阅读说明：本技术 基于upf永磁直线同步电动机的无位置传感器的控制方法 (Position sensorless control method based on UPF permanent magnet linear synchronous motor ) 是由 刘兴华 吕云玲 关建伟 李洁 于 2020-12-04 设计创作，主要内容包括：基于UPF永磁直线同步电动机的无位置传感器的控制方法,包括以下步骤：步骤1,建立PMLSM数学模型,从而得到电动机动态状态估计模型；步骤2,选用比例修正对称采样确定Sigma点集,采用无迹卡尔曼滤波算法求出增益K-(k+1)；步骤3,在粒子滤波的框架下,采用UKF算法求解粒子滤波的重要性密度函数,实时估计电机当前的速度和位置；步骤4,返回步骤2进行下一时刻速度和位置估计；具有推力大、速度快、高效率、牢固性和维修性好的特点。(The control method of the position sensorless based on the UPF permanent magnet linear synchronous motor comprises the following steps: step 1, establishing a PMLSM mathematical model so as to obtain a motor dynamic state estimation model; step 2, selecting proportional correction symmetric sampling to determine a Sigma point set, and calculating gain K by adopting an unscented Kalman filtering algorithm k+1 (ii) a And 3, under the framework of particle filtering, solving an importance density function of the particle filtering by adopting a UKF algorithm, and estimating the motor current in real timeThe speed and position of the front; step 4, returning to the step 2 to estimate the speed and the position at the next moment; the device has the characteristics of high thrust, high speed, high efficiency, good firmness and good maintainability.)

1. The control method of the position sensorless based on the UPF permanent magnet linear synchronous motor is characterized by comprising the following steps:

step 1, establishing a PMLSM mathematical model to obtain a motor dynamic state estimation model;

step 2, selecting proportional correction symmetric sampling to determine a Sigma point set, and calculating gain K by adopting unscented Kalman filtering UKF algorithm_k+1；

Step 3, in the framework of particle filtering, solving an importance density function of the particle filtering PF by adopting a UKF algorithm, and estimating the current speed and position of the motor in real time;

and 4, returning to the step 2 to estimate the speed and the position at the next moment.

2. The UPF permanent magnet linear synchronous motor-based position sensorless control method according to claim 1, wherein in step 1, based on dq system voltage balance equation, electromagnetic torque equation and mechanical motion equation, first, the following mathematical model is obtained:

in the formula:is a differential, i_dAnd i_qD and q axis currents, respectively; omega_eIs the rotor electrical angular velocity, theta is the rotor position angle, R_sThe resistance of the rotor coil; l is the inductance of d and q axes, wherein the inductance of d and q axes is equal and is L; psi_fAn effective flux linkage generated for the permanent magnet; p is a radical of_nThe number of pole pairs of the motor is shown; t is_lThe load resistance when the motor works; j is the moment of inertia; b is the coefficient of friction; u. of_dAnd u_qThe voltages of the d and q axes respectively,

let the motor stator current i_d、i_qAngular rotor speed ω_eAnd rotor position angle theta as state variable, motor stator voltage u_d、u_qConstructing a system state equation for controlling variables, and obtaining a nonlinear measurement equation according to the motor characteristics; since the mathematical model given by equation (1) is deterministic and does not take into account the uncertainty of the measurement, the generalization of this model to a random discrete state space model does require the addition of two noise vectors w_kAnd v_kTo establish a state space representation of a position sensorless system of a permanent magnet linear synchronous motorThe formula is as follows:

in the formula, x_k+1Is the system state vector at time k +1, x_kFor the system state vector at time k, x ═ i_d i_q ω_e θ]^TThe input vector u ═ u_d u_q]^T，u_kIs the input vector at time k, z_kFor the system observation vector at time k, z ═ i_d i_q]^TThe rotor angular velocity and rotor position angular state components are estimators, only the current state components are measurable and detected by the system current sensors, f (-) is an n-dimensional state nonlinear function vector, h (-) is an m-dimensional measurement nonlinear function vector,w_kis process noise, v_kFor measuring noise, it is generally assumed that w_k～N(0,Q),v_kN (0, R) wherein Q and R are each w_kAnd v_kCovariance matrix of (w)_kAnd v_kIndependent of each other and independent of the state variables.

