Online wear state estimation method under semi-observable tool wear

Abstract

A wear state online estimation method under cutter wear semi-observability comprises the steps that firstly, a feature index training set and corresponding state indexes are established based on historical data, then a mapping model between the feature indexes and the state indexes is established, and model parameters are fixed; performing same feature extraction on the sensor data at the current moment, and applying the same feature extraction to the established mapping model; and finally, a more accurate state estimation value is obtained through a state index deviation compensation model. According to the method, the problem that the state estimation value cannot be adaptively adjusted according to individual differences in a traditional method is effectively solved, and online state updating of the multi-source sensor signals is achieved.

Description

Technical Field

[0001] The invention belongs to the technical field of mechanical component state index evaluation, and in particular relates to an online wear state estimation method for a tool under semi-observable wear. Background Art

[0002] In the process of industrial manufacturing and mechanical equipment operation, status indicators are crucial for evaluating equipment health, preventing potential failures, and ensuring the continuity and stability of the production process. Real-time monitoring of these status indicators through multi-sensor fusion technology can effectively optimize maintenance strategies, extend equipment life, and reduce unexpected downtime, thereby significantly improving production efficiency and reducing operating costs.

[0003] At present, most state estimation methods (Guo L, Li W, Zhu Y, et al. Composite disturbance filtering: A novel state estimation scheme for systems with multi-source, heterogeneous, and isomeric disturbances [J]. IEEE Open Journal of the Industrial Electronics Society, 2023.) rely on a large amount of historical data and the mapping relationship between sensor signals and state values ​​for training. However, in actual engineering applications, only a small amount of sensor data can be obtained; in addition, even with enough data, due to factors such as component manufacturing, environment and working conditions, there are large differences in the correspondence between sensor signals and state indicators between different components, resulting in poor adaptability of traditional models on different equipment; at the same time, state indicators such as tool wear values ​​are usually not directly observed and must be indirectly inferred through other signals, which further increases the complexity and uncertainty of state estimation. However, after the traditional model (Wang B, Lei Y, Li N, et al. A hybrid prognostics approach for estimating remaining useful life of rolling element bearings [J]. IEEE Transactions on Reliability, 2018, 69 (1): 401-412.) is trained using historical data, the model parameters cannot be updated online according to the current real-time data, resulting in the gradual failure of the mapping relationship between sensor signals and state values, which significantly affects the accuracy and reliability of state estimation, severely limiting its actual application in complex industrial environments. Summary of the invention

[0004] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide an online estimation method for the wear state of the tool under semi-observable wear, so that the tool component can dynamically adjust the sensor state value mapping model according to its own actual state value, thereby reducing the dependence on the similarity of training data and the amount of historical data; through this online update mechanism, the accuracy of the state estimation of multi-source sensor signals of the tool in industrial production can be significantly improved, and the adaptability of the model under different equipment and working conditions can be improved.

[0005] In order to achieve the above object, the technical solution adopted by the present invention is:

[0006] A wear state online estimation method for tool wear under semi-observable condition comprises the following steps:

[0007] Step 1: Extract features from the multi-dimensional signal data of historical sensors. The multi-dimensional signals include vibration signals, temperature signals, and acoustic emission signals, and obtain the feature training set:

[0008] Step 2: Measure the historical data status index, i.e., tool wear value, and establish a data set corresponding to the characteristic value:

[0009] Step 3: Establish a mapping model between multi-source sensor features and status indicators: process data from multiple samples, each of which contains sensor signals from multiple channels. The sensor signals include vibration signals, current signals, and temperature signals, and each channel contains multiple data points:

[0010] Step 4: Use the current actual sensor data to perform state estimation on the mapping model established in step 3. Assume that the actual data feature is X t , then the current state estimate is Y t , the specific expression is:

[0011] Y t =f t (X t )+∈ t (4)

[0012] Among them, X t is the characteristic value of the data at the current moment, Y t is the estimated value of the current state, ∈ t is the random error generated by the mapping model fitting, f t (·) is the regression function obtained by training with historical data;

[0013] Step 5: Establishment of error compensation model: In the process of establishing the mapping, the preliminary state estimation value is obtained by mapping the sensor signal characteristics and state indicators; the estimation result of the mapping model is corrected using the real state value;

[0014] Step 6: Calculate the actual state at the current time: After obtaining the characteristic value y at the current time m and the eigenvalue y at the next moment m+1 Afterwards, in order to compensate for the time when the true state value cannot be obtained during this period, the compensation value at these moments is approximately calculated using the linear interpolation method. Assuming t m and t m+1 They are y m and m+1 The corresponding time point, the eigenvalue y(t) corresponding to any time point t is calculated by the following formula, t m <t<t m+1 ;

[0015]

[0016] Among them, t m is the last real state time t m The eigenvalue of m+1 is the next real state at time t m+1 The eigenvalue of t is between t m and t m+1 Any time point in between.

