SPM Using Dynamic Process Models

MSPM techniques rely on the model of the process. If the process has significant dynamic variations, state-space and subspace state-space models 

The residuals are used in monitoring with the normalized SPE chart (SPEn) in this example. At time k, SPEN k) is 

The in-control residual mean vector e is almost zero and in-control residual covariance matrix Sg is diagonal. 

The primary source of heat is hot water. The hot water is heated by direct steam injection in the hot water heater. Three PID controllers are used to control product temperature. The first control loop regulates the 

The variables used in process modeling and fault diagnosis implementation are four temperature measurements (°C) and two PID controller 

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To include the information about process d3mamics in the models, the data matrix can be augmented with lagged values of data vectors, or model identification techniques such as subspace state-space modeling can be used (Section 4.5). Negiz and Cinar [209] have proposed the use of state variables developed with canonical variates based realization to implement SPM to multivariable continuous processes. Another approach is based on the use of Kalman filter residuals [326]. MSPM with dynamic process models is discussed in Section 5.3. The last section (Section 5.4) of the chapter gives a brief survey of other approaches proposed for MSPM. 

Autocorrelation in data affects the accuracy of the charts developed based on the iid assumption. One way to reduce the impact of autocorrelation is to estimate the value of the observation from a model and compute the error between the measured and estimated values. The errors, also called residuals, are assumed to have a Normal distribution with zero mean. Consequently regular SPM charts such as Shewhart or CUSUM charts could be used on the residuals to monitor process behavior. This method relies on the existence of a process model that can predict the observations at each sampling time. Various techniques for empirical model development are presented in Chapter 4. The most popular modeling technique for SPM has been time series models [1, 202] outlined in Section 4.4, because they have been used extensively in the statistics community, but in reality any dynamic model could be used to estimate the observations. If a good process model is available, the prediction errors (residual) e k) = y k)—y k) can be used to monitor the process status. If the model provides accurate predictions, the residuals have a Normal distribution and are independently distributed with mean zero and constant variance (equal to the prediction error variance). 

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