IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
1
Using Time-sequential Sampling to Stabilize the
Color and Tone Reproduction Functions of a
Xerographic Printing Process
Teck Ping Sim, Student member, IEEE, Perry Y. Li Member, IEEE and Dongjun Lee Member, IEEE
Abstract
Tone and color reproduction functions (TRC and CRC) characterize how a printer maps a desired tone or color
into the actual printed output. Ideally, the TRC/CRC should be identity maps to achieve tone and color consistency.
The main contribution of this paper is in enabling the stabilization of potentially infinite dimensional TRC/CRC
with limited sensing capabilities. This problem is challenging because: 1) only a small number of tone / color
test patches for sensing can be printed and measured at a time; 2) only a small number of actuators are available
for control. To address the first problem, time-sequential sampling is proposed to enable the time varying TRC or
CRC to be reconstructed based on small number of samples. A periodic Kalman filtering approach is employed for
the reconstruction. Two classic time-sequential sampling sequences are compared based on their spectral aliasing
properties. The second issue is addressed by a curve-fitting TRC stabilization controller based on Linear Quadratic
(LQ) control with integral dynamics. It specifies particular tones or color to be precisely controlled, while allowing
other tones or colors to be close to the desired value. Simulations and experiments show that the proposed TRC/CRC
stabilization system is effective for practical implementation.
Index Terms
Time-sequential sampling, signal reconstruction, Kalman filter, xerographic printing, optimal control, curvefitting, periodic system.
I. I NTRODUCTION
A digital color xerographic (i.e. laser) printer can be considered a mapping between the desired image and the
printed output image. An important performance criterion for a color printer is that any desired colors in the desired
customer image is faithfully reproduced. By ignoring the spatial dimension (such as lines and textures) of print
quality for the moment and focusing on the issue of consistent color reproduction only, a color xerographic printer
can be represented by the color reproduction function: CRC : C → C , desired color 7→ output-color, where C is a
3-dimensional color space. An ideal printer is the one in which the CRC is an identity function.
A digital color xerographic printer generates color by printing and overlaying the Cyan, Magenta, Yellow and
blacK (CMYK) separations. The printing of each color separation is characterized by the tone reproduction curve
(TRC), TRC : [0, 1] → C , desired tone 7→ output-color, where the tone tone of the separation is the solidness of
the primary toner color. For example, a patch with tone = 0.1 for the magenta color separation corresponds to a
light violet whereas tone = 1.0 corresponds to solid magenta color. Physically, the tone of the primary separations
are determined by the pattern and size of the half tone dots printed. Roughly speaking, the denser and bigger
the dots are, the more solid the color. The final color printed is a composition of the colors of the individual
separation. Thus, the so called Image Output Terminal (IOT) portion of the printer can be considered a mapping
IOT : (toneC , toneY , toneM , toneK ) 7→ output-color where (toneC , toneY , toneM , toneK ) are the tones for
the four color separations.
Submitted as a regular paper to the IEEE Trans. Control Systems Technology, March 2005.
Research supported by the National Science Foundation under grant : CMS-0201622
T. P. Sim and P. Y. Li are with the Department of Mechanical Engineering, University of Minnesota. {tpsim,pli}@me.umn.edu
D. J. Lee was with the University of Minnesota. He is currently with the Coordinated Science Laboratory, University of Illinois at
Urbana-Champaign. d-lee@control.csl.uiuc.edu
Please send all correspondence to Professor Perry Y. Li. Department of Mechanical Engineering, University of Minnesota, 111 Church
St. SE, Minneapolis MN 55455. Tel: 612-626-7815, Fax: 612-625-9395.
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
The xerographic printing process is subject to disturbances from many factors including temperature, humidity,
material age and variations etc.. Several xerographic actuators [1] such as laser power, corotron voltage and
development / bias voltages can be used to combat these variations. The goal of the xerographic color control
system is to ensure that the CRC is as close to the identity map as possible. Unlike the control objective for
most processes which is to control or regulate the output of the process, the color control problem consists in
maintaining the process itself to be constant and stable. The difference is because every customer image to be
printed can contain many and any possible colors which the xerographic printer must reproduce correctly all at
once. Moreover, xerographic printers are often used in an on-demand manner in which consecutive customer images
are different.
A similar, but simpler control problem can be formulated for the printing of each color separation. In this case,
the control objective is to maintain and stabilize the tone reproduction curve (TRC) for each separation [2]. If
the manner in which the primary colors are combined is stable and constant, then the output color will also be
consistent when the TRC for each separation has been effectively stabilized.
Both the CRC and TRC control formulations pose significant problems for sensing and control. It is because
the CRC and TRC, as mappings, are potentially infinite dimensional whereas only a small number actuators and
sensors are available. Even when each color coordinate is modestly discretized into 16 steps, the color quality of
163 = 4096 desired colors need to be kept track of for the color control problem, and 16 tones must be kept track
of for the TRC control problem for each separation.
Feedback control for the stabilization of the TRC or CRC requires sensing of the TRC or the CRC. Process sensing
for xerography consists in printing small color patches in the unused areas of the photoreceptor and measuring their
densities or colors. Typically, only 3 to 5 patches are to be printed every few photoreceptor belt cycles. This amounts
to sampling either the TRC or the CRC at 3 to 5 points once in a while. Increasing the number of test patches
increases hardware needs as well as consumable (toner) and productivity. Compared to the large dimensionality
of the TRC or CRC to be controlled, this limited sensing capability is a challenge for feedback control. In this
paper, we focus mainly on the TRC sensing and control problem for simplicity of presentation. However, all the
algorithms and concepts can be applied to the higher dimensional CRC problem where even greater benefits are
expected.
