# Fault diagnosis of sensor based on wavelet frequency band analysis method based on system mathematical model

“In order to accurately distinguish the causes of sensor mutation signals, a wavelet frequency band analysis method based on mathematical model is proposed. Aiming at the measurement and control system in the industrial process, the relationship between the frequency composition of the output mutation signal and the reason of the mutation is analyzed. Using wavelet frequency band analysis technology, the high and low frequency signals are separated, and the energy statistics are carried out. The computer simulation of the classical control system and the experimental results of the constant pressure water supply system show that the method can effectively diagnose whether the sensor fails.

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In order to accurately distinguish the causes of sensor mutation signals, a wavelet frequency band analysis method based on mathematical model is proposed. Aiming at the measurement and control system in the industrial process, the relationship between the frequency composition of the output mutation signal and the reason of the mutation is analyzed. Using wavelet frequency band analysis technology, the high and low frequency signals are separated, and the energy statistics are carried out. The computer simulation of the classical control system and the experimental results of the constant pressure water supply system show that the method can effectively diagnose whether the sensor fails.

In the measurement and control system, the output signal of the sensor is affected by a variety of factors, and often changes abruptly. The values of these abrupt points contain important fault information. Accurately capturing and distinguishing the causes of these abrupt points is the key to sensor fault diagnosis. The literature only relies on the output time series of the sensor to diagnose the fault of the sensor, and the sudden change of the output signal of the sensor is attributed to the fault of the sensor. The method in the literature is to perform wavelet transform on the input and output signals of the control system respectively. When the wavelet function can be regarded as the first derivative of a smooth function, the mutation point of the signal corresponds to the modulus maximum value of its wavelet transform. Detect the mutation point, generate the residual sequence and analyze the sensor failure, and think that the mutation of the sensor output signal is caused by the sensor failure or the mutation of the system input signal. In fact, there are many reasons for the sudden change of the sensor output signal. In addition to the sudden change of the system input and the fault of the sensor itself, there are also process disturbances, actuator faults, controller faults, controlled objects and external electromagnetic field interference. In practical applications, the above sensor fault diagnosis methods have certain limitations. Usually, in industrial process control, the time constant of the controlled object is large and cannot respond to the high frequency components in the sudden change signal. Based on the frequency band analysis technology of wavelet transform, the author discusses and analyzes the reasons for the sudden change of the sensor output signal, and provides a practical analysis method for the fault diagnosis and performance evaluation of the online sensor.

**1 Generation and characteristic analysis of mutation signal**

A typical control system generally consists of four parts: controller (Gc(s)), actuator (Gv(s)), controlled object (Go(g), Gd(s)) and sensor ((Gm(s)) , its block diagram is shown in Figure l.

In the figure, X(s) is the sensor output (that is, the measured value of the controlled parameter of the control system).

The time constant of the dynamic characteristics of most controlled objects in general industrial processes is relatively large. In order to ensure fast and undistorted detection of the output signal, the time constant of the dynamic characteristics of the sensor is relatively small.

The abrupt signal of the system (sensor) means that its output amplitude and frequency suddenly increase or decrease at a larger rate, and the two are mutually dependent.

**1.1 Mutations caused by input R(s)**

In Figure 1, let

Its logarithmic frequency characteristic curve is shown in Fig. 2.

The cut-off frequency ωc≈1 Hz of the combined link can be obtained from the curve.

The characteristics of the high frequency band of the curve (the section with ω>100ωc) are determined by the smaller time constants in Gc(s), Gv(s), and Go(s), due to the distance from ωc and a larger slope to -∞dB The directional attenuation reflects the low-pass filtering characteristics of the combined link, forming the characteristic that the system cannot respond to the high-frequency components in the input signal. The characteristics of the high frequency band have little effect on the transient performance of the system, but it is impossible to reflect the step change of the time domain response. Therefore, there is a delay time. The high frequency band directly reflects the system’s ability to suppress the high frequency components in the input signal. The lower the decibel value, the stronger the suppression ability.

