Look up table to control non-linear xerographic process

An electrostatographic printing machine having an imaging member, operating components, and a control system including a sensor, compensator, and look up table for adjusting the operating components. The sensor signal provides a suitable indication of an operating component condition such as a developer unit or a photoreceptor charging device. A compensator responds to the sensor signal to provide a non-linear adjustment signal and the look up table converts the non-linear adjustment signal to a linear adjustment signal. A device such as a charging corotron or developer power supply responds to the linear adjustment signal to appropriately adjust the operating component.

This invention relates generally to an electrostatographic printing machine 
and, more particularly, concerns a process to adjust a xerographic 
control, in particular, to linearize the control for changing set points. 
The basic reprographic process used in an electrostatographic printing 
machine generally involves an initial step of charging a photoconductive 
member to a substantially uniform potential. The charged surface of the 
photoconductive member is thereafter exposed to a light image of an 
original document to selectively dissipate the charge thereon in selected 
areas irradiated by the light image. This procedure records an 
electrostatic latent image on the photoconductive member corresponding to 
the informational areas contained within the original document being 
reproduced. The latent image is then developed by bringing a developer 
material including toner particles adhering triboelectrically to carrier 
granules into contact with the latent image. The toner particles are 
attracted away from the carrier granules to the latent image, forming a 
toner image on the photoconductive member which is subsequently 
transferred to a copy sheet. The copy sheet having the toner image thereon 
is then advanced to a fusing station for permanently affixing the toner 
image to the copy sheet in image configuration. 
In electrostatographic machines using a drum-type or an endless belt-type 
photoconductive member, the photosensitive surface thereof can contain 
more than one image at one time as it moves through various processing 
stations. The portions of the photosensitive surface containing the 
projected images, so-called "image areas", are usually separated by a 
segment of the photosensitive surface called the inter-document space. 
After charging the photosensitive surface to a suitable charge level, the 
inter-document space segment of the photosensitive surface is generally 
discharged by a suitable lamp to avoid attracting toner particles at the 
development stations. Various areas on the photosensitive surface, 
therefore, will be charged to different voltage levels. For example, there 
will be the high voltage level of the initial charge on the photosensitive 
surface, a selectively discharged image area of the photosensitive 
surface, and a fully discharged portion of the photosensitive surface 
between the image areas. 
The approach utilized for multicolor electrostatographic printing is 
substantially identical to the process described above. However, rather 
than forming a single latent image on the photoconductive surface in order 
to reproduce an original document, as in the case of black and white 
printing, multiple latent images corresponding to color separations are 
sequentially recorded on the photoconductive surface. Each single color 
electrostatic latent image is developed with toner of a color 
complimentary thereto and the process is repeated for differently colored 
images with the respective toner of complimentary color. Thereafter, each 
single color toner image can be transferred to the copy sheet in 
superimposed registration with the prior toner image, creating a 
multi-layered toner image on the copy sheet. Finally, this multi-layered 
toner image is permanently affixed to the copy sheet in substantially 
conventional manner to form a finished color copy. 
As described, the surface of the photoconductive member must be charged by 
a suitable device prior to exposing the photoconductive member to a light 
image. This operation is typically performed by a corona charging device. 
One type of corona charging device comprises a current carrying electrode 
enclosed by a shield on three sides and a wire grid or control screen 
positioned thereover, and spaced apart from the open side of the shield. 
Biasing potentials are applied to both the electrode and the wire grid to 
create electrostatic fields between the charged electrode and the shield, 
between the charged electrode and the wire grid, and between the charged 
electrode and the (grounded) photoconductive member. These fields repel 
electrons from the electrode and the shield resulting in an electrical 
charge at the surface of the photoconductive member roughly equivalent to 
the grid voltage. The wire grid is located between the electrode and the 
photoconductive member for controlling the charge strength and charge 
uniformity on the photoconductive member as caused by the aforementioned 
fields. 