In this paper, time-sequential sampling strategy is proposed to increase the utility of the available feedback
information. Time-sequential sampling was investigated in the 1980’s and 1990’s for video and time varying imaging
applications (e.g. video and tomography) [3], [4], [5], [6], [7], [8]. For time varying images, time-sequential sampling
refers to sampling the image at different spatial locations at different sampling instances. Rastering in television is
an illustration. The benefit is that by trading off temporal bandwidth with spatial bandwidth, the temporal bandwidth
of the time varying image that can be captured faithfully beyond the Nyquist rate determined by the periodicity
of the sampling scheme alone. In another perspective, the sampling rate can be reduced while retaining the same
information content. In our context, time-sequential sampling means that at different sampling instances, different
tones or different colors are sampled. This maximizes the information from the TRC/CRC samples and allows
the time varying CRC or TRC to be captured (and subsequently reconstructed faithfully) even when only a small
number of samples of the CRC / TRC are available at each instant.
In this paper, a periodic Kalman filter is proposed to causally reconstruct any arbitrary time sequentially sampled
TRC or CRC. Using Floquet theory, it is shown that time-sequential sampling and the subsequent reconstruction
consists of the modulation, low pass and demodulation steps. Comparisons between two classic time sequential
sampling sequences (lexicographical and bit-reversed) [3], [4] show that the latter, which produces more uniform
sampling and a more favorable aliasing pattern, out-performs the former.
The small number of available xerographic actuators (m ≈ 3) per color separation poses a challenge to the control
problem even if full information is available. A solution to this control problem has been addressed previously for
the TRC stabilization problem in [2] using a robust curve-fitting technique. In this paper we describe another curve
fitting approach based on linear quadratic control with integrator dynamics [9]. This approach allows the designer
to specify q tones or colors (q < m) which will be precisely regulated while allowing the TRC or CRC to stay
close to the desired values at the other tones or colors.
Simulation and experimental results confirm that the proposed curve fitting controller with time-sequential
sampling has performance comparable with a control that uses full sampling n = M , even when only n = 1
sensor is used at a time.
SIM, LI, AND LEE, USING TIME-SEQUENTIAL SAMPLING TO STABILIZE A XEROGRAPHIC PRINTING PROCESS
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The rest of the paper is organized as follows. In section II, the time sequential sampling approach for tone
reproduction curves and the control objective are formulated. Section III presents the spectral content of timesequentially sampled signals. Section IV presents a Kalman filtering approach to TRC reconstruction from timesequential samples. Section V presents an interpretation of the periodic Kalman reconstruction filter. Section VI
presents the realization of the TRC stabilization controller based on linear quadratic state feedback followercontroller. Section VII presents the simulation and experimental results of the sensing and control realization.
Lastly, section VIII contains some concluding remarks.
II. P ROBLEM F ORMULATION
Let T RC(k) : [0,1] → ℜ be the time-varying TRC of the printing process where k is the time index (or belt or
print cycle index). It maps the desired input tone to the printed output tone. Although potentially infinite dimensional
(since the domain [0, 1] is so), we assume that the T RC can be adequately described by its values at M points,
i.e.


T RC(k)[tone0 ]


..
M
T RC(k) = 
∈ℜ ,
.
T RC(k)[toneM −1 ]
where tonei , i = 1, . . . , M and M >> 1 can be fairly large. As noted in [2], in the presence of xerographic
control inputs and disturbances, the possibly nonlinear TRC can be represented by the static, linear time varying,
uncertain model as follows:
¯
T RC(k) = φ̂(I + ∆(k)Wu )ū(k) + T RC ∗ + d(k)
(1)
where u(k) ∈ ℜm are the xerographic actuators, d(k) ∈ ℜM are the disturbances, and T RC ∗ ∈ ℜM is the nominal
¯
TRC and d(k)
∈ ℜM is a slowly time varying disturbance. Also, ū(k) := u(k) − uo , where uo is the nominal
control input. φ̂ ∈ ℜM ×m is the nominal sensitivity function, ∆(k) ∈ ℜm×m is the multiplicative uncertainty,
Wu ∈ ℜm×m is the matrix of given uncertainty weights. In this paper, we assume that there is no uncertainty in
the model (i.e. ∆(k) = 0 in (1)).
Sensing of the T RC(k) ∈ ℜM at time instant k is achieved by printing and measuring n << M tones in the
form of small test patches. Typically, the same set of n tones is printed at each k and n will be determined by the
number of available sensors, as well as the productivity and materials cost of printing the test patches.
A. Time-Sequential(TS) Sampling
In [10] and here, we propose to print n different tones at different time k . Which tones are printed at what times
are determined by the M − periodic time-sequential (TS) sampling pattern:
α(k) = α(k + M ) = [α1 (k), α2 (k), . . . , αn (k)] ∈ [0, . . . , M − 1]n
where αi (k) are the tone indices so that at time k , the n tones toneαi (k) , i = 1, . . . , n are printed and measured.
α(k) defines an indicator matrix sequence Cα (k) ∈ ℜn×M such that the (i, j)-th element of Cα (k) is a 1 when
j = αi (k) and it is 0 otherwise. In this paper, we focus particularly on the case of n = 1. This allows each of the
M -tones to be sampled in each M −period.
The time sequentially (TS) sampled TRC is therefore given by:
T RCs (k) = Cα (k)T RC(k) ∈ ℜn .
(2)
We consider the following questions:
1) Given T RCs (k), for k = 0, 1, . . . , K , how does one reconstruct T RC(K)? We are particularly interested in
causal reconstructions so that the reconstruction at time K only requires measurements at k ≤ K .
2) How do different sampling sequences affect the reconstruction accuracies? We do this by comparing two
classic sequences [3], [4] where one is intuitively more uniform than the other.
• Lexicographic: Here α(k) := mod(k, M )) so that the order of sampling is according to its index.
α(k) : 0, 1, . . . , M − 1, 0, 1, . . . ,
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
15
Lexico
Bit−revesed
tone indices
10
5
0
Fig. 1.
0
5
10
15
20
25
30
time indices(k)
35
40
45
Lexicographical and bit-reversed time sequential sampling sequence with M = 16 tones.
•
Bit reversed: Here α(k) is given by 1) representing the index k as a binary number, 2) reversing the
order of the significant bits. Thus for M = 16 = 24 ,
α(k) : 0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15, 0, 8, . . . ,
These two sequences α(k) are shown in Fig. 1. It is apparent that the lexicographical sequence does not cover the
tonal-temporal domain as uniformly as the bit-reversed sequence.
