# Permanent Magnet Flux Linkage Identification Method of the Model Predictive Current Control System for PMSM

Authors: Tao Chen; Haihong Xiao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Accurate identification of permanent magnet flux linkage can effectively improve the control performance of model predictive current control system of permanent magnet synchronous motor (PMSM). This paper reviews the existing identification methods of permanent magnet flux linkage of PMSM first and then proposes an identification method of permanent magnet flux linkage based on an improved unscented particle filter, the problem of particle diversity reduction of the traditional unscented particle filtering algorithm is solved, and the identification performance of permanent magnet flux is improved. Simulation research and experimental verification of the proposed method are carried out, and the simulation and experimental results confirm the effectiveness of the proposed method.


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## Body

## 1. Introduction

PMSM has been widely used in the fields of new energy power generation, servo drive, and electric vehicles due to its advantages of high power density, small maintenance, and good control performance. The current control strategy with excellent performance is beneficial to meet the control requirements of fast current response and good dynamic characteristics of the PMSM drive system [1]. Because the model predictive current control (MPCC) algorithm has the advantages of good dynamic current control performance, direct measurement of control variables, simple implementation, and low cost function construction without designing weight coefficients [2], it is widely used in the drive system of PMSM.However, the MPCC algorithm is very sensitive to the change of controlled object parameters, and the mismatch between controlled object parameters and MPCC algorithm control parameters will lead to obvious prediction errors and steady state control errors and even cause algorithm instability [3]. A large number of research achievements show that the changes in permanent magnet flux linkage have the most obvious influence on the control performance of the MPCC system for PMSM [4–6], whereas limited by installation space and heat dissipation conditions, the permanent magnet material of PMSM is usually susceptible to the influence of armature reaction and operation temperature increase, resulting in permanent magnet flux linkage reduction, and then, the control performance of the MPCC system for PMSM is reduced. There are usually three types of methods to solve the above-mentioned problem, and the first method is to compensate for the negative influence caused by permanent magnet flux linkage change as disturbance or to reduce that negative influence by enhancing the robustness of the PMSM drive system [4, 5]. Another type of a method is to combine the MPCC algorithm with intelligent control algorithms to suppress the negative influence of permanent magnet flux linkage change on the PMSM drive system control performance [6], and the third type of a method is to identify permanent magnet flux linkage based on an online identification algorithm [7–13], and then, the parameter mismatch and control performance degradation of the MPCC algorithm for the PMSM drive system caused by permanent magnet flux linkage change are solved; this type of a method can fundamentally solve the mismatch problem between controlled object parameters and control parameters, but it needs to overcome the influences of controlled object nonlinear, measurement noise, and other factors to achieve accurate dynamic identification of permanent magnet flux linkage.In order to realize accurate identification of permanent magnet flux linkage, the least square method [7], Luenberger observer [8], model reference adaptive algorithm [9], immune algorithm [10], genetic algorithm [11], self-learning particle swarm algorithm [12], and adaptive variation difference algorithm [13] have been proposed in the relevant literature. However, the identification results of the Luenberger observer and least square method are susceptible to measurement noise, and the model reference adaptive algorithm is difficult to determine a reasonable adaptive law; the genetic algorithm and particle swarm optimization are model-free identification methods, and their identification results lack determined theoretical support with strict mathematical significance [14].As Bayesian filtering provides a unified framework for solving the state estimation problem of stochastic systems, it can provide a solution with deterministic mathematical significance for permanent magnet flux linkage identification of PMSM. The state estimation method represented by the extended Kalman filter and its improved algorithm has been widely used in permanent magnet flux linkage identification of PMSM [15]. However, the extended Kalman filter is a suboptimal filtering algorithm, and if the estimated system does not satisfy the local linear condition or the state noise and measurement noise do not meet the requirements of a Gaussian distribution, the estimation error increases significantly. Different from the extended Kalman filter, the particle filter uses a weighted sum of a random sample to represent the posterior probability density function of the random system; because of the dispersion of random samples and the wide applicability of the Monte Carlo method to generate a random sample, this algorithm gets rid of the constraints of Gaussian distribution on state estimation [16], but its state estimation process does not consider the latest information of the estimated system, which reduces the estimation accuracy, and this algorithm has serious particle degradation problem.In this paper, an improved unscented particle filter algorithm is proposed to accurately identify the permanent magnet flux linkage of the MPCC control system for PMSM. The importance sampling of the particle filter algorithm is modified first by the unscented Kalman filter algorithm, which is called the traditional UPF algorithm, to solve the particle degradation problem of the particle filter algorithm. Then, the traditional UPF algorithm is improved by the differential evolution (DE) algorithm, which is called the improved UPF algorithm (IUPF), and the IUPF algorithm is implemented in this paper to ensure particle diversity in the course of permanent magnet flux linkage identification and then improve the state estimation performance and achieve the highly accurate identification of permanent magnet flux linkage. Compared with previous studies, the proposed permanent magnet flux linkage identification algorithm overcomes the problem that the identification results of the least square method and Luenberger observer method are sensitive to the measurement noise, and on the other hand, it overcomes the problem of the self-learning algorithm and genetic algorithm that have a large amount of calculation, at the same time, can also effectively solve the problem of the filtering accuracy of the traditional Bayesian filtering algorithm that is affected by non-Gaussian noise or its existing particle degradation and particle diversity reduction, which lays a foundation for the accurate identification of permanent magnet flux linkage.The structure of this paper is as follows: the principle of the model predictive current control algorithm, differential evolution algorithm, and IUP algorithm is presented in Section2; Section 3 presents the PMSM state equation for permanent magnet flux linkage identification; the simulation research and experimental verification are carried out in Section 4 and Section 5, respectively; the sixth section gives the research conclusion.