[0017] The first step is specifically as follows:

[0018] 1.1) When processing sensor signals, feature extraction is first required to convert the historical sensor multi-dimensional signal data into representative features. The effective value feature RMS reflects the overall energy level of the signal by calculating the root mean square value of the signal. The mathematical expression of the effective value is:

[0019]

[0020] Among them, x i represents the sampling value of the signal, and N represents the total number of signal samples;

[0021] 1.2) Perform outlier processing on the obtained features. The outlier processing uses the Pearson correlation coefficient to screen out the signal features of the five channels with the highest correlation with the tool wear value from all signal channels as the feature training set.

[0022] The second step is specifically as follows:

[0023] 2.1) When analyzing and evaluating the tool status, it is necessary to measure the tool wear value in the historical data. The tool wear value is measured after observation under an electron microscope;

[0024] 2.2) Establish a real-time correspondence between sensor signal characteristics and state indicators. When the state indicators cannot be directly measured by sensor signals, use linear interpolation for approximate estimation.

[0025] The third step is specifically:

[0026] 3.1) Feature description: Assume there are M samples, each sample has C channels, each channel contains T feature points, and these data are represented by a three-dimensional tensor X, where The corresponding state indicator is represented by a vector Indicates that Y i is the state indicator of the i-th sample;

[0027] 3.2) Mapping model establishment: Establish a mapping model between sensor signal features and state indicators, integrate all sensor signal features into a feature matrix for training. Assuming that the sensor signal feature matrix is ​​X and the state indicator is Y, the mapping relationship expression is:

[0028] Y=f(X)+∈ (2)

[0029] Where f(·) is the mapping function and ∈ is the error term.

[0030] The step 3.2) is specifically as follows:

[0031] 3.2.1) Feature integration: Flatten or convert the three-dimensional tensor X into a format suitable for model input. Let the shape of X be (M, C, T), and flatten it into a two-dimensional matrix X with a shape of (M, C×T) flat ;

[0032] 3.2.2) Selection of mapping model: Select the Elastic Net Regression model, which can perform feature selection and deal with multicollinearity problems;

[0033]

[0034] Among them, α1 and α2 are regularization parameters, controlling ||β||1 and The regularized weight, X flat and Y flat is the feature matrix and state index used for training;

[0035] 3.2_3) Mapping model evaluation: using training data X flat and Y flat Mapping model training was performed, and the mapping model performance was evaluated using the cross-validation method. Parameters were adjusted and mapping model optimization was performed based on the evaluation results.

[0036] The fifth step is specifically as follows:

[0037] 5.1) Feature establishment: Let Y t is the estimated value of the state at the current moment, is the real state value at the current moment, that is, the actual tool wear value, Δy t It represents the deviation between the real state value and the estimated state value at time t. The specific expression is as follows:

[0038]

[0039] 5.2) Setting of fitting model: Suppose there are n deviation fitting models, and their fitting functions are f1(y, β1), f2(y, β2), f3(y, β3), …, f n (y,β n ), the model parameters are β1, β2, β3, …, β n Assuming that the i-th deviation fitting model has a parameters, its parameter expression is as follows:

[0040] β i =[θ i1 ,θ i2 ,θ i3 ,…,θ ia ] T (9)

[0041] 5_3) Establishment of loss function: Let the deviation value Δy at time t be t The corresponding time axis coordinate is x t , that is, given m data points {(x1, Y1), (x2, Y2), (x3, Y3), …, (x m , Y m )}, use coordinate translation to make (x1, Y1) correspond to (0, 0), the coordinate of the i-th data point after transformation is (x i -x1,Y i -Y1), and use the least squares method to substitute the data points after coordinate translation into the n deviation fitting models to obtain the corresponding parameters β i and the fitted eigenvalues;

