IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
531
Modeling and Control of a Hybrid Two-Component Development Process for Xerography Feng Liu, George T.-C. Chiu, Eric S. Hamby, Member, IEEE, and Yongsoon Eun
Abstract—Development is one of the six key steps in xerographic printing processes. Under certain printing conditions, the “deveopability” of the toner particles tends to degrade resulting in a loss of image quality. Existing process controls have limited authority in compensating for this degradation, and, ultimately, a service operation may be required to install fresh toner. From a customer perspective, this machine maintenance results in productivity loss and added cost, both of which need to be minimized. In this paper, a control oriented model that characterizes “developability loss” is derived from an experimentally validated comprehensive statistical model. The resulting model considers the stress case of printing “low area coverage” documents (e.g., text pages) in a low relative humidity environment, and it maps the development voltage and toner dispensing rate actuators to the developed toner mass per unit area, which is the sensed output and is also used as a surrogate for print quality. Under these operating conditions, system analysis shows that developability loss is unavoidable. Given this result, a constrained time optimal control problem is formulated to determine the dispensing strategy to maximize the printer operating time while maintaining acceptable developability. Numerical solution shows that for the stress operating condition leading to developability loss, the optimal dispensing strategy increased the operating time by 170% compared with a conventional dispensing strategy. Index Terms—Constrained optimal control, electrophotography, system modeling and analysis, xerographic development process.
I. INTRODUCTION EROGRAPHY is the “dry-marking” process used in the majority of laser printers and copiers. A typical xerographic process includes six sequential steps around a photoreceptor: 1) charge; 2) expose; 3) develop; 4) transfer; 5) fuse; and 6) clean, as shown in Fig. 1 [1]. During charging, a high-voltage corona or charge roller is used to induce a uniform charge layer on the surface of a photoreceptor. Pulsed laser or LED light discharges (exposes) selected areas on the photoreceptor surface to
X
Manuscript received September 23, 2008; revised September 25, 2009; accepted January 11, 2010. Manuscript received in final form April 15, 2010. First published May 24, 2010; current version published April 15, 2011. Recommended by Associate Editor G. Stewart. This work was supported in part by the Xerox Foundation and by the National Science Foundation under Award CMS-0201837. F. Liu was with the School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA. He is now with Cummins Inc., Columbus, IN 47201 USA (e-mail:
[email protected]). G. T.-C. Chiu is with the School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail:
[email protected]). E. S. Hamby and Y. Eun are with the Xerox Research Center Webster, Xerox Corporation, Webster, NY 14580 USA (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2010.2048712
Fig. 1. Schematic of the xerographic process.
form a latent charge image. During development, a bias voltage (referred to as development voltage) is applied between the photoreceptor and the development housing to generate an electrostatic field, which forces charged toner particles to move from the toner cartridge to the surface of the photoreceptor to form a toned image. During the transfer step, another electrostatic field is used to transfer the toner particles from the photoreceptor surface to a precharged media (e.g., paper). Heat and pressure are applied during the fusing step to permanently attach the melted toner particles to the media. Finally, residual toner particles on the photoreceptor surface are “cleaned” off with the help of a blade or a brush to prepare the photoreceptor surface for the next xerographic cycle. A more detailed description of the xerographic process can be found in [1]–[3]. In this study, we will focus on the development process. With increasing print speeds and the transition from monochrome to color printing, maintaining uniform and consistent image quality has become an active research area for printing process control [4]. In terms of time scales, most image quality defects fall into one of the following two categories: small time scale defects (seconds) and large time scale defects (tens of minutes or hours). Small time scale defects usually appear within one single printed page, such as the banding defect shown in Fig. 2(a). Large time scale defects, on the other hand, usually appear between multiple printed pages. The page-to-page color shift shown in Fig. 2(b) illustrates a drifting type of disturbance that can occur over hours or even days of printing. Control algorithms have been developed primarily to remedy small time scale defects, while results with respect to the large time scale defects are still quite limited. For example, human visual model-based loop-shaping and repetitive control of the photoreceptor drum velocity was shown to be effective in reducing periodic banding artifacts resulting from periodic gear
1063-6536/$26.00 © 2010 IEEE
532
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
Fig. 2. Examples of image quality defects. (a) Binding defect; (b) page to page color shift defect.
transmission disturbances [5], [6]. The combined use of feedforward and feedback control has been shown to be effective in address banding defects for a class of color laser printers [7]. control has been applied to reduce the impact of In [8], different types of disturbances on tone reproduction in the xerographic process. In these past works, the xerographic process has been modeled as a static mapping, which may be enough to deal with small time scale defects. However, mitigating defects with large time scales is also important, especially in markets with more stringent image quality requirements and/or longer print runs [4]. Researchers working with those large time scale defects have reported comprehensive and precise models based on first principles [9], [10]. Although experimentally validated, these comprehensive models are not readily suited for control system analysis and synthesis, where characterization of the system level dynamics is preferred. In this paper, we develop a control oriented model for toner aging dynamics, a large time scale development system dynamics, that will subsequently be used for system analysis and control strategy design. “Toner aging” refers to a change in the toner charging characteristics that results from subjecting the toner particles to repeated electromechanical stresses in the development subsystem. By repeatedly cycling toner through a set of augers and rollers as well as a “trim” blade (see Fig. 3), the toner experiences many rolling, impact, and shear cycles in addition to the electrostatic forces needed to propagate the toner through the system. If the toner is subjected to too many of these cyclic stresses and/or the magnitude of the stresses becomes too large, then the surface characteristics of the toner change in such a way that the toner particles becomes “sticky” and more difficult to develop. Fig. 4 shows the different surface characteristics between a freshly dispensed (new) toner particle and one that has been in the development sump for some time (aged toner particle). The change in surface characteristics has been correlated to a loss in
Fig. 3. Schematic of a hybrid two component development process.
