Motor multi-parameter identification method based on sampling noise excitation and bias compensation

Abstract

The application discloses a motor multi-parameter identification method based on sampling noise excitation and bias compensation. Firstly, motor voltage and sampling current are acquired, and the sampling current contains sampling noise; then, an information evaluation index is constructed, and the proportional parameter of the current controller is adjusted to improve the identification information quantity; then, shaft voltage feedforward compensation is constructed to suppress electromagnetic torque ripple; then, the least square method is used to recursively obtain parameter initial values and covariance matrices based on a discrete linear parameterized model; then, parameters containing bias are estimated online, and sampling noise variance is calculated by using estimation residual; finally, unbiased parameter estimation is obtained by bias compensation iterative update, so as to output online estimation values of the stator resistance, direct-axis inductance and permanent magnet flux linkage. L d , quadrature-axis inductance L q and permanent magnet flux linkage.

Description

Technical Field

[0001] This invention relates to the field of motor control technology, and more specifically to a method for identifying multiple parameters of a motor based on sampling noise excitation and bias compensation. Background Technology

[0002] Currently, permanent magnet synchronous motors ( Permanent Magnet Synchronous Motor , PMSM With its superior performance such as high efficiency, high power density, and fast dynamic response, it has been widely used in key fields such as industrial automation, new energy vehicles, rail transportation, and aerospace. PMSM Precise control depends on the accuracy of the voltage-current model and the torque-current model, while the stator resistance ( ), direct-axis inductor ( L d ), quadrature axis inductance ( L q ), permanent magnet flux ( Core parameters such as these are the foundation for building the above model, and their accuracy directly determines the dynamic performance, steady-state accuracy, efficiency optimization effect, and field weakening speed extension capability of motor control.

[0003] However, in engineering practice PMSM Online parameter identification has long faced two irreconcilable core contradictions, making it difficult for existing technologies to balance identification accuracy, system stability, and industrial applicability.

[0004] The first type of contradiction is the problem of observability and insufficient excitation: When multiple parameters are identified online at the same time, they are usually limited by insufficient excitation during motor operation and strong coupling characteristics between parameters. The amount of information in parameter identification is insufficient, which can easily lead to divergent identification results or low accuracy. Currently, most of these contradictions are addressed by high-frequency voltage injection method, frequency scanning method, specific voltage vector sampling method, or by increasing the amount of information through shutdown / low-speed testing. However, the above methods will mostly increase the complexity of the system or disturb the mechanical performance.

[0005] The second type of contradiction concerns the estimation bias and reliability problem. In the industrial field, algorithms such as least squares (LS) or recursive least squares (RLS) are widely used. However, in actual industrial environments, noise interference is unavoidable in the current sampling stage, causing the regression matrix and output vector to be simultaneously contaminated by noise during the identification process, forming a typical error-variable ( Errors - in - Variables , EIV The structure of the LS / RLS algorithm can introduce an inherent estimation bias, which may lead to the risk that the identification result appears to converge but actually converges to the wrong value. This seriously affects the reliability of parameter identification, which in turn leads to the deterioration of motor control performance and may even cause the control system to become unstable.

[0006] A deeper analysis of the root causes of the above-mentioned defects reveals that, firstly, to improve identifiability, existing technologies mostly rely on external excitation or operating condition switching; however, in industrial drives, injection and switching can introduce torque disturbances, noise and NVH problems, and increase the difficulty of parameter tuning; secondly, the error-variable structure leads to LS / RLS bias, because sampling noise not only contaminates the output y(k) but also enters the regression matrix x(k) through difference and product terms, resulting in a non-negligible bias term; thirdly, the noise covariance varies significantly under different operating conditions. If the noise variance is calibrated offline only, the bias compensation will fail when the online operating conditions change, leading to insufficient estimation reliability.

[0007] In view of this, this application has conducted in-depth research on this basis, resulting in this case. Summary of the Invention

[0008] The purpose of this invention is to provide a multi-parameter identification method for motors based on sampling noise excitation and bias compensation, which can achieve high-precision convergence of multiple parameters with low computational load under actual working conditions, thereby improving model reliability and control performance.

