Cluster Analysis for Multivariable Process Control E. L. Sutanto and K. Warwick Department of Cybernetics, University of Reading, P.O. Box 225, Whiteknights, Reading, Berkshire, RG6 6AY, U.K. emails:
[email protected] [email protected]
Published in Proceedings of the American Control Conference 1995 Vol. 1, pp. 749-750 Seattle, June 1995
Abstract This paper describes the novel use of cluster analysis in the field of industrial process control. The severe multivariable process problems encountered in manufacturing have often led to machine shutdowns, where the need for corrective actions arises in order to resume operation. Production faults which are caused by processes running in less efficient regions may be prevented or diagnosed using a reasoning based on cluster analysis. Indeed the internal complexity of a production machinery may be depicted in clusters of multidimensional data points which characterise the manufacturing process. The application of a Mean-Tracking cluster algorithm (developed in Reading) to field data acquired from a high-speed machinery will be discussed. The objective of such an application is to illustrate how machine behaviour can be studied, in particular how regions of erroneous and stable running behaviour can be identified.
Keywords Cluster analysis, Classification, Multivariable analysis, MeanTracking Cluster Algorithm.
1. Introduction Within the manufacturing process, there are many complex conditions which are difficult to model. Indeed, to model such complexity within the framework of model-based control theory is neither plausible nor sensible as any unsatisfied assumptions requested by the theory may lead to unsatisfactory control being achieved; if control is achieved at all. Nevertheless, the problem of obtaining some control to reach improved efficiency in process performance still needs to be considered; calling for suitable solutions to be found. Techniques to provide some degree of solution to the problem under discussion, have risen from various fields of research such as Statistical Process Control [1], Expert Systems [2] and Fuzzy Control Theory [3], [4] with applications to image processing equipment, camcorders and washing machines. Although each field has contributed significantly to the solution domain in one form or another, there is much room for improvement and novel work to be explored; the latter of which may include a combination of related fields to produce a hybrid system. Cluster analysis, currently seen as a disparate field to those in the control world, may indeed part-contribute an effective reasoning strategy to a much larger system [5], [6]. Although cluster analysis in general has seen through a number of diverse applications [7], [8], its application in the manufacturing industries has not been fully exploited. The work described in this paper is mainly concerned with the results achieved from the use of cluster analysis in an open-loop industrial application; the cluster algorithm of which has been designed and developed by [9]. The so called Mean-Tracking cluster algorithm is not only novel in design, but has the ability to locate multiple clusters in multidimensional space created by plotting the values of measured machine variables against each other at each point in time. For the case where the data space under analysis characterises the manufacturing process, the clusters that exist within it may carry meanings for machine diagnoses and predictions. Indeed, under identical manufacturing conditions, repeated measurements of the machine variables will create repeated patterns (subject to variations due to random measurement noise). Regions of the multidimensional space which are visited frequently are highly populated by the points representing the measurement samples, and these form ‘clusters’ in the space. It makes sense therefore to locate these clusters so that a study into machine behaviour can be carried out. It is hoped that this exercise will provide a better under-
be profitable to industry in the long term, but also creates an academic interest in the field of multivariable control.
2. Mean-Tracking Cluster Algorithm Summary Within a data space defined by N Ðn = {Di ; Di ∈ℜ : i = 1, . . , n } where Di = {(di,j); (i,j) ∈N× N }, n is the number of data samples N and ℜ is an N-dimensional Euclidean Space, a search window is placed, whose lower and upper limits [lower, upper] are [C(xj) - a(xj) , C(xj) + a(xj)] for j = 1 ... N (1) where
C(xj) is the centre of the search window on the xj axis a(xj) is the predefined length interval on the xj axis.
The centre of gravity (c.o.g) of the search window is then determined by calculating the mean value of all the vectors which lie within the boundary given in (1). The window is then recentred at this value and the procedure is repeated until the c.o.g converges to some constant. Once all the clusters have been found, a grouping procedure is carried out on each individual cluster centre. This is done by grouping each data point to its nearest cluster, given that the distance between them is close enough, thus a data point D is said to be closest to cluster S and is valid to be grouped to it iff Mk = max {Ck,j - dj - λj : j = 1 ... N } M = min {Mk ≤ 0 : k = 1, . . , noc } S = {k : M = Mk} λj : predefined interval length of the Manhattan grouping boundary on the xj axis, and noc: number of clusters. If M = min {∅ } where ∅ indicates a null vector, then D is classified as an outlier and is left ungrouped.