Exactly the same approach can be applied to a color reproduction function. In this case the sampling sequence
is: α(k) ∈ C n where C is the 3-dimensional color space. The benefits of time-sequential sampling will be even
more important since the dimensionality of the CRC is larger.
B. Control Objective
The control objective is for the TRC to match the desired nominal TRC at each tonei , i = 1, 2, ..., M , as k → ∞:
T RC(k)[tonei ] → T RC ∗ (k)[tonei ]
(3)
where T RC ∗ (k) is the desired TRC.
Since there are fewer actuators than the number of tones to be controlled (m << M ), it is typically not
¯ . One possibility is to require that (3)
possible for (3) to hold for all M tones in the presence of disturbance d(t)
is satisfied only at m instead of all M tones. Theoretically, this can be achieved using integral control for constant
or slowly varying disturbances. However, as shown in [2], the integral control approach may lead to mis-behavior
at the unspecified tones and poses robustness problem in the presence of model uncertainty. Instead, a curve-fitting
approach is proposed in [2] to minimize the 2-norm error of the TRC over the entire tone range. In this paper, we
extend the curve fitting approach by allowing q < m tones to be precisely controlled.
III. S PECTRAL C ONTENT
OF
T IME - SEQUENTIALLY
SAMPLED
TRC
Consider a tonal-temporal signal ω(tone, t) ∈ ℜ (where tone ∈ [0, 1] is the tone and t ∈ ℜ is actual time) such
as a TRC. Let Ω(u, f ) be its power-spectral density (u[cycle/tone] is the tonal frequency, f [Hz] is the temporal
Ω(u,f)
SIM, LI, AND LEE, USING TIME-SEQUENTIAL SAMPLING TO STABILIZE A XEROGRAPHIC PRINTING PROCESS
5
40
20
0
8
6
M/2A=8
4
2
0
u [cyc/tone]
−2
1/(2MT)=3/64
0.4
0.2
−4
−6
−0.2
0.6
0
f[Hz]
−0.4
−8
−0.6
−0.8
Fig. 2.
Spectral content of a typical commercial printer with M = 16 tones,A = 1 and T = 6/9s
frequency). Allebach [3] shows that the power-spectral density Ωs (u, f ) of the time-sequentially sampled signal is
given by the original power-spectral density, aliased by its weighted tonal and temporal frequency translates:
X
Ωs (u, f ) =
|Qmp |2 Ω(u − m/A, f − p/B)
(4)
m,p
Qmp =
M −1
2π
1 X
exp−j M (mα(l)+pl)
M
(5)
l=0
where m, p ∈ Z , B = M T is the periodicity of the sampling sequence with T being the sampling period, and
A/M is the tonal resolution with A being the tone range (A = 1 in our case). Notice that the frequency translates
are given by (m/A, p/B)) when Qmp 6= 0. Also, both the frequency translates and the weights depend on the
sampling sequence.
If the TRC has a tonal frequency support less than M/(2A), and a temporal frequency support less than 1/(2B) =
1/(2M T ), then no-aliasing occurs. The original signal can (theoretically) be reconstructed perfectly via an ideal
low pass spatial/temporal filter. M/(2A) and 1/(2B) are therefore the tonal and temporal Nyquist frequencies
when t and tone are treated as fixed respectively. Note that when M is large to satisfy its tonal (color) frequency
requirements, 1/(2B) can be very small for time-sequentially sampled signal.
For a typical commercial printer, the spectral content of the tonal-temporal signal is shown in Fig. 2. This is
obtained by repeatedly printing and measuring the printer’s TRC for a long period. For the number of tonal sample
of M = 16, the TRC is tonally over-sampled. However, even with the aggressive sampling period of T = 6/9s, the
TRC would clearly be temporally under-sampled for time sequential sampling with n = 1 (worse if M increases).
Time sequential sampling sequences make trade-offs between the temporal and tonal capabilities. Their effectiveness
will be related to the aliasing patterns that they generate (Qmp in Eq.(5)).
For the lexicographic sequence and the bit-reversed sequence with M = 16, the aliasing patterns (Qmp ) for
the lexicographical sequence and the bit-reversed sequence are shown in Fig. 3. The aliasing weights for the
lexicographical sequence are concentrated on a line, whereas for the bit-reversed case, the weights are more
uniformly distributed, even though the sums of Qmp for both sequences are the same. The consequence of this result
is that for signals with high spatial/temporal bandwidth, the lexicographical sampling strategy will have more severe
aliasing compared to the bit-reversed sampling strategy. Hence the reconstruction from the bit-reversed sampling
sequence will be more tolerant to high spatial/temporal bandwidth signal.
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
Spatio−temporal aliasing pattern: Bit−reversed
1
1
0.8
0.8
0.6
0.6
|Qmp|
|Qmp|
Spatio−temporal aliasing pattern: Lexicographic
0.4
0.4
0.2
0.2
0
0
10
m(spatial freq.)
1
0
−10
−0.5
−1
0.5
0
0
−10
Fig. 3.
10
1
0.5
0
m(spatial freq.)
p(temporal freq.)
−0.5
−1
p(temporal freq.)
Aliasing weights |Qmp | for lexicographical (left) and bit-reversed sequences (right).
IV. R ECONSTRUCTION
VIA A
K ALMAN F ILTER
Assuming that no aliasing occurs, time-sequentially sampled signals are traditionally reconstructed by low pass
temporal-spatial filtering based on the assumed signal spectral support. This approach, however, is problematic in
feedback control application for the following reasons:
• The zero-phase ideal low pass filter cannot be implemented causally.
• When the reconstructed TRC is used for feedback control, the frequency content of the TRC will be significantly
increased due to the control input. This induces undue aliasing, while the inherent process dynamics actually
remains the same. However, it is not apparent how to incorporate the known information about the control
input in the low-pass reconstruction technique.
These difficulties can be resolved using an alternate approach based on Kalman filtering.
