A new fault-relevant KPCA algorithm is proposed. Then the fault detection approach is proposed based on the fault-relevant KPCA algorithm. The proposed method further decomposes both the KPCA principal space and residual space into two subspaces. Compared with traditional statistical techniques, the fault subspace is separated based on the fault-relevant influence. This method can find fault-relevant principal directions and principal components of systematic subspace and residual subspace for process monitoring. The proposed monitoring approach is applied to Tennessee Eastman process and penicillin fermentation process. The simulation results show the effectiveness of the proposed method.

Process monitoring and fault diagnosis are important for the safety and reliability of industrial processes [

PCA is one of the most widely used linear techniques for transforming data into a new space. It divides data information into the significant patterns, such as linear tendencies or directions in model subspace, and the uncertainties, such as noises or outliers located in residual subspace.

KPCA is one nonlinear version of PCA. It can efficiently compute PCs in a high-dimensional feature space using nonlinear kernel functions. The core idea of KPCA is to first map the data space into a feature space using a nonlinear mapping and then carry out the PCA operation in the feature space. KPCA divides the data into a systematic subspace and a residual subspace and uses

In this paper, to improve the KPCA model, a fault-relevant KPCA algorithm is proposed and the approach of process monitoring based on the new fault-relevant KPCA algorithm is proposed for fault detection. The proposed method further decomposes both the KPCA principal space and residual space into two subspaces by checking the influences by process disturbances. The basic objective for further subspace decomposition is to separate the part which is influenced greatly by the fault from the part that is not clearly fault-relevant, that is, to find the fault-relevant directions and fault-relevant principal components. Then a new monitoring method is proposed based on the fault-relevant directions. Compared with traditional statistical techniques, the fault subspace is separated based on the fault-relevant influence.

The remaining sections of this paper are organized as follows. Section

For the traditional PCA algorithm, some faults may not influence all the principal directions; that is, to a given fault, some principal directions are not relevant. KPCA algorithm is the method for nonlinear data extended from PCA algorithm, so it has the same characteristics mentioned above [

The purpose of the proposed algorithm is to get the fault-relevant principal directions in the systematic subspace and those of the residual subspace. With the obtained fault-relevant principal directions and a new set of data, the scores of new data can be gotten. Therefore, the

In KPCA, the training samples

For

Using kernel trick

Then, the calculation is equivalent to resolving the eigen problem of (

The scores

Now PCs of training data are gotten the PCs that are relevant to faults will be found next as follows.

First, a fault process space

Then, the fault-relevant PCs of fault data

And the fault-relevant PCs of normal data

In this way, some largest fault-relevant directions of normal data and fault data are revealed, respectively. Define the ratio of the fault-relevant PC variances between fault case and normal case as follows:

The largest value of

The

Then, the PCs of normal case in the residual subspace can be calculated as follows:

Similarly, the PCs of fault case in the residual subspace can be calculated as follows:

Following the ways of (

Then the fault-relevant residual subspace of fault case is

The largest value denotes the direction along which there are the largest changes of squared errors from normal status to fault case. Keep those fault-relevant residual directions with values of larger than 1 which are the fault-relevant directions with increased squared errors. The final number of dimensions of fault-relevant residual subspace is

There exist a number of kernel functions. According to Mercer’s theorem of functional analysis, there exists a mapping into a space where a kernel function acts as a dot product if the kernel function is a continuous kernel of a positive integral operator. Hence, the requirement on the kernel function is that it satisfies Mercer’s theorem. Theoretically, all functions that satisfy Mercer’s theorem can be utilized, while there are several most widely used kernel functions such as Gaussian function

The fault-relevant KPCA-based monitoring method is similar to that using KPCA. The Hotelling’s

In (

In (

In (

In (

In (

The confidence limit of

The confidence limit of

Acquire normal operating data and several different known fault data sets.

Given a set of

Carry out centering in the feature space for

Solve the eigenvalue problem

The main thought of on-line monitoring is that

Obtain new data for each sample.

Given the

Mean center the test kernel vector

For the test data

Calculate the monitoring statistics of four subspaces of the test data in

Monitor whether

The proposed fault-relevant KPCA method was applied to fault detection and diagnosis in benchmark simulations of Tennessee Eastman process and penicillin fermentation process and compared with the conventional KPCA model.