3. Implementation The implementation of the Mean-Tracking algorithm on real data acquired from high-speed machinery has been carried out. As confidentiality has been imposed upon discussion of the nature of the application, the freedom to convey some aspects in detail is limited. Parameter settings for the application are given as follows; n = 10373; N = 7; a(xj) = 1.036σ(xj) / 2; λ = 1.08σ(xj) / 2 where σ(xj) is the standard deviation of xj : this is found to be a reasonable size for the search window through the experience acquired from applying the algorithm to various data sets. Multiple clusters are located by placing a number of search windows, some of which may locate clusters already found (duplicate clusters) whilst others may locate spurious ones (further discussion to be published). A 2-dimensional cluster search is shown in figure 1 for explanation purposes, and as it is difficult to visualise searches in higher dimensions, numerical iterations to cluster detection are given in figure 2. Diagnosis based on the clusters follows in figure 3. Here, it can be seen that although cluster 0 contains only 87 points (equal to approximately 44 minutes of real time), the cluster is error free thus suggesting a good operating region (it is feasible to find a number of error-free clusters, the regions of which are equally acceptable). Other clusters however, are erroneous to varying degrees, some which contain several error types. Such clusters do not only convey the possible correlation between errors, but that they also provide regions for which machine operation should be avoided.
points to determine on-line machine behaviour and the trajectory path of a running machinery towards a defined cluster may assist in the prediction of future machine behaviour.
4. Conclusions It can be seen that a new dimension has been added to control analysis, by the introduction of cluster analysis to an industrial application. Indeed, clusters of data points may not only have captured the complex correlation between variables, but also between error types. The Mean-Tracking cluster algorithm is efficient in locating N-dimensional clusters; clusters from which not only good operating regions can be determined, but also modes of failure can be identified. It is felt therefore, that progress has been made in solving open loop problems such as the one described in this paper. The cluster algorithm does not only provide relevant reasoning to the application concerned, but its versatility allows its implementation to reach a wide range of industrial production applications regardless of internal complexity or dimensionality.
figure 1. Two-Dimensional Clustering
5. Acknowledgements The authors would like to acknowledge the collaboration of the industrial partner with the project and the invaluable financial support of the Engineering and Physical Sciences Research Council in the U.K.
6. References [1] W. E. Deming, "Out of the Crisis: Quality, Productivity and Competitive Position", Cambridge University Press, 1986. [2] J. Girst, "The Practical Application of Rule and Knowledge Based System Techniques to Industrial Processes", Quality Forum, Vol. 18, No. 1, pp. 25 - 32, March 1992. [3] J. Yen and R. Langari, eds., "Industrial Applications of Fuzzy Control and Intelligent Systems", Von Nostrand Reinhold, 1993. [4] H. Tagaki, "Survey: Fuzzy Logic Applications to Image Processing Equipment", Industrial Applications of Fuzzy Control and Intelligent Systems, ed. J. Yen and R. Langari, Von Nostrand Reinhold, 1993. [5] L. Kitainik, S. Orlovski and M. Roubens, "Expert System Ficckas - Fuzzy Information, Cluster, Choice and Knowledge Acquisition System", Fuzzy Sets and Systems, Vol.58, No. 1, pp. 105 - 118, 1993. [6] B. W. Ginsberg and W. J. Whiten, "Expert System Development using Ellipsoid-Base Clustering", Minerals Engineering, Vol. 6, No. 1, pp. 31 - 40, 1993. [7] W. Vogt and D. Nagel, "Cluster Analysis in Diagnosis", Clinical Chemistry, Vol. 38, No. 2, pp. 182 - 198, 1992. [8] R. Huth, I. Nemesova and N. Klimperova, "Weather Categorisation based on the Average Linkage Clustering Technique - An Application to European Midlatitudes", International Journal of Climatology, Vol. 13, No. 8, pp. 817 - 835, 1993. [9] E. L. Sutanto and K. Warwick, "Cluster Analysis: An Intelligent System for the Process Industries", Cybernetics and Systems ’94, Vol. I, pp. 327 - 334, Robert Trappl eds., Word Scientific 1994.
Figure 2. N-Dimensional Cluster Iterations
Figure 3. Cluster Diagnosis