Control of the field strength and the uniformity of the charge on the 
photoconductive member is very important because consistently high quality 
reproductions are best produced when a uniform charge having a 
predetermined magnitude is obtained on the photoconductive member. If the 
photoconductive member is not charged to a sufficient level, the 
electrostatic latent image obtained upon exposure will be relatively weak 
and the resulting deposition of development material will be 
correspondingly decreased. As a result, the copy produced by an 
undercharged photoconductor will be faded. If, however, the 
photoconductive member is overcharged, too much developer material will be 
deposited on the photoconductive member. The copy produced by an 
overcharged photoconductor will have a gray or dark background instead of 
the white background of the copy paper. In addition, areas intended to be 
gray will be black and tone reproduction will be poor. Moreover, if the 
photoconductive member is excessively overcharged, the photoconductive 
member can become permanently damaged. 
A useful tool for measuring voltage levels on the photosensitive surface is 
an electrostatic voltmeter (ESV) or electrometer. The electrometer is 
generally rigidly secured to the reproduction machine adjacent the moving 
photosensitive surface and measures the voltage level of the 
photosensitive surface as it traverses an ESV probe. The surface voltage 
is a measure of the density of the charge on the photoreceptor, which is 
related to the quality of the print output. In order to achieve high 
quality printing, the surface potential on the photoreceptor at the 
developing zone should be within a precise range. 
In a typical xerographic charging system, the amount of voltage obtained at 
the point of electrostatic voltage measurement of the photoconductive 
member, namely at the ESV, is less than the amount of voltage applied at 
the wire grid of the point of charge application. In addition, the amount 
of voltage applied to the wire grid of the corona generator required to 
obtain a desired constant voltage on the photoconductive member must be 
increased or decreased according to various factors which affect the 
photoconductive member. Such factors include the rest time of the 
photoconductive member between printing, the voltage applied to the corona 
generator for the previous printing job, the copy length of the previous 
printing job, machine to machine variance, the age of the photoconductive 
member and changes in the environment. 
One way of monitoring and controlling the surface potential in the 
development zone is to locate a voltmeter directly in the developing zone 
and then to alter the charging conditions until the desired surface 
potential is achieved in the development zone. However, the accuracy of 
voltmeter measurements can be affected by the developing materials (such 
as toner particles) such that the accuracy of the measurement of the 
surface potential is decreased. In addition, in color printing there can 
be a plurality of developing areas within the developing zone 
corresponding to each color to be applied to a corresponding latent image. 
Because it is desirable to know the surface potential on the photoreceptor 
at each of the color developing areas in the developing zone, it would be 
necessary to locate a voltmeter at each color area within the developing 
zone. Cost and space limitations make such an arrangement undesirable. 
In a typical charge control system, the point of charge application and the 
point of charge measurement is different. The zone between these two 
devices loses the immediate benefit of charge control decisions based on 
measured voltage error since this zone is downstream from the charging 
device. This zone may be as great as a belt revolution or more due to 
charge averaging schemes. This problem is especially evident in aged 
photoreceptors because their cycle-to-cycle charging characteristics are 
more difficult to predict. Charge control delays can result in improper 
charging, poor copy quality and often leads to early photoreceptor 
replacement. Thus, there is a need to anticipate the behavior of a 
subsequent copy cycle and to compensate for predicted behavior beforehand. 
Various systems have been designed and implemented for controlling 
processes within a printing machine. For example, U.S. Pat. No. 5,243,383 
discloses a charge control system that measures first and second surface 
voltage potentials to determine a dark decay rate model representative of 
voltage decay with respect to time. The dark decay rate model is used to 
determine the voltage at any point on the imaging surface corresponding to 
a given charge voltage. This information provides a predictive model to 
determine the charge voltage required to produce a target surface voltage 
potential at a selected point on the imaging surface. 
U.S. Pat. No. 5,243,383 discloses a charge control system that uses three 
parameters to determine a substrate charging voltage, a development 
station bias voltage, and a laser power for discharging the substrate. The 
parameters are various difference and ratio voltages. 
Process loops are designed to keep control of the electrostatics and the 
development system. They track setpoints for developed mass per unit area 
on the paper. To achieve the tracking of setpoints actuator parameters, 
grid voltage, laser power and donor voltages are varied in a controlled 
way with the help of compensator algorithms. These algorithms use the 
measured voltages on the photoreceptor and the toner mass. The process in 
the prior art, generally, is non-linear for the complete range over which 
the printer is expected to operate. 