Let ∆T RC(k) := T RC(k)−T RC ∗ be the TRC error. Neglecting model uncertainty (∆(k) = 0), the xerographic
plant given in (1) becomes:
¯
∆T RC(k) = φ̂ū(k) + d(k).
(6)
To exhibit its tonal spectral contents, the TRC disturbances are modeled by its discrete fourier transform (DFT) so
that:
¯ = G · xd (k)
d(k)
(7)
where k ∈ Z + is the index, G ∈ ℜM ×M is a matrix of Fourier basis functions and xd (k) ∈ ℜM is the vector of
Fourier coefficients representing the tonal frequency content of the disturbance. Substituting (7) into (6), we have:
∆T RC(k) = φ̂ū(k) + G · xd (k).
The disturbance dynamics, xd (k) in (7) is modeled as a pink noise having a compact tonal-temporal spectral support
as generated by the following dynamics:
xw (k + 1) = Aw xw (k) + Bw w(k)
(8)
xd (k) = Cw xw (k) + Dw w(k)
where w(k) is a white process noise. The matrix Aw , Bw , Cw and Dw are obtained from a bank of low-pass
butterworth filters that filter each spatial channel (each Fourier coefficient) with temporal cutoff frequencies corresponding to an ellipse. This gives an approximation of a compact ellipsoidal tonal-temporal spectral support. From
the definition given in (2), we can define the signal,
y(k) := ∆T RCs (k) − Cα (k)φ̂ū(k)
= [Cα (k)GCw ] · xw (k) + [Cα (k)GDw ] · w(k) + n(k)
(9)
SIM, LI, AND LEE, USING TIME-SEQUENTIAL SAMPLING TO STABILIZE A XEROGRAPHIC PRINTING PROCESS
7
where both w(k) and n(k) are zero-mean white noise sequences with covariance Rww and Rnn respectively. y(k)
in (9) can be treated as the measurement for the observer.
Under very general conditions, the M -periodic linear system given by (7) (8) and (9) is observable and therefore
b of the disturbance d(k)
¯
admits a M -periodic Kalman filter to form the estimate d(k)
[11]:
x̂w (k + 1) =Ac (k)x̂w (k) + Bc (k)y(k)
b =G̃ · x̂w (k)
d(k)
where
(10)
Ac (k) = Aw (I − L(k)Cα (k)G̃)
Bc (k) = Aw L(k)Cα (k)
G̃ = GCw
and L(k) is the periodic Kalman filter gain obtained by solving the periodic Riccati equation:
T
P̄ (k + 1) = Aw [P̄ (k) − L(k)Cα (k)G̃P̄ (k)]ATw + Bw Rww Bw
h
i−1
L(k) = P̄ (k)G̃T CαT (k) Rnn + Cα (k)G̃P̄ (k)G̃T CαT (k)
P̄ (k) = P̄ (k + M )
(11)
The estimate of the TRC error ∆T RC(k) can be reconstructed as:
b + φ̂ū(k)
d
∆T
RC(k) = d(k)
(12)
For further analysis and derivation of time-sequential sampling with reconstruction using Kalman filter, the readers
are referred to [10].
V. A NALYSIS
OF
TS S AMPLING
AND
R ECONSTRUCTION
VIA A
K ALMAN F ILTER
We now describe our analysis of the Kalman filter (10). Since the steady state Kalman filter is M −periodic, it
is difficult to talk about its frequency response. Instead, we decompose its input/output relationship using Floquet
theory for periodic systems [12] and to analyze how the spectrum is transformed. According to Floquet theory, the
transition matrix Φ(k, k0 ) for the M − periodic discrete time system (10) can be written as:
Φ(k, k0 ) = P (k)Λk−k0 P −1 (k0 )
where Λ := Φ(M, 0)1/M is a constant matrix and P (k) ∈ ℜM ×M is a M −periodic matrix sequence given by:
(
Φ(k, 0)Λ−k for k ∈ [0, M − 1]
P (k) =
.
P (k − M ) when k ≥ M
Using this decomposition and applying it to the convolution formula for (10), we have:


k−1

 X
′
¯ ′ )] .
b = [G̃P (k)G̃−1 ] · G̃
Λk−k −1 G̃−1 [G̃P (k ′ + 1)−1 Bc (k ′ ) · d(k
d(k)

 ′
(13)
k =k0
¯
Eq. (13) shows that, in the original disturbance coordinates, the Kalman filter operates on the d(k)
in three steps:
−1
1) multiplication by the periodic static operator - y(k) → G̃P (k + 1) Bc (k) · y(k).
2) time invariant linear filtering by the filter - y(k) → G̃x(k) with x(k + 1) = Λx(k) + G̃−1 y(k)
3) multiplication by another periodic static operator - y(k) → G̃P (k)G̃−1 y(k).
The three steps are illustrated for a low single tonal-temporal frequency disturbance input signal (at one tone
u = 4cyc/tone, f = 0.01Hz ) in Fig. 4. Step 1 modulates the input signal by duplicating the weighted copies
of the original signal according to the spectrum of G̃P (k ′ + 1)−1 Bc (k ′ ). Step 2 represents a low pass filtering
according to the matrix Λ which eliminates nearly all but the original signal frequency and one extra copy. Step 3
corresponds to a demodulation due to the multiplication by G̃P (k)G̃−1 . Interestingly, the demodulation folds the
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
u = 4, f = 0.010000
PSD
4000
Orginal signal
spectra
2000
0
−1.5
−1
−0.5
0
f
0.5
1
1.5
PSD
200
Modulated
signal
spectra
100
0
−1.5
−1
−0.5
0
f
0.5
1
1.5
PSD
4000
Low−pass signal
spectra
2000
0
−1.5
−1
−0.5
0
f
0.5
1
1.5
PSD
4000
Demodulated
signal
spectra
2000
0
−1.5
−1
−0.5
0
f
0.5
1
1.5
PSD : Power Spectral Density
Fig. 4. Transformation of a sinusoidal temporal-spatial (u = 4cyc/tone, f = 0.01Hz) disturbance input through the three processes
in the Kalman filter. From top to bottom are the temporal spectra for a particular tone: 1) original disturbance signal; 2) transformed by
G̃P (k + 1)−1 Bc (k); 3) after filtering by Λ; 4) demodulation by G̃P (k)G̃−1 .
high frequency copy of the signal after step 2 back to the low frequency, resulting in a reconstruction with minimal
error.
