A tool wear state online estimation method under tool wear semi-observability

Abstract

An online estimation method for tool wear state under semi-observable conditions is proposed. First, a training set of feature indices and corresponding state indices are established based on historical data. Then, a mapping model between the feature indices and state indices is constructed, and the model parameters are fixed. Next, the sensor data at the current moment undergoes the same feature extraction and is applied to the established mapping model. Finally, a more accurate state estimate is obtained through a state index deviation compensation model. This invention effectively solves the problem that traditional methods cannot adaptively adjust state estimates based on individual differences, and realizes online state updates for multi-source sensor signals.

Description

Technical Field

[0001] This invention belongs to the technical field of mechanical component condition index evaluation, specifically relating to an online estimation method for wear state under semi-observable tool wear. Background Technology

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

[0003] Currently, 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 practical engineering applications, only a small amount of sensor data is often available. In addition, even with sufficient data, due to factors such as component manufacturing, environment, and operating conditions, the correspondence between sensor signals and state indicators varies greatly between different components, resulting in poor adaptability of traditional models to different devices. At the same time, state indicators such as tool wear values ​​are usually not directly observable and must be inferred indirectly through other signals, which further increases the complexity and uncertainty of state estimation. Traditional models (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.) cannot update their parameters online based on current real-time data after training with historical data. This causes the mapping relationship between sensor signals and state values ​​to gradually fail, which significantly affects the accuracy and reliability of state estimation and severely limits its practical application in complex industrial environments. Summary of the Invention

[0004] To overcome the shortcomings of the prior art, the purpose of this invention is to provide an online estimation method for tool wear state under semi-observable conditions, enabling tool components to dynamically adjust the sensor state value mapping model according to their own true state values, thereby reducing the dependence on the similarity of training data and the amount of historical data. Through this online update mechanism, the accuracy of tool state estimation from multiple sensor signals in industrial production can be significantly improved, and the adaptability of the model under different equipment and working conditions can be enhanced.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

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

[0007] Step 1: Extract features from historical sensor multidimensional signal data. The multidimensional signals include vibration signals, temperature signals, and acoustic emission signals, resulting in a feature training set.

[0008] Step 2: Measure the historical data status indicators, i.e., tool wear values, and establish a dataset corresponding to the feature values:

[0009] Step 3: Establish a mapping model between multi-source sensor features and state indicators: Process data from multiple samples, each sample containing sensor signals from multiple channels, including vibration signals, current signals, and temperature signals, with each channel containing multiple data points.

[0010] Step 4: Use the current actual sensor data to perform state estimation on the mapping model established in step 3, assuming the actual data features are X. t Then the estimated state value at the current moment is Y. t The specific expression is:

[0011] Y t =f t (X t )+∈ t (4)

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

[0013] Step 5, Establishment of the error compensation model: In the process of establishing the mapping, preliminary state estimates are obtained by mapping sensor signal characteristics and state indicators; the estimation results of the mapping model are corrected using the actual state values.

[0014] Step 6, calculate the actual state at the current time: obtain the feature value y at the current time. m and the eigenvalue y at the next time step m+1 Subsequently, to compensate for the values ​​at times when the true state values ​​could not be obtained, linear interpolation was used to approximate the compensation values ​​at these times, assuming t m and t m+1 They are y m and y m+1 The corresponding time point, the eigenvalue y(t) at any time point t is calculated using the following formula, t m <t<t m+1 ;

[0015]

[0016] Among them, t m It is the time t of the previous real state. m eigenvalues, y m+1 The next real state time t m+1 The eigenvalues, t is the eigenvalues ​​between t m and t m+1 Any point in time between.