The paradigm of the printing process, in fact, is non-linear, time varying, 
noisy and unfortunately, multivariable. Such systems are generally hard to 
control. On the other hand, using the assumption of linearity, process 
loops can be designed using modern multivariable linear control 
techniques. The linearized version of the nonlinear system gives good 
results at one operating point about which the system is approximately 
linear. Outside of that point, however, the control system performance 
will be different, which results in loss print quality. For designing 
control algorithms, it would be useful if the nonlinear process would be 
converted to a linear process at different operating points. This can be 
done in accordance with the present invention by artificially generating 
inverse system functions. 
It would be desirable, therefore, to provide a linear approach to control, 
in particular, in which the linearization is done by using estimated 
lookup tables. The lookup tables would be obtained from experimental data 
once during a setup process. The look up table would act like an 
additional gain table in a multivariable control system. New values would 
be accessed from the table each time the operating point moves, thus 
preserving the linearity. 
It is an object of the present invention, therefore, to be able to linearly 
adjust a xerographic system requiring multiple changes in various system 
integrators and compensators. It is another object of the present 
invention to be able to convert a non-linear response system to a linear 
response system over a wide range of operating variables. It is another 
object of the present invention to provide a look up table that linearizes 
control responses to changing parameters. 
SUMMMARY OF THE INVENTION 
The present invention relates to an electrostatographic printing machine 
having an imaging member operating components, and a control system 
including a sensor, compensator, and look up table for adjusting the 
operating components. The sensor signal provides a suitable indication of 
an operating component condition such as a developer unit or a 
photoreceptor charging device. A compensator responds to the sensor signal 
to provide a non-linear adjustment signal and the look up table converts 
the non-linear adjustment signal to a linear adjustment signal. A device 
such as a charging corotron or developer power supply responds to the 
linear adjustment signal to appropriately adjust the charging device or 
developer unit.

For a general understanding of the features of the present invention, 
reference is made to the drawings wherein like references have been used 
throughout to designate identical elements. A schematic elevational view 
showing an exemplary electrophotographic printing machine incorporating 
the features of the present invention therein is shown in FIG. 1. It will 
become evident from the following discussion that the present invention is 
equally well-suited for use in a wide variety of printing systems 
including ionographic printing machines and discharge area development 
systems, as well as other more general non-printing systems providing 
multiple or variable outputs such that the invention is not necessarily 
limited in its application to the particular system shown herein. 
Turning initially to FIG. 1, before describing the particular features of 
the present invention in detail, an exemplary electrophotographic copying 
apparatus will be described. The exemplary electrophotographic system may 
be a multicolor copier, as for example, the recently introduced Xerox 
Corporation "5775" copier. To initiate the copying process, a multicolor 
original document 38 is positioned on a raster input scanner (RIS), 
indicated generally by the reference numeral 10. The RIS 10 contains 
document: illumination lamps, optics, a mechanical scanning drive, and a 
charge coupled device (CCD array) for capturing the entire image from 
original document 38. The RIS 10 converts the image to a series of raster 
scan lines and measures a set of primary color densities, i.e. red, green 
and blue densities, at each point of the original document. This 
information is transmitted as an electrical signal to an image processing 
system (IPS), indicated generally by the reference numeral 12, which 
converts the set of red, green and blue density signals to a set of 
colorimetric coordinates. The IPS contains control electronics for 
preparing and managing the image data flow to a raster output scanner 
(ROS), indicated generally by the reference numeral 16. 
A user interface (UI), indicated generally by the reference numeral 14, is 
provided for communicating with IPS 12. UI 14 enables an operator to 
control the various operator adjustable functions whereby the operator 
actuates the appropriate input keys of UI 14 to adjust the parameters of 
the copy. UI 14 may be a touch screen, or any other suitable device for 
providing an operator interface with the system. The output signal from UI 
14 is transmitted to IPS 12 which then transmits signals corresponding to 
the desired image to ROS 16. 