[0042] The mean square error is selected as the fitting error metric function. For n deviation fitting functions, the optimal function is determined by comparing the size of the loss function. The smaller the loss function value, the better the fitting function effect. The minimum loss function is the required fitting function, that is, it meets the following conditions:

[0043]

[0044] Where i represents the deviation fitting model number (i=1, 2, 3, ..., n), and k represents the data point number (k=1, 2, 3, ..., m);

[0045] 5.4) Bias compensation model: Since the fitting effect still has a certain degree of deviation, the deviation caused by fitting will fluctuate within a certain range. Assume that the selected fitting function is fb 魄 (·);

[0046] y m+1 =f best (x m+1 ,θ best )+ε (12)

[0047] ε is the error term, which represents the deviation of the model fitting. It is assumed that its distribution follows a Gaussian distribution with a mean of 0 and a variance of

[0048] The deviation fitting function in step 5.2) includes the following functions:

[0049] (1) Linear function: The form is f(△y t )=p1△y t +b function, where p1 and b are constants;

[0050] (2) Exponential function: A function of , where p2 is a positive constant and p2≠1;

[0051] (3) Power function: A function of , where p3 is a constant;

[0052] Compared with the prior art, the present invention has the following beneficial effects:

[0053] The present invention proposes a state estimation method for online updating of fused multi-source sensor data. By establishing a deviation compensation model, the deviation caused by the mapping model is minimized, which has the advantage of improving the accuracy of state estimation. A feature index training set and a corresponding state index are established through historical data; and a mapping model of the feature index and the state index is established, and finally a more accurate state index is obtained by using a state index deviation value compensation model; the present invention realizes the online update of the state index estimated by multi-source sensor signals, and overcomes the problem that the previous method cannot adaptively adjust the state estimation value according to individual differences. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 The figure is a flow chart of the method of the present invention.

[0055] Figure 2 This is a diagram showing the application results of an embodiment of the present invention to a tool data set. DETAILED DESCRIPTION

[0056] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0057] Reference Figure 1 , an online estimation method of tool wear state under semi-observable wear, comprising the following steps:

[0058] Step 1: Extract features from the multi-dimensional signal data of historical sensors. The multi-dimensional signals include vibration signals, temperature signals, and acoustic emission signals, and obtain feature training sets: specifically:

[0059] 1.1) When processing sensor signals, feature extraction is first required to convert the historical sensor multi-dimensional signal data into representative features. The effective value feature RMS reflects the overall energy level of the signal by calculating the root mean square value of the signal. The mathematical expression of the effective value is:

[0060]

[0061] Among them, x i represents the sampling value of the signal, and N represents the total number of signal samples;

[0062] 1.2) Perform outlier processing on the obtained features. The outlier processing is to use the Pearson correlation coefficient to screen out the signal features of the five channels with the highest correlation with the tool wear value from all signal channels as the feature training set;

[0063] Step 2: Measure the tool wear value, the historical data status indicator, and establish a data set corresponding to the characteristic value:

[0064] 2.1) When analyzing and evaluating the tool status, it is necessary to measure the tool wear value in the historical data. The tool wear value is measured after observation under an electron microscope;

[0065] 2.2) Establish a real-time correspondence between sensor signal characteristics and state indicators. When the state indicators cannot be directly measured by sensor signals, use linear interpolation for approximate estimation;

[0066] Step 3: Establish a mapping model between multi-source sensor features and status indicators: process data from multiple samples, each sample contains sensor signals from multiple channels, the sensor signals include vibration signals, current signals and temperature signals, and each channel contains multiple data points:

[0067] 3.1) Feature description: Assume there are M samples, each sample has C channels, each channel contains T feature points, and these data are represented by a three-dimensional tensor X, where The corresponding state indicator is represented by a vector Indicates that Y i is the state indicator of the i-th sample;

[0068] 3.2) Mapping model establishment: Establish a mapping model between sensor signal features and state indicators, integrate all sensor signal features into a feature matrix for training. Assuming that the sensor signal feature matrix is ​​X and the state indicator is Y, the mapping relationship expression is:

[0069] Y=f(X)+∈ (2)

[0070] Where f(·) is the mapping function and ∈ is the error term;