The modulation effect of step 1 can also be thought of as modulating the spectral content of the map G̃P (k +
−1
1) Bc (k) by the input signal. The pattern in which the single tonal-temporal frequency of the disturbance signal is
modulated by this map is determined by the aliasing weights (Qmp ) of the sampling pattern shown in Fig. 3. Figs.
5-6 show the spectral contents of the signal after the first transformation for different single tonal-temporal input
frequencies for the lexicographic and bit-reversed time sequential sampling sequences. As the temporal frequency
increases (left to right), copies of the spectral content of the map approach each other on the temporal frequency
domain (horizontally); and similarly as the tonal frequency increases (bottom to top), copies of the spectrum of
the map approach each other on the tonal frequency domain (vertical). Notice that for the lexicographical sampling
sequence, the frequency replications are very distinct and follows a well structured pattern; whereas for the bitreversed sampling sequence, the frequency replications are much more diffused. Therefore for the class of signal with
a compact ellipsoidal tonal-temporal spectral support, the modulation process (step 1) with spectral support larger
than the corresponding Nyquist frequencies would result in a very distinct and sudden overlap with the lexicographic
sampling sequence; whereas for the bit-reversed sequence, the overlap process is much more diffused.
VI. TRC S TABILIZATION C ONTROLLER
The high dimensionality of the TRC coupled with limited actuation does not permit us to control all M color
tones. An optimal control approach is proposed to ensure the M -tones are close to the nominal TRC in a leastsquared sense. Integrator dynamics are also imposed on the optimal control formulation to ensure precise control
at certain q -tones (q ≤ m). This feature would be useful when it is desirable to have certain q-tones to coverage
exactly to the desired tones. This however will come at the expense of reducing the freedom to curve fit the TRC
SIM, LI, AND LEE, USING TIME-SEQUENTIAL SAMPLING TO STABILIZE A XEROGRAPHIC PRINTING PROCESS
W = 0.00Hz
W=0.03Hz
W=0.08Hz
0
0
0
0
5
5
5
5
9
W=0.12Hz
u
U = 10
10
0
0
0.5
f
10
1
0
0
0.5
f
1
10
0
0
0.5
f
1
10
0
0.5
f
1
0
0.5
f
1
0
u
U=6
5
10
5
0
0.5
f
10
1
5
0
0.5
f
1
0
0.5
f
1
0
u
U=1
10
5
0
0.5
f
1
10
5
10
Fig. 5. Tonal-temporal spectral content for various single tone input frequencies (u[cyc/tone], f [Hz]) after the first transformation of the
reconstruction filter for the Lexicographic TS sampling scheme.
W = 0.00Hz
W=0.03Hz
W=0.08Hz
0
0
0
0
5
5
5
5
W=0.12Hz
u
U = 10
10
0
0
0.5
f
10
1
0
0
0.5
f
1
10
0
0
0.5
f
1
10
0
0.5
f
1
0
0.5
f
1
0
u
U=6
5
10
5
0
0.5
f
10
1
5
0
0.5
f
1
0
0.5
f
1
0
u
U=1
10
5
0
0.5
f
1
10
5
10
Fig. 6. Tonal-temporal spectral content for various single tone input frequencies (u, f ) after the first transformation of the reconstruction
filter for the Bit-reversed TS sampling scheme.
10
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
onto the desired TRC. Hence the optimal control problem is to find the control u(k) based on the measured TRC,
T RC(k), such that the following quadratic performance index (QPI), J is minimized:
1X T
1X
J=
∆I (k)QI ∆I (k) +
∆T RC T (k)Q∆T RC(k)
(14)
2
2
k∈Z
where ∆I (k) ∈
by:
ℜq
k∈Z
is the integrated error at the q < m tones to be controlled precisely, whose dynamics are given
∆I (k + 1) = ∆I (k) + CI ∆T RC(k)
(15)
where CI ∈ ℜq×M is the indicator matrix for the selected q -tones to fulfill (3). QI ∈ ℜq×q and Q ∈ ℜM ×M are
the weighting matrices. The linear-quadratic state feedback follower-controller for the given QPI for system (6) can
¯
be solved by using the backward-sweep solution [13] as given in the appendix. Assuming the disturbance d(k)
is
available (it will be replaced by its estimate), the optimal control is given by:
¯
ū(k) = −K1 ∆I (k) + K2 d(k)
(16)
where
−1 T
K1 =Zww
Zxw
−1
K2 = − Zww
(φ̂T CiT SB CI + φ̂T Q − φ̂T CIT FB )
are the feedback and feedforward gains respectively, with Zww := φ̂T Qφ̂ + φ̂T CIT SB CI φ̂; Zxw := SB CI φ̂. SB is
obtained from the solution of the discrete algebraic Riccatti equation.
I T SB I − SB − SB CI φ̂ · (φ̂T Qφ̂ + φ̂T CiT SB CI φ)−1 (SB CI φ̂)T + QI = 0
and,
−1
−1 T T −1
(φ̂T CIT SB CI + φ̂T Q) − SB CI )
φ̂ CI ) · (Zxw Zww
FB = (Zxw Zww
b
To implement (16), the Kalman filter estimate of the TRC disturbance based on time-sequential sampling, d(k)
¯ . ∆I (k) is obtained from (15) where the Kalman estimate
in (10), is used in lieu of the actual disturbance d(k)
of the TRC error based on time-sequential sampling, ∆Td
RC(k) in (12), is used in lieu of the actual TRC error,
∆T RC(k).
Figure 7 shows the schematic of the TRC stabilization controller with time sequential sampling. The well known
separation principle [13] allows the controller and estimator to be designed separately yet used together.