ROS 16 includes a laser with rotating polygon mirror blocks. The ROS 16 
illuminates, via mirror 37, a charged portion of a photoconductive belt 20 
of a printer or marking engine, indicated generally by the reference 
numeral 18 Preferably, a multi-facet polygon mirror is used to illuminate 
the photoreceptor belt 20 at a rate of about 400 pixels per inch. The ROS 
16 exposes the photoconductive belt 20 to record a set of three 
subtractive primary latent images thereon corresponding to the signals 
transmitted from IPS 12. One latent image is to be developed with cyan 
developer material, another latent image is to be developed with magenta 
developer material, and the third latent image is to be developed with 
yellow developer material. These developed images are subsequently 
transferred to a copy sheet in superimposed registration with one another 
to form a multicolored image on the copy sheet which is then fused thereto 
to form a color copy. This process will be discussed in greater detail 
hereinbelow. 
With continued reference to FIG. 1, marking engine 18 is an 
electrophotographic printing machine comprising photoconductive belt 20 
which is entrained about transfer rollers 24 and 26, tensioning roller 28, 
and drive roller 30. Drive roller 30 is rotated by a motor or other 
suitable mechanism coupled to the drive roller 30 by suitable means such 
as a belt drive 32. As roller 30 rotates, it advances photoconductive belt 
20 in the direction of arrow 22 to sequentially advance successive 
portions of the photoconductive belt 20 through the various processing 
stations disposed about the path of movement thereof. 
Photoconductive belt 20 is preferably made from a polychromatic 
photoconductive material comprising an anti-curl layer, a supporting 
substrate layer and an electrophotographic imaging single layer or 
multi-layers. The imaging layer may contain homogeneous, heterogeneous, 
inorganic or organic compositions. Preferably, finely divided particles of 
a photoconductive inorganic compound are dispersed in an electrically 
insulating organic resin binder. Typical photoconductive particles include 
metal free phthalocyanine, such as copper phthalocyanine, quinacridones, 
2,4-diamino-triazines and polynuclear aromatic quinines. Typical organic 
resinous binders include polycarbonates, acrylate polymers vinyl polymers, 
cellulose polymers, polyesters, polysiloxanes, polyamides, polyurethanes, 
epoxies, and the like. 
Initially, a portion of photoconductive belt 20 passes through a charging 
station, indicated generally by the reference letter A. At charging 
station A, a corona generating device 34 or other charging device 
generates a charge voltage to charge photoconductive belt 20 to a 
relatively high, substantially uniform voltage potential. The corona 
generator 34 comprises a corona generating electrode, a shield partially 
enclosing the electrode, and a grid disposed between the belt 20 and the 
unenclosed portion of the electrode. The electrode charges the 
photoconductive surface of the belt 20 via corona discharge. The voltage 
potential applied to the photoconductive surface of the belt 20 is varied 
by controlling the voltage potential of the wire grid. 
Next, the charged photoconductive surface is rotated to an exposure 
station, indicated generally by the reference letter B. Exposure station B 
receives a modulated light beam corresponding to information derived by 
RIS 10 having a multicolored original document 38 positioned thereat. The 
modulated light beam impinges on the surface of photoconductive belt 20, 
selectively illuminating the charged surface of photoconductive belt 20 to 
form an electrostatic latent image thereon. The photoconductive belt 20 is 
exposed three times to record three latent images representing each color. 
After the electrostatic latent images have been recorded on photoconductive 
belt 20, the belt is advanced toward a development station, indicated 
generally by the reference letter C. However, before reaching the 
development station C, the photoconductive belt 20 passes subjacent to a 
voltage monitor, preferably an electrostatic voltmeter 33, for measurement 
of the voltage potential at the surface of the photoconductive belt 20. 
The electrostatic voltmeter 33 can be any suitable type known in the art 
wherein the charge on the photoconductive surface of the belt 20 is 
sensed, such as disclosed in U.S. Pat. Nos. 3,870,968; 4,205,257; or 
4,853,639, the contents of which are incorporated by reference herein. 
A typical electrostatic voltmeter is controlled by a switching arrangement 
which provides the measuring condition in which charge is induced on a 
probe electrode corresponding to the sensed voltage level of the belt 20. 
The induced charge is proportional to the sum of the internal capacitance 
of the probe and its associated circuitry, relative to the 
probe-to-measured surface capacitance. A DC measurement circuit is 
combined with the electrostatic voltmeter circuit for providing an output 
which can be read by a conventional test meter or input to a control 
circuit, as for example, the control circuit of the present invention. The 
voltage potential measurement of the photoconductive belt 20 is utilized 
to determine specific parameters for maintaining a predetermined potential 
on the photoreceptor surface, as will be understood with reference to the 
specific subject matter of the present invention, explained in detail 
hereinbelow. 