[0009] To achieve the above objectives, the solution of the present invention is: a multi-parameter identification method for motors based on sampling noise excitation and bias compensation, comprising the following steps:

[0010] Step 1: Obtain the sampling current of the permanent magnet synchronous motor in the synchronous rotating coordinate system. , and control the output voltage , The sampling current contains sampling noise;

[0011] Step 2: Construct information evaluation indicators and adjust the proportional parameters in the current controller. To meet the information evaluation index under stability constraints, the sampling noise is used by the current controller to generate an effective excitation for identification, thereby increasing the amount of identification information.

[0012] The constructed information evaluation index is as follows: In the formula, for The information evaluation index of the axis, where det is used to calculate the determinant of the matrix. for Fisher information matrix of the axis, for The variance of the sampling noise of the shaft sampling current. The frequency contained in the input signal. For the input signal at The corresponding signal amplitude at that location, for Shaft inductor, for Shaft inductor, For stator resistance, Electric angular velocity;

[0013] The stability constraint is as follows: >0;

[0014] Step 3, the current controller output Axis control output voltage and according to Fluctuation of shaft control output voltage Build Shaft voltage feedforward compensation To suppress the electromagnetic torque pulsation caused by step 2 and output Shaft control output voltage;

[0015] Step 4: Construct a discrete linear parameterized model, and use the least squares method to recursively obtain the parameter estimates and covariance matrix based on the discrete linear parameterized model;

[0016] Step 5: Estimate the bias parameters online using the least squares method, and calculate the sampling noise variance using the estimated residuals;

[0017] Step 6: Obtain the unbiased parameter estimate by iteratively updating according to the bias compensation, and output the stator resistance. Direct-axis inductor L d quadrature axis inductance L q and permanent magnet magnetic flux The online estimate.

[0018] In step 2, the current controller is a PI controller, and the transfer function of the PI controller is: In the formula, Pass functions to the PI controller. For the Laplace transform, The proportional parameter of the PI controller, These are the integral parameters of the PI controller;

[0019] The power spectral density dominated by the proportional term is: In the formula, This is the lower bound of the output power spectral density of the PI controller. For frequency, The variance of the current sampling noise;

[0020] Among them, increasing the value in the transfer function of the PI controller is adopted. Parameters to improve Axis control output voltage The power spectral density.

[0021] In step 3, The shaft control output voltage is:

[0022] ,

[0023] In the formula, for Shaft output voltage, To increase The subsequent PI controller, for Control error of shaft sampling current; for Reference value for shaft sampling current;

[0024] The fluctuation value of the shaft control output voltage is: In the formula, for time Shaft output voltage fluctuation value, for time shaft voltage, for time The expected estimate of the axis control voltage within a short observation window;

[0025] in, Using a recursive summation form with a forgetting factor, we obtain the following formula:

[0026] ,

[0027] ;

[0028] In the formula, The forgetting factor is greater than 0 and less than 1. for exist The summation of forgetting moments.

[0029] The formula for shaft voltage feedforward compensation is: In the formula, for time Shaft feedforward compensation voltage, for time Shaft current sampling value, for Shaft inductor, for Shaft inductor, It is a permanent magnet flux linkage.

[0030] In step 4, the discrete linear parameterized model is:

[0031] ,

[0032] ;

[0033] In the formula, Electric angular velocity, For stator resistance, for Output matrix at time step; for The input matrix at time t, its superscript Indicates transpose; for The electric angular velocity at time -1 for Time and -1 hour The change in shaft current The target parameter vector.

[0034] In step 5, the formula obtained by recursion using the least squares method is:

[0035] ,

[0036] ,

[0037] ,

[0038] ;

[0039] In the formula, For parameter estimation vectors; for Gain matrix at time step for The covariance matrix at time t, for The covariance matrix at time t, for time The denominator of the matrix, It is a 2-dimensional identity matrix.

[0040] The residual energy recursion matrix is ​​constructed as follows:

[0041] ,

[0042] In the formula, The residual energy recursion matrix is... For the amount of data required for estimation, for The residuals calculated using the recursive least squares method at time points. for Output matrix at time step;

[0043] The formula for calculating the estimated residual is as follows: In the formula, for The estimated residuals are calculated using the recursive least squares method at each time step.