Fig. 4. Photomicrograph of (left) fresh toner and (right) aged toner.
toner charging [11], [12], which, in turn, leads to an increase in the electrostatic field strength needed to develop a given amount of toner on the photoreceptor. The resulting impact of the toner aging dynamics on the development voltage required to achieve target toner levels on the photoreceptor is generically referred to as “developability loss.” Two types of external disturbances drive the toner aging dynamics: customer job type and the environment in which the machine operates. For instance, customer print jobs with “low area coverage1” subject the toner in the development sump to long residence times and therefore many electromechanical stress cycles, which can lead to developability loss. Running low area coverage jobs on a machine in a low humidity environment (e.g., Denver, CO, in the winter) can exacerbate developability loss because of the impact of humidity on the toner tribo charging characteristics. Acting together, these disturbances, through the toner aging dynamics can lead to developability loss that ultimately results in saturation of the development bias voltage. Once this occurs, the printer is no longer able to maintain color consistency from print-to-print, and the machine would need to be taken offline for service. 1“Area coverage” refers to the portion of a page surface that is covered with toner. Low area coverage documents, such as a page of text, use relatively small amounts of toner; whereas, a high area coverage document, such as a photograph, uses relatively large amounts of toner.
LIU et al.: MODELING AND CONTROL OF A HYBRID TWO-COMPONENT DEVELOPMENT PROCESS FOR XEROGRAPHY
In this paper, we will derive a control oriented model of the toner aging dynamics from the first-principle based stochastic model of [11] and [12]. The control oriented model is able to characterize the toner aging dynamics as well as explain the lack of control authority using existing control actuation. Given the limited control authority, a constrained optimal control problem is formulated to maximize the operating time while maintaining an acceptable level of toner developability. The rest of this paper is organized as follows. In Section II, the hybrid two-component development process is introduced and the comprehensive first-principle-based model of [11], [12] is briefly recaptured. The model reduction approach used to derive the control oriented model is discussed in Section III. Analysis results based on the control oriented model are given in Section IV. In Section V, an operating time maximization problem is formulated and solved numerically. Finally, concluding remarks are given in Section VI. Some detailed proofs are included in the Appendix.
II. PROCESS DESCRIPTION AND A FIRST PRINCIPLE BASED FULL-ORDER MODEL The development process is a key step in determining xerographic productivity and image quality. Based on the composition of the materials used, development systems can be divided into single-component and two-component. In single component development, the developer material only contains toner particles and donor roll is used to transport the toner to the photoreceptor. In a two-component development, the developer material contains toner and carriers, and a magnetic roller is used to form toner-carrier brushes that transport toner to the photoreceptor. For a two-component process, toner concentration , which is defined as the ratio of the toner mass to that , i.e., , is frequently used as an inof the carrier mass dicator of the charging status [3]. In most processes, the carrier can be assumed to be constant. mass A hybrid two component development approach was introduced in [13], where a donor roller and a magnetic roller are both used. This approach has been used in various products for its high reliability and low operating cost [14]–[16], since it can significantly alleviate the physical contact between the magnetic brush and the photoreceptor. Fig. 3 illustrates a typical hybrid two component development process, where uncharged toner and carrier particles are dispensed from the dispenser to the sump area. In the sump area, the toner and carrier particles are mixed and transported through a set of augers. The mixing enables tribo-charge between the particles and induces charges on the toner and carrier particles. Additives are also dispensed to control the charging process and the particle flow. The carrier particles attached with toner particles are attracted to a magnetic roller (Mag Roll in Fig. 3). The toner particles are then detached from the carrier particles and developed onto the donor roll. As the donor roller rotates and aligns with the development electrostatic field, the field strength detaches the charged toner particles and developed them onto the photoreceptor surface. The developed mass per area (DMA) on the photoreceptor is a measure of the effectiveness of the development process. It
533
Fig. 5. Solid area development curve.
has been shown to be highly correlated with the color appearance on the print image. It is thus desired to keep the DMA at a constant level under different operating conditions to maintain color consistency. A. Linearized Solid Area Development Curve Model The effect of various development process parameters on the DMA can be modeled by a static, nonlinear solid area development curve [1]
where is the voltage for the onset of development, is the slope of the developability curve at the onset of development, is the maximum achievable toner mass, and is the applied development voltage (control input), see Fig. 5. To maintain color consistency, it is desired to regulate the DMA at a . In the neighborhood of , target value, denoted as ) of the nona local linear approximation (with respect to linear solid area development curve can be written as
(1) where
is the development bias voltage needed to achieve and is the local slope of the development curve at , as shown in Fig. 5. Experiment data suggest that the is highly correlated to toner mass, relative hulocal slope midity, and material properties of the toner particles, i.e., (2) where is the toner mass in the development area, is the relative humidity (assumed to be constant in this investigation), is the material property of toner particles on the donor and roller, as will be defined as in the next section. It has also been in (2) is monotonic increasing in each of its observed that variables while holding the other two constant, i.e.,
Experiment data also suggest that the required voltage to achieve a desired toner mass can be modeled as a monotonic decreasing function with respect to the local slope , i.e., (3)
534
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
By combining (2) and (3), it can be shown that the required will decrease whenever , , or increases, voltage as will be shown in Fig. 9. It is also worth noting that the control can be easily manipulated in practice, while the mainput and ) can only be changed in a relatively terial properties ( slower manner through material flow. B. Discretized Toner Aging Dynamics The development process is essentially a transport process of charged particles in an electrostatic field. The amount of particles being transported depends on the filed strength, particle characteristics, and the amount of particles in the electrostatic field. To model the transportation of toner particles and their charge characteristics within the development system, a probabilistic approach to characterize the toner particle properties has been proposed in [11] and [12], based on control volume analysis. Specifically, a toner development probability is assigned to every toner particle at time that have been in the development area for time (residence time). The time that toner particles reside in the development area is characterized that represents the fraction of by a toner age distribution toner particles with residence time at time . The amount of toners in the development area at any given time is captures by the toner mass . characterizes the amount The toner age distribution of time toner particles spend in the development process. Typiis not an uniform distribution cally, the age distribution with respect to the residence time . For a hybrid two component system, the toner age distribution is different in the sump area and on the donor roll, which are essentially the two control volumes for the analysis. Therefore, the toner age distributions and will represent the toner age distribution for in the sump area and on the donor roll, respectively. The deof a toner particle characterizes velopment probability the probability of a toner particle being transported from one location to another, e.g. from the sump area to the donor roll or from the donor roll to the photoreceptor. It depends on the charge density of the toner particle (charge per unit mass) and the adhesion force between toner particles, both of which can be characterized by the amount of additives that adhere to the toner surface. It is assumed in [11] and [12] that fresh toner particles from the dispenser can bring extra additives into the sump area. Part of the fresh additives is assumed to be available to all the toners, instantly and uniformly, which implies that the developis effected by the dispenser without any ment probability transport lag. Under these assumptions, the evolution of the de, the toner age velopment probability of toner particles and ) and the toner mass , distributions ( were derived in [11] and [12] as a set of coupled nonlinear difference equations. can be The evolution of development probability written as
Fig. 6. Structure of the hybrid two component development system. (a) Toner mass flow of the hybrid two component development system; (b) signal flow of the hybrid two component development system.
