi Ye
52
Feedforward control
Objective function with input delay
25
24
23
22
21
20
19
0
1
2
3
time delay (μs)
4
5
6
Figure 4-11: Values of objective function J (θopt , d).
4-3-1
Simulation results
In this section, simulation results are given. First, the worst-case as depicted in Fig.4-8 (c)
is used here to show the integrated sensor signal when all five channels are simultaneously
actuated without an input time-delay considered. Then, the aforementioned odd and even
systems are adopted together with the input time-delay between neighboring two channels.
The integrated sensor signals are also shown to make a comparison. Fig.4-12 shows the simulation results of the integrated sensor signal for even and odd channel system.
As we can see, with an optimal time-delay applied, the pressure signal inside the directchannel is much more close to the a reference trajectory than that without such a time-delay,
both in even channel system (a) and odd channel system (b). This can also be proved by
the sum of square errors between the reference signal and the optimal response. For the odd
system, the value reduced from 23.417 to 19.374, an improvement of 17.3% can be obtained.
For the even system, the value with an input delay is 16.8% smaller than that of without the
optimal time-delay, reduced from 23.417 to 19.491.
Shenxi Ye
Master of Science Thesis
4-3 MIMO control
53
Integrated sensor signal (Vs)
(a) Even channel system
2
with input delay
reference signal
without input delay
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
30
35
40
45
50
Integrated sensor signal (Vs)
(b) Odd channel system
2
reference signal
with input delay
without input delay
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
Figure 4-12: Cross-talk compensation with input delay 4.4 µs.
Basically, we can combine the optimization problem concerning with the input parameter θ
and time-delay d together. And then we can get a new optimal actuation input pulse, as
shown in Fig.4-13, and a new optimal time-delay. Furthermore, we compare the integrated
sensor signals with this pulse and time-delay to previous ones, here, we take the even channel
system for an example.
Master of Science Thesis
Shenxi Ye
54
Feedforward control
Integrated sensor signal (Vs)
(a)
2
response with new optimal pulse
reference signal
response with old optimal pulse
1.5
1
0.5
0
−0.5
−1
0
5
10
15
20
25
30
35
40
45
50
(b)
30
old optimal pulse
new optimal pulse
Voltage (V)
20
10
0
−10
0
5
10
15
20
25
Time (μs)
30
35
40
45
50
Figure 4-13: Cross-talk compensation with new optimal input and input delay.
As we can see, there is very slight change between these two results, which can also be
verified by the sum of square errors between the reference signal yref (k) and the integrated
sensor signal y(k) in these two conditions, 19.491 and 19.879 for old optimal pulse and new
optimal pulse, respectively. So we just keep the optimal pulse which we got from SISO case
the optimal time-delay from MIMO case to test the experimental setup.
Shenxi Ye
Master of Science Thesis
4-3 MIMO control
4-3-2
55
Experimental results
Here, we also use the DoD curve to show the improvement in cross-talk compensation. Fig.414 and Fig.4-15 shows the DoD curve for even and odd channel system, respectively. With
the optimal actuation pulse applied, as we can see, if all five channels are simultaneously
actuated, the DoD curve is lower than that of only the direct-channel is actuated. The drop
velocity has a decrease of 0.5 m/s along the jetting frequencies. The reason for that is the
cross-talk, as we discussed in section 2-3-2. After the time-delay is used, as we supposed,
the performance can be improved. The DoD curves of five-channel systems are quite close
(less than 0.1 m/s) with that of only direct-channel system, both for even and odd channel
systems.
Even channel system
10
without cross−talk
with cross−talk; with time−delay
with cross−talk; without time−delay
9
Drop velocity (m/s)
8
7
6
5
4
3
2
10
15
20
25
30
35
DoD frequency (KHz)
40
45
50
Figure 4-14: DoD curves for even channel system with time-delay.
Master of Science Thesis
Shenxi Ye
56
Feedforward control
Odd channel system
10
without cross−talk
with cross−talk; with time−delay
with cross−talk; without time−delay
9
8
Drop velocity (m/s)
7
6
5
4
3
2
10
15
20
25
30
DoD frequency (KHz)
35
40
45
50
Figure 4-15: DoD curves for odd channel system with time-delay.
