A Fault Detection and Data Reconciliation Algorithm in Technical Processes with the Help of Haar Wavelets Packets ^†

The gross error detection and replacement (GEDR) for many methods based on models or statistics is an important issue in the control field. For instance, a parameter estimation (PE) and data reconciliation (DR) can be taken into consideration very often. At the present time, only a priori noise knowledge based techniques in GEDR are used in industrial applications to detect the outliers that are based on statistical methods. It should be noted that many techniques on the basis of statistical methods could be applied in this context. It becomes more and more essential to clean the data coming from a distributed control system (DCS) because applications in the field of production rely on a great quantity of raw data. Soft sensoring, data reconciliation and parameter estimation need “clean data” [1]. A denoising method based on techniques described in [2] is shown in [3]. This method presented in [4] is based on other methods considered in [2]. The above-mentioned method compares the information theory criterion. A robust version of the Hampel Filter has been developed in [5,6,7]. A new method of de-noising, which is based on methods already introduced in [2], is presented in [3]. It is based on the comparison of an information theory criterion that is the “description length” of the data. In [4,8], this theoretical approach is applied. The description length is calculated for different subspaces of the basis and the method suggests choosing the noise variance and the subspace for which the description length of the data is at a minimum. In the work, based on [4], it is shown that the de-noising process can be done simultaneously. The approach presented in [4,8] is based on “minimum description length” and that’s why it seems to be unsuitable for online detection because it needs a relatively large amount of data. Simultaneous approaches often suffer from the fact that they are based on iterations, which, in turn, impose heavy CPU load requirements on the applications. As a consequence, these approaches often become infeasible for real-time applications, and, therefore, have not been widely used in industry. Real-time approaches dealing with outlier detection and/or de-noising, and featuring easy means of implementation, have also been developed as in [9]. In general, for removing the noise, a lower threshold should be set. This article intends to create an approach treating the outlier detection with denoising in real time without using an a priori noise knowledge technique. Reference [10] demonstrates a procedure that is median function-based, and it is applied in a wavelet-based algorithm. The wavelet-based algorithm proposed in this work is of particular quality due to the adopted a priori noise knowledge free technique. Nevertheless, some misclassifications that may occur are briefly shown at the end of this contribution. The advantage of the presented algorithm consists of its working independently, so no a priori knowledge of the noise level is necessary. Noise level algorithms are considered to be complex and need a great data volume (see [2,3,8]). The advantage of this technique shows that, using a limited stored data, a denoising and fault detection is possible. Figure 1 presents a block diagram of a GEDR package (PCK) interacting with gEST. This paper proposes an algorithm to differentiate discrete signals from their outlier observations using a library of Haar wavelet basis. The algorithm is totally general and can be used in many industrial applications. The effectiveness of the proposed method consists of applying the concept of Lipschitz for using as few samples as possible of the measured signal, and, in the meantime, highlighting the difference between the outliers and the desired signal. The goal is to extract coherent features from the measured signal that will be saved in terms of variance of the Lipschitz constant of the desired signal and outlier signals (or incoherent signal) of which the variance of the Lipschitz constant lies outside this confidential interval. In this sense, the proposed method represents a statistical method that calculates the variance of the Lipschitz constant of the observed signal. The algorithm that has been developed is utilized as a filter to extract features for training neural networks, and it is currently integrated in the inferential modelling platform of the unit for Advanced Control and Simulation Solutions within ABB’s industry division. Industrial applications in fault detection, in which coherent and incoherent signals are univocally visible, are shown.

Wavelets are mathematical functions that are based on time frequency. They analyze physical situations with the signal containing discontinuities and sharp spikes better than traditional Fourier Methods. Wavelets are very often applied in mathematics, quantum physics and electrical engineering. Intersection between these subjects in the last decade caused increasing wavelet applications such as image compression, turbulence, human vision and radar. Currently, Digital Signal Processors implement sampled signals. Only approximated values of the signal projection coefficients can be often calculated by selecting a particular signal approximation. A great number of wavelet functions that have different features exist, and, a short time ago, polynomial structures started to be applied for creating wavelets. In [13], a wavelet having a form of a cubic spline is presented in pre-filtering applications. Haar wavelets have a wide application field in industry. Representation of discrete signals using Haar wavelets is very compact due to the use of microprocessors (see [14]). A brief overview on the Haar wavelets function ${\psi }_{(d,j,n)}\left(t\right)={\psi }_j(2^{d}t-n)$ is considered with a support of size $2^{-d}$ of the Nyquist frequency. Furthermore, d is a scale parameter, j is a phase parameter and n is a time translation parameter. Here, the pyramidal packet is represented by the indices $(d,j,n)$, d is the level of the tree (scaling parameter), j is the frequency cell (oscillation parameter) and n the time cell (localization parameter). Basically, the Haar basis has the following two properties:

• the ${\psi }_{(d,j,n)}^h\left(t\right)$ are orthonormal,
• any ${\mathcal{L}}^2\left(\Re \right)$ function $f\left(t\right)$ can be approximated, up to arbitrarily low precision, by a finite linear combination of the ${\psi }_{(d,j,n)}^h\left(t\right)$.

In particular, to be more precise, coefficients $w_{(d,j,n)}$ (the weight coefficients) are calculated as follows:

For $j=0$, it is possible to define the following coefficients: where $f\left(t\right)$ is the required signal, I is the considered time interval and ${\psi }_{(d,0,n)}^h\left(t\right)$ is the well-known mother function. To conclude: where $s_{(d,n)}=w_{(d,0,n)}$. It is shown that there are just two independent parameters and that parameter d characterizes the level of the tree. The wavelet packets under consideration are derived from the Haar mother wavelet.

