Abstract: To accurately distinguish the causes of sudden signal changes in sensors, a wavelet frequency band analysis method based on a mathematical model is proposed. For measurement and control systems in industrial processes, the relationship between the frequency composition of the output sudden signal and the cause of the change is analyzed. Using wavelet frequency band analysis, high and low frequency signals are separated and energy statistics are performed. Based on the change in the energy ratio of high and low frequency signals, the cause of the sudden signal is determined. Computer simulation of a classical control system and experimental results of a constant pressure water supply system show that this method can effectively diagnose whether a sensor has malfunctioned. Keywords: Online sensor, sudden signal analysis, high and low frequency components, wavelet frequency band analysis. In measurement and control systems, the output signal of sensors is affected by various factors and often undergoes sudden changes. These sudden point values ​​contain important fault information. Accurately capturing and distinguishing the causes of these sudden points is key to sensor fault diagnosis. Previous literature relies solely on the sensor's output time series to diagnose sensor faults, attributing all sudden changes in the sensor's output signal to sensor malfunction. The literature approach involves performing wavelet transforms on the input and output signals of the control system separately. When the wavelet function can be considered as the first derivative of a smooth function, the signal's abrupt change corresponds to the modulus maxima of its wavelet transform. This allows for the detection of abrupt changes, the generation of residual sequences, and the analysis of sensor faults. It is assumed that abrupt changes in the sensor output signal are caused by either sensor faults or abrupt changes in the system input signal. In reality, many factors can cause abrupt changes in sensor output signals, including system input abrupt changes, sensor faults, process disturbances, actuator faults, controller faults, interference from the controlled object, and external electromagnetic fields. In practical applications, the aforementioned sensor fault diagnosis methods have certain limitations. Typically, in industrial process control, the time constant of the controlled object is large, making it unable to respond to high-frequency components in abrupt signals. Based on wavelet transform frequency band analysis, this paper explores and analyzes the causes of abrupt changes in sensor output signals, providing a practical analytical method for online sensor fault diagnosis and performance evaluation. 1. Generation and Characteristic Analysis of Abrupt Signals A typical control system generally consists of four parts: a controller (Gc(s)), an actuator (Gv(s)), a controlled object (Go(g), Gd(s)), and a sensor (Gm(s)). Its block diagram is shown in Figure 1. In the figure, X(s) represents the sensor output (i.e., the measured value of the controlled parameter of the control system). In general industrial processes, most controlled objects have relatively large time constants in their dynamic characteristics. To ensure rapid and distortion-free detection of their output signals, the time constants of the sensor's dynamic characteristics are relatively small. An abrupt signal in the system (sensor) refers to a sudden increase or decrease in its output amplitude and frequency at a large rate, with the two being mutually dependent. 1.1 Abrupt Signals Caused by Input R(s) In Figure 1, let its logarithmic frequency response curve be as shown in Figure 2. From the curve, the cutoff frequency ωc ≈ 1 Hz of this combined element can be obtained. The characteristics of the high-frequency band of the curve (the section where ω>100ωc) are determined by the smaller time constant among Gc(s), Gv(s), and Go(s). Because it is far from ωc and decays at a large slope towards -∞dB, it reflects the low-pass filtering characteristics of this combined circuit, resulting in the system's inability to respond to high-frequency components in the input signal. The characteristics of the high-frequency band have little impact on the system's transient performance, but reflect that the time-domain response cannot change abruptly. Therefore, a delay time exists. The high-frequency band directly reflects the system's ability to suppress 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 cutoff frequency ωc is small. Therefore, when the input R(s) changes abruptly, the frequency distribution of the abrupt response signals of the object's output Xo(s) and the sensor's output X(s) is low and the bandwidth is narrow. 1.2 Abrupt changes caused by controller and actuator faults and process disturbances Using the same analysis method, the same conclusion can be drawn: the frequency distribution of the abrupt output response signal caused by controller and actuator faults and process disturbances is low and the bandwidth is narrow. 