[0044] The variance of the sampling noise estimated online is:

[0045] (twenty one);

[0046] In the formula, for Variance estimation of sampling noise for shaft sampling current; for Real parameter vector at any given time;

[0047] in, The intermediate variable matrix is ​​derived, where, .

[0048] In step 6, the formula for calculating the unbiased parameter estimate is:

[0049] ;

[0050] In the formula, , for The estimation of the true parameter vector at each time step, therefore ,in, This is an estimated value for the stator resistance. This is an estimated value for the direct-axis inductance. This is an estimated value for the quadrature-axis inductance. This is an estimated value for the magnetic flux linkage of the permanent magnet.

[0051] By employing the above method, the present invention has the following beneficial effects: In current closed-loop control, the sampling noise is regarded as a broadband random excitation source, and the proportional parameter in the current controller is adjusted... To improve observability and information content; and, to avoid Improvement leads to Increased shaft output voltage fluctuations lead to torque pulsation. A design based on the desired short window is employed. Shaft output voltage fluctuation and calculate Shaft feedforward compensation To counteract torque disturbances and suppress torque pulsation, a discrete linear parameterized model is established. Initial parameter values ​​are obtained using recursive least squares with multiple inputs and multiple outputs. Then, the noise variance is estimated online, and the bias expression derived from the error-variable structure is used to iteratively compensate for the biased parameters estimated by the least squares method, resulting in unbiased parameter estimates. Compared with existing technologies, this invention can achieve high-precision convergence of multiple parameters with low computational cost under actual operating conditions, improving model reliability and control performance. Attached Figure Description

[0052] Figure 1 This is a system control block diagram of the present invention. Detailed Implementation

[0053] To further explain the technical solution of the present invention, the present invention will be described in detail below through specific embodiments.

[0054] A method for identifying multiple parameters of a motor based on sampling noise excitation and bias compensation is proposed. This method is applicable to the control system of a permanent magnet synchronous motor. The control system includes a current controller, which is a conventional dual closed-loop PI controller.

[0055] like Figure 1 As shown, the multi-parameter identification method for motors includes the following steps.

[0056] Step 1: Obtain the sampling current of the permanent magnet synchronous motor in the synchronous rotating coordinate system. , and control the output voltage , The sampling current contains sampling noise.

[0057] To elaborate, obtaining the control voltage of a permanent magnet synchronous motor, q Shaft current and rotor mechanical angle information, Shaft control voltage and q Shaft control voltage ,through Clark , Park coordinate transformation Shaft sampling current , q Shaft sampling current and electric angular velocity .

[0058] It should be noted that the acquisition of the control voltage, current and rotor mechanical angle information mentioned above is a conventional technical method in this field, and therefore will not be described in detail.

[0059] Step 2: Construct information evaluation indicators and adjust the proportional parameters in the PI controller. To meet the information evaluation index under stability constraints, the sampling noise is used by the PI controller to generate an effective excitation that can be used for identification, thereby improving the amount of identification information.

[0060] To elaborate, the information evaluation index is established using the D-optimal criterion. This information evaluation index is as follows:

[0061] (1);

[0062] In the formula, for The information evaluation index of the axis, where det is used to calculate the determinant of the matrix. for Fisher information matrix of the axis, for The variance of the sampling noise of the shaft sampling current. The frequency contained in the input signal. For the input signal at The corresponding signal amplitude at that location, for Shaft inductor, for q Shaft inductor, For stator resistance, Electric angular velocity, This represents the total number of peak frequencies contained in the signal. Indicates the first Each frequency.

[0063] It should be noted that in formula (1), and All are determined by the power spectral density of the input signal; the higher the power spectral density, The larger it is, the more it improves. The power spectral density of the shaft input signal can be effectively improved. That is, to improve The information content in the shaft output signal is increased, thereby improving the estimation effect of parameters.

[0064] Furthermore, the aforementioned information satisfaction evaluation indicators are as follows: det(M) > 0, meaning that only when det(M) is greater than 0 can the parameters in the permanent magnet synchronous motor be identified, and The larger the size, the more information it contains, and the better the recognition effect.