dispensed toners and denoting a small time period. In (4), is the toner dispensing rate that characterizes the amount of added toners and denotes the time constant for the natural decay of the development probability and is the toner mass at time instance . The first term on the right-hand side of (4) depicts the effect of newly dispensed fresh toner on toner developability. The second term depicts the natural decay of developability due to aging. The non-negative is due to the fact that dispenser can only constraint on add more materials into the system. The volume in the developer around the sump area and the Mag Roll is defined as a control volume [the block on the righthand side of Fig. 6(a)], where the mass conservation is maintained. The evolution of toner mass in this control volume can be derived based on mass conservation, i.e., (5) where is the throughput rate that represents the amount of toner being developed onto the photoreceptor and is the toner waste removal rate. The second term on the right-hand side of (5) indicates the net toner mass change in the sump area over time . Equations (4) and (5) imply that without and , the developadding fresh toners, i.e., will eventually decay to a small value ment probability that practically no toners can be developed onto the photoreceptor. The evolution of toner age distribution in the sump area can be derived based on the mass balance of the control volume
(6)
(4) with where with denoting the fraction of freshly dispensed additives that came with the
LIU et al.: MODELING AND CONTROL OF A HYBRID TWO-COMPONENT DEVELOPMENT PROCESS FOR XEROGRAPHY
Equation (6) keeps track of the mass evolution of toner particles of age at time instance . The age of these toner particles beat time instance . Therefore, the evolution comes to of toner age distribution follows the mass evolution in (6), where the denominator of the second term on the right-hand side characterizes the average toner developability over all age distribution in the sump area. It can also be observed that, with only affects respective to age distribution, dispensing rate . the age distribution with residence time 0, i.e., , the Similar to the toner age distribution in the sump evolution of toner age distribution on the donor roll can be modeled based on another control volume, namely the volume around the developer roller, i.e., the block on the lefthand side of Fig. 6(a). Due to the use of some metering blade similar to the one used on the magnetic roller in Fig. 3, the toner . The mass in this control volume is assumed to be a constant can therefore be written as evolution of
(7) Similar to (6), the denominator of the second term on the righthand side of (7) characterizes the out-flow’s average toner developability over all age distribution on the donor roll. And the denominator of the third term on the right-hand side of (7) characterizes the in-flow’s average toner developability over all age distribution from the sump. The development model described in [11] and [12] essentially keeps track of age distributions and development probability using (6) and (7), and calculates the averaged toner developaat each time step using the following bility in the sump expression:
535
area (DMA) on the photoreceptor is the output. The available (voltage control inputs are the development bias voltage between the magnetic roll and the photoreceptor) and the dispensing rate of toner particles from the dispenser . Note that the evolution of the process is captured by (4)–(7), where the dimension of the discretized age distribution and development probability will continue to increase as time increases. This results in an infinite dimensional system that is not suitable for system analysis and controller synthesis. In the next section, we will development a simplified model more suitable for system analysis and controller design. III. CONTROL ORIENTED MODEL In this section, a control oriented state space model for the development process will be derived from the comprehensive model described in the previous section [17]. Ordinary differential equations (ODEs) to describe the evolution of states will be established to replace (4)–(7) of the previous section. Choice of inputs and outputs of the development process in the control oriented model will remain the same as in the previous section: the control inputs to the process include the development bias and the dispensing rate , the exogenous voltage disturbance input to the process is the combined effect of the , throughput rate together with the waste rate, since it is largely determined by customer usage (e.g., area coverage of a print job, printing speed and paper size), and the process output DMA. The approach we adopt to reduce the number of states associ, , and developated with toner age distributions is to examine time evolution of a set of ment probability aggregated developabilities over all age distributions at specific spatial locations of the process. More specifically, we propose to keep track of the developabilities of the toners in the sump and at the donor roll, which we will refer to as the sump state and the donor state , respectively, as defined by (8) and (9). Without loss of generality, the developability states are normalized to between 0 and 1 with an initial value of 1 representing the state of fresh toner particles. The toner mass in the will still be retained as an internal state that charsump acterizes the toner mass change in the flow of material.
(8) A. Evolution of the State Variables With the age distribution evolution on the donor roll, the avercan be comaged toner developability on the donor roll puted by
1) Evolution of the Toner Mass: Following (5), the time derivative of the toner mass in the sump can be written as (10)
(9) Given (4)–(9), the development process model can be summarized as a cascade structure shown in Fig. 6, where Fig. 6(a) depicts the toner mass flow of the development system and Fig. 6(b) is the signal flow diagram. The block on the left-hand side of Fig. 6(b) is the static solid area development curve (2), and nonlinear in which is linear in development voltage the local slope . The block on the right-hand side of Fig. 6(b) characterizes the toner aging dynamics. The developed mass per
2) Evolution of the Sump State: Based on the definition of as in (8), the evolution of for a short sump state duration can be written as
(11)
536
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
Fig. 7. Block diagram of the control oriented model.