3. The UPF permanent magnet linear synchronous motor-based position sensorless control method according to claim 1, wherein in the step 2, a proportional correction symmetric sampling is selected to determine a Sigma point set, and an Unscented Kalman Filter (UKF) algorithm is adopted to obtain a gain K_k+1The method comprises the following specific steps:

step 2.1, initialize, initialized parameter has state variable x₀Sum covariance matrix P₀；

In the formula (I), the compound is shown in the specification,is when k is 0, from the known initial distribution p (x)₀) Extract N particles, i.e. the original set of particles Andmean and covariance for initializing the particle set, respectively, E representing the expectation for the variables;

step 2.2, performing unscented transformation, determining a Sigma point set by selecting proportional correction symmetric sampling, taking L as 2n for symmetric sampling, wherein n represents a system state dimension, so that the number of Sigma points is 2n +1, and the symmetrically sampled Sigma points and weight coefficients thereof can be represented as follows:

in the formulaIs the state estimation result of the state quantity at the kth time, χ_i,k|kThe method is characterized in that the ith Sigma point is constructed by a corresponding sampling strategy according to the estimation result of the motor state quantity at the previous moment, and lambda is alpha²(n+κ)-n,α∈[0.0001,1]The scale factor is used for changing the value of alpha to adjust the distribution distance between a Sigma point and an average value and reduce the prediction error; k is redundancy and is set to 0 during state estimation; beta is a parameter related to the prior distribution of the state vector, which is optimal for a gaussian distribution, beta 2,is the ith column of the square root of the matrix,is the mean weighted value;is a covariance weighting value;

step 2.3, time update and prediction

The Sigma sampling points obtained by the UT sampling strategy are subjected to f (-) nonlinear transformation by a system state equation (2), and then 2n +1 Sigma points can be predicted in one step:

χ_k|k-1＝f(χ_k-1,u_k-1)+w_k-1 (5)

χ_k|k-1for the Sigma sample points from time k-1 to time k, obtained by UT sampling, f (-) is an n-dimensional state non-linear function vector, χ_k-1Sigma sample point at time k-1, u_k-1Is the input vector at time k-1, w_k-1In order to be a noise of the process,

multiplying the transformed Sigma point sets by the weights corresponding to the Sigma point sets respectively to obtain a one-step state prediction value

In the formula (I), the compound is shown in the specification,is a one-step predicted value from the time k-1 to the time k,the transformed Sigma point sets are multiplied by their respective corresponding weights,

using the same weighting process to obtain a one-step prediction variance matrix P of states_x,k|k-1：

In the formula, P_x,k|k-1Is a one-step predictive variance matrix of the states from time k-1 to time k,weighting the difference value between the Sigma point from the moment k-1 to the moment k and a one-step predicted value,is a covariance weight, Q_k-1Is process noise w_k-1The covariance matrix of (a) is determined,

for the observation equation in the formula (2), the newly obtained Sigma point set { χ }is subjected_i,k|k-1Making a non-linear change:

z_i,k|k-1＝h(χ_i,k|k-1) (8)

in the formula, z_i,k|k-1Is the observed value of the set of particles from time k-1 to time k, h (-) is a vector of a m-dimensional measurement nonlinear function, χ_i,k|k-1The Sigma sample points from time k-1 to time k sampled by the UT,

a transformed value z obtained by equation (8)_i,k|k-1Obtaining a system measurement prediction value using weighted summationAnd system measurement variable variance P_z,k|k-1Sum cross covariance P_xz,k|k-1：

In the formula (I), the compound is shown in the specification,is an estimated value of a system observation vector from the time k-1 to the time k,is to the transformed value z_i,k|k-1The weighted sum process is performed and the sum is calculated,

in the formula, P_z,k|k-1Is the variance of the system measurement variable,weighting the measurement error from the time k-1 to the time k,is a covariance weight, R_kIs a measurement noise v_kThe covariance matrix of (a) is determined,

in the formula, P_xz,k|k-1Is the cross-covariance of the two,weighting the system error and the measurement error,is a covariance weighting value;

step 2.4, updating measurement;

establishing a time updating equation:

in the formula (I), the compound is shown in the specification,for an a posteriori estimate of the state vector at time k +1,the difference between the measured variable and its prediction for a prior estimate of the state vector at time k +1Called innovation or residue of the measurement process, reflecting the degree of inconsistency between the predicted value and the true value, the matrix K_k+1Called residual gain, which acts to minimize the covariance of the a posteriori estimation error of equation (14),

posterior error:

the posteriori estimation error covariance:

order toThe following were obtained:

in the formula, K_k+1Is the residual gain, P_xz,k+1|kIs the cross-covariance of the two,is the inverse of the variance of the system measurement variables.