[0065] Furthermore, in the synchronous coordinate system, the actual sampled current (i.e., the input signal) is expressed as:

[0066] (2);

[0067] In the formula, The obtained sampled current contains the actual current and the sampled noise; This represents the actual current. This is sampling noise, which can be considered to be wide stationary with zero mean; This represents the variance of the sampling noise. It's worth noting that, due to... d Shaft sampling current The impact on the torque of the permanent magnet synchronous motor is small, therefore, in this embodiment, we choose... Shaft sampling current The input is fed into the PI controller as the adjustment target.

[0068] In this embodiment, when the current controller is a PI controller, the transfer function of the PI controller is: (3), where, Pass functions to the PI controller. For the Laplace transform, The proportional parameter of the PI controller, These are the integral parameters of the PI controller.

[0069] In this way, the sampling noise can be tuned by the PI controller. Shaft control voltage The random stimulus used for identification is embodied in formulas (2) and (3), which can be used to derive formula (4), that is... The power spectral density of the shaft control voltage is:

[0070] (4);

[0071] Among them, the lower bound of the power spectral density, dominated by the proportional term, is used to simplify online parameter tuning, i.e., the power spectral density is optimized as follows: (5), where, for The lower bound of the output power spectral density of the shaft current PI controller. For frequency.

[0072] As can be seen from formula (4), in a PI controller, by increasing the transfer function... Parameters can be effectively improved Shaft output voltage The power spectral density is increased, thereby increasing the amount of information contained in the input and output signals and improving the identifiability of parameters.

[0073] Step 3, PI controller output Axis control output voltage and according to Axis control output voltage Fluctuation Build Shaft voltage feedforward compensation This is to suppress the electromagnetic torque pulsation caused by step 2 and to compensate for it. Axis control output voltage .

[0074] To elaborate, the PI controller outputs the corrected value after step 2. Axis control output voltage ,Should The shaft control output voltage is:

[0075] (6),

[0076] In the formula, for Shaft control output voltage; ,for Control error of shaft sampling current; for Shaft sampling current Reference values; among which, To increase The PI controller following the parameters is a single unit, and is an abbreviation for "modified PI controller".

[0077] To suppress due to Electromagnetic torque pulsation caused by shaft control output voltage fluctuations is defined as follows: The fluctuation value of the shaft control output voltage is: (7); where, for time The fluctuation value of the shaft control output voltage. for time Shaft control output voltage, for time The expected estimate of the axis control voltage within a short observation window.

[0078] Then, by replacing the expectation in equation (7) with the average of the accumulated values ​​within the observation window, and using a recursive summation method, we can obtain the following formula:

[0079] (8),

[0080] (9);

[0081] In the formula, The forgetting factor is greater than 0 and less than 1. for exist The summation of forgetting moments.

[0082] Furthermore, the above The current fluctuation caused by the fluctuation of the shaft control output voltage is:

[0083] (10);

[0084] In the formula, for time The fluctuation value of the shaft sampling current; The sampling period.

[0085] Similarly, the above The current fluctuation caused by the shaft output voltage fluctuation is:

[0086] (11);

[0087] In the formula, for time Shaft current fluctuation value.

[0088] The torque formula for the aforementioned permanent magnet synchronous motor is: (12); where, For electromagnetic torque, This represents the number of pole pairs of the motor.

[0089] In this embodiment, to ensure The shaft output voltage fluctuation will not cause torque fluctuation or torque jitter, therefore reverse injection is required. The shaft output voltage, obtained by using formulas (10)-(12) while keeping the torque constant, can be derived. Shaft voltage feedforward compensation In other words, the compensation voltage calculated by the following formula (13) will be superimposed on the PI controller. Shaft control output voltage This is to eliminate torque jitter.

[0090] in, The formula for shaft voltage feedforward compensation is:

[0091] (13).

[0092] in, The shaft control output voltage is: (14), where, For the original Axis control output voltage (i.e., before compensation) The axis control / output voltage is also directly output by the PI controller. (axis voltage) for time Shaft feedforward compensation voltage, for time Shaft current sampling value, for Shaft inductor, for Shaft inductor, It is a permanent magnet flux linkage.

[0093] Step 4: Construct a discrete linear parameterized model, and then, based on the discrete linear parameterized model, use the multi-input multi-output least squares method to recursively obtain the parameter estimation vector and covariance matrix.