[0071] The step 3.2) is specifically as follows:

[0072] 3.2.1) Feature integration: Flatten or convert the three-dimensional tensor X into a format suitable for model input. Let the shape of X be (M, C, T), and flatten it into a two-dimensional matrix X with a shape of (M, C×T) flat ;

[0073] 3.2.2) Selection of mapping model: Select the Elastic Net Regression model, which can perform feature selection and deal with multicollinearity problems;

[0074]

[0075] Among them, α1 and α2 are regularization parameters, controlling ||β||1 and The regularized weight, X flat and Y flat is the feature matrix and state index used for training;

[0076] 3.2.3) Mapping model evaluation: using training data X flat and Y flat Conduct mapping model training, use cross-validation method to evaluate mapping model performance, and adjust parameters and optimize mapping model based on the evaluation results;

[0077] Step 4: Use the current actual sensor data to perform state estimation on the mapping model established in step 3. Assume that the actual data feature is X t , then the current state estimate is Y t , the specific expression is:

[0078] Y t =f t (X t )+∈ t (4)

[0079] Among them, X tis the characteristic value of the data at the current moment, Y t is the estimated value of the current state, ∈ t is the random error generated by the mapping model fitting, f t (·) is the regression function obtained by training with historical data;

[0080] Step 5: Establishment of error compensation model: In the process of establishing the mapping, the preliminary state estimation value is obtained by mapping the sensor signal characteristics and state indicators; the estimation result of the mapping model is corrected using the real state value; specifically:

[0081] 5.1) Feature establishment: Let Y t is the estimated value of the state at the current moment, is the real state value at the current moment, that is, the actual tool wear value, △yt represents the deviation between the real state value and the estimated state value at time t, and the specific expression is as follows:

[0082]

[0083] 5.2) Setting of fitting model: Suppose there are n deviation fitting models, and their fitting functions are f1(y, β1), f2(y, β2), f3(y, β3), …, f n (y,β n ), the model parameters are β1, β2, β3, …, β n Assuming that the i-th fitting model has a parameters, its parameter expression is as follows:

[0084] β i =[θ i1 ,θ i2 ,θ i3 ,…,θ ia ] T (9)

[0085] 5_3) Establishment of loss function: Let the deviation value △y at time t be t The corresponding time axis coordinate is x t , that is, given m data points {(x1, Y1), (x2, Y2), (x3, Y3), …, (x m , Y m )}, use coordinate translation to make (x1, Y1) correspond to (0, 0), the coordinate of the i-th data point after transformation is (x i -x1,Y i -Y1), and use the least squares method to substitute the data points after coordinate translation into the n deviation fitting models to obtain the corresponding parameters β i and the fitted eigenvalues;

[0086] The mean square error is selected as the fitting error metric function. For n deviation fitting functions, the optimal function is determined by comparing the size of the loss function. The smaller the loss function value, the better the fitting function effect. The minimum loss function is the required fitting function, that is, it meets the following conditions:

[0087]

[0088] Where i represents the deviation fitting model number (i=1, 2, 3, ..., n), and k represents the data point number (k=1, 2, 3, ..., m);

[0089] 5.4) Bias compensation model: After selecting the optimal deviation fitting model, the corresponding function and model parameters are fixed accordingly. However, since there is still a certain degree of deviation in the fitting effect, the deviation caused by the fitting will fluctuate within a certain range. Assume that the selected fitting function is fbes t (·);

[0090] y m+1 =f best (x m+1 ,θ best )+ε (12)

[0091] ε is the error term, which represents the deviation of the model fitting. It is assumed that its distribution follows a Gaussian distribution with a mean of 0 and a variance of

[0092] The deviation fitting function in step 5.2) includes the following functions:

[0093] (1) Linear function: The form is f(△y t )=p1Δy t +b, where p1 and b are constants; (2) Exponential function: A function of , where p2 is a positive constant, and p21;

[0094] (3) Power function: A function of , where p3 is a constant;

[0095] Step 6: Calculate the actual state at the current time: After obtaining the characteristic value y at the current time m and the eigenvalue y at the next moment m+1 Afterwards, in order to compensate for the time when the true state value cannot be obtained during this period, the compensation value at these moments is approximately calculated using the linear interpolation method. Assuming t m and t m+1 They are y m and m+1 The corresponding time point, the eigenvalue y(t) corresponding to any time point t is calculated by the following formula, t m <t<tm+1 ;

[0096]

[0097] Among them, y m is the last real state time t m The eigenvalue of m+1 is the next real state at time t m+1 The eigenvalue of t is between t m and t m+1 Any time point in between.