[0017] The first step is specifically as follows:

[0018] 1.1) When processing sensor signals, feature extraction is the first step, transforming historical multidimensional sensor signal data into representative features. The RMS (Round Mean Square) feature reflects the overall energy level of the signal by calculating its root mean square value. The mathematical expression for the RMS is:

[0019]

[0020] Where, x i The value represents the sampled value of the signal, and N represents the total number of signal samples;

[0021] 1.2) Outlier processing is performed on the obtained features. Outlier processing is performed by using the Pearson correlation coefficient to select the signal features of the five channels with the highest correlation to the tool wear value from all signal channels, and use them as the feature training set.

[0022] The second step is specifically as follows:

[0023] 2.1) When analyzing and evaluating the condition of the cutting tool, it is necessary to measure the tool wear value in the historical data. The tool wear value is obtained by observation with an electron microscope.

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

[0025] The third step is specifically as follows:

[0026] 3.1) Feature Description: Assume there are M samples, each sample has C channels, and each channel contains T feature points. These data are represented by a three-dimensional tensor X, where... The corresponding state indicators are represented by vectors. It means that Y i It is the state index 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 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] Step 3.2) specifically refers to:

[0031] 3.2.1) Feature Integration: Flatten or convert the 3D tensor X into a format suitable for the model input. Let the shape of X be (M, C, T), flatten it into a 2D matrix X of shape (M, C×T). flat ;

[0032] 3.2.2) Selection of mapping model: Elastic Net Regression is selected, which can perform feature selection and handle multicollinearity problems;

[0033]

[0034] Where α1 and α2 are regularization parameters, controlling ||β||1 and α2. The weights for regularization, X flat and Y flat These are the feature matrix and state metrics used for training;

[0035] 3.2_3) Mapping Model Evaluation: Using Training Data X flat and Y flat Train the mapping model, evaluate its performance using cross-validation, and adjust the parameters and optimize the mapping model based on the evaluation results.

[0036] The fifth step is specifically as follows:

[0037] 5.1) Feature Establishment: Let Y t This is the estimated state value at the current moment. The actual state value at the current moment, i.e., the actual tool wear value, Δy t The deviation between the actual state value and the estimated state value at time t is expressed as follows:

[0038]

[0039] 5.2) Setting the Fitting Model: 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 Given a vector, assuming the i-th bias 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) Establishing the loss function: Let the deviation value at time t be Δy 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 By using coordinate translation, (x1, Y1) corresponds to (0, 0). The coordinates of the i-th data point after the transformation are (x1, Y1) / (0, 0). i -x1, Y i -Y1), using the least squares method, the data points after coordinate translation are substituted into n deviation fitting models to obtain the corresponding parameter β. i and the fitted feature values;

[0042] We choose mean squared error as the fitting error metric. For n bias fitting functions, we determine the optimal function by comparing them with the magnitude of the loss function. The smaller the loss function value, the better the fitting function. The minimum loss function is the desired fitting function, which satisfies the following condition:

[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 bias, the bias caused by the 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, representing the bias in the model fit. It is assumed to follow a Gaussian distribution with a mean of 0 and a variance of 0.

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

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

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

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

[0052] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0053] This invention proposes a state estimation method that integrates multi-source sensor data for online updates. By establishing a deviation compensation model, the method minimizes the deviation introduced by the mapping model, thereby improving the accuracy of state estimation. A training set of feature indicators and corresponding state indicators are established using historical data. A mapping model between the feature indicators and state indicators is then established. Finally, a state indicator deviation compensation model is used to obtain more accurate state indicators. This invention achieves online updates of state indicators estimated from multi-source sensor signals, overcoming the problem of previous methods being unable to adaptively adjust state estimates based on individual differences. Attached Figure Description

[0054] Figure 1 This is a flowchart of the method of the present invention.