Remark 1 A robust static controller was previously proposed in [2] which also uses the curve-fitting approach (i.e.
it tries to minimize the overall TRC error). A major issue addressed in [2] is that fixed TRC sampling is assumed,
and the number of tones that are measured is small compared to the dimensionality of the TRC (n << M ). The
proposed robust control ensures that the TRC behaves adequately even at unmeasured tones in the presence of
uncertainty. With the use of time-sequential sampling, the apparent number of measured tones can be significantly
increased (→ M ), thus relaxing this difficulty. Nevertheless, the time-sequential sampling can also be used with
the robust static control law in [2] to take advantage of its robustness feature.
VII. S IMULATION
AND
E XPERIMENTS
A. Simulation
The behavior of the xerographic plant is simulated by taking the model of the form of (1) based on TRC
experimental data from a commercial printer using different xerographic inputs. The nominal sensitivity matrix φ̂
in (1) is obtained by least-square fitting of the experimental data into the linear model. The behavior of the system
¯
without plant perturbation is considered i.e. ∆(k) = 0. The disturbance d(k)
dynamics is simulated from the pink
noise model given in (8) with the temporal cutoff frequency, f of the butterworth filter at each spatial channel, u
given by an ellipse:
(u/U )2 + (f /W )2 = 1
SIM, LI, AND LEE, USING TIME-SEQUENTIAL SAMPLING TO STABILIZE A XEROGRAPHIC PRINTING PROCESS
Cα (k)T RC ∗
n(k)
d̄(k)
11
T RCs (k)
¯ + T RC ∗
T RC(k) = φ̂ū(k) + d(k)
Cα (k)
∆T RCs (k)
ū(k)
y(k) = ∆T RCs (k) − Cα (k)φ̂ū(k)
y(k)
ˆ
ū(k) = −K 1 ∆I (k) + K2 d(k)
Fig. 7.
x̂w (k + 1)
ˆ
d(k)
=
Ac (k)x̂w (k) + Bc (k)y(k)
=
∆ T RC(k)
=
G̃.x̂w (k)
ˆ + φū(k)
d(k)
∆I (k + 1)
=
∆I (k) + CI ∆ T RC(k)
Kalman estimator of time-sequentially sampled TRC and TRC stabilization controller mechanization
¯ . In our study, we used a
where U [cyc/tone] and W [Hz] give the highest tonal and temporal frequencies in d(k)
sampling interval of T = 0.4s and the tonal range is tonei ∈ [0,1]. With M = 16, this gives a tonal temporal
Nyquist frequencies of (uN , fN ) = (8.0cycles/tone, 0.078Hz).
We consider two cases of sampling: full sampling where all M -measurement points are used at each sampling
instant, k and time sequential sampling where only n = 1 tone is sampled at each sampling instant according to a
prescribed sampling pattern i.e. lexicographical or bit-reversed sampling sequences. As our primary interest is to
analyze the effect of disturbances with different sampling schemes, we assume the measurement noise n(k) ≈ 0.
The controller that minimizes (14) with one (q = 1) integral fix point at tone2 is obtained from (16). The
performance weighting matrices QI and Q in (14) are identities. The time-normalized quadratic performance index,
QPIL = Jk∈[l0 ,l0 +L] /L is taken as the measure of performance of the TRC stabilization controller.
¯ within the range of
Simulations were carried out at different tonal-temporal support frequencies (U, W ) for d(k)
{(U, W ) | 1 ≤ U ≤ uN , 0.01 ≤ W ≤ 3fN } [cyc/tone,Hz] using the optimal TRC stabilization controller. As shown
in Figure 8, both the (lexicographical) time-sequential sampling and the full sampling have comparable QPIL TRC
error (less than 0.15) for the range of input frequencies tested. This means that we are able to achieve as good
TRC stabilization by just sampling one TRC tone at each time step as by sampling all M = 16 tones. Figure 8
also shows the expected response with increasing tonal-temporal support frequencies of the disturbance signal – the
higher the tonal-temporal support frequencies, the higher the QPIL TRC error. The higher tonal-temporal disturbance
frequencies the harder it is for the TRC controller to curve-fit the measured TRC to the desired TRC. Figure 9
compares the difference in QPIL TRC error between lexicographical and bit-reversed sampling sequences. This
shows clearly that control using the bit-reversed time-sequential sampling sequence out-performs control using
the lexicographic sequence at all signal supports. The better signal reconstruction of the bit-reversed sampling
sequence as reported in [10] translates to better TRC stabilization control performance. Hence the bit-reversed
sampling sequence is preferred.
12
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
Full Sampling
Bit−reversed Time Sequential Sampling
8
8
0.12
6
0.06
U(spatial freq.)
0.1
0.08
5
4
3
2
0.02
2
0.2
0.25
1
0.14
0.12
0.1
0.08
0.0
0.0 504
0.0 448761
0.0 3928 1
336 2
92
3
42
57
00
0.
0.04
0.1
0.15
W(temporal freq.)
1
82
72 231
0.0 67 641
0.0.061 051
0 056
0.
0
0 .07
0 .06 44
0. .063 915 75
05 8 7
85 38
19
2
10
28 12
0.0 225 2
2
0 .0 69
01
0.
2
33
11
0.0
U(spatial freq.)
7
3
0.05
2
87 3
8
12 31
0. .12 754195 6
51 59 1
0 1 1 3
09 89 00 11
1
0. 0.1.106077 0. 0.0 84 84
0 7
0 .10
0. 0.0
0
2
70 1
11 7
0.111 39 49
0. 10607 57 31
0.101 .09904112 4
0. 00.0 85 79
0 9
0. .07
0
45
72 27
03 9 8
0. 31 60
0
0. 026 289
0. 21
0
0.
338
4
1
Fig. 8.
0.0053
5
1
597
0.01 52
0.0106
6
01
532
0.0 7882
4
0 .0 2 5 6 4
4
0 .0
7
0.14
0.06
0.04
0.02
0.05
0.1
0.15
W(temporal freq.)
0.2
0.25
QPIL of TRC error using full and lexicographic time-sequential sampling at different tonal-temporal frequencies support (U, W )
−3
Difference Between Leixographic and Bit−reversed Sampling
x 10
8
4
7
3.5
U(spatial freq.)