To further the mathematical simplification, the following approximation is used.A natural choice is , since is the “average value” of for over all toner age distribution development probability as indicated in (8). This approximation partially ignores the on , i.e., the toner age dependent dependence of selectivity of the development process (from the sump to the donor roll), as will be explored again in the model verification subsection. With this approximation and (4)–(6), (11) can be rewritten as
(12) where has the same definition as in (4). Note that in (12), the first term on the right hand side represents the effect of newly dispensed toner particles, which only has new additives with residence time zero. The second term represents the effect of new additives on old toners and the third term characterizes the decay of the effectiveness of “old additives” on old toners. The time derivative of can be derived as (13) The impact of dispensing on the sump state can be explained , i.e., no disusing (13), when the dispensing rate pensing, the sump state decays exponentially, when , the rate of the decay can be reduced. The amount of reduction is proportional to and inversely proportional to . More free additives, i.e., the amount of toner in the sump larger , results in larger reduction. Equation (13) also shows that the dispenser is more effective in reducing the decay rate for is small, i.e., the older toner particles, where the sump state term is larger. On the other hand, the more toner in the , the less effective a constant sump, i.e., larger toner mass dispensing rate will have in reducing the decay rate. 3) Evolution of the Donor State: Based on the definition of donor state as in (9), following similar approach and approximation used for developing the sump state. The time can be written as derivative of the donor state
where the value of needs to be determined based on a more complicated simulation models or experiment. Note that when deriving (14) we assume negligible exchange of toners between the donor roll and the sump, i.e., the toner flow happens in a “one-way” manner, although material exchange does exist even at zero throughput rate. Structure-wise, (14) is very similar to (13). Without dispensing supply, the donor state also exhibit an exponential decay response. Compare to its impact to the sump state dynamics, see (13), dispensing rate is less effective to with a gain factor of in (14) comthe donor state in (13). This observation agrees with the pared with “step-by-step” mass transfer nature of the process. It is also interesting to note that the second term on the right-hand side of (14) is a proportional control on the error between the donor state and the sump state. Note that the proportional gain is . This implies higher proportional to the throughput rate proportional control is associated with higher throughput rate that tends to reduces the difference between the donor state and the sump state. Similarly the effect of this self adjusting mechanism is reduced in low throughput operations. Without any integral action, for lower throughput operation, the donor state may be quite different from the sump state. This will be further discussed in later sections. Combining the evolution of all the states, the hybrid twocomponent development system can be modeled as a system with three internal states , one output (DMA) and two control inputs , as shown in Fig. 7. The state equations are
(15) where and . The corresponding output equation is given by combining (1) and (2) (16) where
(14)
(17)
LIU et al.: MODELING AND CONTROL OF A HYBRID TWO-COMPONENT DEVELOPMENT PROCESS FOR XEROGRAPHY
537
development probability, defined in (8) and (9), the close match for these two states suggests that the finite dimensional control oriented model is able to capture the main transient dynamics of the infinite dimensional discretized numerical model. Compared with the sump state, more discrepancy can be observed in the donor state. This is because the age based selectivity from the donor roll to the photoreceptor is more significant than that from the sump to the donor roll, i.e., the assumption after (11) does not hold as well for the donor state as for the sump state. Comparisons using different area coverage as well as different toner concentrations give similar results for area coverage less than 10%. IV. SYSTEM ANALYSIS
Fig. 8. Comparison between the control oriented model and the full order model.
B. Model Verification To verify that the control oriented model described in (15)–(17) is able to capture the characteristics of the hybrid two-component development system, simulation results from the control-oriented model are compared with simulation results from the comprehensive discretized numerical model. We and donor state compared the time history of sump state derived following (4)–(9) with the time history following (15)–(17). In the numerical computation, the dispensing is used to maintain a constant level of toner mass, i.e., and the waste rate is assumed to be zero. Fig. 8 shows the comparison and donor state between the two of the sump state models for the following operating condition. Both the sump state and the donor state start with initial values of 1, which corresponds to fresh toner when simulation starts. A constant 4% toner concentration and a nominal target DMA are maintained as in typical practice. The relative humidity is less than 20%. The throughput rate starts with a value corresponding to 200 min, is switched roughly 2% area coverage, and at to a value corresponding to roughly 6% area coverage. In Fig. 8, the solid blue line is the response from the control oriented model [(15)–(17)] and the dashed red line is the response from the discretized numerical model [(4)–(9)]. During the first 200 minutes of the simulation, the sump state and the donor state both decays as a result of the low throughput rate, and the donor state decays faster than the sump state as mentioned before. Result in this period can be regarded as the “free response” of the states under a constant input. At about 200 min, both the sump state and the donor state reach their steady state values and the throughput rate is tripled, since the toner concentration is maintained at constant, the corresponding dispensing rate is also tripled, causing the increasing in the sump and the donor state until reaching their steady-state values, and result in this period can be regarded as the “step response” due to the step change of input. Since the overall property of the toner particles in the sump and on the donor roll are captured by the averaged
In the previous section, we have derived a control oriented model for the development process. In this section, the control oriented model will be analyzed to explain certain experimental observations. More specifically, we will focus on the developability loss phenomenon observed in low throughput printing, where a monotonic increasing development voltage is required to achieve the same DMA level. For a given initial condition, the acceptable operating region for the development process is bounded by extremum toner mass levels as well as maximum development voltage. We will demonstrate that the state trajectory of (15)–(17) will eventually leave the acceptable operating region in finite time for all continuous state feedback control involving the dispensing rate . Noting that although , only has the system has two control inputs control authority on the toner aging dynamics, while only effects the output DMA. Before diving into detailed analysis, several concepts need to be defined. For a system defined in (15)–(17), an acceptable operating region is defined as a compact subset of the state space such that the desired output DMA can be maintained and state within their maxwhile keeping the input imum and minimum bounds, i.e.,
(18) where and denote the lower and upper bound for the , respectively. denotes the upper bound for the alstate lowable control input. Using (17), the last condition in (18) can be written into bounds on state , i.e., (19) is the lower bound of for the state trajectory to where stay within the acceptable operating region. Therefore, an acceptable operating region can be redefined as (see Fig. 9)
(20) With defined by (20), given the system dynamics and a fixed dispensing strategy , if we define an initial state
538
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
To show that the state trajectory of (15)–(17) will eventually leave an acceptable operating region, it is equivalent to show . For future reference, we also define as that
(23)
Fig. 9. Projected acceptable operating region in the
M 0
phase plane.