4. The UPF permanent magnet linear synchronous motor-based position sensorless control method according to claim 1, wherein the step 3 specifically comprises: in the framework of particle filtering, the UKF algorithm is adopted to solve the importance density function of the particle filtering PF, the current speed and position of the motor are estimated in real time,

step 3.1, importance sampling

The estimated value of each particle at the moment is obtained by using UKF algorithmSum error covarianceThe important density function is

In the formula: n (-) is a Gaussian function; subscripts 0: k and 1: k represent time 0 to k and time 1 to k, respectively,is a probability density function, and N (·) represents obeying to a normal distribution, the sampling particles in the important density function are:

order toCalculating the weight of each particle:

normalizing the weight:

step 3.2, resampling

i) Effective particle volume N_effIt can be measured whether the algorithm is degraded, but this effective particle capacity cannot be obtained by strict calculation, and its estimated value can be approximated as follows:

in the formula, N_effIs the effective particle capacity of the particles,a normalized weight defined for equation (19);

ii) determining whether resampling is required:

if N is present_eff≤N_th，N_thrThe set threshold, i.e. the set number of valid samples, is generally N/3, which indicates that the weight of the particle has been degraded seriously, N_effThe smaller the sample size is, the more serious the degradation phenomenon is, resampling is needed, namely, particles with small weight are removed, and the original weighted sample is takenMapping to equal-weight samplesI.e. the processed particles are mapped to equal-weight N particles,

if N is present_eff＞N_thDirectly proceed to the next step.

5. The UPF permanent magnet linear synchronous motor-based position sensorless control method according to claim 1, wherein the step 4 specifically comprises: returning to the step 2 to estimate the speed and the position at the next moment, outputting a mean value estimation and a variance estimation,

Technical Field

The invention belongs to the technical field of electric servo transmission, and particularly relates to a control method of a position sensorless linear synchronous motor based on UPF permanent magnets.

Background

The Permanent Magnet Linear Synchronous Motor (PMLSM) has the characteristics of large thrust, high speed, high efficiency, good firmness and maintainability and the like, and is widely applied to occasions such as numerical control machines, industrial robots and the like. In order to realize accurate control of the rotating speed of the permanent magnet synchronous motor, speed feedback is an indispensable link, and therefore obtaining accurate rotor information is the core problem of the whole control system. In the PMLSM driving system, the conventional rotor position information is acquired by using a grating scale as a position sensor, and 3 functions of magnetic pole position detection, speed detection and system positioning are completed. However, the existence of the position sensor increases the system cost, reduces the operation reliability, limits the use range, limits the motion characteristics of the linear motor by the position sensor and increases the system size. Therefore, the research of a position-sensorless control system with high accuracy and strong robustness becomes the first choice of the motor speed regulation control system. Then how do the position sensor motor commutate? How to adjust the speed? The position and speed of the permanent magnet linear synchronous motor are estimated. The current estimation method mainly comprises extended Kalman filtering and unscented Kalman filtering:

1) extended Kalman Filter (EKF) algorithm. Extended Kalman Filter (EKF) is representative of conventional non-linear estimation, and Gopinath et al [1] apply extended Kalman Filter in a position sensorless permanent magnet synchronous motor drive system. EKF converts the non-linear problem into linearity by first order linearization of the Taylor expansion of the non-linear function. Thus, EKF has disadvantages: firstly, when the high-order term of the nonlinear function Taylor expansion cannot be ignored, the linearization makes the system generate larger error, even makes the filter unstable; ② it is difficult to derive Jacobian matrix, sometimes difficult to implement. And thirdly, the given noise covariance matrix has great influence on the EKF filtering process, and if the noise covariance matrix is not properly selected, the convergence of the filtering process is seriously influenced.