The development station C includes four individual developer units 
indicated by reference numerals 40, 42, 44 and 46. The developer units are 
of a type generally referred to in the art as "magnetic brush development 
units". Typically, a magnetic brush development system employs a 
magnetizable developer material including magnetic carrier granules having 
toner particles adhering triboelectrically thereto. The developer material 
is continually brought through a directional flux field to form a brush of 
developer material. The developer material is constantly moving so as to 
continually provide the brush with fresh developer material. Development 
is achieved by bringing the brush of developer material into contact with 
the photoconductive surface. 
Developer units 40, 42, and 44, respectively, apply toner particles of a 
specific color corresponding to the compliment of the specific color 
separated electrostatic latent image recorded on the photoconductive 
surface. Each of the toner particle colors is adapted to absorb light 
within a preselected spectral region of the electromagnetic wave spectrum. 
For example, an electrostatic latent image formed by discharging the 
portions of charge on the photoconductive belt corresponding to the green 
regions of the original document will record the red and blue portions as 
areas of relatively high charge density on photoconductive belt 20, while 
the green areas will be reduced to a voltage level ineffective for 
development. The charged areas are then made visible by having developer 
unit 40 apply green absorbing (magenta) toner particles onto the 
electrostatic latent image recorded on photoconductive belt 20. Similarly, 
a blue separation is developed by developer unit 42 with blue absorbing 
(yellow) toner particles, while the red separation is developed by 
developer unit 44 with red absorbing (cyan) toner particles. Developer 
unit 46 contains black toner particles and may be used to develop the 
electrostatic latent image formed from a black and white original 
document. 
In FIG. 1, developer unit 40 is shown in the operative position with 
developer units 42, 44 and 46 being in the non-operative position. During 
development of each electrostatic latent image, only one developer unit is 
in the operative position, while the remaining developer units are in the 
non-operative position. Each of the developer units is moved into and out 
of an operative position. In the operative position, the magnetic brush is 
positioned substantially adjacent the photoconductive belt, while in the 
non-operative position, the magnetic brush is spaced therefrom. Thus, each 
electrostatic latent image or panel is developed with toner particles of 
the appropriate color without commingling. 
After development, the toner image is moved to a transfer station, 
indicated generally by the reference letter D. Transfer station D includes 
a transfer zone, generally indicated by reference numeral 64, defining the 
position at which the toner image is transferred to a sheet of support 
material, which may be a sheet of plain paper or any other suitable 
support substrate. A sheet transport apparatus, indicated generally by the 
reference numeral 48, moves the sheet into contact with photoconductive 
belt 20. Sheet transport 48 has a belt 54 entrained about a pair of 
substantially cylindrical rollers 50 and 52. A friction retard feeder 58 
advances the uppermost sheet from stack 56 onto a pre-transfer transport 
60 for advancing a sheet to sheet transport 48 in synchronism with the 
movement thereof so that the leading edge of the sheet arrives at a 
preselected position, i.e. a loading zone. The sheet is received by the 
sheet transport 48 for movement therewith in a recirculating path. As belt 
54 of transport 48 moves in the direction of arrow 62, the sheet is moved 
into contact with the photoconductive belt 20, in synchronism with the 
toner image developed thereon. 
In transfer zone 64, a corona generating device 66 sprays ions onto the 
backside of the sheet so as to charge the sheet to the proper magnitude 
and polarity for attracting the toner image from photoconductive belt 20 
thereto. The sheet remains secured to the sheet gripper so as to move in a 
recirculating path for three cycles. In this manner, three different color 
toner images are transferred to the sheet in superimposed registration 
with one another. Each of the electrostatic latent images recorded on the 
photoconductive surface is developed with the appropriately colored toner 
and transferred, in superimposed registration with one another, to the 
sheet for forming the multi-color copy of the colored original document. 
One skilled in the art will appreciate that the sheet may move in a 
recirculating path for four cycles when undercolor black removal is used. 
