Unmeasured disturbances can severely undermine the quality of MPC models determined from plant tests. Unmeasured disturbances include input changes (e.g., feed composition changes) and changes to the process (e.g., an operator changes the tuning parameters of the regulatory controllers or an operator opens a by-pass around an exchanger). If plant test data that is corrupted by unmeasured disturbances is used to train an MPC model, significant mismatch between the MPC model and the process can result.

In general, there are three sources of errors between an MPC model and the process: process noise, process nonlinearity and unmeasured disturbances. Process noise results from sensor noise and the coupling between the regulatory control loops. Because MPC is based on a linear representation of the process, process nonlinearity contributes to mismatch between the model predictions and the process measurements. It was assumed for this work that the process noise level was moderate, i.e., the regulatory control loops were properly tuned and that excessively noisy sensors were repaired or replaced. The process nonlinearity was compensated for by using piecewise linearization, i.e., the process gain is constant over a limited operating region and with a piecewise connection from one linearization region to another. By using piecewise linearization and assuming moderate process noise, the primary source of mismatch between a proper process model and the process data will be due to unmeasured disturbances. On the other hand, when corrupted data is used to identify an MPC model, the process model will not be proper.

Two methods were developed and tested for identifying significant levels of unmeasured disturbances within a set of plant test data. The first method, which is called the response-to-error ratio (RER), is based on using the model to predict the change in a CV based on the measured change in an MV. Then the RER is given by the predicted change in the CV divided by the error between the predicted and measured value of the CV. The RER can be applied to each data point. The larger the RER, the better the performance of a model.

The second method uses a Chi-squared statistic which is a function of the normalized residual between the model and the process data and the covariance matrix of the residual. Therefore, Chi-squared should be small for each data point for an uncorrupted set of training data.

A detailed simulator of a FCC unit has been used to evaluate the efficiency of these two approaches for identifying unmeasured in MPC plant test data. The FCC model (Han and Chung) was treated as the plant in this study, using the reactor riser temperature and the O2 in the flue gas as the CVs and the valve position for the slide valve between the regenerator and the reactor and the air flow rate to the regenerator as the MVs. In addition, the composition of the feed to the reactor was used as an unmeasured disturbance.

First, the FCC model was tested using a series of step tests to identify step response models for the MPC controller for the case without unmeasured disturbances or process noise. In addition, piecewise linearization was applied to reduce the effect of process nonlinearity. The resulting models were assumed to be the “true” models in this case. Next, the individual effects of nonlinearity, process noise and unmeasured disturbances on resulting model fidelity were studied. Using moderate levels of process noise, we were able to identify the point where the magnitude of unmeasured disturbances began to affect the fidelity of the process models obtained from the test data.

Then sets of test data were generated with varying degrees of the unmeasured disturbance, some above and below the threshold for effect on the MPC models. For each set of test data, approximately 15-20% of the test data was corrupted with unmeasured disturbances. The RER method and the Chi-squared method were applied to each set of test data to test their ability to identify the corrupted test data. For each set of test data, all the test data including the corrupted portion was used to identify the MPC model. Then this model was used to evaluate the RER and the Chi-squared value for each data point in the data set. It was found that the RER was able to identify a poor model, but was unable to identify the corrupted portion of the test data. On the other hand, the Chi-squared method was able to identify the corrupted data and even at unmeasured disturbance levels that did not significantly corrupt the resulting MPC models. In addition, the effect of the percentage of corrupted data on the ability of the Chi-squared method to identify the corrupted data was studied.

Han, I.S. and C.B. Chung, “Dynamic Modeling and Simulation of a Fluidized Catalytic Cracking Process. Part 1: Process Modeling”, Chem Eng. Sci., Vol 56, pp 1951-1971 (2001)