, the corresponding operating time can be defined as (21), shown at the bottom of the page, where is defined in (20) and the system dynamics is defined in (15). In other words, is the time the state trajectory leaves , the acceptable operating and a fixed initial region for the first time. Given can be defined condition , the maximal operating time is the set of as (22), shown at the bottom of the page, where is assumed all allowable dispensing inputs. In this research, to be the set of all non-negative state feedback based control law, which is also continuous, i.e.,
and
is continuous
where is the maximal value of due to practical limitations such as motor speed, etc. The non-negative requirement above is from the fact that dispenser can only add more materials into the system, while the continuous requirement is due to the practical limitation of actuators, and this also guarantees that the resulting closed-loop system represented by continuous ordinary differential equations.
and It is worth noting that the difference between is that the bound is on in (22) and on in (23). of the state space is said to In system analysis, a subset be positively invariant with respect to the system dynamics, if , , for all [18]. A subset of the state space is said to be controlled invariant with respect to the system dynamics, if there exists a continuous feedback control which assures the existence and uniqueness of the solution of the closed-loop system and it is such that is positively invariant [19]–[21]. For the development process modeled in (15)–(17), we have the following invariant property. is conClaim 1: The set . trolled invariant for any positive dispensing Proof: See the Appendix. Remark 1: In practice, since all printers start with the initial , this claim guarantees condition for any . Referring to the definitions of and in (22) and (23), we have the following corollary. Corollary 1: Given and initial condition , for all . Corollary 1 indicates that to show , it suffices to , i.e., if the sump state is less than the alshow lowable lower bound, the donor state must be less than the allowable lower bound. It is worth noting that in the system dyand the sump state namics of (15), the toner mass state are not explicitly affected by the donor state (also shown in Fig. 10). This observation together with Claim 1 allows us to show by showing , while the later argument can be accomplished by working with toner mass
(21)
or
(22)
LIU et al.: MODELING AND CONTROL OF A HYBRID TWO-COMPONENT DEVELOPMENT PROCESS FOR XEROGRAPHY
539
Fig. 10. Structure of the toner aging dynamics.
state and sump state only. From now on, we only focus on the first two equations in (15), although the non-negative control input can still depends on all three states. To phase facilitate our proof, two sets can be defined in the plane
(24)
(25) and a non-intersecting relationship between these two sets can be summarized by the following claim. Claim 2: Assuming zero waste rate and a constant , with two subsets defined in (24) and throughput rate [defined in (19)] holds for all (25), if , then . is a superset of equilibrium Remark 2: By definition, toner mass state and sump state [see (10) and (13)]. Although for different control input strategies, equilibrium states may change, they are included in , which is valid for all , including all . can be apprecidispensing rate ated as follows: The statement that the projection of (3-D) state trajectory onto phase plane stays in forever is equiv. Therefore, Claim 2 is a alent to the statement that sufficient condition for the nonexistence of equilibrium of the reduced dynamics in . Proof: See the Appendix. Remark 3: It is also interesting to note that for all throughput , the set defined in (24) is a branch of a hyperbola represented by , where . in the phase plane is therefore The location of determined by the developer material property ( and ) and . As the throughput decreases, the set throughput rate shifts down to approach the -axis as shown in Fig. 11. Therefore, for lower throughput rate, is more likely to be satisfied as shown in Fig. 11. With the invariant property of Claim 1 and the equilibrium property of Claim 2, the “finite time escaping” property of the dynamics of (15)–(17) can be summarized by the following theorem. and the Theorem: Assuming zero waste rate closed-loop system is locally Lipschitz continuous, for any constant throughput rate , with and defined in
Fig. 11.
X (C ) and 1(M ; M ; V
).
(24) and (25), if for all , then , holds. for any Proof: See the Appendix. , by checking the property of the Remark 4: For any and dynamics on a superset of closed-loop equilibrium, this theorem gives a sufficient condition for the “finite time escaping” of the three-state dynamics modeled in (15)–(17). V. OPERATING TIME MAXIMIZATION For low throughput printing, the system analyses in the previous section showed that developability loss is unavoidable using dispensing and development voltage as control actuators. These conclusions match empirical observations [12]. Given the limited available actuation options and the fact that loss of developability is unavoidable, two types of solutions may be adopted: 1) adding additional control actuators or 2) maximizing the operating time with the available control input. In this section, we will focus on how to find an appropriate dispensing strategy that can maximize the operating time. A. Problem Formulation Assume that the maximum allowable development voltage and acceptable toner concentration levels are given as shown in Fig. 9. Furthermore, assume that the development voltage is adjusted appropriately to maintain DMA at a desired target value with a small constant throughput. The task of maximizing the operating time can be stated as: for given initial states, , and , find a dispensing strategy that maximizes operating time until the system fails to maintain constant while keeping the development voltage and toner mass below their prescribed maximum and minimum bounds [22]. This problem can be formulated mathematically as follows. , , and as in (18), Given the same definition of let the DMA target be given by , and assume constant , and , find a throughput rate . For given for that dispensing strategy maximizes a cost function (26) subject to the state dynamics in (15) (27)
540
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
and
TABLE I SIMULATION PARAMETERS
(28) (29) (30) (31) (32) (33) An analytical solution to the above problem is very involved [23], the objective here is not to solve this optimal control problem analytically, instead, we will focus on the guidelines for designing appropriate dispensing strategies provided by this optimal control problem. Therefore, a numerical approximation [24] will be used to solve the problem formulated above. To make the numerical optimization problem more tractable, the above optimal control problem can be translated to an equivalent problem as follows. Claim 3: The optimal control problem formulated in (26)–(33) is equivalent to (34) subject to (29)–(33) and (35) (36) (37) is an arbitrarily chosen positive number. where Proof: See the Appendix. Remark 5: Noticing that after replacing (28) with (35) and (36) and replacing (27) with (37), Claim 3 essentially converts the original free final time problem with constraint on state variables into a fixed final time problem with fixed final state of and . This conversion can substantially reduce the computation load by replacing searching all possible final state along the and . constraint with searching over fixed final state of B. Numerical Discretization To make the optimization process more manageable, we can use the redefined control input . With piecewise constant discretization, the augmented constrained optimal control problem formulated in (34)–(37) and (29)–(33) can be converted to the following static optimization problem. The objective is to minimize the inverse of the cost function (34), (38) subject to dynamics (39) with given initial condition and final condition in (35) and (36) together with constraints given in (29)–(31) and (40)
