2.3 Summary of dPCA based time-series segmentation
3.2.2 Example with synthetic data of a polymerization process
Identification of highly nonlinear process models is more complex task than the previously presented illustrative example. Due to the complex nonlinear effect of parameters it is really necessary to support the parameter estimation procedure by information rich data regarding to the estimated parameters. Polymerization processes and their first principle models are highly suitable for representing the characteristics of nonlinear process models. The task is to automatically determine information rich segments that are applicable to the identify the parameters of the white box model described in the following subsection.
Description of the process
A continuously stirred tank reactor (CSTR)is considered in which a free radical polymerization reaction of methyl-metacrylate using azo-bis-isobutironitril (AIBN) as initiator and toulene as solvent. The number-average molecular weight (NAMW) is used for qualifying the product and process state. The polymerization process can be described by the following model equations, [67]:
dCm The mathematical model of the simple input simple output process consists of four states (Cm, CI, D0, D1) and four nonlinear differential equations, where the manipulated input is the inlet initiator flowrate and the output is the NAMW defined by the ratio ofD1/D0.
The mathematical model of the multiple-input multiple-output process consists of six states (Cm, CI, T, D0, D1, Tj) and six nonlinear differential equations. By assuming an isotherm operation mode the process model could be reduced to four differential equations by neglecting Eq(3.23) and Eq(3.26), which still yields a highly nonlinear process but an easier way to investigate the proposed methodology.
Time-series segmentation scenarios and results
The process data that shall be segmented is depicted in Figure 3.5.
10 20 30 40 50 60 70 80 90 100 110
Cooling water flowrate Time (h)
Figure 3.5: Process data used in the polymerization reactor example
Firstly all kinetic parameters of the model defined in Eq(3.28) are considered unknown and involved in the segmentation (identification) procedure.
As the state equations are built up as complex combinations of model parameters, the model parameter estimation is quite difficult. Also due to this complexity the derivation of sensitivities from state equation is complicated, so the finite difference method is chosen to generate the sensitivities. The sample time in this case is 0.03h. In sensitivity calculation, the simulation time for sensitivity calculation (tsim) is chosen to be 100 time samples, which is longer than the dominant time constant of the process (which is almost 1h). See the definition of simulation time in Eq. 3.12.
The first step of the segmentation procedure is the selection of the minimal resolution. In this particular case initial segments consist of 1000 samples. Using the bottom-up segmentation algorithm six segments were determined. The results of the time-series segmentation is summarized in Figure 3.6.
The first segment has the lowest information content and the fifth is the richest in this aspect. An identification procedure is performed using the data with the lowest and the highest informative segments to confirm this difference. In the identification
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Output
Time (h)
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34 36 38 40
Time (h)
log(E) criteria of the segments
1 2 3 4 5 6
Figure 3.6: Result of segmentation for supporting to the identification of all kinetic parameters
process the following cost function is minimized:
Aminr,Er
N
X
i=1
(100·(˜yi,T −yi,T)2+ (˜yi,N AM W −yi,N AM W)2) (3.29) wherey˜i,T, yi,T are output and calculated temperature values inithsample time,
˜
yi,N AM W, yi,N AM W mean the same in terms of NAMW. The identification scenarios are performed using MATLAB and its fmincon function.
3 3.5 4 4.5 5 5.5 6
Figure 3.7: Results of identification scenarios (full line - original data, dashed line - the worst scenario, dashdot line - the best scenario)
Figure 3.7 shows informative results, the model identified from the segment with highest information content gives much better performance than model identified based on the worst segment. In this example all parameters were taken into account.
Two further examples were designed to check the selectivity of the method respect to the parameter-set:
1. just theErparameters from Eq. 3.28 are considered as unknown and involved in the identification procedure.