6
3
2.5
5
2
4
1.5
3
1
0.5
2
0
1
0.05
0.1
0.15
W(temporal freq.)
0.2
0.25
Fig. 9. Difference in QPIL TRC error (QPILLexico − QPILBit-reverse ) of lexicographical and bit-reversed time sequential sampling for different
tonal-temporal support frequencies (U, W ) [cyc/tone, Hz].
B. Experiment
The proposed TRC stabilization system was also experimentally tested on a Xerox Phaser 7700 xerographic
printer. Currently we do not have direct access to the xerographic actuators. To evaluate the TRC stabilization
controller with full and time sequential samplings, a virtual printer model is used to generate the response (color
image) due to changes in the actuator inputs. The virtual printer model is the one used in the simulation study
above. The output response from the virtual printer is then printed using the physical printer. By calibrating the
printer such that it is an identity map at the nominal condition, we can capture the effect of the actual disturbances
on the performance of the TRC stabilization system. The output response is in the form of a single colorant wedge
of 21 different tones i.e. M = 21. A pseudo-bit-reversed sequence is used in this case. The disturbances was
artificially induced by introducing a transparency in the optical path of the laser. Sensing of the color wedge is
performed using a spectrophotometer.
Figure 10, 11 and 12 show the effectiveness of the proposed TRC stabilization system using both full and time
SIM, LI, AND LEE, USING TIME-SEQUENTIAL SAMPLING TO STABILIZE A XEROGRAPHIC PRINTING PROCESS
13
Full Sampling
1.2
1
tone−out
0.8
k=49
0.6
k=0
0.4
0.2
0
−0.2
0
0.2
0.4
0.6
0.8
1
tone−in
0.1
qpi(k)
QPI =0.1186
L
0.05
0
0
10
20
30
time−indices
40
50
Fig. 10. Response of TRC stabilization control subjected to induced disturbance with full sampling. The curve marked with * is the desired
TRC (qpi(k) = 12 ∆TI (k)QI ∆I (k) + 21 ∆T RC T (k)Q∆T RC(k))
sequential samplings. The TRC stabilization using all the sampling approaches result in the convergence of the
TRC to the nominal TRC with each time step. Considering that only one tone is sampled at each time step, the
time sequential sampling perform well compared to that achieved using full sampling (M = 21).
VIII. C ONCLUSIONS
This paper addressed two main problems in realizing a practical TRC/CRC stabilization controller in maintaining
consistency in color reproduction. The first problem of under actuation is resolved using a curve-fitting optimal
control approach. The proposed TRC stabilizing control makes use of all available measurement/reconstruction
data and allows certain tones to be regulated to converge to the corresponding desired tones. The second problem
relating to limited sensing capability is resolved using time sequential sampling. Time sequential sampling increases
the printers’ ability to sense the high dimensional time varying tone or color reproduction functions (TRC/CRC),
which is required for the proposed TRC stabilization controller. A Kalman filter based method has been developed
and shown to be effective to reconstruct the original function from the time-sequential samples.
Both simulation and experimental results shows the effectiveness of the proposed approach for TRC stabilization.
In particular, comparable control performance is obtained using time sequential sampling in place of full sampling.
The bit-reversed sequence is also found to yield better signal reconstruction as has been reported in [10]. In general,
the efficiency of the time-sequential sampling scheme depends both on the signal model and the sampling strategy
as demonstrated using Floquet theory. Time sequential sampling substantially lowers the TRC sensing requirements
and this is important in actual implementation where available print area should be devoted to customer images
and not to printing sensor patches. Time sequential sampling can also be thought of as active-sensing. As such, it
can have many other applications as well.
The next step would be to implement and test the sensing and stabilization scheme for the CRC problem. Benefits
of time-sequential sampling is expected to be even greater because of the higher dimensionality. In addition, while the
bit-reversed sampling sequence out performs lexicographical sampling sequence, it is still not optimal. Procedures
for designing optimal sampling sequences would therefore be desirable.
14
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
Lexicographical Sampling
1.2
1
0.8
k=49
tone−out
k=0
0.6
0.4
0.2
0
−0.2
0
0.2
0.4
0.6
0.8
1
tone−in
0.2
qpi(k)
QPI =0.2992
L
0.1
0
0
10
20
30
time−indices
40
50
Fig. 11. Response of TRC stabilization control subjected to induced disturbance with time sequential sampling (Lexicographical sequence).
The curve marked with * is the desired TRC
Bit−reversed Sampling
1.2
1
0.8
k=49
tone−out
k=0
0.6
0.4
0.2
0
−0.2
0
0.2
0.4
0.6
0.8
1
tone−in
0.2
qpi(k)
QPI =0.1902
L
0.1
0
0
10
20
30
time−indices
40
50
Fig. 12. Response of TRC stabilization control subjected to induced disturbance with time sequential sampling (Bit-reversed sequence).
The curve marked with * is the desired TRC
SIM, LI, AND LEE, USING TIME-SEQUENTIAL SAMPLING TO STABILIZE A XEROGRAPHIC PRINTING PROCESS
15
R EFERENCES
[1] L. B. Schein, Electrophotography and Development Physics. Springer-Verlag, 1992.
[2] P. Y. Li and S. A. Dianat, “Robust stabilization of tone reproduction curves for xerographic printing process,” IEEE Transactions on
Control Systems Technology, vol. 9, no. 2, pp. 407–415, 2001.
[3] J. P. Allebach, “Analysis of sampling-pattern dependence in time sequential sampling of spatiotemporal signals,” Journal of Optical
Society of America, vol. 71, no. 1, pp. 99–105, January 1981.
[4] ——, “Design of antialiasing patterns for time-sequential sampling of spatiotemporal signals,” IEEE Trans. Acoustic, Speech, and
Signal Processing, vol. ASSP-32, no. 1, pp. 137–144, 1984.
[5] N. P. Willis and Y. Bresler, “Lattic-theoretic analysis of time-sequential sampling of spatiotemporal signals - part 1,” IEEE Trans. on
Information Theory, vol. 43, no. 1, pp. 190–207, January 1997.