[0055] Figure 2 This is a diagram showing the application results of the tool dataset in an embodiment of the present invention. Detailed Implementation

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

[0057] Reference Figure 1 A method for online estimation of tool wear state under semi-observable tool wear includes the following steps:

[0058] Step 1 involves feature extraction from historical sensor multidimensional signal data. The multidimensional signals include vibration signals, temperature signals, and acoustic emission signals, resulting in a feature training set. Specifically:

[0059] 1.1) When processing sensor signals, feature extraction is the first step, transforming historical multidimensional sensor signal data into representative features. The RMS (Round Mean Square) feature reflects the overall energy level of the signal by calculating its root mean square value. The mathematical expression for the RMS is:

[0060]

[0061] Where, x i The value represents the sampled value of the signal, and N represents the total number of signal samples;

[0062] 1.2) Outlier processing is performed on the obtained features. Outlier processing is performed by using the Pearson correlation coefficient to select the signal features of the five channels with the highest correlation to the tool wear value from all signal channels, and using them as the feature training set.

[0063] Step 2 involves measuring historical data status indicators, specifically tool wear values, and establishing a dataset corresponding to these feature values. Specifically:

[0064] 2.1) When analyzing and evaluating the condition of the cutting tool, it is necessary to measure the tool wear value in the historical data. The tool wear value is obtained by observation with 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 the sensor signals, use linear interpolation for approximate estimation.

[0066] Step 3: Establish a mapping model between multi-source sensor features and state indicators: Process data from multiple samples, each sample containing sensor signals from multiple channels, including vibration signals, current signals, and temperature signals. Each channel contains multiple data points. Specifically:

[0067] 3.1) Feature Description: Assume there are M samples, each sample has C channels, and each channel contains T feature points. These data are represented by a three-dimensional tensor X, where... The corresponding state indicators are represented by vectors. It means that Y i It is the state index 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 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] Step 3.2) specifically refers to:

[0072] 3.2.1) Feature Integration: Flatten or convert the 3D tensor X into a format suitable for the model input. Let the shape of X be (M, C, T), flatten it into a 2D matrix X of shape (M, C×T). flat ;

[0073] 3.2.2) Selection of mapping model: Elastic Net Regression is selected, which can perform feature selection and handle multicollinearity problems;

[0074]

[0075] Where α1 and α2 are regularization parameters, controlling ||β||1 and α2. The weights for regularization, X flat and Y flat These are the feature matrix and state metrics used for training;

[0076] 3.2.3) Mapping model evaluation: using training data X flat and Y flat Train the mapping model, evaluate its performance using cross-validation, and adjust the parameters and optimize the 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, assuming the actual data features are X. t Then the estimated state value at the current moment is Y. t The specific expression is:

[0078] Y t =f t (X t )+∈ t (4)

[0079] Among them, X tY represents the feature value of the data at the current time. t The estimated state at the current time, ∈ t f represents the random error generated during the fitting of the mapping model. t (·) represents the regression function obtained by training with historical data;

[0080] Step 5, Establishing the Error Compensation Model: During the mapping process, preliminary state estimates are obtained by mapping sensor signal characteristics and state indicators; the estimation results of the mapping model are then corrected using the actual state values; specifically:

[0081] 5.1) Feature Establishment: Let Y t This is the estimated state value at the current moment. The actual state value at the current moment is Δyt, which represents the actual tool wear value. Δyt represents the deviation between the actual state value and the estimated state value at time t. The specific expression is as follows:

[0082]

[0083] 5.2) Setting the Fitting Model: 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 Given a vector, assuming the i-th fitted 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) Establishing the 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 By using coordinate translation, (x1, Y1) corresponds to (0, 0). The coordinates of the i-th data point after the transformation are (x1, Y1) / (0, 0). i -x1, Y i -Y1), using the least squares method, the data points after coordinate translation are substituted into n deviation fitting models to obtain the corresponding parameter β. i and the fitted feature values;

[0086] We choose mean squared error as the fitting error metric. For n bias fitting functions, we determine the optimal function by comparing them with the magnitude of the loss function. The smaller the loss function value, the better the fitting function. The minimum loss function is the desired fitting function, which satisfies the following condition:

[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 bias fitting model, the corresponding function and model parameters are fixed. However, since the fitting effect still has a certain degree of bias, the bias 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, representing the bias in the model fit. It is assumed to follow a Gaussian distribution with a mean of 0 and a variance of 0.