After the last transfer operation, the sheet transport system directs the 
sheet to a vacuum conveyor, indicated generally by the reference numeral 
68. Vacuum conveyor 68 transports the sheet, in the direction of arrow 70, 
to a fusing station, indicated generally by the reference letter E, where 
the transferred toner image is permanently fused to the sheet. The fusing 
station includes a heated fuser roll 74 and a pressure roll 72. The sheet 
passes through the nip defined by fuser roll 74 and pressure roll 72. The 
toner image contacts fuser roll 74 so as to be affixed to the sheet. 
Thereafter, the sheet is advanced by a pair of rolls 76 to a catch tray 78 
for subsequent removal therefrom by the machine operator. 
The last processing station in the direction of movement of belt 20, as 
indicated by arrow 22, is a cleaning station, indicated generally by the 
reference letter F. A lamp 80 illuminates the surface of photoconductive 
belt 20 to remove any residual charge remaining thereon. Thereafter, a 
rotatably mounted fibrous brush 82 is positioned in the cleaning station 
and maintained in contact with photoconductive belt 20 to remove residual 
toner particles remaining from the transfer operation prior to the start 
of the next successive imaging cycle. 
A prior art diagrammatic representation of the system currently under 
practice for most xerographic print engines is shown in FIG. 2. Block 102 
represents the charging and exposure systems. The block 104 representing 
compensators usually contains suitable integrators such as 106, 108 with 
some weighting. Here V.sub.h represents the voltage on the unexposed 
photoreceptor and V.sub.1, represents the voltage after the exposure. 
V.sup.t.sub.h and V.sup.t.sub.l are the desired states for the voltages 
V.sub.h and V.sub.l and E.sub.h is the error generated by subtracting the 
V.sup.t.sub.h values with those measured by the ESV. Similarly, E.sub.l is 
the error generated by subtracting the V.sup.t.sub.l values with those 
measured by the ESV. U.sub.g and U.sub.l are the control signals to vary 
the grid voltage and laser power respectively. 
When the setpoint changes, there is a large error created by the system. 
Within a few prints V.sub.h and V.sub.l settle to new target values 
depending on the integrator weights. The difficult problem is in tuning 
the controller weights to trace the V.sub.h and V.sub.l target values so 
that the best print quality is preserved even if the electrostatic system 
drifts with time. The problem becomes even more difficult when there are 
many gains involved in the controller. 
In accordance with the present invention, linearization techniques are 
first discussed for electrostatic control. After that similar techniques 
are extended for implementing control for tracking Area Coverage or DMA 
setpoints. 
Linearization lookup tables are obtained from a small signal model 
disclosed in pending D/95541 Serial No. (not yet assigned) incorporated 
herein. If B.sub.11, B.sub.12 and B.sub.22 are the slopes of the curves of 
photoreceptor voltage versus grid voltage and laser power at given 
operating points on the curves, then the small signal model is written as: 
##EQU1## 
In the small signal model shown in Equation 1. V.sub.h =voltage on 
unexposed photoreceptor 
V.sub.l =voltage on photoreceptor after exposure, 
U.sub.g =control signal to vary grid voltage, and, 
U.sub.l =control signal to vary laser power 
Equation 1 also contains the input matrix B to describe the model of the 
electrostatic system. To have the model valid for the full operating 
region, feedforward lookup tables are implemented as shown in pending 
D/95541. With this scenario the linearization of the system involves 
merely finding the inverse of the B matrix. This can be written in terms 
of the constituent elements as follows: 
##EQU2## 
From suitable curves, the parameters of the B matrix can be extracted at 
one operating point. They are shown below: 
##EQU3## 
The elements B.sub.11i, B.sub.12i, B.sub.21i, B.sub.22i form an estimated 
lookup table for linearizing the non-linear system around one operating 
point. Similarly, when we move to another operating point over the curve, 
new elements of the B.sup.-1 matrix are obtained. The change in operating 
points are initiated when a change takes place in the target value. 
Likewise, satisfactory numbers of data points are initiated when a change 
takes place in the target value. Likewise, satisfactory numbers of data 
points are selected to describe the complete operating region. Having all 
the elements of the B.sup.-1 matrix the overall system used for controller 
design is transformed algebraically into a linear design, fully or 
partially. This will enable the application of linear control techniques. 
After implementing the linearization look up table, the overall system for 
designing controllers becomes linear. 