2) Unscented Kalman Filter (UKF) algorithm. The UKF algorithm approximately obtains the statistical property of the state quantity after nonlinear transformation through Unscented Transformation (UT), avoids errors caused by EKF normal linearization, omits the calculation of a Jacobian matrix, and has good convergence, estimation precision higher than that of an EKF method and stability. However, the UKF is only suitable for a system in which noise is white gaussian noise, when the posterior probability density of the state is non-gaussian, the filtering performance of the UKF may be degraded, and in a severe nonlinear system, the UKF may adversely affect the state estimation.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention aims to provide a position sensorless control method based on a UPF permanent magnet linear synchronous motor, which solves the problem of particle degradation in an algorithm and has the characteristics of high thrust, high speed, high efficiency, high firmness and good maintainability.

In order to achieve the purpose, the invention adopts the technical scheme that: the control method of the position sensorless based on the UPF permanent magnet linear synchronous motor comprises the following steps:

step 1, establishing a PMLSM mathematical model so as to obtain a motor dynamic state estimation model;

step 2, selecting proportional correction symmetric sampling to determine a Sigma point set, and calculating gain K by adopting unscented Kalman filtering UKF algorithm_k+1；

Step 3, in the framework of particle filtering, solving an importance density function of the particle filtering PF by adopting a UKF algorithm, and estimating the current speed and position of the motor in real time;

and 4, returning to the step 2 to estimate the speed and the position at the next moment.

In the step 1, according to a dq system voltage balance equation, an electromagnetic torque equation and a mechanical motion equation, firstly, the following mathematical model can be obtained:

in the formula:is a differential, i_dAnd i_qD and q axis currents, respectively; omega_eIs the rotor electrical angular velocity, theta is the rotor position angle, R_sThe resistance of the rotor coil; l is the inductance of d and q axes, wherein the inductance of d and q axes is equal and is L; psi_fAn effective flux linkage generated for the permanent magnet; p is a radical of_nThe number of pole pairs of the motor is shown; t is_lThe load resistance when the motor works; j is the moment of inertia; b is the coefficient of friction; u. of_dAnd u_qThe voltages of the d and q axes respectively,

let the motor stator current i_d、i_qAngular rotor speed ω_eAnd rotor position angle theta as state variable, motor stator voltage u_d、u_qConstructing a system state equation for controlling variables, and obtaining a nonlinear measurement equation according to the motor characteristics; since the mathematical model given by equation (1) is deterministic and does not take into account the uncertainty of the measurement, the generalization of this model to a random discrete state space model does require the addition of two noise vectors w_kAnd v_kThus, the state space expression of the position sensorless system of the permanent magnet linear synchronous motor is established as follows:

in the formula, x_k+1Is the system state vector at time k +1, x_kFor the system state vector at time k, x ═ i_d i_q ω_e θ]^TThe input vector u ═ u_d u_q]^T，u_kIs the input vector at time k, z_kFor the system observation vector at time k, z ═ i_d i_q]^TThe rotor angular velocity and rotor position angular state components are estimators, only the current state components are measurable and detected by the system current sensors, f (-) is an n-dimensional state nonlinear function vector, h (-) is an m-dimensional measurement nonlinear function vector,w_kis process noise, v_kFor measuring noise, it is generally assumed that w_k～N(0,Q),v_kN (0, R) wherein Q and R are each w_kAnd v_kCovariance matrix of (w)_kAnd v_kIndependent of each other and independent of the state variables.

Step 2, a Sigma point set is determined by adopting proportional correction symmetric sampling, and the gain K is solved by adopting an unscented Kalman filter UKF algorithm_k+1The method comprises the following specific steps:

step 2.1, initialize, initialized parameter has state variable x₀Sum covariance matrix P₀；