[6] ——, “Lattic-theoretic analysis of time-sequential sampling of spatiotemporal signals - part 2: Large space-bandwidth product
asymptotics,” IEEE Trans. on Information Theory, vol. 43, no. 1, pp. 208–220, January 1997.
[7] M. A. Rahgozar and J. P. Allebach, “Motion estimation based on time-sequentially sampled imagery,” IEEE Trans. on Image Processing,
vol. 4, no. 1, January 1995.
[8] N. P. Willis and Y. Bresler, “Optimal scan for time-varying tomography I: Theoretical analysis and fundamental limitations,” IEEE
Trans. on Image Processing, vol. 4, no. 5, pp. 642–653, May 1995.
[9] T. P. Sim and P. Y. Li, “Stabilization of color/tone reproduction curves using time-sequential sampling,” in Proceedings of the 20th
International Congress on Digital Printing Technologies (NIP20), Salt Lake City, November 2004, pp. 204–209.
[10] P. Y. Li, T. P. Sim, and D. J. Lee, “Time-sequential sampling and reconstruction of tone and color reproduction functions for xerographic
printing,” in Proceedings of the 2004 American Control Conference, Boston, United States, June 2004, pp. 2630–2635.
[11] A. E. Bryson, Applied Linear Optimal Control. Cambridge University Press, 2002.
[12] F. Callier and C. A. Desoer, Linear System Theory. Springer Verlag, 1991.
[13] J. D. Powell, G. F. Franklin, and M. Workman, Digital Control of Dynamic Systems. Addison-Wesley, 1998.
A PPENDIX
Derivation of the optimal control for TRC stabilization is as follows. From (15):
∆I (k + 1) = ∆I (k) + CI ∆T RC(k)
From (6):
¯
∆T RC(k) = φ̂ū(k) + d(k)
The QPI is given by:
N −1
1
1X T
J = ∆TI (N )QI ∆I (N ) +
[∆I (k)QI ∆I (k) + ∆T RC T (k)Q∆T RC(k)]
2
2
(17)
k=0
The integrator dynamics (15) and plant (6) are adjoint to the quadratic performance index (17) with a Lagrange
multiplier vector sequence λT (k + 1) as follows:
J¯ = J +
N
−1
X
¯ − ∆I (k + 1)] + λT (0)[∆I0 − ∆I (0)]
λT (k + 1)[∆I (k) + CI φū(k) + CI d(k)
k=0
Define the discrete Hamiltonian
1
¯ + d(k)Q
¯
¯
¯
H(k) := [∆TI (k)QI ∆I (k) + ūT (k)φ̂T Qφ̂ū(k) + ūT (k)φ̂T Qd(k)
φ̂ū(k) + d(k)Q
d(k)]
2
¯
+ λT (k + 1)[∆I (k) + CI φ̂ū(k) + CI d(k)]
then:
1
J¯ = ∆TI (N )QI ∆I (N ) +
2
N
−1
X
[H(k) − λT (k)∆I (k)]
k=0
(18)
− λT (N )∆I (N ) + λT (0)∆I0
¯
Consider infinitesimal changes δ J¯ due to infinitesimal changes in δ(∆I (k)), δ(∆I (N )),δ(∆I0 ),δ(ū(k)) and δ(d(k))
away from their optimal values:
δ J¯ = ∆TI (N )QI δ(∆I (N )) − λT (N )δ(∆I (N ))+
T
λ (0)δ(∆I0 ) +
N
−1
X
¯
[(H∆I (k) − λT (k))δ(∆I (k)) + Hū (k)δ(ū(k)) + Hd¯(k)δ(d(k))]
k=0
(19)
16
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (SUBMITTED, MARCH 2005)
The necessary conditions are given by:
1) λT (k) ≡ H∆I (k), then:
λ(k) ≡ λ(k + 1) + QI ∆I (k)
(20)
where λT (N ) = QI ∆I (N )
2) For a given initial condition ∆I0 , δ(∆I0 ) = 0.
3) Hū (k) ≡ 0 and Hd¯(k) ≡ 0, then:
¯
−φ̂T Qφ̂ū(k) = φ̂T CIT λ(k + 1) + φ̂T Qd(k)
(21)
Consider the backward sweep solution of the following form:
λ(k) = SB (k)∆I (k) − λB (k)
(22)
¯
λ(k) = (SB (k + 1) + QI )∆I (k) + Zxw ū(k) − λB (k + 1) + SB (k + 1)CI d(k)
(23)
From condition (20) we have:
From condition (21) we have:
T
¯
0 = Zxw
∆I (k) + Zww ū(k) − φ̂T CIT λB (k + 1) + [φ̂T CIT SB (k + 1)CI + φ̂T Q]d(k)
(24)
where Zww := φ̂T Qφ̂ + φ̂T CIT SB (k + 1)CI φ̂; Zxw := SB (k + 1)CI φ̂. Substituting (24) into (23) and comparing it
with the form given by (22) gives:
−1 T
SB (k) = SB (k + 1) + QI − Zxw Zww
Zxw
(25)
where SB (N ) = QI
−1 T T
λB (k) = [I − Zxw Zww
φ̂ CI ]λB (k + 1)+
−1
¯
[Zxw Zww
(φ̂T CIT SB (k + 1)CI + φ̂T Q) − SB (k + 1)CI ]d(k)
(26)
where λB (N ) = 0
At steady state, we obtained the discrete algebraic Riccatti equation from (25):
I T SB I − SB − SB CI φ̂(φ̂T Qφ̂ + φ̂T CIT CI φ̂)−1 (SB CI φ̂)T + QI = 0
The solution of the discrete algebraic Riccatti equation gives SB and from (26) the optimal control is derived:
−1 T
−1
¯
ū(k) = −Zww
Zxw ∆I (k) − Zww
(φ̂T CIT SB CI + φ̂T Q − φ̂T CIT FB )d(k)
−1 φ̂T C T )−1 (Z Z −1 (φ̂T C T S C + φ̂T Q) − S C
where FB = (Zxw Zww
xw ww
B I
I
I B I