Performance metrics for web-forming processes
Highlights
► This study proposes performance metrics for two-dimensional web-forming processes. ► Simulated disturbance data illustrate the properties of the performance metrics. ► Industrial implementation to a paper mill is described.
Introduction
Modern research in the field of CPA began when Harris presented the minimum variance benchmark derived from the theoretical background of the minimum variance controller [1]. Ever since, a number of methods have been developed for measuring the performance of control loops. For a review of different metrics, see [2], [3], [4]. Proprietary software for evaluating controller performance is on the market [4], [5], [6], and several industrial applications have been reported [4], [7], [8], [9], [10], [11], [12].
Most performance metrics concentrate on one-dimensional measurements. This paper proposes several performance metrics to be used in evaluating the performance of two-dimensional web-forming processes. Such processes are found, for instance, in paper, plastic film and steel industries.
In a two-dimensional web process the product is a planar object being relatively thin when compared to the width and the length of the object. The direction of product movement is called the MD, whereas the direction perpendicular to the MD is the CD. Material flow is provided to the headbox, which is the device responsible of distributing the material in the cross direction. Because of the configuration of the web process, the CD and MD are also called the spatial and temporal dimensions, respectively.
As a result of the different physical configuration of the web-forming process in CD and MD directions, the dynamical properties of the product in CD and MD directions tend to be unalike: the variations in the MD are due to variations, e.g., in the input to the headbox, whereas the variations in the CD are caused for instance by the headbox and the devices following the headbox. An example in Fig. 1 illustrates the differences between CD and MD dynamics by showing a set of moisture data from a paper machine. Although corrupted by measurement noise, the overall profile in the CD remains the same. In the MD, however, there are fast variations in the measurements.
Fig. 2 illustrates a measurement arrangement that uses a scanner traversing the web with constant speed. Since the web travels in the MD, the relative path of the scanner with respect to the web is of zig-zag shape. There exist also measurement arrangements that use a variable-speed scanner to improve accuracy, and full-web width scanners that measure the entire CD simultaneously.
Because of the zig-zag movement of the scanner, the raw profiles are a combination of three components [13]:
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The machine direction (MD) component. Variations occurring along the sheet travel direction of the machine and affecting the whole width of the machine. MD variations are expected to affect the whole width of the sheet similarly.
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The cross-direction (CD) component. Variations occurring perpendicularly to the sheet travel direction of the machine. Generally speaking, CD variations are slower than MD variations, and can be assumed to be nearly time-invariant. CD variations will persist in successive profiles.
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The random component. Random variations that occur neither along nor across the machine. These variations are due to measurement noise etc.
The process of separating the MD, CD, and random components is referred as CD-MD separation. The problem that needs to be addressed in CD-MD separation is that certain pure CD variations, pure MD variations and variations that are a function of both CD and MD position, produce the same signal when measured with a traversing scanner [14], [15]. The separation is necessary because the control system will correct variations in a different manner whether they are CD or MD variations by origin. Literature related to CD-MD separation includes [14], [16], [17], [18].
This paper is organized as follows: Performance metrics for two-dimensional web processes are presented in Section 2. Sections 3 and 4 give an simulation example and an industrial implementation example, respectively. Conclusions in Section 5 wind up the paper.
Section snippets
Two-dimensional performance metrics
In this section, several performance indices for two-dimensional web processes are proposed. The algorithms were designed with certain disturbance classes in mind. The objective was to develop such indices that react to a certain disturbance and that are insensitive to others.
It is assumed that constant-speed scanner measurements are available, sampled in the CD and the MD evenly in the time domain. The sampling rates in CD and MD need not to be the same. Denote the measurements with p(k, i),
Simulation example
The following simulation example shows how the indices react to the disturbances defined in Tables Table 1, Table 2. The disturbances have been selected such that they represent actual process disturbances that occur in the papermaking process.
The simulated two-dimensional data are obtained by computingwhere k and i are row and column numbers, respectively, and d1 … d7 are the disturbances defined in Table 1. A graphical representation of
Industrial implementation
To test the two-dimensional metrics in an industrial setting, a quality monitoring system was implemented to a paper machine. In the papermaking process, several variables affecting the quality of the paper are measured online from the web, including moisture, basis weight, caliper, opacity, and ash content. Other variables, including tensile, tear, and bonding strengths as well as stiffness, are impossible to measure online, since the measurement may require breaking the paper. These
Conclusion
In this paper, a number of performance measures for two-dimensional web processes were proposed.
The list of algorithms presented is not exhaustive: it is possible to modify the proposed algorithms and to develop new ones. Median filtering was selected in several algorithms instead of using the mean in order to have a faster response to step changes in the CD. In some applications the mean, or frequency domain filtering, might be more appropriate.
The measurement setup that was used in the
Appendix: Windowed median filter
The median of a dataset is the value that is in the middle of the dataset if the data are ordered. If the number of data points is even, the median is the mean of the two values that are closest to the middle of the dataset. Median may be preferable to mean if there are outliers in the data. Outliers affect the mean more than they affect the median.
The windowed median filter uses a sliding window to filter profile measurements (Fig. 9). Using a window size of ( is odd), the filter slides the
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