[0094] To elaborate, the aforementioned discrete linear parameterized model is as follows:

[0095] ,

[0096] (14);

[0097] In the formula, Electric angular velocity, For stator resistance, for Output matrix at time step; for The input matrix at time t, its superscript Indicates transpose; for The electric angular velocity at time -1 for Time and -1 hour The change in shaft current; The target parameter vector is the vector consisting of the parameters to be estimated.

[0098] Furthermore, using the conventional multiple-input multiple-output least squares (RLS) method, the following formula is obtained recursively:

[0099] (15),

[0100] (16)

[0101] (17)

[0102] (18);

[0103] In the formula, The parameter estimation vector is the parameter vector estimated by the least squares method, which is the parameter vector with bias. for Gain matrix at time step for The covariance matrix at time t, for The covariance matrix at time t, for time The denominator of the matrix, It is a 2-dimensional identity matrix. Formula (15) is also called the error-variable structure.

[0104] Step 5: Estimate the bias parameters online using the least squares method, and calculate the sampling noise variance using the estimated residuals.

[0105] To elaborate, we construct the residual energy recursive matrix. That is, the equivalent cumulative squared residual, the residual energy recursion matrix is:

[0106] (19);

[0107] In the formula, The residual energy recursion matrix is... The amount of data required for estimation; for The residuals calculated using the recursive least squares method at time step 1, where, for The output matrix at time step.

[0108] The formula for calculating the estimated residual is as follows: (20), where, for The estimated residuals are calculated using the recursive least squares method at each time step.

[0109] Furthermore, the variance of the sampling noise estimated online is:

[0110] (twenty one);

[0111] In the formula, for Variance estimation of sampling noise for shaft sampling current; for The true parameter vector at time step is, in this embodiment, the unbiased estimator estimated from the previous time step. Replacement.

[0112] The intermediate variable matrix is ​​derived, where, .

[0113] Step 6: Iteratively update the unbiased parameter estimates according to the bias compensation to output the stator resistance. Direct-axis inductor L d quadrature axis inductance L q and permanent magnet magnetic flux The online estimate.

[0114] In this embodiment, the biased parameters are estimated using a conventional unbiased estimator iteratively compensated least squares method. Specifically, the biased parameters are iteratively updated to obtain unbiased parameter estimates. The formula for calculating these unbiased parameter estimates is as follows:

[0115] (twenty two);

[0116] In the formula, , for The estimation of the true parameter vector at each time step, therefore ,in This is an estimated value for the stator resistance. This is an estimate of the direct-axis inductance. This is an estimated value for the quadrature-axis inductance. This is an estimated value for the flux linkage of the permanent magnet.

[0117] It is worth mentioning that the aforementioned stator resistor Direct-axis inductor L d quadrature axis inductance L q and permanent magnet magnetic flux The online estimates are output to the corresponding positions in the control system in the conventional manner, which will not be described in detail here.

[0118] This invention discloses a multi-parameter identification method for motors based on sampling noise excitation and bias compensation. It uses sampling noise as an intrinsic excitation and improves the information content through parameter adjustment of the aforementioned current controller, thereby reducing disturbances to mechanical performance and parameter tuning costs. This solves the problems of insufficient excitation, limited observability, and unsuitability for injection in multi-parameter identification. Furthermore, it employs torque ripple suppression in permanent magnet synchronous motors. q Shaft feedforward compensation is used to address the problem that increasing the gain of the current controller in the traditional way will enhance the excitation and introduce torque ripple. In addition, online noise covariance estimation and unbiased iterative compensation are used to solve the problem that the traditional LS / RLS has inherent bias under sampling noise and that the compensation fails due to the change of noise statistics with operating conditions.

[0119] The above description is only a preferred embodiment of this invention. Any equivalent changes and modifications made within the scope of the claims of this invention shall fall within the scope of the claims of this invention.

