(16e) Fault Diagnosis of a Benchmark Fermentation Process Based On Probabilistic Qualitative Analysis of Trends

Fault
Diagnosis of a benchmark fermentation process based on Probabilistic
Qualitative Analysis of Trends

K.
Villez, V. Venkatasubramanian

Qualitative analysis of trends consists of the task to segment a time
series into contiguous time windows in which the first and/or second
derivative of the trend underlying to the (noisy) data is considered
to be unique. As such, one identifies so called episodes in which the
trend is of a selected type: A – convex antitonic (monotone
decrease); B – convex isotonic (monotone increase); C –
concave isotonic; D – concave antitonic; etc. The resulting
segmentation is referred to as a qualitative representation.

Such an analysis has been proposed frequently for the purpose of
fault diagnosis of batch and continuous processes. Indeed, because
process knowledge is often formulated or obtained easily in
qualitative terms by communication with plant operators. However,
recent research (Villez et al., 2011) points out that at least some
of these existing methods are hardly robust to realistic noise
levels. All existing methods are either based on recursive schemes or
the application of heuristic rules which offer computational
advantages at the loss of accuracy. For this reason, a spline based
method has recently been proposed for improved accuracy (Villez et
al., 2012a). In this line of research, focus is given to accuracy and
statistical validity, while ignoring computational efforts as of yet.
In essence, this spline method fits a spline subject to a supposed
qualitative representation. This is a convex Second Order Cone
Problem (SOCP) as long as the type and location of the episodes are
known in advance and the spline function is of order four (cubic
splines) or less (Nesterov, 2000; Papp, 2011). In Villez et al.
(2012a) it is proved and demonstrated that the location of the
episodes can be identified in a globally optimal fashion, given a
qualitative sequence. Such qualitative sequence is the ordered series
of episode types.

Given this method, the remaining problem is to identify the proper
qualitative sequence. In Villez et al. (2012b), a probabilistic
strategy is developed. To this end, each considered fault is
associated with a unique qualitative sequence and a prior likelihood.
Given a time series, the conditional likelihood (the likelihood of
the data conditional to a qualitative sequence) is approximated with
the maximum likelihood. This maximum likelihood follows from
maximization of the spline function fit by searching the optimal
locations of the episodes as given by the proposed qualitative
sequences. By use of Bayes' rule, fault diagnosis boils down to the
selection of the maximum a posteriori (MAP) likelihood qualitative
sequence. This means that the qualitative sequence leading to the MAP
likelihood, following global optimizition of the episode locations,
is selected.

The provided method is applied to the simulated benchmark simulation
model of [Birol]. This model was set up for benchmarking of fault
diagnosis methods, amongst others. In Villez et al. (2011), noisy
Penicillin concentration profiles were analyzed with an existing,
wavelet-based method. Profiles of three conditions were selected,
each of which have distinct qualitative representations based on
visual inspection. In Figure 1, one can see that Normal Operational
Conditions (NOC) correspond to a BC sequence. Fault 2 correspond to
BCBC sequence. Fault 3 correspond to BCDA a sequence. The results in
Villez et al. (2011) indicated extreme sensitivity to noise. The
shown results are the initial results obtained with the newly
proposed spline-based method and are obtained in the contex of
larger, comparitive study of techniques for qualitative analysis.

Figure 1: Simulated
contration profiles for different conditions in the benchmark
fermentation model.

Figure 2 shows the maximum likelihood fits for one batch with Fault 2
and for each of the proposed qualitative sequences. One can see that
the spline fits for each shape constraints can hardly be
distinguished.

Figure 2: Maximum likelihood
fits for a batch with Fault 2. The fits of different sequences are
hard to distinguish.

Figure 3 shows the a posteriori likelihoods (normalized) which are
obtained for a batch with Fault 2 and for three qualitative sequences
(BC/BCBC/BCDA). Prior likelihoods for each of the qualitative
sequences were set equal, implying complete ignorance. One can see
that the maximum a posteriori (MAP) selection of a qualitative
sequence is correct. Indeed, the BCBC sequence is selected. Note that
the difference in likelihood is rather subtle as would be expected
from the almost equal fits.

Figure 3: A posteriori
likelihoods for three batches and three qualitative sequences. The
maximum a posteriori (MAP) likelihoods lead to a correct matching.

REFERENCES

Birol, G., Undey, C., Cinar, A., "A modular simulation package
for fed-batch fermentation: penicillin production", Comput.
Chem. Eng., 2002, 26:1553-1565.

Villez, K., Rosén, C., Duchesne, C., Anctil, F., Vanrolleghem,
P. A., "Qualitative representation of trends (QRT): Extended
method for proper inflection point recognition." Comput.
Chem. Eng., 2011 (Submitted)

Villez, K., Rengaswamy, R., Venkatasubramanian, V., "Generalized
qualitative shape constrained spline fitting", Technometrics,
2012 (Submitted)

Nesterov, Y., "Squared functional systems and optimization
problems", In: Frenk, H., Roos, K., Terlaky, T., Zhang, S.
(eds.) High performance optimization, applied optimization, vol. 33,
pp. 405–440, Kluwer Academic Publishers, Dordrecht, 2000.

Papp, D., "Optimization models for shape-constrained function
estimation problems involving nonnegative polynomials and their
restrictions", M.Sc. thesis, Rutgers University, 2011.