[0098] In order to prove the effectiveness of the method of the present invention, this embodiment uses the vibration signal characteristics of the degradation trajectory of 4 tools, combined with the method of the present invention to carry out state estimation. The effective value (RMS) of the vibration signal is selected as the input feature of the model, and the method verification method uses cross-validation, that is, one tool is selected for testing, and the other tools are trained. Each tool has about 200 sensor data sampling time points. The vibration data of 8 sampling channels are extracted for characteristic values, and outlier processing is performed. A small amount of real state values ​​are processed by linear interpolation, which correspond to the characteristic values ​​of 200 sampling moments respectively; an elastic network regression model is selected to train the real state values ​​and characteristic values; linear functions, exponential functions, and power functions are selected as fitting models for deviation values, and a compensation model for deviations is established. The selected points are 3 consecutive points, that is, three consecutive points are used to fit the fourth point, and the obtained state compensation value is compensated to all sampling moments before the next real value appears through the linear difference method, and a comparison effect diagram before and after the model correction is obtained, as shown in FIG. Figure 2 As shown, it can be seen that the corrected state estimate is closer to the true value than the state estimate without any correction, that is, it has higher accuracy after correction.

[0099] In order to evaluate the performance of the method of the present invention in the whole process, the effective value of the deviation between the true value and the approximate fitting value is selected as the evaluation index. It can be seen that the method of the present invention has a good performance on this data set, which shows the effectiveness of the method of the present invention in state assessment. It should be pointed out that the adjustments and modifications made to the method of the present invention without departing from the concept of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A wear state online estimation method for tool wear under semi-observable conditions, characterized in that: The following steps are involved: Step 1: Extract features from the multi-dimensional signal data of historical sensors. The multi-dimensional signals include vibration signals, temperature signals, and acoustic emission signals, and obtain the feature training set: Step 2: Measure the historical data status index, i.e., tool wear value, and establish a data set corresponding to the characteristic value: Step 3: Establish a mapping model between multi-source sensor features and status indicators: process data from multiple samples, each of which contains sensor signals from multiple channels. The sensor signals include vibration signals, current signals, and temperature signals, and each channel contains multiple data points: Step 4: Use the current actual sensor data to perform state estimation on the mapping model established in step 3. Assume that the actual data feature is X t , then the current state estimate is Y t , the specific expression is: Y t =f t (X t )+∈ t (4) Among them, X t is the characteristic value of the data at the current moment, Y t is the estimated value of the current state, ∈ t is the random error generated by the mapping model fitting, f t (·) is the regression function obtained by training with historical data; Step 5: Establishment of error compensation model: In the process of establishing the mapping, the preliminary state estimation value is obtained by mapping the sensor signal characteristics and state indicators; the estimation result of the mapping model is corrected using the real state value; Step 6: Calculate the actual state at the current time: After obtaining the characteristic value y at the current time m and the eigenvalue y at the next moment m+1 Afterwards, in order to compensate for the moments during which the true state value cannot be obtained, the compensation values ​​at these moments are approximately calculated using the linear interpolation method. Assuming t m and t m+1 They are y m and m+1 The corresponding time point, the eigenvalue y(t) corresponding to any time point t is calculated by the following formula, t m <t<t m+1 ; Among them, y m is the last real state time t m The eigenvalue of m+1 is the next real state at time t m+1 The eigenvalue of t is between t m and t m+1 Any time point in between.

2. The wear state online estimation method according to claim 1, characterized in that: The first step is specifically as follows: 1.1) When processing sensor signals, feature extraction is first required to convert the historical sensor multi-dimensional signal data into representative features. The effective value feature RMS reflects the overall energy level of the signal by calculating the root mean square value of the signal. The mathematical expression of the effective value is: Among them, x i represents the sampling value of the signal, and N represents the total number of signal samples; 1.2) Perform outlier processing on the obtained features. The outlier processing uses the Pearson correlation coefficient to screen out the signal features of the five channels with the highest correlation with the tool wear value from all signal channels as the feature training set.