Claims

1. A method for multi-parameter identification of motors based on sampling noise excitation and bias compensation, characterized in that, Includes the following steps: Step 1: Obtain the sampling current of the permanent magnet synchronous motor in the synchronous rotating coordinate system. , and control the output voltage , The sampling current contains sampling noise; Step 2: Construct information evaluation indicators and adjust the proportional parameters in the current controller. To meet the information evaluation index under stability constraints, the sampling noise is used by the current controller to generate an effective excitation for identification, thereby increasing the amount of identification information. The constructed information evaluation index is as follows: In the formula, for The information evaluation index of the axis, where det is used to calculate the determinant of the matrix. for Fisher information matrix of the axis, for The variance of the sampling noise of the shaft sampling current. The frequency contained in the input signal. For the input signal at The corresponding signal amplitude at that location, for Shaft inductor, for Shaft inductor, For stator resistance, Electric angular velocity; The stability constraint is as follows: >0; Step 3, the current controller output Axis control output voltage and according to Fluctuation of shaft control output voltage Build Shaft voltage feedforward compensation To suppress the electromagnetic torque pulsation caused by step 2 and output Shaft control output voltage; Step 4: Construct a discrete linear parameterized model, and use the least squares method to recursively obtain the parameter estimates and covariance matrix based on the discrete linear parameterized model; Step 5: Estimate the bias parameters online using the least squares method, and calculate the sampling noise variance using the estimated residuals; Step 6: Obtain the unbiased parameter estimate by iteratively updating according to the bias compensation, and output the stator resistance. Direct-axis inductor L d quadrature axis inductance L q and permanent magnet magnetic flux The online estimate.

2. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 1, characterized in that: In step 2, the current controller is a PI controller, and the transfer function of the PI controller is: In the formula, Pass functions to the PI controller. For the Laplace transform, The proportional parameter of the PI controller, These are the integral parameters of the PI controller; The power spectral density dominated by the proportional term is: In the formula, This is the lower bound of the output power spectral density of the PI controller. For frequency, The variance of the current sampling noise; Among them, increasing the value in the transfer function of the PI controller is adopted. Parameters to improve Axis control output voltage The power spectral density.

3. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 2, characterized in that: In step 3, The shaft control output voltage is: ， In the formula, for Shaft output voltage, To increase The subsequent PI controller, for Control error of shaft sampling current; for Reference value for shaft sampling current; The fluctuation value of the shaft control output voltage is: In the formula, for time Shaft output voltage fluctuation value, for time shaft voltage, for time The expected estimate of the axis control voltage within a short observation window; in, Using a recursive summation form with a forgetting factor, we obtain the following formula: ， ； In the formula, The forgetting factor is greater than 0 and less than 1. for exist The summation of forgetting moments.

4. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 3, characterized in that: The formula for shaft voltage feedforward compensation is: In the formula, for time Shaft feedforward compensation voltage, for time Shaft current sampling value, for Shaft inductor, for Shaft inductor, It is a permanent magnet flux linkage. The sampling period.

5. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 4, characterized in that: In step 4, the discrete linear parameterized model is: ， ； In the formula, Electric angular velocity, For stator resistance, for Output matrix at time step; for The input matrix at time t, its superscript Indicates transpose; for The electric angular velocity at time -1 for Time and -1 hour The change in shaft current The target parameter vector.

6. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 5, characterized in that: In step 5, the formula obtained by recursion using the least squares method is: ， ， ， ； In the formula, For parameter estimation vectors; for Gain matrix at time step for The covariance matrix at time t, for The covariance matrix at time t, for time The denominator of the matrix, It is a 2-dimensional identity matrix.

7. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 6, characterized in that: The residual energy recursion matrix is ​​constructed as follows: ， In the formula, The residual energy recursion matrix is... For the amount of data required for estimation, for The residuals calculated using the recursive least squares method at time points. for Output matrix at time step; The formula for calculating the estimated residual is as follows: In the formula, for The estimated residuals are calculated using the recursive least squares method at each time step.

8. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 7, characterized in that: The online estimated variance of the sampling noise is: （21）； In the formula, for Variance estimation of sampling noise for shaft sampling current; for Real parameter vector at any given time; in, The intermediate variable matrix is ​​derived, where, .

9. The method for multi-parameter identification of a motor based on sampling noise excitation and bias compensation according to claim 8, characterized in that: In step 6, the formula for calculating the unbiased parameter estimate is: ； In the formula, , for The estimation of the true parameter vector at each time step, therefore ,in, This is an estimated value for the stator resistance. This is an estimated value for the direct-axis inductance. This is an estimated value for the quadrature-axis inductance. This is an estimated value for the magnetic flux linkage of the permanent magnet.

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