In the formula (I), the compound is shown in the specification,is when k is 0, from the known initial distribution p (x)₀) Extract N particles, i.e. the original set of particles Andmean and covariance for initializing the particle set, respectively, E representing the expectation for the variables;

step 2.2, performing unscented transformation, determining a Sigma point set by selecting proportional correction symmetric sampling, taking L as 2n for symmetric sampling, wherein n represents a system state dimension, so that the number of Sigma points is 2n +1, and the symmetrically sampled Sigma points and weight coefficients thereof can be represented as follows:

in the formulaIs the state estimation result of the state quantity at the kth time, χ_i,k|kThe method is characterized in that the ith Sigma point is constructed by a corresponding sampling strategy according to the estimation result of the motor state quantity at the previous moment. λ ═ α²(n+κ)-n,α∈[0.0001,1]The scale factor is used for changing the value of alpha to adjust the distribution distance between a Sigma point and an average value and reduce the prediction error; κ is the amount of redundancy, and is typically set to 0 during state estimation; beta is a parameter related to the prior distribution of the state vector, which is optimal for a gaussian distribution, beta 2,is the ith column of the square root of the matrix,is the mean weighted value;is a covariance weighting value;

step 2.3, time update and prediction

The Sigma sampling points obtained by the UT sampling strategy are subjected to f (-) nonlinear transformation by a system state equation (2), and then 2n +1 Sigma points can be predicted in one step:

χ_k|k-1＝f(χ_k-1,u_k-1)+w_k-1 (5)

χ_k|k-1for the Sigma sample points from time k-1 to time k, obtained by UT sampling, f (-) is an n-dimensional state non-linear function vector, χ_k-1Sigma sample point at time k-1, u_k-1Is the input vector at time k-1, w_k-1In order to be a noise of the process,

multiplying the transformed Sigma point sets by the weights corresponding to the Sigma point sets respectively to obtain a one-step state prediction value

In the formula (I), the compound is shown in the specification,is a one-step predicted value from the time k-1 to the time k,the transformed Sigma point sets are multiplied by their respective corresponding weights,

using the same weighting process to obtain a one-step prediction variance matrix P of states_x,k|k-1：

In the formula, P_x,k|k-1Is a one-step predictive variance matrix of the states from time k-1 to time k,weighting the difference value between the Sigma point from the moment k-1 to the moment k and a one-step predicted value,is a covariance weight, Q_k-1Is process noise w_k-1The covariance matrix of (a) is determined,

for the observation equation in the formula (2), the newly obtained Sigma point set { χ }is subjected_i,k|k-1Making a non-linear change:

z_i,k|k-1＝h(χ_i,k|k-1) (8)

in the formula, z_i,k|k-1Is the observed value of the set of particles from time k-1 to time k, h (-) is a vector of a m-dimensional measurement nonlinear function, χ_i,k|k-1The Sigma sample points from time k-1 to time k sampled by the UT,

a modification obtained by the formula (8)Change of value z_i,k|k-1Obtaining a system measurement prediction value using weighted summationAnd system measurement variable variance P_z,k|k-1Sum cross covariance P_xz,k|k-1：

In the formula (I), the compound is shown in the specification,is an estimated value of a system observation vector from the time k-1 to the time k,is to the transformed value z_i,k|k-1The weighted sum process is performed and the sum is calculated,

in the formula, P_z,k|k-1Is the variance of the system measurement variable,weighting the measurement error from the time k-1 to the time k,is a covariance weight, R_kIs a measurement noise v_kThe covariance matrix of (a) is determined,

in the formula, P_xz,k|k-1Is the cross-covariance of the two,is to weight the systematic error and the measurement errorIn order to solve the problems that,is a covariance weighting value;

step 2.4, updating measurement;

establishing a time update equation

In the formula (I), the compound is shown in the specification,for an a posteriori estimate of the state vector at time k +1,the difference between the measured variable and its prediction for a prior estimate of the state vector at time k +1Called innovation or residue of the measurement process, reflecting the degree of inconsistency between the predicted value and the true value, the matrix K_k+1Called residual gain, which acts to minimize the covariance of the a posteriori estimation error of equation (14),

posterior error:

the posteriori estimation error covariance:

order toThe following were obtained:

in the formula, K_k+1Is the residual gain, P_xz,k+1|kIs the cross-covariance of the two,is the inverse of the variance of the system measurement variables.