1.3 Sudden Changes Caused by Strong External Electromagnetic Field Interference It is generally believed that sensors can resist various high-frequency electronic (radio, not considered here) interferences. Strong external electromagnetic field interference generally does not cause changes in the output Xo(s) of the controlled object; it often directly causes changes in the sensor output signal X(s) through circuit coupling, and is generally a pulse signal. 1.4 Sudden Changes Caused by Sensor Failure Sensor faults are classified into sudden faults and incipient faults. This paper only analyzes sudden faults. Sudden sensor faults mainly include: deviation faults, pulse faults, drift faults, and periodic faults. Regardless of the type, sudden faults directly lead to abrupt changes in the sensor output signal X(s). Because these sudden faults are caused by abrupt changes in the parameters of internal components of the sensor, the frequency band of the output X(s) response to the abrupt change signal is relatively wide, containing not only low-frequency components but also certain high-frequency components. This is a significant characteristic that distinguishes the sensor output X(s) response to abrupt changes caused by abrupt changes in the input signal, controller faults, actuator faults, and process disturbances. This is the theoretical basis for distinguishing the causes of abrupt changes and diagnosing sensor faults in this paper. 1.5 Abrupt Changes Caused by Controlled Object Faults When a controlled object malfunctions, the spectrum of the abrupt change signal is closely related to the sensor's input frequency band. When the sensor's input frequency band is wide, the abrupt change signal will contain high-frequency components. However, sensors used in general industrial processes have narrow input frequency bands, and the abrupt change signal generally does not contain high-frequency components. Various causes of abrupt changes and their signal characteristics are shown in Table 1. 2 Frequency Band Analysis Based on Wavelet Transform Wavelet analysis in a narrow sense refers only to multi-resolution analysis, while in a broader sense it includes both multi-resolution analysis and wavelet packet analysis, as shown in Figure 3. The thick solid line in Figure 3 represents the multi-resolution decomposition process. Wavelet packet decomposition is a generalization of multi-resolution decomposition using wavelet transform. Multi-resolution decomposition only decomposes the scale space V, while wavelet packet decomposition further decomposes the undecomposed wavelet space Wj from multi-resolution decomposition. Because wavelet space partitioning corresponds to frequency band partitioning, wavelet packet decomposition can achieve higher frequency resolution. Most conventional frequency band analysis is based on wavelet packet analysis, but while improving frequency resolution, it also increases algorithm complexity. Based on the needs of the practical problem, the authors chose a method based on multi-resolution analysis, which can meet the requirements. 2.1 Frequency Band Analysis Method Let the bandwidth of signal X(t) be [0, f], and the number of decomposition levels be N. Then, after multi-resolution decomposition, the signal frequency range corresponding to each space is analyzed for signals within different frequency bands. Typically, the signal can be decomposed at a certain scale based on the frequency range of interest, thereby extracting information within the corresponding frequency band. If the energy of the signal within each frequency band is statistically analyzed to form a feature vector reflecting the signal energy, this is called frequency band energy analysis. 2.2 Wavelet Frequency Bands and Energy Integration The theoretical basis of wavelet frequency band analysis technology is the Parseval energy integration equation. For discrete orthogonal wavelet transform, the Parseval equation is: where x(t) is the signal to be analyzed; Let be the wavelet transform coefficients. Equation (1) corresponds the energy of the signal in the time domain to the energy in the wavelet expansion domain. In this way, the changes in the constituent frequencies of the signal x(t) can be studied based on the changes in the wavelet coefficients in each frequency band. 2.3 Analysis Steps Based on the analysis results in Table 1, the band analysis based on multi-resolution analysis is implemented as follows: ① Using the prior knowledge of the system mathematical model, determine the cutoff frequency ωc of the object, and use 0～10ωc as the system bandwidth; ② Determine a suitable sampling frequency to ensure that the electromagnetic interference signal can be collected. If the sampling frequency is f, then the analysis frequency is ③ Determine a suitable wavelet decomposition level N so that 0～10ωc is exactly included in the low-frequency space VN, and divide the entire analysis space into a relative low-frequency space and a high-frequency space. Except for the low-frequency space VN where the system bandwidth is located, the other spaces WN, WN-1, and Wl are merged into the high-frequency space; ④ Select a suitable wavelet function for multi-resolution decomposition, and calculate the energy of the signal in the corresponding space (band) according to the wavelet coefficients obtained by decomposition according to Equation (1), forming a two-dimensional vector