A set of Haar functions are presented in Figure 4. The localisation of the Haar functions through the tree is accomplished by means of the parameter tuple, $(d,j,n)$, as shown in Figure 4. Moreover, there is a relation of parameter j to the frequency shift, a relation of parameter n to the time shift, and the supplementary parameter d is necessary to address the level of the tree directly depending on the number of the examined samples. It means that if eight samples are to be considered, the level of the tree will be three. Wavelet functions are constructed like a tree, so $d=1$ is the highest degree of improvement concerning the time. Figure 4 demonstrates four wavelet functions at $d=1$ and the four wavelet functions for $d=2$. The appropriate coefficients, $w(d,j,n)$ representing the wavelet functions, for a tree consisting of $d=3$ is shown in the right part of Figure 5. $w(1,0,0..3)$ indicates the coefficients of the first level on the extreme left with time shifts 0 through 3. One of the motivations to choose Haar wavelets is that they are the most structurally simple wavelets to be applied. This means that, with a window of four samples, we can already achieve very good results. This is not the same for other wavelet families because of higher vanishing moments, and some wavelet families are available for application starting with a window of 16 samples. This represents a drawback. In fact, using a long window, the actuality of the information is lost. Another reason to use Haar wavelet family is that, because of the application in micro-controllers, the discrete shape of the signals can be represented basically as a combination of Haar functions. This represents an advantage because the compression of the signal into the Haar basis is a very compact one and can be very useful in the case of localisation of disturbances or identification of dominant or subdominant harmonics.

Fault evaluation is an important problem both in theory and in practice. Works dedicated to this topic are considered in [15,16]. Some various methods for choosing fitting threshold values have been introduced. The presently applied algorithms used to evaluate the faults with the help of wavelets could be summed up as follows:

• Obtain the coefficients by applying the Haar wavelet transform to the signal affected by faults;
• Threshold those elements in the wavelet coefficients, which are considered to refer to faults;
• Replace the fault in the considered sequence.

The most important point in this approach is represented by the threshold step, which is when wavelet coefficients are supposed to refer to the fault. The fault (outlier) is determinate in case of the local Lipschitz constant being external with respect to the calculated boundary. Two confidential constants are denoted by parameters $c_1$ and $c_2$. A short version of this algorithm is also published in [17], but an analysis in terms of soft, hard and very hard thresholding is done in this paper.

The evaluation of the procedure is made by means of artificial data in which the positions of the outliers are given.

As already mentioned, the developed algorithm is currently integrated into the inferential modelling platform of the unit responsible for Advanced Control and Simulation Solutions within ABB’s (Asea Brown Boveri) industry division. Experimental results using sensor measurements of temperature, and pressure in a Distillation Column are presented in this session. In particular, a process is taken into consideration, and this is represented by a dryer section within a paper mill, and it uses steam at different pressures for the drying. The measured quantities are, as already mentioned, temperature, pressures, flow rates, moisture, and levels, and they include process variables, set point variables, manipulated variables, and disturbances. Figure 7 shows the result obtained from the offline mode to validate the presented algorithm. Figure 7a shows data with artificial outliers. In Figure 7b, all of the outliers are correctly detected, and no misclassifications occur using the proposed algorithm with no priori knowledge of the noise. It is possible to notice that almost $100\%$ of outliers are detected in the data in the right way. The rate of the incorrect outliers detected in the data can be higher in the first part of the classification because of the initialisation part of the algorithm in which a statistic is built using the local Lipschitz approach already discussed. The initial standard deviation of the local Lipschitz constant is very small when the algorithm starts. The starting phase can take some samples, normally 15–20 samples. It is the weak point of the procedure being a stochastic one, and, in some cases, it can be not robust. The most difficult situation is if the standard deviation of the local Lipschitz constant is small. The algorithm considered in [5] and the one presented in this contribution are compared. The upper parts of Figure 8, Figure 9 and Figure 10 show the results using an algorithm developed in [5] (median filter technique) based on an application of the wavelet filter in the distillation column case. They are compared with the proposed algorithm in which they are shown in the lower parts of the same figures. To summarise the results, it is possible to notice that the proposed algorithm with respect to that proposed in [5] results to be “more aggressive” in the sense that, presumably, it eliminates data that does not belong to the fault set. This is due to the fact that, with respect to the algorithm developed in [5], which is based on a median filter technique, this new method is statistically based, and it is sensible for the chosen threshold (soft, hard and very hard threshold).

More in detail, in Figure 8, a data set from the distillation column case has been contaminated with the outliers. As it can be seen, the MAD algorithm performs much better than the wavelet based one. The MAD algorithm removes all of the outliers and does not remove any measurements that are not outliers. On the other hand, the wavelet based approach removes some of the measurement noise and, more seriously, also some of the signals.

Misclassifications of the outliers can be particularly found in the first part of the data, as it can be seen from Figure 7. The detected rate of the faulty outliers in the data expands in the first part of the classification. The sampling period is usually requested in a linear case to be one-tenth of the "dominating" system time constant. A part of misclassification problems are presented below. It can be clarified by the fact that the chosen sample time of the algorithm is too long if we compare it with the time constant (dynamics) of the considered signal. This algorithm demonstrates other problems, as it can be seen from Figure 11. The algorithm has a problem with following the signal changes when they occur very quickly. In this case, the algorithm can erroneously interpret the changes as sequences of outliers.

This work is devoted to the Gross Error Detection applying a wavelet signal based approach. To be more precise, it describes a signal-based algorithm, which can be implemented in industry. This algorithm can be used for solving various problems. Identification of inductance and resistance of an electrical systems is treated in this contribution. A priori knowledge of the noise level is not required for the presented algorithm. Computer simulations and industrial real cases were also treated in this work.