The step 3 specifically comprises: in the framework of particle filtering, the UKF algorithm is adopted to solve the importance density function of the particle filtering PF, the current speed and position of the motor are estimated in real time,

step 3.1, importance sampling

The estimated value of each particle at the moment is obtained by using UKF algorithmSum error covarianceThe important density function is:

in the formula: n (-) is a Gaussian function; subscripts 0: k and 1: k represent time 0 to k and time 1 to k, respectively,is a probability density function, and N (·) represents obeying to a normal distribution, the sampling particles in the important density function are:

order toCalculating the weight of each particle:

normalizing the weight:

step 3.2, resampling

i) Effective particle volume N_effIt can be measured whether the algorithm is degraded, but this effective particle capacity cannot be obtained by strict calculation, and its estimated value can be approximated as follows:

in the formula, N_effIs the effective particle capacity of the particles,a normalized weight defined for equation (19);

ii) determining whether resampling is required:

if N is present_eff≤N_th(N_thrIs a set threshold value, i.e. a set number of effective samples, which is generally N/3), it indicates that the weight of the particle has been degraded seriously, N_effThe smaller the sample size is, the more serious the degradation phenomenon is, resampling is needed, namely, particles with small weight are removed, and the original weighted sample is takenMapping to equal-weight samplesI.e. the processed particles are mapped to equal-weight N particles,

if N is present_eff＞N_thDirectly proceed to the next step.

The step 4 specifically comprises: returning to the step 2 to estimate the speed and the position at the next moment, outputting a mean value estimation and a variance estimation,

the invention has the beneficial effects that:

according to the invention, on the basis of a PF algorithm, the UKF is used as a required density function to generate the predicted particles, and the measurement information of the current moment is effectively utilized in each step of iterative calculation, so that the distribution of the generated sampled particles is closer to the real posterior distribution, and the particle requirement for describing the posterior distribution is effectively reduced. Meanwhile, the UPF reserves the flexibility of the PF algorithm, namely the filtering precision can be adjusted by changing the number of particles, and theoretically, the estimated value can be infinitely close to the true value along with the increase of the number of particles. The UPF consists of an importance sampling part and a resampling part, the weight of effective particles is increased, and the calculation efficiency is further improved.

Compared with the prior art, the invention discloses a position sensorless control method of a Permanent Magnet Linear Synchronous Motor (PMLSM) based on Unscented Particle Filter (UPF), and belongs to the technical field of electric servo transmission. The Permanent Magnet Linear Synchronous Motor (PMLSM) in the method has the characteristics of high thrust, high speed, high efficiency, good firmness and good maintainability, is widely applied to occasions such as numerical control machines, industrial robots and the like, and speed feedback is an indispensable link for realizing the accurate control of the rotating speed of the permanent magnet synchronous motor, so that the core problem of the whole control system is to obtain accurate rotor information. The PMLSM driving system generally adopts a grating ruler as a position sensor to complete 3 functions of magnetic pole position detection, speed detection and system positioning. The position sensor increases the system cost and the system size, is easily influenced by the external environment, directly reduces the observation accuracy, and is even not suitable for installing the sensor in some high-pressure and high-temperature occasions. Therefore, the research on the position-sensorless control system with high accuracy and strong robustness has important significance in the speed regulation control system of the motor and some special occasions.

Drawings

FIG. 1 shows the variable i_dAnd (4) state estimation diagrams.

FIG. 2 shows the variable i_qAnd (4) state estimation diagrams.

Fig. 3 is a state change graph of the angular velocity ω.

Fig. 4 is a state change graph of the position angle θ.

Fig. 5 is a flow chart of the UPF algorithm of the present invention.

Detailed Description

The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

A first part:

the invention relates to a control method of a position sensorless based on a UPF permanent magnet linear synchronous motor, which comprises the following steps:

step 1, establishing a PMLSM mathematical model so as to obtain a motor dynamic state estimation model;

step 2, selecting proportional correction symmetric sampling to determine a Sigma point set, and calculating gain K by adopting unscented Kalman filtering UKF algorithm_k+1；

Step 3, in the framework of particle filtering, solving an importance density function of the particle filtering PF by adopting a UKF algorithm, and estimating the current speed and position of the motor in real time;

and 4, returning to the step 2 to estimate the speed and the position at the next moment.

