Misleading process information can be generated if there is a bias change, drift or high levels of noise in measurements reported by some of the sensors. Erroneous information often causes decisions and actions that are unnecessary, resulting in the deterioration of product quality, safety and profitability. Identifying failures such as a broken thermocouple is relatively easy since the signal received from the sensor has a fixed and unique value. Incipient sensor failures that cause drift, bias change or additional noise are more difficult to identify and may remain unnoticed for extended periods of time. Auditing sensor behavior can warn plant personnel about incipient sensor faults and initiate timely repair and maintenance. Many approaches have been proposed for sensor fault detection and diagnosis using statistical methods, model-based fault diagnosis techniques (Section 8.3) such as parity space [189, 292], state estimators  and parameter change detection , artificial intelligence applications such as neural networks  and knowledge-based systems. In this section, sensor auditing methods based on functional redundancy generated by PLS or canonical variate state space (CVSS) models (Section 4.3.2) are presented. Integration of these statisti cal methods with knowledge-based systems to discriminate between sensor faults and process disturbances is discussed in . Another method for FDD of sensors and discrimination of faults from process upsets relies on changes in correlation between data from various sensors . Sensor FDD by wavelet decompositions of data followed by calculation of signal features and nonparametric statistical test has also been reported ,
The use of the mean and variance of residuals between measured and estimated sensor readings based on PCA and PLS models was proposed in late 1980s . The authors cautioned the users about the corruption of estimates when erroneous sensor data were used when multiple sensors were faulty. Negiz and Cinar  developed a sequential PLS model development and sensor FDD method to reduce the effect of multiple faulty sensors and to discriminate between sensor bias, drift and noise. These methods use a data sequence to compare the mean and variance of data batches. They can be implemented to run repeatedly at frequent intervals and warn plant personnel about incipient sensor faults to initiate timely repair and maintenance. Both methods are based on interpreting the magnitudes of the mean and variance of the residuals between a data batch and their prediction from a process model. The PLS-based method is useful for process data with milder autocorrelation, while the CVSS-based version is more appropriate for processes with significant dynamic changes and autocorrelation in data. Industrial continuous processes have a large number of process variables and are usually operated for extended periods at fixed operating points under closed-loop control, yielding process measurements which are autocorrelated, cross correlated, and colinear. A CVSS model would be appropriate for modeling data generated from such processes. Once an accurate statistical description of the in-control variability of a continuous process is available, the next step is the design and implementation of the sensor monitoring SPM procedure. Multipass PLS-Based Sensor FDD Method. The multipass PLS algorithm was developed for detecting simultaneous multiple sensor abnormalities. This is achieved by eliminating successively the corrupted measurements from both the calibration and test data sets and identifying a different smaller PLS submodel.
Assume that there are p sensors to be monitored and the calibration data set is of length N. The mean and the variance of the residuals for each variable is computed through the N x p residuals block matrix R. Once the PLS (calibration) model is identified for the in-control data set, the statistics for the residuals are computed for setting the null hypothesis. Then, a test data block of size Nt x p is formed from new process measure ments. The residual statistics for the test sample are then generated by using the PLS calibration model. The statistical test compares the residuals statistics of the test sample with the statistics of the calibration set for detecting any significant departures.
Denote by R„ the ith N x 1 residual vector column from the N x p residual block matrix R. The statistic for testing the null hypothesis of the equality of means from two normal populations with equal and unknown variances is
where R.ite3t and R.lmodel denote the maximum likelihood estimates of the residual means for the variable i in the test sample and the calibration set, crPi is the pooled standard deviation of the two residual populations for the i-th variable, N and Nt denote the sizes of the calibration and testing populations, and ijv+ivt-2 is the ¿-distribution with N + Nt —2 degrees of freedom .
The statistic for testing the null hypothesis of the equality of variances from two normal populations with unknown means is 
^model where F/vt-i,jv-i is the F distribution with respective degrees of freedom. The level of the test for all the testing statistics is chosen to be 5% and two sided. This part of the procedure is similar to that given by .
The algorithm takes action when either the mean or variance of the residuals are out of the statistical limits (based on t and F probability distributions) for a particular variable. Since the corrupted variable affects the predictions of the remaining ones, false alarms might be generated unless the corrupted variable is taken out from both the calibration and test data blocks. The information loss due to taking the variable out of both the calibration and the test sample sets is not significant since the testing procedures are based on the iid assumption of the residuals and not on the minimum prediction error criterion by the model. The algorithm discards the variable with the highest corruption level by looking at the ratios of its residual variance and its residual mean to their statistical limits which are based on Eqs. 8.42-8.43.
Excluding variables and computing a new PLS model for the remaining variables is the key step of the sensor auditing and fault detection algorithm. The likelihood for all of the process sensors to become simultaneously faulty is extremely small. After several successive steps, if the mean and variance of the remaining residuals still indicate significant variation, then it is more likely that a disturbance is active on the system causing the in-control variability to change.
Multipass CVSS-Based Sensor FDD Method. A multipass CVSS technique similar to the multipass PLS algorithm is developed for detecting multiple sensor failures. This is achieved by eliminating successively the corrupted measurements from both the calibration and test data sets and identifying a different CVSS submodel. The algorithm discards the variable with the highest corruption level by looking at the ratios of its residual variance and its residual mean to their statistical limits which are based on Eqs. 8.42-8.43. Excluding variables and computing a new CV realization for the remaining variables, the algorithm proceeds in a manner similar to the PLS-based version. The application of the method for FDD of the sensors of a high-temperature short-time pasteurization process is reported in .
Real-time Sensor FDD by Statistical Methods. A sensor FDD that checks sensor status at each sampling instant can be developed by using T2 and squared prediction error (SPE) charts. Once these charts indicate an out-of-control status, discrimination between sensor faults and disturbances should be made and the faulty sensor should be isolated. One approach used for discrimination of sensor faults and disturbances is the redundant sensor voting system  that utilizes backward elimination for sensor identification . The backward elimination is similar to the multipass PLS approach, but remodeling is implemented at each time instant the SPE limit is violated. In this approach, once the SPE limit is violated at a specific sampling time, every sensor is sequentially removed from the model matrix and the control limit is recalculated. If the ratio SPE/SPEiimit < 1, the search terminates and the sensors eliminated up to that point are declared faulty. Otherwise, the search continues by eliminating another sensor from the model. This approach has significant computational burden. In addition, sensor faults that do not inflate the SPE statistic cannot be detected. Incorporation of T2 charts and use of correlation coefficient criterion were proposed to improve this method .
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