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

[0093] (1) Linear function: of the form f(Δy) t )=p1Δy t (1) Functions of type +b, where p1 and b are constants; (2) Exponential functions: in the form of... The function of , where p2 is a positive constant and p21;

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

[0095] Step 6, calculate the actual state at the current time: obtain the feature value y at the current time. m and the eigenvalue y at the next time step m+1 Subsequently, to compensate for the values ​​at times when the true state values ​​could not be obtained, linear interpolation was used to approximate the compensation values ​​at these times, assuming t m and t m+1 They are y m and y m+1 The corresponding time point, the eigenvalue y(t) at any time point t is calculated using the following formula, t m <t<tm+1 ;

[0096]

[0097] Among them, y m It is the time t of the previous real state. m eigenvalues, y m+1 The next real state time t m+1 The eigenvalues, t is the eigenvalues ​​between t m and t m+1 Any point in time between.

[0098] To demonstrate the effectiveness of the method of this invention, this embodiment utilizes the vibration signal characteristics of the degradation trajectories of four cutting tools, combined with the method of this invention, to perform state estimation. The effective value (RMS) of the vibration signal is selected as the input feature of the model. Cross-validation is used for method verification, i.e., one tool is used for testing, and other tools are used for training. Each tool has approximately 200 sensor data sampling points. Feature values ​​are extracted from the vibration data of eight sampling channels, and outlier handling is performed. A small number of true state values ​​are linearly interpolated and mapped to the feature values ​​of the 200 sampling times. An elastic network regression model is used to train the true state values ​​and feature values. Linear functions, exponential functions, and power functions are selected as fitting models for the deviation values ​​to establish a deviation compensation model. Three consecutive points are selected, i.e., three consecutive points are used to fit the fourth point. The obtained state compensation value is then used to compensate for all sampling times before the next true value appears using a linear difference method. A comparison of the model before and after correction is obtained, as shown in the figure. Figure 2 As shown, the corrected state estimate is closer to the true value than the uncorrected state estimate, meaning that the corrected estimate has higher accuracy.

[0099] To evaluate the performance of the method of the present invention throughout the process, the effective value of the deviation between the true value and the approximate fitted value was selected as the evaluation index. It can be seen that the method of the present invention performs well on this dataset, indicating its effectiveness in state evaluation. It should be noted that any adjustments and modifications made to the method of the present invention without departing from the inventive concept should also be considered within the scope of protection of the present invention.

Claims

1. A method for online estimation of tool wear state under semi-observable tool wear, characterized in that, Includes the following steps: Step 1: Extract features from historical sensor multidimensional signal data. The multidimensional signals include vibration signals, temperature signals, and acoustic emission signals, resulting in a feature training set. Step 2: Measure the historical data status indicators, i.e., tool wear values, and establish a dataset corresponding to the feature values: Step 3: Establish a mapping model between multi-source sensor features and state indicators: Process data from multiple samples, each sample containing sensor signals from multiple channels, including vibration signals, current signals, and temperature signals, with each channel containing multiple data points. Step 4: Use the current actual sensor data to perform state estimation on the mapping model established in step 3, assuming the actual data features are X. t Then the estimated state value at the current moment is Y. t The specific expression is: Y t =f t (X t )+∈ t (4) Among them, X t Y represents the feature value of the data at the current time. t The estimated state at the current time, ∈ t f represents the random error generated during the fitting of the mapping model. t (·) represents the regression function obtained by training with historical data; Step 5, Establishment of the error compensation model: In the process of establishing the mapping, preliminary state estimates are obtained by mapping sensor signal characteristics and state indicators; the estimation results of the mapping model are corrected using the actual state values. Step 6, calculate the actual state at the current time: obtain the feature value y at the current time. m and the eigenvalue y at the next time step m+1 Subsequently, to compensate for the values ​​at times when the true state values ​​could not be obtained, linear interpolation was used to approximate the compensation values ​​at these times, assuming t m and t m+1 They are y m and y m+1 The corresponding time point, the eigenvalue y(t) at any time point t is calculated using the following formula, t m <t<t m+1 ; Among them, y m It is the time t of the previous real state. m eigenvalues, y m+1 The next real state time t m+1 The eigenvalues, t is the eigenvalues ​​between t m and t m+1 Any point in time between.