The well-known TE process has been widely used for testing various process monitoring and fault diagnosis methods [

As a complex chemical process, TE process provides a superior simulation platform to validate the proposed method. In this study, fifty-two variables, including 41 process measurement variables and 11 manipulated variables, are used. Four hundred and eighty normal samples are used for model identification. Fifteen known faults as described in Downs and Vogel’s work are considered. Faults 1–7 are associated with step changes in different process variables, for example, in the

Based on KPCA algorithm, the normal process space is decomposed into a systematic subspace and a residual subspace first. Then some fault-relevant directions or principal components are picked up from the systematic subspace with the help of information extracted from fault data. In this article, Fault 1, Fault 7, and Fault 13 are used to develop different monitoring models. In the models built with these faults, all the principal components in the residual subspace are fault relevant, so that the

For Fault 1, Figure

Monitoring results of the Tennessee Eastman process based on (a) fault-relevant KPCA and (b) KPCA in the case of Fault 1.

Fault-relevant KPCA

KPCA

For Fault 7, the results in Figure

Monitoring results of the Tennessee Eastman process based on (a) fault-relevant KPCA and (b) KPCA in the case of Fault 7.

Fault-relevant KPCA

KPCA

For Fault 13, as shown in Figure

Monitoring results of the Tennessee Eastman process based on (a) fault-relevant KPCA and (b) KPCA in the case of Fault 13.

Fault-relevant KPCA

KPCA

In summary, the proposed method pays more attention to the fault-relevant process variations and separates them from the fault-irrelevant variations for monitoring. Comparatively, KPCA model treats them together. For Fault 1, Fault 7, and Fault 13, the monitoring results show that the fault-relevant KPCA based monitoring performance is better than that based on KPCA model. For Fault 13, the proposed method based on monitoring performance is not worse than that based on KPCA.

The choosing of the kernel parameter is important for KPCA and other kernel methods, which would affect their performances. Similarly, the kernel parameter is also an influential factor in this method and its monitoring. Changing of the kernel parameter, the shape of the

In this section, the proposed method is applied to the monitoring of a well-known benchmark process, penicillin fermentation process. A flow diagram of the penicillin fermentation process is given in Figure

Penicillin fermentation process.

Trajectories of nine variables from a nominal batch run.

The models are constructed using the proposed method. KPCA is then tested against monitoring of fault batches. Fault 1 is implemented by introducing a 10% step increase in the Aeration rate at 100 h and retaining until 300 h. Fault 2 is implemented by introducing a 2% step increase in the Aeration rate at 100 h and retaining until 300 h. Fault 3 is implemented by introducing a 10% step increase in the agitator power at 100 h and retaining until 300 h. The monitoring results are shown in Figures

Monitoring results of the penicillin fermentation process based on (a) fault-relevant KPCA and (b) KPCA in the case of Fault 1.

Fault-relevant KPCA

KPCA

Monitoring results of the penicillin fermentation process based on (a) fault-relevant KPCA and (b) KPCA in the case of Fault 2.

Fault-relevant KPCA

KPCA

Monitoring results of the penicillin fermentation process based on (a) fault-relevant KPCA and (b) KPCA in the case of Fault 3.

Fault-relevant KPCA

KPCA

In this article, the fault-relevant KPCA algorithm is proposed to decompose the process variations from the fault-relevant perspective. By further decomposing the KPCA subspaces, the underlying process information can be more comprehensively looked into, which is helpful to the detection of abnormal changes. Fault-relevant principal components extracted from KPCA systematic subspace and residual subspace are used to monitor the process. With fault-relevant principal components, instead of with all principal components which may not be influenced by the disturbances, better monitoring results are gotten. The case study on TEP and penicillin fermentation process is performed to show the performance of the fault-relevant KPCA algorithm for process monitoring. In general, swifter and more sensitive fault detection is reported in comparison with the conventional KPCA method.

The work is supported by China’s National 973 program (2009CB320602 and 2009CB320604) and the NSF (60974057 and 61020106003).