At the level of unmeasurable states the results of process models are used to check the consistency of instrumentation output. Wang and Stepanopoulos (1983) give a methodology for applying statistical hypothesis testing for error detection. This technique involves using mass balance equations to describe a fermentation. The output from these balances, taken together with the output from an observer, overdefines the fermentation. One compares the output from this model with the output from sensors, and tests the resulting residuals to check for bias in their values. If the test is passed, the residuals can be used to update the balance equation.
A methodology for the identification of errors on the detection of inconsistent results is also given by Wang and Stepanopoulos. This method works by deleting one measurement at a time and checking the consistency of the remaining measurements and their associated equations. This type of book-keeping redundancy is also
applicable to information which is initially produced by dynamic models, such as rates of material utilisation, and production.
Wang and Stepanopoulos's technique is interesting in that it incorporates all the elements of a basic fault detection and identification (FDI) system. It is relatively easy to implement and is based on directly established relationships. However, it does rely in part on off-line data such as biomass and product concentrations. Some of these may be estimatable but there is a limit to this before the system becomes
unobservable. The method also requires unbiased models which can track the fermentation parameters of interest.
A range of techniques is available for using dynamic equations describing a process for fault analysis. The usual approach is to couch the set of state equations in some form of observer. In this project this was done through the use of a Kalman filter (Litchfield 1979, Montague et al. 1986) A detailed description of a Kalman filter as applied to a penicillin fermentation is given in chapter 3 section 3. Willsky (1976) suggests the use of failure sensitive filters, the basis of these being that the
as the computed error covariances and filter gain becomes small. This effect can be overcome by either weighting new data, or fixing the filter gain. The consequence of these alterations is that the filter is more sensitive to abrupt changes, at the cost however of degrading its performance with respect to process noise.
Faults are detected by comparing the state estimates produced by a sensitised filter, with the estimates produced by an optimal filter. This technique, while identifying that a fault has occurred, will not however give information on the nature of the fault. In addition, since the penicillin fermentation process is non-linear, an extended
Kalman filter is required, and it cannot thus be said to be optimal. This undermines the basis of Willsky's method as linearisation errors mean that both filters will be subject to some bias.
An alternative approach described by Willsky is the use of multiple hypothesis filter detectors. In this approach, a bank of filters based on various hypotheses concerning the behaviour of the system, given the occurrence of faults, is established, together with a filter which describes the fully operating system. The estimates from each filter are compared to obtain a probability that the models contained therein are correct. The need to run multiple filters across a large number of fermentation batches may impose unreasonable computation time constraints, and require a large modelling effort in complex systems such as penicillin fermentations. Also this technique is vulnerable to frequent process changes. Every time the process is changed, new models both describing correct operation and faulty operation will be necessary.
The use of multiple failure sensitive filters does solve the fault identification problem. This approach could be very valuable in an environment where accurate fault
diagnosis was important, for example, it could be used to selectively boost the reliability of sub components of the fermentation process such as the aeration system. In this case a consistent pattern of events on the onset of a fault could be expected, such as a drop in dissolved oxygen, and evolved carbon dioxide. This would make possible the establishment of some filter models which were insensitive to process changes. The lack of general applicability of this technique resulted in its not being used in this project.
An alternative way of looking at the output from a Kalman filter is the analysis of the residuals, or innovations, defined as the difference between the actual process, and the
estimated output from modelling, and previous data. A property of a sequence of innovations is that its elements are independent, hence a whiteness test can be used for fault detection (Mehra 1970, Mehra and Peschon 1971). This test is based on an autocorrelation function for a stationary process, alternatively a sample correlation coefficient can be used as a whiteness test. A second test that may be employed is to compute the mean of the innovation sequence, which should be zero if there is no erroneous bias present. The third test that Mehra suggests is to test that the
covariance of the innovation sequence has a Chi-Square distribution. This technique could usefully be extended by a comparison of unobserved states with either off-line data, or modelled states. An issue with the use of whiteness tests in the context of this work is that the process under consideration is non linear, the equations used to describe it have to be linearised to make the Kalman filter work, this means that the residuals will inevitably be biased.
Another approach to the use of innovations, suggested by Willsky, is the generalised likehhood ratio approach, which is based on the modelled effects of a fault on the innovations obtained. From these models, fault signatures are obtained, and these are used to compute the likelihood of a failure having occurred. This technique requires an ability to model the unsteady state behaviour of the system, and also suffers the problem of the requirement of a growing bank of signatures, with increasing system complexity.
Turning the problem around, Chow and Willsky (1984) suggest the use of parity vectors, which are formed by comparison of system inputs and outputs. In the absence of failures, in a noisy system, the parity vector is a zero mean random vector, which can be formulated independent of system states. Upon a failure the parity vector becomes biased, and moreover the bias is a function of the specific failure.
Thus the parity vector may be used as the signature canying residual for a fault detection system. The property of being independent of state estimation makes this an attractive approach where available models are poor. This method is theoretically elegant, but requires a greater degree of system data than is used in the fermentation system under consideration.