3. The wear state online estimation method according to claim 1, characterized in that: The second step is specifically as follows: 2.1) When analyzing and evaluating the tool status, it is necessary to measure the tool wear value in the historical data. The tool wear value is measured after observation under an electron microscope; 2.2) Establish a real-time correspondence between sensor signal characteristics and state indicators. When the state indicators cannot be directly measured by sensor signals, use linear interpolation for approximate estimation.

4. The wear state online estimation method according to claim 1, characterized in that: The third step is specifically: 3.1) Feature description: Assume there are M samples, each sample has C channels, each channel contains T feature points, and these data are represented by a three-dimensional tensor X, where The corresponding state indicator is represented by a vector Indicates that Y i is the state indicator of the i-th sample; 3.2) Mapping model establishment: Establish a mapping model between sensor signal features and state indicators, integrate all sensor signal features into a feature matrix for training. Assuming that the sensor signal feature matrix is ​​X and the state indicator is Y, the mapping relationship expression is: Y=f(X)+∈ (2) Where f(·) is the mapping function and ∈ is the error term.

5. The wear state online estimation method according to claim 4, characterized in that: The step 3.2) is specifically as follows: 3.2.1) Feature integration: Flatten or convert the three-dimensional tensor X into a format suitable for model input. Let the shape of X be (M, C, T), and flatten it into a two-dimensional matrix X with a shape of (M, C×T) flat ; 3.2.2) Selection of mapping model: Select the Elastic Net Regression model, which can perform feature selection and deal with multicollinearity problems; Among them, α1 and α2 are regularization parameters, controlling ||β||1 and The regularized weight, X flat and Y flat is the feature matrix and state index used for training; 3.2.3) Mapping model evaluation: using training data X flat and Y flat Mapping model training was performed, and the mapping model performance was evaluated using the cross-validation method. Parameters were adjusted and mapping model optimization was performed based on the evaluation results.

6. The wear state online estimation method according to claim 1, characterized in that: The fifth step is specifically as follows: 5.1) Feature establishment: Let Y t is the estimated value of the state at the current moment, is the real state value at the current moment, that is, the actual tool wear value, Δy t It represents the deviation between the real state value and the estimated state value at time t. The specific expression is as follows: 5.2) Setting of fitting model: Suppose there are n deviation fitting models, and their fitting functions are f1(y,β1),f2(y,β2),f3(y,β3),…,f n (y,β n ), the model parameters are β1, β2, β3, …, β n Assuming that the i-th deviation fitting model has a parameters, its parameter expression is as follows: b i =[θ i1 ,i i2 ,i i3 ,…,θ ia ] T (9) 5.3) Establishment of loss function: Let the deviation value Δy at time t be t The corresponding time axis coordinate is x t , that is, given m data points {(x1,Y1),(x2,Y2),(x3,Y3),…,(x m ,Y m )}, use coordinate translation to make (x1, Y1) correspond to (0, 0), the coordinates of the i-th data point after transformation are (x i -x1,Y i -Y1), and use the least squares method to substitute the data points after coordinate translation into the n deviation fitting models to obtain the corresponding parameters β i and the fitted eigenvalues; The mean square error is selected as the fitting error measurement function. For n deviation fitting functions, the optimal function is determined by comparing the size of the loss function. The smaller the loss function value, the better the fitting function effect. The minimum loss function is the required fitting function, that is, it meets the following conditions: Where i represents the deviation fitting model number (i=1,2,3,…,n), and k represents the data point number (k=1,2,3,…,m); 5.4) Bias compensation model: Since the fitting effect still has a certain degree of deviation, the deviation caused by fitting will fluctuate within a certain range. Assume that the selected fitting function is f best (·); y m+1 =f best (x m+1 ,θ best )+ε (12)ε is the error term, which represents the deviation of the model fitting. It is assumed that its distribution follows a Gaussian distribution with a mean of 0 and a variance of 7. The wear state online estimation method according to claim 6, characterized in that: The deviation fitting function in step 5.2) includes the following functions: (1) Linear function: The form is f(Δy t )=p1Δy t +b function, where p1 and b are constants; (2) Exponential function: A function of , where p2 is a positive constant and p2≠1; (3) Power function: A function of , where p3 is a constant.

Images