IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 9, NO. 2, MARCH 2001 407 Robust Stabilization of Tone Reproduction Curves for the Xerographic Printing Process Perry Y. Li and Sohail A. Dianat Abstract—The problem of stabilizing a single color tone repro- duction curve of a xerographic printing engine is considered. This problem is critical to ensure high-fidelity color printing. The con- trol system uses a small number of actuators and a small number of measurements to stabilize the potentially high-dimensional tone re- production curve. The goal is to minimize the overall least squares deviation of the tone reproduction curve from the ideal nominal one in the face of disturbances like material changes, temperature, humidity, and uncertainty in the system description. The control design consists of the steps of 1) determining a robust optimally performing static controller and 2) realizing the controller that uti- lizes only past measurements. Numerical simulations and experi- ments validate the efficacy of the controller. Index Terms—Curve fitting, half-toning, imaging, optimal con- trol, printing, robust control, tone reproduction curves, xerography. I. INTRODUCTION F OR A COLOR printer/copier to attain good color rendering quality, the image output terminal (IOT) must be capable of producing the desired tone, i.e., the solidness, of each of the four primary color separations (cyan, magenta, yellow, black) as requested. To render a primary color with a desired tone, a dig- ital printer or copier first translates the desired continuous tone image into one of many binary bitmap patterns (halftone im- ages), each labeled by its halftone density, using a halftoning al- gorithm. Given the halftone image, the IOT of the printer/copier then physically lays down the appropriate amount of toner on the output medium (paper or the photoreceptor) (Fig. 1). The de- sired result is that the toner image on the output medium should approximate the desired continuous tone image. A tone repro- duction curve (TRC) of the IOT is a characterization of this latter physical process and determines the amount of toner that would be deposited on the output media (i.e., paper or photoreceptor) when a halftone image of a certain half-tone density is given. Thus, the TRC is a mapping , so that represents the developed toner area coverage on the photore- ceptor, when a halftone image of density is presented. In a xerographic printing process [3], the TRC is subject to both con- trolled and uncontrolled operating conditions. Variation in un- controlled operating conditions, such as temperature, humidity, toner age, and charge density, etc., can cause the TRC to vary so that the IOT can produce unpredictable output images at various Manuscript received September 27, 1999; revised June 5, 2000 and October 13, 2000. Recommended by Associate Editor N. Sundararajan. This work was performed in part at the Wilson Center for Research and Technology, Xerox Corporation, Webster, NY. P. Y. Li is with the Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: pli@me.umn.edu). S. A. Dianat is with the Electrical Engineering Department, Rochester Insti- tute of Technology, Rochester, NY 14623 USA (e-mail: sadeee@rit.edu). Publisher Item Identifier S 1063-6536(01)01806-1. Fig. 1. Image path in a digital printing system. times with the same input halftone image. Thus, maintaining the TRC constant, or the stabilization of the TRC, is necessary to avoid having to retune the half-toning algorithm, and to allow the same halftone image to be reused over time. The objective of this paper is to develop a TRC stabilizing controller for the xe- rographic process so that the TRC remains close to the nominal curve despite variations in uncontrolled operating conditions. The TRC stabilization problem is interesting in that while the TRC is a potentially infinite dimensional object (it is a function of ), only a small number of actuators are available for con- trol. Also, only samples of the TRC at a small number of tones, not the entire TRC, are available for feedback. Consequently, the control must take caution that in the process of maintaining the TRC at one tone does not degrade its performance at another tone. In addition, the xerography process is nonlinear and un- certain, and the manufactured units on the production line vary from unit to unit. These challenges are further complicated by the fact that both the effects and the sources of the disturbances cannot be characterized easily. Because the xerographic process is essentially a static process, we are able, in this paper, to represent the nonlinear, uncertain process using a set of uncertain static linear models. Moreover, since xerographic disturbances do not vary very quickly, the controller performance is evaluated in the steady state with respect to the tolerance on the model uncertainties and constant disturbances. Thus, the results in [4] can be applied to compute the optimal static controller. However, the static controller cannot be directly implemented because the system outputs are not available for feedback until the next sampling period. Therefore, we propose a procedure to derive a dynamic realization of the static controller that ensures stability, and achieves the same steady state performance as the optimal static controller. The rest of this paper is organized as follows. In Section II, the xerographic control process is briefly described. The TRC stabilization problem is formulated in Section III. In Section IV, the controller design methodology is presented. Section V con- tains simulation results. Experimental results are presented in Section VI. Section VII contains concluding remarks. II. THE XEROGRAPHIC PROCESS CONTROL SYSTEM We briefly describe the xerographic control system in this section. For details on the xerographic process, readers are di- 1063–6536/01$10.00 © 2001 IEEE