(41) (42) where is the time step-length in discretization and is the number of time steps. C. Simulation Results and Discussion In this subsection, three cases of different dispensing strategies will be simulated using the control oriented model. For Case 1, the dispensing strategy is to maintain TC at a specific target level. This strategy is used in most printing products. Cases 2 and 3 relax the TC regulation requirement and maintain TC within a given range (i.e., the toner mass is bounded by the ). For Case 2, an intuitive dispensing strategy interval is used to extend the operating time: use the maximal dispensing at the onset of developability loss. This is because for rate a fixed target DMA, dispensing counteracts the effects of developability loss by reducing the required development voltage. However, increasing dispensing results in increased TC (see Fig. 17 in the Appendix). Therefore, the strategy that we consider for Case 2 is a concatenation of the maximal dispensing and a modulated dispensing to maintain the at . In Case 3, the optimal dispensing strategy to maximize the operating time as formulated in (26)–(33) is used. System parameters used in the simulation studies are summarized in Table I. The throughput rate corresponds to a constant 2% area coverage, which represents a common area coverage for variable data printing applications that contain mostly text. A relative humidity of less than 20% is representative of a customer environment in an arid location, e.g., Arizona and/or a typical working environment during the winter months. The throughput rate and relative humidity conditions used in the simulation study represent a known “stress condition” for xerographic print engines. The initial condition of the donor state corresponds to a development voltage at 75% of its maximum . The initial condition of the sump state is assumed to level, be equivalent to that of the donor state, and the initial condition of toner mass is assumed to be at its nominal value. For the three cases, the phase plots of donor state and toner ) are given in Fig. 12. The developmass (proportional to ment voltage time histories are plotted in Fig. 13. Normalized dispensing profiles for Cases 1 and 2 are plotted in Fig. 14, and normalized dispensing profiles for Cases 1 and 3 are plotted in Fig. 15, where the dispensing for Case 1 has been normalized to 1. In Case 1, if zero waste rate is assumed, the dispensing rate is equal to the throughput rate (the solid blue lines in Figs. 14 and 15). The corresponding toner mass is maintained at its nom-
LIU et al.: MODELING AND CONTROL OF A HYBRID TWO-COMPONENT DEVELOPMENT PROCESS FOR XEROGRAPHY
541
Fig. 14. Normalized optimal dispensing strategy of Cases 1 and 2.
Fig. 12. State trajectories corresponding to different dispensing strategies. Fig. 15. Normalized optimal dispensing strategy of Cases 1 and 3.
by solving a constrained optimal control problem using the control oriented development model. The resulting solution reduces the developability in the beginning by starving the toner supply and then dispensing to maintain acceptable DMA with maximum development voltage. This strategy further increases the operating time, achieving 170% improvement from Case 1. VI. CONCLUSION
Fig. 13. Time histories of development voltages corresponding to different dispensing strategies.
inal value (i.e., TC is constant) as shown in Fig. 12 , which results an operating time of 1071 s as shown in Fig. 13. In Case 2, the donor state is maintained almost at a constant level before the toner mass state reaches its upper bound (path in Fig. 12), which results an operating time of 2690 s as shown in Fig. 13. This is a 150% increase compared with Case 1. In Case 3, the dispensing strategy that maximizes operating time is a concatenation of zero dispensing with a controlled dispensing rate associated with operating the development voltage at its maximum value, i.e., path in Fig. 12. The development voltage history in Fig. 13 shows an operating time of 2902 s, a 170% increasing from Case 1. From Figs. 14 and 15, it can be observed that there is an order of magnitude difference between the dispensing profiles of Cases 1 and 3, before the state trajectories reach the corresponding state boundaries. The use of extreme control values agrees with some necessary conditions of the time optimal control problem formulated above and is further explored in [26] and [27]. From the three simulation scenarios, we can conclude that the operating time can be extended with an enlarged operating region by relaxing the fixed TC level (Case 1) to an allowable TC range (Case 2). A 150% increase of the operating time can be achieved using an intuitive dispensing strategy. However, to take full advantage of the extended TC range, an optimal dispensing strategy to maximize the operating time can be obtained
In this work, a control oriented model was developed to study the behavior of a hybrid two-component xerographic development process. Numerical simulation shows that, for the properties of interest, the toner aging dynamic response from the control oriented model matches that from an experimentally verified complex comprehensive model. For the process model and operating conditions under consideration, acceptable operating region analyses have indicated that the loss of developability is a direct result of low throughput rate and low relative humidity operating condition and therefore the saturation of development voltage is unavoidable for all continuous state feedback involving dispensing strategies. In this sense, this paper presents the first analytical results that successfully predicted the loss of developability in a xerographic development process. Given the inherent system limitations, an operating time maximization problem has been formulated and numerically solved. Three different cases of simulation results show that the operating time level to vary can be substantially increased by allowing the in an allowable range, and the non-intuitive optimal dispensing control strategy further illustrates the importance and utility of the control-oriented model. APPENDIX A. Proof of Claim 1 Claim 1 can be shown using the vector field of (15) at the boundary as shown in Fig. 16.
542
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
The system dynamics in (27) can be rewritten as (A2) This is the form of (37). The final time can be translated to the -framework as
in the -framework
(A3) and all other constraints in (29)–(33) still hold in the -framework, this completes the proof of Claim 3.
Fig. 16. Positively invariant set f0 g.
