(615d) Effect of Soft Sensor Dynamics On Process Monitoring

Soft sensors are a powerful tool for acquiring information about important, difficult to measure variables from other variables, which can be more readily measured (Fortuna et al., 2005; Kadlec et al., 2009; Lin et al., 2009). State estimation techniques play a key role for soft sensor design and various types of filters have been developed. Among them, observer-based filters such as the traditional Kalman filter have found wide-spread application (de Assis and Maciel 2000; Soroush, 1998).

As soft sensors are starting to replace some plant measurements, it is important to adapt techniques, such as fault detection, to variables that are computed from soft sensors. Probably the simplest fault detection technique is to compute upper and lower control limits for the variables to be monitored and check if the variables' values are within the bounds (Isermann, 1997). For variables that are directly measured, these upper and lower control limits are given as a function of the covariance matrix of the measured variables. However, the situation is more complex if the variables are determined from soft sensors as the covariance matrix of the predicted variables is affected by the soft sensor design and is not equal to the covariance matrix of the original measurements. Instead, the covariance matrix of the estimated variables has to be computed if limits for the predicted variables are to be determined. For a linear system and a Kalman filter, the update of the covariance matrix of the estimated states can be directly computed, as the covariance matrix at the current time point only depends on the current measurements and the covariance matrix at the previous time point. However, this is not true for general estimators, especially if the system under investigation is nonlinear, where the current covariance matrix is dependent on the covariance matrices at all the previous time steps. As a result, it is non-trivial to compute the covariance matrix for a general case and, accordingly, to determine if process variables are within desired limits.

It is the objective of this work to determine appropriate control limits for variables that are monitored using a soft sensor and use this information for fault detection. The emphasis is on computing the covariance matrix of the estimated states given the state space model of the system and the measurement models. The derivation is made possible by the assumption that the process statistics will be constant at steady state, including the covariances of the state estimates and correlations between estimates and measurements. However, neither a specific filter gain nor a certain number of measurements are required for this work. The standard deviation calculated by the presented technique can be used directly to produce control boundaries for fault detection. Multivariable process monitoring is also possible using this approach as the entire covariance matrix of the estimated states is computed. The results returned by the presented technique are comparable to the result that a Monte Carlo technique should asymptotically approach, however, significantly less computational effort is required.

References

de Assis, A.J., Maciel, R., 2000. Soft sensors development for on-line bioreactor state estimation, Computers & Chemical Engineering 24 (2-7), 1099-1103.

Fortuna, L., Graziani, S., Xibilia, M.G., 2005. Soft sensors for product quality monitoring in debutanizer distillation columns, Control Engineering Practice 13 (4), 499-508.

Isermann, R., 1997. Supervision, fault-detection and fault-diagnosis methods - An introduction, Control Engineering Practice 5 (5), 639-652.

Kadlec, P., Gabrys, B., Strandt, S., 2009. Data-driven soft sensors in the process industry, Computers & Chemical Engineering 33 (4), 795-814.

Lin, B., Recke, B., Knudsen, J.K.H., Jorgensen, S.B., 2005. A systematic approach for soft sensor development, Computers & Chemical Engineering 31 (5-6), 419-425. Soroush, M., 1998. State and parameter estimations and their applications in process control, Computers & Chemical Engineering 23 (2), 229-245.