(b) Stage 2. Fa: Inflow air rate, Tn: Inflow air temperature, mh20: Product moisture, %, dr: Drying rate.
Figure 6.57. AHPCA based on-line monitoring of process stages under faulty operation.
points (run length of 12 observations) are observed in the T2 plot with the local model for phase 1 than the number the overall model produces (six observations),
The case illustrates that local models provide the capability to detect earlier small trends and departures from NO that will be propagated to the next phase and eventually cause significant deviation, thus allowing process operators to improve their operations.
Online monitoring was performed in each processing stage based on adaptive hierarchical PCA (AHPCA). For this multistage process, AH-PCA is limited to stages due to interstage discontinuity. To overcome this problem, different AHPCA models are developed for each stage. Different weightings can also be applied to better account for changing phase structure.
To illustrate online monitoring, a case is generated where a small drift in impeller speed is introduced (dashed line) in the first stage and a step increase (dashed line) in inflow air rate in the second stage. Each AHPCA model successfully detected and diagnosed the problem online in each stage for the overall process (Fig. 6.57).
Multiscale MPCA (MSMPCA) is a combination of MPCA with wavelet decomposition. Traditional MPCA is applied at a single time scale by representing data with the same time-frequency localization at all locations. MPCA may not be suitable for processes which include measurements with different sampling rates and measurements whose power spectrum changes with time.
Another important factor is that an MPCA model will still include imbedded random noise although the random noise is reduced by selecting only the significant components. This random noise may cause failure in detecting small deviations from normal operating conditions of process. In order to improve the performance of MPCA, the random noise should be extracted from the signal in an enhanced manner. A possible solution to this shortcoming of MPCA is to apply wavelet transformation to the signal before developing MPCA model. The role of wavelet decomposition here is similar to that of filtering the signal to separate the errors. Examples of this pretreatment by filtering data such as exponential smoothing and mean filtering can be found in literature .
The algorithm for MSMPCA is
1. Use a historical data set of the past batches
2. Choose a wavelet function
4. Apply 1-D wavelet decomposition to each variable trajectory in historical data which are the unfolded, mean-centered and scaled version of a three-way array
5. Develop MPCA models for the coefficients at each scale for the past batches
6. Reconstruct the models in a recursive manner at each scale to form the model for all scales together
7. Apply the same 1-D wavelet decomposition on each variable trajectory of the batch to be monitored
8. Identify the scales that violate the detection limits as important scales
9. Reconstruct the new data by including only the important scales
10. Check the state of the process by comparing the reconstructed data with detection limits
The data set representing normal operation is decomposed to wavelet coefficients for each variable trajectory. MPCA models are developed at each scale. The overall MPCA model for all scales is obtained by reconstructing the decomposed reference data. Wavelet decomposition is applied to new batch data using the same wavelet function. For each scale, T2 and SPE values of the new batch are compared with control limits computed based on reference data. The scales that violate the detection limits are considered as important scales for describing the critical events in current data. Inverse wavelet transform is applied recursively to the important scales to reconstruct the signal. The new batch is considered to be out-of-control if T2 and/or SPE values of the reconstructed signal violate the control limits.
Selecting the number of scales of the wavelet decomposition is important. The optimum scale number gives maximum separation between the stochastic and deterministic components of the signal. If the scale number is chosen too small, the signal will still have noise. On the other hand, if the scale number is too large, the coarser scales will have too few a data to form an accurate model. The number of scales should be determined according to
Wavelet coefficients data sets
Data set size
Approximation coefficients at scale m = 3, A3 Detail coefficients at scale m = 3, D3 Detail coefficients at scale m — 2, D2 Detail coefficients at scale m = 1, Di
(40 x 1337) (40 x 1337) (40 x 2674) (40 x 5348)
the dimension of data used. For selection of scales the following formula can be used:
where i is the number of scales and n is the number of observations.
Example. MSMPCA based SPM framework is illustrated for a simulated data set of fed-batch penicillin production presented in Section 6.4.1. Two main steps of this framework are model development stage using a historical reference batch database that defines normal operation and process monitoring stage that uses the model developed for monitoring of a new batch.
MSMPCA model development stage: A reference data set of equalized/synchronized (Figures 6.42 and 6.43), unfolded and scaled 40 good batches (each batch contains 14 variables 764 measurements resulting in a three-way array of size X(40 x 14 x 764) is used. After unfolding by preserving the batch direction (I), the unfolded array becomes X(40 x 10696)). Each variable trajectory in X is decomposed into its approximation and detail coefficients in three scales using Daubechies 1 wavelet family that is chosen arbitrarily. Although Eq. 6.118 suggests four scales, the decomposition level of three is found sufficient in this case. Since the original signals can be reconstructed from their approximation coefficients at coarsest level and detail coefficients at each level, those coefficients are stored for MPCA model development (Table 6.12). Then, MPCA models with five PCs are developed at each scale and MV control limits are calculated.
Process monitoring stage: MPCA models developed at each scale are used to monitor a new batch. A faulty batch with a small step decrease on glucose feed between measurements 160 and 200 is mean-centered and scaled similarly to the reference set and 1-D wavelet decomposition is performed on variable trajectories using Daubechies 1 wavelets. This three-level decomposition is illustrated for penicillin concentration profile (variable 8 in the data set, — ao) in Figure 6.58. Note that, the effect of the step change on this variable becomes more visible as one goes to coarser
scales. Starting and end points of the fault are more apparent in the detail coefficient (d^) of the third scale since detail coefficients are sensitive only to changes and this sensitivity increases in coarser scales. SPE on each scale is calculated based on MPCA models on scales. An augmented version of SPE values at all scales is presented in Figure 6.59. The 99% control limit is violated at scale m = 3 on both its approximation and detail coefficients. There are also some violation at scale two but no violation is detected at the first scale hence this scale is eliminated. Fault detection performances of conventional MPCA and MSMPCA are also compared in Figure 6.60. The lower portion of this figure represents SPE of the approximation coefficients. The first out-of-control signal is detected at point 162 and returning to NO is detected at point 208 at that scale on SPE whereas conventional MPCA detects first out-of-control signal at 165th measurement and returning point to NO at 213th measurement. In addition, MSMPCA-based SPE contains no false alarms but conventional MPCA has 16 false alarms after the process returns to NO. The advantage of MSMPCA stems from combined used of PCA and wavelet decomposition. The relationship between the variables is decorrelated by MPCA and the relationship between the stochastic measurements is decorrelated by the wavelet decomposition. MV
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