In the step 1, according to a dq system voltage balance equation, an electromagnetic torque equation and a mechanical motion equation, firstly, the following mathematical model can be obtained:

in the formula:is a differential, i_dAnd i_qD and q axis currents, respectively; omega_eIs the rotor electrical angular velocity, theta is the rotor position angle, R_sThe resistance of the rotor coil; l is the inductance of d and q axes, wherein the inductance of d and q axes is equal and is L; psi_fAn effective flux linkage generated for the permanent magnet; p is a radical of_nThe number of pole pairs of the motor is shown; t is_lThe load resistance when the motor works; j is the moment of inertia; b is the coefficient of friction; u. of_dAnd u_qThe voltages of the d and q axes respectively,

let the motor stator current i_d、i_qAngular rotor speed ω_eAnd rotor position angle theta as state variable, motor stator voltage u_d、u_qConstructing a system state equation for controlling variables, and obtaining a nonlinear measurement equation according to the motor characteristics; since the mathematical model given by equation (1) is deterministic and does not take into account the uncertainty of the measurement, the generalization of this model to a random discrete state space model does require the addition of two noise vectors w_kAnd v_kThus, the state space expression of the position sensorless system of the permanent magnet linear synchronous motor is established as follows:

in the formula, x_k+1Is the system state vector at time k +1, x_kFor the system state vector at time k, x ═ i_d i_q ω_e θ]^TThe input vector u ═ u_d u_q]^T，u_kIs the input vector at time k, z_kFor the system observation vector at time k, z ═ i_d i_q]^TThe rotor angular velocity and rotor position angular state components are estimators, only the current state components are measurable and detected by the system current sensors, f (-) is an n-dimensional state nonlinear function vector, h (-) is an m-dimensional measurement nonlinear function vector,w_kis process noise, v_kFor measuring noise, it is generally assumed that w_k～N(0,Q),v_kN (0, R) wherein Q and R are each w_kAnd v_kCovariance matrix of (w)_kAnd v_kIndependent of each other and independent of the state variables.

Step 2, a Sigma point set is determined by adopting proportional correction symmetric sampling, and the gain K is solved by adopting an unscented Kalman filter UKF algorithm_k+1The method comprises the following specific steps:

step 2.1, initialize, initialized parameter has state variable x₀Sum covariance matrix P₀；

In the formula (I), the compound is shown in the specification,is when k is 0, from the known initial distribution p (x)₀) Extract N particles, i.e. the original set of particles Andmean and covariance for initializing the particle set, respectively, E representing the expectation for the variables;

and 2.2, performing unscented transformation, and determining a Sigma point set by using proportional correction symmetric sampling. L ═ 2n is taken for symmetric sampling, n represents the system state dimension, so the number of Sigma points is 2n +1, then the symmetric sampling Sigma points and their weight coefficients can be represented as:

in the formulaIs the state estimation result of the state quantity at the kth time, χ_i,k|kThe method is characterized in that the ith Sigma point is constructed by a corresponding sampling strategy according to the estimation result of the motor state quantity at the previous moment. λ ═ α²(n+κ)-n,α∈[0.0001,1]The scale factor is used for changing the value of alpha to adjust the distribution distance between a Sigma point and an average value and reduce the prediction error; κ is the amount of redundancy, and is typically set to 0 during state estimation; beta is a parameter related to the prior distribution of the state vector, which is optimal for a gaussian distribution, beta 2,is the ith column of the square root of the matrix,is the mean weighted value;is a covariance weighting value;

step 2.3, time update and prediction

The Sigma sampling points obtained by the UT sampling strategy are subjected to f (-) nonlinear transformation by a system state equation (2), and then 2n +1 Sigma points can be predicted in one step:

χ_k|k-1＝f(χ_k-1,u_k-1)+w_k-1 (5)

χ_k|k-1for the Sigma sample points from time k-1 to time k, obtained by UT sampling, f (-) is an n-dimensional state non-linear function vector, χ_k-1Sigma sample point at time k-1, u_k-1Is the input vector at time k-1, w_k-1In order to be a noise of the process,

multiplying the transformed Sigma point sets by the weights corresponding to the Sigma point sets respectively to obtain a one-step state prediction value

In the formula (I), the compound is shown in the specification,is a one-step predicted value from the time k-1 to the time k,the transformed Sigma point sets are multiplied by their respective corresponding weights,

using the same weighting process to obtain a one-step prediction variance matrix P of states_x,k|k-1：