2. just the Ar parameters from Eq. 3.28 are involved in the identification procedure. 1
As first, just the determination of values of exponential parameters (Er) is examined when preexponential (kr) parameters are fixed on previously determined values. The segments with different information content are differentiated regarded to the exponential parameters. In Figure 3.10 the result of the segmentation is depicted with calculated the information content in each segments using the E criteria values. To be able to differentiate the segments they are marked with numbering.
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Output
Time (h)
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13
log(E) criteria of the segments
1 2 3 4 5 6
Figure 3.8: Result of segmentation for supporting the identification of exponential parameters
As it is depicted in Figure 3.8 the 3rd segment has the highest and the 1st has the lowest information content, respectively. Similarly to the previous scenario
1see results of segmentation scenarios in Appendix, Table A.3, Table A.4, Table A.5
where all the kinetic parameters are involved in the identification procedure a new identification process is performed to demonstrate the differences in information content. The results are summarized in Table A.4 and in Figure 3.9.
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Figure 3.9: Result of identification of the exponential parameters (full line - original data, dotted line - best case, dashed line - worst case)
The value of the cost function is significantly reduced comparing the best case of this scenario and the previous case. This proves that some historical data segments have higher information content than the other ones.
The identification of the preexponential parameters (kr) is examined to further improve prediction performance in the next scenario. In this case the exponential parameters are fixed in the value of the best case of the previous scenario. In Figure 3.10 the result of the segmentation is depicted with the information content in each segments using the E criteria.
As Figure 3.10 and Figure 3.8 show the result of the two scenarios are the same, but the information content of the segments are different as the identification point of view is changed. It shows that different segments of historical data are suitable for identification of different model parameters. Similarly to the scenarios above, an identification procedure is performed in this case too. The richest segment in information (related to the identification of the preexponential parameter) is5thand the poorest is the 2nd as it is depicted in Figure 3.8. Results of this identification scenario is summarized in Table A.5 and Figure 3.11.
As Figure 3.11 shows the best case of the recent scenario approaches the original data quite well, since the difference is minimal (as it is shown in Table A.5).
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Output
Time (h)
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34
log(E) criteria of the segments
1 2 3 4 5 6
Figure 3.10: Result of segmentation for supporting the identification of preexponential parameters
Figure 3.11: Identification result focusing to preexponential parameters (full line -original data, dotted line - best case, dashed line - worst case)
Table3.1:Resultofidentificationscenariosofdeterminingallkineticparameters ScenarioSegmentcost 1000samplesTypeofparameterr=pr=fmr=Ir=tcr=td Original--kr1.77·109 1.0067·1015 3.792·1018 3.8223·1010 3.1457·1011 Er1.8283·104 7.4478·104 1.2877·105 2.9442·103 2.9442·103 Best5th 7.71·107kr2.13·109 1.75·1015 2.33·1018 2.53·1010 1.57·1011 Er2.07·104 8.16·104 1.25·105 3.95·103 2.33·103 Worst1st 3.38·108kr2.85·109 9.09·1014 7.58·1018 4.42·1010 2.77·1011 Er2.18·104 7.97·104 1.29·105 2.11·103 2.36·103
Table3.2:Resultofidentificationscenariosofdeterminingexponentialparameters ScenarioSegmentcost 1000samplesTypeofparameterr=pr=fmr=Ir=tcr=td Best5th 4.0538·103 Er1.8626·104 7.4832·104 1.2917·105 5.7232·103 3.4302·103 Worst2nd 1.9573·106 Er1.9978·104 7.7165·104 1.2775·105 4.4439·103 5.2659·103
Table3.3:Resultofidentificationscenariosofdeterminingpreexponentialparameters ScenarioSegmentcost 1000samplesTypeofparameterr=pr=fmr=Ir=tcr=td Best5th 165kr1.6281·109 9.2605·1014 4.3787·1018 6.6971·1010 2.4946·1011 Worst2nd 1.6794·107 kr2.1630·109 1.6956·1015 2.3419·1018 2.6473·1010 1.5729·1011