Before implementing the linear look up table, the state-space model of the 
system is set forth to: 
##EQU4## 
After implementing the inverse B matrix table the new state space model of 
the system cancels the B matrix. Due to numerical approximation in the 
lookup table, one would not get an exact cancellation. Those small effects 
can be cured by robust controllers. The new state space model of the 
system becomes equal to: 
##EQU5## 
In equation 7, matrices A and I are identity matrices. The B matrix is now 
mathematically converted to become the identity matrix, I. As can be seen, 
this type of approach holds good only when the B matrix is invertible. In 
our xerographic printing system, models for electrostatics contained 
invertible B matrices for the full operating range. 
In FIG. 3, a technique to implement the elements of estimated look up table 
110 including elements B.sub.21i, B.sub.12i, B.sub.11i, and B.sub.22i is 
shown in diagrammatic form. The actuator signals .DELTA.U.sub.g and 
.DELTA.U.sub.l are passed through lookup table 110 and then added to the 
feedforward actuator signals U.sub.go and U.sub.lo at summing nodes 114 
and 116 to generate U.sub.g and U.sub.l, to control charging and exposure 
systems illustrated at 112. This type of formulation basically turns out 
to be one type of controller with gains obtained directly from the 
measurements on the electrosatic subsystem rather than by conventional 
trial and error methods of the past. 
Look up tables 118 and 120 are formed from system charging and photo 
induced discharge curves or equations. Look up tables 118 and 120 place 
the system in a correct operating range, but look up table 110 provides 
precise, linear control for a given operating range. Operating alone, look 
up table 110 provides precise, linear control in a given operating range 
such as direct, linear control of the charging and exposive system 112. 
Operating in conjunction with feed forward look up tables 118 and 120, a 
control is provided by look up table 110 that puts the system at a correct 
operating point and also produces linearizes the system within that 
operating point. 
The technique described above also applies to development systems for 
control. For development control, because of three different area coverage 
(or DMA) measurements, there are nine elements in the matrix. The small 
signal model for developability control is written as: 
##EQU6## 
Where .DELTA.V.sub.h, .DELTA.V.sub.l and .DELTA.V.sub.d are the small 
control signals expected to change first level V.sub.h and V.sub.l target 
values and the donor voltage, V.sub.d. They correspond to small signals 
.DELTA.U.sub.1, .DELTA.U.sub.2, and .DELTA.U.sub.3, FIG. 4 describing 
implementation of the estimated lookup table for linearizing a non-linear 
system for development control. Also .DELTA.D.sub.1, .DELTA.D.sub.2, and 
.DELTA.D.sub.3 are small deviations around the operating point D.sub.1o, 
D.sub.2.sbsb.o and D.sub.3o of the Area Coverage or DMA targets. 
In FIG. 4 the linearization lookup table is shown by 130. The elements of 
the B matrix are extracted from the model curves to generate a linearizing 
look up table, called an estimated lookup table. The matrix is given by: 
##EQU7## 
The elements of B.sub.11i, B.sub.12i, B.sub.33i are implemented in a 
similar way as that shown for the first level electrostatic control in 
FIG. 3. With reference to FIG. 4, signals derived from Multi Input/Output 
compensator 124 in response to signals from ETACs or OCD sensors measuring 
toner mass, and D1, D2, and D3 represent these different DMA measurements. 
These nominal actuator values are linearized by look up table 130 to 
control subsystem 128. An option is also to provide signals from feed 
forward look up table 126 to summing nodes 132 to place the control in a 
correct operating range as well as to provide linearization. 
With the implementation of the linearization look up table, the system can 
be modeled with state space equation of the type shown in equation 7. With 
this approach, the controller gains are fixed. When the Area Coverage or 
DMA setpoints change, the operating points also change. For a new 
operating point, new sets of inverse B matrices are used. In this way the 
system as seen by the controller remains linear and is immune to changes 
in the operating points. 
It is, therefore, apparent that there has been provided in accordance with 
the present invention, a charge control system that fully satisfies the 
aims and advantages hereinbefore set forth. While this invention has been 
described in conjunction with a specific embodiment thereof, it is evident 
that many alternatives, modifications, and variations will be apparent to 
those skilled in the art. Accordingly, it is intended to embrace all such 
alternatives, modifications and variations that fall within the spirit and 
broad scope of the appended claims.