In the formula, P_x,k|k-1Is a one-step predictive variance matrix of the states from time k-1 to time k,weighting the difference value between the Sigma point from the moment k-1 to the moment k and a one-step predicted value,is a covariance weight, Q_k-1Is process noise w_k-1The covariance matrix of (a) is determined,

for the observation equation in the formula (2), the newly obtained Sigma point set { χ }is subjected_i,k|k-1Making a non-linear change:

z_i,k|k-1＝h(χ_i,k|k-1) (8)

in the formula, z_i,k|k-1Is the observed value of the set of particles from time k-1 to time k, h (-) is a vector of a m-dimensional measurement nonlinear function, χ_i,k|k-1The Sigma sample points from time k-1 to time k sampled by the UT,

a transformed value z obtained by equation (8)_i,k|k-1Obtaining a system measurement prediction value using weighted summationAnd system measurement variable variance P_z,k|k-1Sum cross covariance P_xz,k|k-1：

In the formula (I), the compound is shown in the specification,is an estimated value of a system observation vector from the time k-1 to the time k,is to the transformed value z_i,k|k-1The weighted sum process is performed and the sum is calculated,

in the formula, P_z,k|k-1Is the variance of the system measurement variable,weighting the measurement error from the time k-1 to the time k,is a covariance weight, R_kIs a measurement noise v_kThe covariance matrix of (a) is determined,

in the formula, P_xz,k|k-1Is the cross-covariance of the two,weighting the system error and the measurement error,is a covariance weighting value;

step 2.4, updating measurement;

establishing a time update equation

In the formula (I), the compound is shown in the specification,for an a posteriori estimate of the state vector at time k +1,the difference between the measured variable and its prediction for a prior estimate of the state vector at time k +1Called innovation or residue of the measurement process, reflecting the degree of inconsistency between the predicted value and the true value, the matrix K_k+1Called residual gain, which acts to minimize the covariance of the a posteriori estimation error of equation (14),

posterior error:

the posteriori estimation error covariance:

order toThe following were obtained:

in the formula, K_k+1Is the residual gain, P_xz,k+1|kIs the cross-covariance of the two,is the inverse of the variance of the system measurement variables.

The step 3 specifically comprises: in the framework of particle filtering, the UKF algorithm is adopted to solve the importance density function of the particle filtering PF, the current speed and position of the motor are estimated in real time,

step 3.1, importance sampling

The estimated value of each particle at the moment is obtained by using UKF algorithmSum error covarianceThe important density function is:

in the formula: n (-) is a Gaussian function; subscripts 0: k and 1: k represent time 0 to k and time 1 to k, respectively,is a probability density function, and N (·) represents obeying to a normal distribution, the sampling particles in the important density function are:

order toCalculating the weight of each particle:

normalizing the weight:

the weight of each particle is obtained, and the normalization is to divide each particle by the total particle;

step 3.2, resampling

i) Effective particle volume N_effIt can be measured whether the algorithm is degraded, but this effective particle capacity cannot be obtained by strict calculation, and its estimated value can be approximated as follows:

in the formula, N_effIs the effective particle capacity of the particles,a normalized weight defined for equation (19);

ii) determining whether resampling is required:

if N is present_eff≤N_th(N_thrIs a set threshold value, i.e. a set number of effective samples, which is generally N/3), it indicates that the weight of the particle has been degraded seriously, N_effThe smaller the sample size is, the more serious the degradation phenomenon is, resampling is needed, namely, particles with small weight are removed, and the original weighted sample is takenMapping to equal-weight samplesI.e. the processed particles are mapped to equal-weight N particles,

if N is present_eff＞N_thDirectly proceed to the next step.

The step 4 specifically comprises: and returning to the step 2 to estimate the speed and the position at the next moment, and outputting the mean value estimation and the variance estimation.

Referring to FIG. 1, is the variable i_dA state estimation graph; referring to FIG. 2, variable i_qA state estimation graph; referring to fig. 3, a state change graph of the angular velocity ω; referring to fig. 4, a state change graph of the position angle θ.

Fig. 1 and 2 show state estimates of motor stator currents obtained by the UPF algorithm. We can observe that the estimated current is very close to the actual current and the dynamic response is very fast. The estimation results are then introduced into a discrete formula to obtain the variation graphs of ω and θ of fig. 3 and 4. Due to i_dAnd i_qThe simulation results are closer to the actual situation, so the state estimates and actual values of ω and θ are also closer. As can be seen from the figure, the UPF is very effective for state estimation of the permanent magnet linear synchronous motor.