Since the time constant To of general industrial objects is generally large, the cut-off frequency ωc is small, so when the input R(s) is abruptly changed, the frequency distribution of the response mutation signal of the object output Xo(s) and the sensor output X(s) is relatively small. low, and the frequency band is narrow.

**1.2 Abrupt changes caused by controller and actuator faults and sudden changes in process disturbances**

Using the same analysis method, the same conclusion can be drawn: the frequency distribution of the output response sudden change signal caused by the fault of the controller and the actuator and the sudden change of the process disturbance is lower and the frequency band is narrower.** 1.3 Sudden mutation caused by external strong electromagnetic field interference**

It is generally believed that the sensor can resist various high-frequency Electronic (radio, not considered here) interference. The external strong electromagnetic field interference generally does not cause the change of the output Xo(s) of the controlled object. It is often coupled through the circuit and directly causes the sensor output signal X(s) to change, and it is usually a pulse signal.

**1.4 Sudden mutation caused by sensor failure**

Sensor faults are divided into abrupt faults (abrupt) and slowly changing faults (incipient). The author only analyzes sudden faults. The sudden faults of sensors mainly include: deviation faults, pulse faults, drift faults and periodic faults. Regardless of the sudden faults, it will directly lead to the sudden change of the sensor output signal X(s). Since these sudden faults are caused by the sudden change of the parameters of the internal components of the sensor, the output X(s) has a wider frequency band in response to the sudden change signal, including not only low frequency components, but also certain high frequency components, which is different from the input signal. The significant characteristics of the sensor output X(s) response to the abrupt signal caused by the sudden change of signal, controller fault, actuator fault and sudden change of process disturbance are the theoretical basis for distinguishing the cause of sudden change and carrying out sensor fault diagnosis in this paper.

**1.5 Mutation caused by the failure of the controlled object**

When the controlled object fails, the spectrum of the mutation signal is closely related to the input frequency band of the sensor. When the input frequency band of the sensor is wide, the mutation signal will contain high-frequency components, but the input frequency band of the sensor used in general industrial processes is narrow. The mutation signal generally does not contain high frequency components. The causes of various mutations and their signal characteristics are shown in Table 1.

**2 Frequency band analysis based on wavelet transform**

The narrow wavelet analysis only refers to multi-resolution analysis, while the generalized wavelet analysis includes multi-resolution analysis and wavelet packet analysis. The relationship between them is shown in Figure 3.

The thick solid line in Figure 3 is the multi-resolution decomposition process. The wavelet packet decomposition is a generalization of the multi-resolution decomposition of the wavelet transform. The multi-resolution decomposition only decomposes the scale space V, that is,

The wavelet packet decomposition further decomposes the undecomposed wavelet space Wj in the multi-resolution decomposition. Because the wavelet space division corresponds to the frequency band division, the wavelet packet decomposition can obtain higher frequency resolution. The usual frequency band analysis is mostly based on wavelet packet analysis, but it increases the frequency resolution and the complexity of the algorithm at the same time. Starting from the needs of practical problems, the author chooses a method based on multi-resolution analysis, which can meet the requirements.

**2.1 Frequency band analysis method**

Suppose the frequency bandwidth of the signal X

①Using the prior knowledge of the mathematical model of the system, determine the cut-off frequency ωc of the object, and take 0~10ωc as the system bandwidth;

② Determine the appropriate sampling frequency to ensure that the electromagnetic interference signal can be collected. If the sampling frequency is f, the analysis frequency

③ Determine the appropriate number of wavelet decomposition layers N, so that 0~10ωc is just included in the low-frequency space VN, and divide the entire analysis space into relative low-frequency space and high-frequency space. Except for the low-frequency space VN where the system bandwidth is located, the rest of the space WN , WN-1, Wl merge into high frequency space;

(4) Select the appropriate wavelet function for multi-resolution decomposition, and calculate the energy of the signal in the corresponding space (frequency band) according to the obtained wavelet coefficients according to formula (1), and form a two-dimensional vector e=[e1, which represents the energy of the signal in the space, e2]where e1 represents the energy of the low-frequency signal, and e2 represents the energy of the high-frequency signal;

⑤ Normalize the two-dimensional vector e=[e1, e2]representing the space energy, that is

Perform feature analysis. e01 represents the ratio of the energy of the low frequency signal to the total energy, and e02 represents the ratio of the energy of the high frequency signal to the total energy.