This also refers to the fact that the cross-talk in multi-channel system can be reduced by
applying such an optimal time-delay between the neighboring channels, with a respect of
drop-consistency in velocity.
Shenxi Ye
Master of Science Thesis
Chapter 5
Conclusions and recommendations
The ability to deposit various types of material onto a substrate in certain patterns has made
inkjet technology a very important technology. For document printing, a typical design of a
piezoelectric inkjet printhead comprises a large array of piezo-actuated channels. The corresponding actuation pulse can be tuned to get the required drop-on-demand results. Generally,
there are two operational issues in practical that would deteriorate the printing performance,
that is, residual pressure oscillations and cross-talk. The former one relates to the fact that
the ink in a pressure is not at rest state immediately after jetting a droplet, while cross-talk
results from the fact that one channel would be influenced when its neighboring channels are
actuated simultaneously. These two phenomena would limit the productivity as well as the
drop consistency considerably, and then the printing performance of such a piezoelectric inkjet
printhead. In this report, a systems and control approach is adopted to solve these limitations.
First is the modeling of such a printhead so as to provide good insight. Unfortunately,
the internal physical relationships are quite difficult to get a clear recognition. So we use the
system identification method to estimate models. With some assumptions, the task could
be simplified to identify three models including one for direct channel and two for the first
two neighboring channels. With the experimental setup, we choose the piezo-unit as both
the actuator and the sensor because of its so-called ‘self-sensing’ capability. Then we have
the piezo-sensor signal, which is proportional to the deviation of the channel pressure, as the
output signal. Together with the actuation input signals, we can get good models with the
system identification toolbox in Matlab.
Given the restrictions of feedback control, an optimization-based feedforward control method
is used to tune the shape of the actuation pulse. With the identified direct-channel model, the
result of a standard positive trapezoidal pulse adding a negative pulse could follow a reference
trajectory quite well. This optimal actuation pulse would damp the ink channel pressure oscillations quickly. However, even with this optimal pulse, the resulted cross-talk is still very
significant. Furthermore, a time-delay between the neighboring channels are considered to
compensate that. By the optimization toolbox in Matlab, this off-line nonlinear optimization
problem can be solved.
Master of Science Thesis
Shenxi Ye
58
Conclusions and recommendations
Finnally, the optimal actuation pulse and input delay are implemented on printhead setup.
The experimental results demonstrate that the research objectives are achieved by a considerable improvement of the printing performance with respect to the productivity and dropconsistency, e.g. the maximum velocity variation of the DoD curve in direct-channel can be
reduced by 66.7%.
Several recommendations are provided for further increase of the performance quality.
First, more accurate models can be obtained by the adjustments of more precise equilibrium of the bridge circuit for ‘self sensing’ character. Second, the piezo-unit sensor signal is
now collected by subtracting the output signals of filled-channel and empty-channel. Some
problems, like the temperature difference or empty-channel with some ink left, would influence
the quality of the sensor signal. So a structure design of the printhead can be investigated.
That is, in a printhead, one ink channel can be settled in empty state all the time. This would
also save a lot of efforts since we can collect the sensor signal with only one experiment. Third,
the uncertainty in the parameters of the estimated models needs to be considered since there
exists a rather large model variance when the identification procedure is based on piezo-unit
sensored signal. A robust feedforward control can be used to extend the optimization-based
technique.
Shenxi Ye
Master of Science Thesis
Bibliography
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Shenxi Ye
Master of Science Thesis
Glossary
List of Acronyms
DoD
drop-on-demand
PEM
prediction-error identification method
SMI
subspace model identification
SISO
single input single output
MIMO
multi input multi output
I/O
input and output
ETFE
empirical transfer function estimate
OE
output error
LCD
liquid crystal display
LED
light emitting diode
3D
three dimensional
Master of Science Thesis
Shenxi Ye
62
Shenxi Ye
Glossary
Master of Science Thesis
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