MPC Study for a Dual

Control Problem Daniel R. Sa er II Francis J. Doyle III 1 Department of Chemical Engineering The University of Delaware Newark DE 19716 Apostolos Rigopoulos Philip Wisnewski Corporate Research and Development Weyerhaeuser WTC 1B20 PO Box 2999 Tacoma WA 98477-2999 Abstract This paper utilizes recent advances in paper machine technology, speci cally full array sensors and multiple headboxes, to implement model predictive control of basis weight in the crossdirection. An identi cation technique is applied to a dualheadbox paper machine benchmark problem. The identi ed model is used within a linear programming based model predictive controller. The study compares wet and dry end full array sensors on the benchmark problem. Performance of various linear programming formulations are compared for a nominal case and one in which shrinkage occurs in the drying process. Introduction A recent article by Pikulik et al. [11] pointed out that paper machine operations are improving at an ever-growing rate: \sophisticated automatic control will result in further improvements in product quality and machine e ciency." Mention is also made to increases in machine speed, width, actuator number and sensor number. Older machines are also being retro tted with new sensors that have the ability to measure the entire width of the sheet at each sampling interval. Several researchers have begun considering Model Predictive Control (MPC) as a feasible alternative to current control schemes for Cross-Directional (CD) control of paper machines [1, 3, 5, 12, 14, 15]. The key challenges of using MPC for CD control are time delays between the headbox and the measurement area, production rate and grade transition changes, plantmodel mismatch due to paper shrinkage or shifting, and the need to compute control moves for a large number of actuators in a short period of time. Advantages to using MPC include the ability to handle constraints on the rate, magnitude, and bending moment of the actuators and the ability to compensate for the interactions between the di erent actuator positions in the CD as well as interactions of multiple headboxes in the Machine Direction (MD). In this paper, MPC is explored as a control methodology that can utilize new types of sensor technologies to control paper machines with multiple headboxes while at the same time producing a control scheme that is both e cient and yields high performance. An identi cation study of a dual headbox paper machine benchmark problem will be described that considers shrinkage of the sheet during drying to produce an improved 1To whom correspondence should be addressed (e-mail: [email protected]) model. A variety of Linear Programming (LP) formulations of MPC are then discussed and compared in size and complexity. Finally, these formulations are tested and compared on the benchmark problem with full array sensors on either the wet end or the dry end of the paper machine. The Paper Machine Benchmark The benchmark problem studied in this paper is a proprietary Matlab and Simulink model of dual-headbox paper machine. Measurement of cross direction variations with full array wet end sensors at the end of each of the fourdrinier tables or a full array dry end sensor located at the end of the machine can be selected (Figure 1). Each of these sensor banks was modeled to contain roughly 400 independent sensor locations distributed evenly across the machine. The objective in each case is to control the CD variation of basis weight in the presence of sustained disturbances. Like many other sheet and lm processes, the simulation model assumes rst-order-plus-time-delay dynamics between the actuator and sensor with a Toeplitz symmetric input-output interaction matrix. Prior to dry-end sensing, the outputs of the model can be nonlinearly remapped, based on the percentage of shrinkage that may occur. Important time delays are included in Figure (1). The model assumes that no signi cant shrinkage has occurred prior to the wet end measurements. In trials where wet end sensor measurements were used, both sensors collected data on the process, thus giving twice as many measurements as in the dry end trials. The number of actuators and nominal weight of material added at each headbox were unequal. The rst headbox, also referred to as the primary headbox, was modeled to deliver most of the material and have roughly 50 actuators on its slice lip, while the other headbox, the secondary headbox, delivered the remaining material and had fewer actuators, roughly 40, across the sheet.