D. Proof of the Theorem At the boundary of the set, we have at at at at at at These results indicate that at the boundary of the set, the state trajectory points inside of the set, and this is true for all possible . B. Proof of Claim 2 The equilibrium of the derivatives in (15), i.e.,
and the
dynamics has zero time
(corollary1), for the theorem to hold, Since using the and dynamics. it suffices to show With the assumption that the closed-loop dynamics is Lipschitz continuous, the existence and uniqueness of the solution is guaranteed. The proof of will use the concept of nonwandering point and nonwandering set for the closed-loop three-state autonomous system. A point is called nonwandering with respect to the system dynamics, if for any neighborhood of , and any , there exists a , such that , where denotes the state trajectory at time with initial condition , i.e., , and denotes all state trajectories starting in . The set of all such nonwandering points forms the nonwandering set. It can also be shown that if the state trajectory is bounded, the non-wandering set is a compact, nonempty and invariant set [25]. in the phase plane, the For each point direction of the (closed-loop) state trajectory is determined by . A local (may translate along the trajecthe dispensing tory) four-quadrant coordinate and a (state and dispensing decan be defined pendent) tangent angle of trajectory as in Fig. 17
in quad. I and IV
Canceling the term in above equations, the equilibrium can be shown to satisfy the equality constraint in (24). Thereholds on , then the set is fore, if empty.
in quad. II
(A4)
in quad. III
C. Proof of Claim 3 Without loss of generality, we assume the time-optimal state trajectory ends with a different point on one of the state boundaries [defined in (28)] as shown in Fig. 9, we can always further drive the state trajectory to [defined in (35) and (36)]. The existence of such a dispensing profile is due to the fact that the trajectory can always go in the “right-down” direction, which can be shown using similar approach adopted in (A7) and (A8) of next subsection. To convert the free final time optimal control problem of (26)–(35) to a fixed final time problem, a scaled time framework can be introduced using an extra variable (A1)
. The slope of the state trajectory can be defined as
(A5) Note that the so defined slope in (A5) is a monotonic increasing function of the dispensing rate , since (A6) for all some small
, (when changes from , changes from
to to
for for
LIU et al.: MODELING AND CONTROL OF A HYBRID TWO-COMPONENT DEVELOPMENT PROCESS FOR XEROGRAPHY
Fig. 17. Local quadrants and tangent angle in the projected operating region.
Fig. 18. Projection of an “almost” closed trajectory onto the plane.
M 0
phase
some small ). Therefore, for each point in the projected operating region, the maximal (over all ) can be derived as
543
, where , i.e., an open ball centered at with a much smaller radius and we chose , according exists, such to the definition of nonwandering point, a that . In particular, there exists a such , i.e., the state trajectory starting that will return to the much smaller ball after from . If we denote , we can show leaving that , i.e., the projection of state trajectory , onto phase plane is almost a closed curve . Since can be made as small as we wish, the state trajectory can be made arbitrarily close to a closed phase plane, therefore the velocity concurve in the straints at can be made arbitrarily close to those at . However, when traveling along an “almost closed” curve, the tangent to continuously, thus C2 is violated. angle changes from The proof is complete. Remarks: Note that in this proof, it is assumed that the projection of the state trajectory does not intersect with itself on the phase plane, if intersection happens (possible due to the input’s dependence on ), the above argument can be applied to the point of intersection. Another remark here is although condition C1 is not explicitly used in the proof, it is necessary in the sense that it rules out the case where the trajectory returns along the leaving path. The last remark about this proof is that since the input also depends on , theorems for two-state dynamics (Poincare–Bendixson theorem, etc.) cannot be applied directly. REFERENCES
(A7) and the minimal
(over all
Combining (23) and (24) with have
) can be derived as
and
(A8) derived above, we
(A9) Equations (A7)–(A9) indicates that for the state trajectory of the closed-loop dynamics to remain in , the following two condiphase tions need to be satisfied simultaneously in the plane: ; (C1) . (C2) With these preliminary results, the theorem can be proved , an equivalent with contradiction. Assume that all . Since is compact, statement is that is bounded, a nonwandering point exists, note it is possible that . Since is not an equiliband a , such rium (by Claim 2), there exists a as shown in Fig. 18. However, since that is an nonwandering point, if we chose its neighborhood as
[1] L. B. Shein, Electrophotography and Development Physics. New York: Springer-Verlag, 1988. [2] D. M. Pai and B. Springett, “Physics of electrophotography,” Rev. Modern Phys., vol. 65, pp. 163–211, Jan. 1993. [3] C. B. Duke, J. Noolandi, and T. Thieret, “The surface science of xerography,” Surface Sci., vol. 500, pp. 1005–1023, Mar. 2002. [4] E. Hamby and E. Gross, “A control-oriented survey of xerographic systems: Basic concepts to new frontiers,” in Proc. Amer. Control Conf., 2004, pp. 2615–2629. [5] C.-L. Chen and G. T.-C. Chiu, “Banding reduction in electrophotographic process using human contrast sensitivity function shaped photoreceptor velocity control,” J. Imag. Sci. Technol., vol. 47, pp. 209–223, 2003. [6] C.-L. Chen and G. T.-C. Chiu, “Incorporating human visual model and spatial sampling in banding artifact reduction,” in Proc. Amer. Control Conf., 2004, pp. 2622–2627. [7] C.-L. Chen and G. T.-C. Chiu, “Banding artifact reduction for a class of color electrophotographic printers with under-actuated motor/gear configuration,” IEEE Trans. Control Syst. Technol., vol. 16, no. 4, pp. 577–588, Jul. 2008. [8] P. Li and S. Dianat, “Robust stabilization of tone reproduction curves for the xerographic printing process,” IEEE Trans. Control Syst. Technol., vol. 9, no. 2, pp. 407–415, Mar. 2001. [9] R. J. Nash, J. T. Bickmore, W. H. Hollenbaugh, and C. L. Wohaska, “Xerographic response of an aging conductive developer,” J. Imag. Sci. Technol., vol. 40, pp. 347–353, 1996. [10] M. Anzai and N. Hoshi, “Prediction of deposit toner mass by improved toner flow model for dual component magnetic brush development,” in Proc. IS&T NIP 14 Int. Conf. Digit. Print. Technol., 1998, pp. 462–465. [11] P. Ramesh, “Modeling and control of toner material state in two component development systems,” J. Imag. Sci. Technol., vol. 53, no. 4, p. 041206, 2009. [12] P. Ramesh, “Quantification of toner aging in two component development systems,” in Proc. IS&T NIP 21 Int. Conf. Digital Print. Technol., 2005, pp. 544–547. [13] A. J. Liebman, “Toner loading for touchdown donor,” U.S. Patent 3 929 098, Dec. 30, 1975. [14] R. Lux and H. Yuh, “Is image-on-image color printing a privileged printing architecture for production digital printing applications?,” in Proc. IS&T NIP 20 Int. Conf. Digit. Print. Technol., 2004, pp. 323–327.