2. The online wear condition 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 the first step, transforming historical multidimensional sensor signal data into representative features. The RMS (Round Mean Square) feature reflects the overall energy level of the signal by calculating its root mean square value. The mathematical expression for the RMS is: Where, x i The value represents the sampled value of the signal, and N represents the total number of signal samples; 1.2) Outlier processing is performed on the obtained features. Outlier processing is performed by using the Pearson correlation coefficient to select the signal features of the five channels with the highest correlation to the tool wear value from all signal channels, and use them as the feature training set.

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

4. The online wear condition estimation method according to claim 1, characterized in that, The third step is specifically as follows: 3.1) Feature Description: Assume there are M samples, each sample has C channels, and each channel contains T feature points. These data are represented by a three-dimensional tensor X, where... The corresponding state indicators are represented by vectors. It means that Y i It is the state index 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 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 online wear condition estimation method according to claim 4, characterized in that, Step 3.2) specifically refers to: 3.2.1) Feature Integration: Flatten or convert the 3D tensor X into a format suitable for the model input. Let the shape of X be (M, C, T), flatten it into a 2D matrix X of shape (M, C × T). flat ; 3.2.2) Selection of mapping model: Elastic Net Regression is selected, which can perform feature selection and handle multicollinearity problems; Where α1 and α2 are regularization parameters, controlling ||β||1 and α2. The weights for regularization, X flat and Y flat These are the feature matrix and state metrics used for training; 3.2.3) Mapping model evaluation: using training data X flat and Y flat Train the mapping model, evaluate its performance using cross-validation, and adjust the parameters and optimize the mapping model based on the evaluation results.

6. The online wear condition estimation method according to claim 1, characterized in that, The fifth step is specifically as follows: 5.1) Feature Establishment: Let Y t This is the estimated state value at the current moment. The actual state value at the current moment, i.e., the actual tool wear value, Δy t The deviation between the actual state value and the estimated state value at time t is expressed as follows: 5.2) Setting the Fitting Model: 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 Given a vector, assuming the i-th bias 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) Establishing the loss function: Let the deviation value at time t be Δy. 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 By using coordinate translation, (x1, Y1) corresponds to (0, 0). The coordinates of the i-th data point after the transformation are (x1, Y1) / (0, 0). i -x1,Y i -Y1), using the least squares method, the data points after coordinate translation are substituted into n deviation fitting models to obtain the corresponding parameter β. i and the fitted feature values; We choose mean squared error as the fitting error metric. For n bias fitting functions, we determine the optimal function by comparing them with the magnitude of the loss function. The smaller the loss function value, the better the fitting function. The minimum loss function is the desired fitting function, which satisfies the following condition: 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 bias, the bias caused by the 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, representing the deviation of the model fit. It is assumed that its distribution follows a Gaussian distribution with a mean of 0 and a variance of .

7. The online wear condition estimation method according to claim 6, characterized in that, The deviation fitting function in step 5.2) includes the following functions: (1) Linear function: of the form f(Δy) t )=p1Δy t A function of +b, where p1 and b are constants; (2) Exponential function: in the form of The function of p2, where p2 is a positive constant and p2≠1; (3) Power function: in the form of The function is denoted by p3, where p3 is a constant.

Images