**3 Simulation analysis**

The author conducted a simulation experiment on the typical system shown in Figure 1. In normal working state, the values of Gc(s), Gv(s), and Go(s) are the same as before,

At different moments when the system is stable, make R(s) and D1(s) change in unit step respectively; D2(s) changes from 0 to a pulse signal with an amplitude of 1 or a periodic signal of 0.2sin100πt; objects and sensors The characteristic transfer function is switched between the normal value and the fault value to simulate the 5 causes and 6 forms of the sudden change of the output signal, and collect the data of each sudden change process. Regardless of the signal mutation caused by the reason, the high-frequency signal component is generated instantaneously and disappears quickly. Therefore, in the total energy of the collected signal, the high-frequency component accounts for a small proportion. In order to improve the detection sensitivity, the collected data is subjected to “DC” processing, that is, the sampling data is compared with the 10 points before the signal mutation The average value is subtracted. In addition, a zero-mean white noise with a variance of 0.003 was added to the sampled data. The sampling frequency of the system is f=200Hz, and the analysis frequency is fo=100 Hz. The db4 wavelet is selected to decompose the signal in three layers, so that the signal frequency range of the low-frequency space is 0-12.5 Hz, and the signal frequency range of the high-frequency space is 12.5～100 Hz, and the high-frequency coefficients obtained from the analysis are subjected to hard threshold denoising, and then the energy ratio statistics are carried out according to formula (1). The results are shown in Table 2.

In Table 2, the proportion of the low-frequency component of the mutation signal caused by external electromagnetic field interference is small, the reason is the result of removing “DC”; the proportion of the high-frequency component of the mutation signal caused by the fault of the controlled object is very small, the reason is that The input frequency band of the sensor used in this simulation is only a dozen Hz. The simulation results in Table 2 are consistent with the theoretical analysis results in Table 1, which shows the effectiveness of this method.

**4 Experimental study**

The experimental study is carried out with a constant pressure water supply system, as shown in Figure 4, the pressure sensor is LDG-S type. After testing, the transfer function of generalized object G(s)=l/(0.22s+1). Adjuster parameter setting value: proportionality P=142%, integral time ti=3 s, differential time td=2 s, thus it can be estimated that the low frequency frequency is less than 4 Hz. At different times when the system is stable, adjust the given value to simulate a given input mutation signal, adjust the zero point to simulate the sensor constant deviation fault, adjust the regulator ratio to simulate the regulator fault, frequently start and stop the surrounding motors to simulate the sudden change of the sensor output caused by the electromagnetic field, and collect experiments under various conditions data. The sampling frequency of the system is f=128 Hz, and the analysis frequency is fo=64 Hz. The db4 wavelet is used to decompose the signal in four layers, so that the signal frequency range of the low frequency space is 0～4 Hz, and the signal frequency range of the high frequency space is 4～4 Hz. 64 Hz, and the high-frequency coefficients obtained from the analysis are subjected to hard threshold denoising, and then the energy ratio statistics are carried out according to formula (1). 1 is consistent with the theoretical analysis results, which shows the effectiveness of this method.

**5 Conclusion**

The sudden change signal output by the sensor contains very important fault information, and the frequency composition of the sudden change signal is different for different reasons. For the controlled object with a large time constant, there are generally only low-frequency components in the sensor mutation signal caused by given input changes, disturbance changes, controller faults and actuator faults. In the mutation signal caused by the failure of the controlled object, there are generally only low-frequency components. The mutation signal caused by external electromagnetic field interference is generally a pulse signal, which contains low frequency components and more high frequency components. In addition to the low-frequency components, the mutation signal caused by the sensor deviation fault also contains a small amount of high-frequency components.

The wavelet frequency band analysis method based on the mathematical model of the system proposed in this paper does not require high accuracy of the mathematical model, can effectively diagnose the fault of the sensor, and provides a new idea for the fault detection and performance evaluation of the sensor.

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