Research into a raft of multivariate statistical process control methodologies for linear, non-linear, steady-state and dynamic, batch and continuous systems has been widely reported in the literature over the past two decades [1,2,4]. As a consequence of the on-line, real time implementation of these technologies, it is possible to detect changes in process behavior that may otherwise not have been identified if univariate statistical process control methodologies had been applied. In particular for dynamic systems, the univariate methods of CUSUM and EWMA and their multivariate counterparts, MCUSUM and MEWMA are not applicable for systems that comprise a large number of highly correlated variables [1]. This limitation has materialized in the development of methodologies formulated around the statistical projection techniques of principal component analysis and projection to latent structures. Consequently, the most commonly implemented monitoring metrics are those of Hotelling's T² and the squared prediction error [1,2,4].

 

More recently, to address the dynamic nature of processes, Ku et al. [5] proposed dynamic principal component analysis (DPCA). A data matrix is formulated, similar to that used in time series modelling, and PCA is then applied to the augmented matrix. However a fundamental issue is the sensitivity of the projection based methods to the presence of outliers. A consequence of this additional complication is that a number of robust principal component analysis approaches have been proposed including a methodology proposed by Hubert et al. [3]. The concept is based on projection pursuit and the minimum covariance determinant.  More specifically, the robust principal component analysis approach is as follows. First a measure of the level to which an observation is categorised as an outlier is quantified according to:

 

(1)

 

where  is the projection of an observation onto the univariate space defined through the minimum covariance determinant.  and  are the univariate minimum covariance determinant estimators of location and scale respectively. The observations with the smallest value of  are retained and the resulting covariance matrix is used to calculate the loadings. Utilising the loadings associated with the largest eigenvalues, estimates of the retained observations are obtained and an estimate of the robust distance between the observations and the minimum covariance determinant location estimate, , is calculated according to:

 

(2)

 

If this distance is smaller than , where p is the number of analysed variables and  is the significance level, the data point is given a weight of unity, otherwise, the weight is zero. The scores and estimates of the retained observations are then attained from the loadings calculated from those observations with a weight of unity.

 

In the reported research, robust and dynamic principal component analysis are combined thereby drawing together the dynamic nature of the process and the potential presence of outliers. The approach is tested using a simulation of a multivariate autoregressive system and on a polyvinyl acetate CSTR where known faults are introduced. In the case of the polyvinyl acetate CSTR several faults are introduced. The faults pertain to disturbances in the temperatures of the feed and of the cooling water, changes in the flows of feed and initiator, disturbances in the controller setpoint, a step change of 10% in the magnitude of the initiator density and also a 10% step change in the monomer volume fraction. All faults materialise in significant changes in the molecular weight of the product. The analysed variables are the flow rates of feed and initiator, the temperatures of the reactor, the feed and cooling water and the output of the controller. All these variables are measurable with instrumentation present in any plant. The two metrics, Hotelling's T² and the SPE obtained from the application of PCA, robust PCA, dynamic PCA and the proposed robust dynamic DPCA are used to determine the presence of faults.

 

References

 

1. Castro, D., Deteccion y diagnostico de fallas mediante tecnicas estadisticas multivariables y clasificacion de patrones usando Support Vector Machines, in Facultad de Ingenieria. 2002, Universidad de Los Andes: Merida, Venezuela.
2. Chiang, L.H., E.L. Russell, and R.D. Braatz, Fault Detection and Diagnosis in Industrial Systems. Advanced Textbooks in Control and Signal Processing. 2001, London: Springer.
3. Hubert, M. and Engelen, S. Robust PCA and classification in biosciences. Bioinformatics. 2004. 30: p. 1728-1736.
4. Kourti, T., MacGregor, J., Process Analysis, monitoring and diagnosis using multivariate projection methods. Chemometrics and Intelligent Laboratory Systems, 1995. 28: p. 3-21.
5. Ku, W., R.H. Storer, and C. Georgakis, Disturbance detection and isolation by dynamic principal component analysis. Chemometrics and Intelligent Laboratory Systems, 1995. 30: p. 179-196.