e=[e1, e2] representing the energy of the signal in the space, 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 energy of the space, 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 authors conducted a simulation experiment on the typical system shown in Figure 1. Under normal operating conditions, the values ​​of Gc(s), Gv(s), and Go(s) remained the same. At different stable moments in the system, R(s) and D1(s) were subjected to unit step changes; D2(s) was changed from 0 to a pulse signal with an amplitude of 1 or a periodic signal of 0.2sin100πt; the characteristic transfer functions of the object and sensor switched between normal and fault values ​​to simulate five causes and six forms of sudden changes in the output signal, and data for each sudden change process were collected. Regardless of the cause of the signal sudden change, its high-frequency signal component was generated instantaneously and disappeared quickly. Therefore, the high-frequency component accounted for a relatively small proportion of the total energy of the collected signal. To improve the detection sensitivity, the collected data was de-DC processed, i.e., the sampled data was subtracted from the average value of the 10 points before the signal sudden change. Additionally, zero-mean white noise with a variance of 0.003 was added to the sampled data. The system's sampling frequency f = 200 Hz and analysis frequency fo = 100 Hz were used. The db4 wavelet was selected to decompose the signal into three layers. Thus, the signal frequency range in the low-frequency space is 0 to 12.5 Hz, and the signal frequency range in the high-frequency space is 12.5 to 100 Hz. Hard threshold denoising was performed on the high-frequency coefficients obtained from the analysis. Then, the energy ratio was statistically analyzed according to formula (1). The results are shown in Table 2. In Table 2, the proportion of low-frequency components of the sudden signal caused by external electromagnetic interference is relatively small. This is because of the result of removing "DC". The proportion of high-frequency components of the sudden signal caused by the fault of the controlled object is very small. This is because the input frequency band of the sensor used in this simulation is only a few tens of Hz. The consistency between the simulation results in Table 2 and the theoretical analysis results in Table 1 shows the effectiveness of this method. 4 Experimental Study An experimental study was conducted using a constant pressure water supply system. As shown in Figure 4, the pressure sensor is of type LDG-S. The transfer function of the generalized object was determined to be G(s) = 1/(0.22s + 1). The regulator parameters were set as follows: proportional gain P = 142%, integral time ti = 3 s, and derivative time td = 2 s. From this, the low-frequency range frequency was estimated to be less than 4 Hz. At different stable system moments, the setpoint was adjusted to simulate sudden changes in the input signal, the zero point was adjusted to simulate a constant deviation fault in the sensor, the regulator proportional gain was adjusted to simulate a regulator fault, and surrounding motors were frequently started and stopped to simulate sudden changes in sensor output caused by electromagnetic fields. Experimental data were collected under various conditions. The system's sampling frequency f = 128 Hz and analysis frequency fo = 64 Hz were used. The db4 wavelet was used to decompose the signal into four layers. Thus, the signal frequency range in the low-frequency space is 0-4 Hz, and the signal frequency range in the high-frequency space is 4-64 Hz. Hard threshold denoising was performed on the high-frequency coefficients obtained from the analysis. Then, the energy ratio was statistically analyzed according to equation (1). The results are shown in Table 3. As can be seen from Table 3, the experimental results are consistent with the simulation results in Table 2 and the theoretical analysis results in Table 1, which shows the effectiveness of this method. 5 Conclusion The sudden change signal output by the sensor contains very important fault information. The frequency composition of the sudden change signal is different depending on the cause of the sudden change. For controlled objects with a large time constant, the sensor sudden change signal caused by changes in given input, changes in interference, controller faults, and actuator faults usually only contains low-frequency components. The sudden change signal caused by the fault of the controlled object usually only contains low-frequency components. The sudden change signal caused by external electromagnetic field interference is generally a pulse signal, containing low-frequency components and more high-frequency components. The abrupt change signal caused by sensor deviation faults contains not only low-frequency components but also a small amount of high-frequency components. The wavelet frequency band analysis method based on the system mathematical model proposed in this paper does not require high precision of the mathematical model and can effectively diagnose sensor faults, providing a new approach for sensor fault detection and performance evaluation.