544
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 3, MAY 2011
[15] R. Clark and D. Craig, “Xerox nuvera technology for image quality,” in Proc. IS&T NIP 21 Int. Conf. Digit. Print. Technol., 2005, pp. 671–674. [16] M. Hirsch, “Some fundamental performance aspects of the Xerox iGen3 development system,” in Proc. IS&T NIP 22 Int. Conf. Digit. Print. Technol., 2006, pp. 398–401. [17] F. Liu, G. T.-C. Chiu, E. S. Hamby, Y. Eun, and P. Ramesh, “Control oriented modeling of a hybrid two-component development process for xerography,” in Proc. IS&T Int. Congr. Image Sci., 2006, pp. 87–90. [18] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 2002. [19] F. Blanchini, “Set invariance in control,” Automatica, vol. 35, pp. 1747–1767, 1999. [20] H. Nijmeijer and A. V. Schaft, “Controlled invariance for nonlinear systems,” IEEE Trans. Autom. Control, vol. AC-27, no. 4, pp. 904–914, Aug. 1982. [21] H. Nijmeijer and A. V. Schaft, “Controlled invariance for nonlinear systems: Two worked examples,” IEEE Trans. Autom. Control, vol. AC-29, no. 4, pp. 361–364, Apr. 1984. [22] F. Liu, G. T.-C. Chiu, E. S. Hamby, and Y. Eun, “Control analysis of a hybrid two-component development process,” in Proc. IS&T NIP 22 Int. Conf. Digit. Print. Technol., 2006, pp. 564–567. [23] A. E. Bryson and Y.-C. Ho, Applied Optimal Control. Waltham, MA: Blaisdell Publishing Company, 1969. [24] J. Liang, Y. Chen, M. Meng, and R. Fullmer, “Solving tough optimal control problems by network enabled optimization server (NEOS),” presented at the IEEE Intel. Autom. Conf., Hong Kong, Dec. 15–17, 2003. [Online]. Available: http://mechatronics.ece.usu.edu/yqchen/ paper/03C15_IEEE_CIAC2003_IAC-M24.pdf [25] K. E. Petersen, Ergodic Theory. Cambridge, U.K.: Cambridge University Press, 1983. [26] F. Liu, G. T.-C. Chiu, E. S. Hamby, and Y. Eun, “Optimal dispensing strategy for a two component xerographic development process,” in ASME Dyn. Syst. Control Conf., Ann Arbor, MI, 2008, pp. 1–8. [27] E. Gross and P. Ramesh, “Xerographic printing system performance optimization by toner throughput control,” J. Imag. Sci. Technol., vol. 53, no. 4, p. 041207, 2009.
Feng Liu received the B.S. degree from Tsinghua University, Beijing, China, in 1998, the M.S. degree from Iowa State University, Ames, Iowa, in 2002, and the Ph.D. degree in mechanical engineering from Purdue University, West Lafayette, IN, in 2009. Since 2009, he has been with the Cummins Inc., Columbus, IN, where he is currently a Technical Specialist with the Cummins Technical Center and has been engaged in projects focusing on fuel economy optimization, diesel engine controls and diagnostics. Dr. Liu is a member of ASME.
George T.-C. Chiu received the B.S. degree in mechanical engineering from the National Taiwan University, Taipei, Taiwan, in 1985 and the M.S. and Ph.D. degrees in mechanical engineering from the University of California at Berkeley, in 1990 and 1994, respectively. Since 1996, he has been with the School of Mechanical Engineering, Purdue University, West Lafayette, IN, where he is currently a Professor. Before joining Purdue, he was with the Hewlett-Packard Company. His current research interests include modeling and control of digital imaging and printing systems, motion and vibration control and perception, human motor control and digital fabrication. He has authored and coauthored more than 100 journal and refereed conference papers and 3 patents. Prof. Chiu is a member of ASME, IEEE, and the Society for Image Science and Technology (IS&T). He is an Associate Editor for the Journal of Electronic Imaging and the IFAC Journal of Control Engineering Practice.
Eric S. Hamby (M’97) received the B.S. degree in aerospace engineering from the University of Kansas, Lawrence, in 1990, the S.M. degree in aeronautics and astronautics from Massachusetts Institute of Technology, Cambridge, in 1992, and the Ph.D. degree in aerospace engineering from the University of Michigan, Ann Arbor, in 1998. Since 1998, he has been with the Xerox Corporation, where he is currently a Principal Scientist in the Xerox Innovation Group, Webster, NY. He is a certified Black Belt in Design for Lean Six Sigma and has led a variety of projects ranging from applied control in xerographic marking to data mining to machine prognostics and health management. He holds 27 U.S. patents and has published more than 20 journal and refereed conference papers. His research interests include machine prognostics and health management, building data driven models and applying probabilistic methods in system analysis and design.
Yongsoon Eun received the B.A. degree in mathematics, the B.S. degree and the M.S.E. degree in control and instrumentation engineering, all from Seoul National University, Seoul, Korea, in 1992, 1994, and 1997, respectively, and the Ph.D. degree in electrical engineering and computer science from the University of Michigan, Ann Arbor, in 2003. Since 2003, he has been with Xerox Innovation Group in Webster, NY, where he is currently a Senior Research Scientist. He has worked on a number of subsystem technologies in the xerographic marking process, and recent activities focus on the image registration of inkjet marking technology. To date, he has published more than 30 journal and refereed conference papers and holds 4 U.S. patents. His current research interest is in describing, both in quantitative and qualitative terms, the impact of multitasking individuals on organizational productivity.