## 2. Principle of the MPCC Algorithm, DE Algorithm, and IUPF Algorithm

### 2.1. MPCC Algorithm

#### 2.1.1. Prediction Model

In thed-q axis synchronous rotation coordinate system, the PMSM dynamic current equation can be expressed as(1)diddt=−RsLdid+ωeLqLdiq+1Ldud,diqdt=−RsLqiq−ωeLdLqid+1Lquq−ωeLqψf.where ud,q, id,q, and Ld,q represent d-q axis stator voltage, d-q axis stator current, and d-q axis stator inductance, respectively, ωe is rotor electric angular speed, Rs is stator resistance, ψf is permanent magnet flux linkage, and d/dt represents a differential operator.If the current sample period is short enough, the discrete model of equation (1) can be expressed by the Taylor series and expressed in the following equation:(2)diddt=idk+1−idkTs,diqdt=iqk+1−iqkTs.In equation (2), Ts represents the sampling period, id(k), iq(k), and id(k + 1) and iq(k + 1) represents the d-q axis stator current at the moment of (k)Ts and (k + 1)Ts, respectively. Equation (1) is discretized by equation (2) to obtain the discrete current predictive model of PMSM, which is expressed as follows:(3)idpk+1=1−TsRsLdidk+TsudkLd+TsωekLqLdiqk,iqpk+1=1−TsRsLqiqk+TsuqkLq−TsωekLdLqidk−TsωekψfLq.In equation (3), idpk+1 and iqpk+1 represent (k + 1)thd-q axis predictive current, respectively.


#### 2.1.2. Cost Function Design

The three-phase PMSM drive system powered by a two-level inverter has eight voltage vectors. In order to achieve high-performance stator current tracking of PMSM drive systems with the MPCC algorithm, a reasonable cost function should be defined, and the voltage vector with the minimum cost function should be taken as the optimal voltage vector of the PMSM drive system in the next sampling period.In this paper, the expression of the defined cost function is(4)gi=id∗−idpk+12+iq∗−iqpk+12+fidpk+1,iqpk+1.In equation (4), i = 0,1, …, 7; id∗ and iq∗ represent d axis and q axis reference current, respectively, and fidpk+1,iqpk+1 is the nonlinear equation, which is expressed in the following equation:(5)fidpk+1,iqpk+1=∞,idpk+1>idmax,or,iqpk+1>iqmax,0,idpk+1<idmax,and,iqpk+1<iqmax.In equation (5), idmax is the d-axis current limiting value, and iqmax represents the q-axis current limiting value. According to equation (5), if the predictive current generated by the voltage vector exceeds idmax and iqmax, the cost function is infinite, and its corresponding voltage vector will be discarded. If the predictive current is within the allowable range, the value of equation (5) is 0, and then, the cost function of (4) only has the first two terms; the optimal voltage vector with the minimum cost function is calculated, and then, it is applied to the next control cycle of the PMSM drive system.


### 2.2.  The  Principle of the DE Algorithm

The DE algorithm is a global optimization algorithm, and its implementation process is as follows [17]:(1)
Initializing population. The initialization expression is(6)xj,i0=xj,imin+r,0,1xj,imax−xj,imin,wherexj,imax and xj,imin represent search space upper limit value and lower limit value, respectively, xj,i0 represents jth dimension of the ith individual in the 0th generation, and rand(0,1) represents a random number evenly distributed between (0,1).(2)
The mutation process. The mutation strategy can be expressed as(7)vig+1=xr3g+Fxr1g−xr2g,wherexrig represent rith individual of the gth population; F is the zoom factor, which is generally a small positive number between 0 and 2, and it needs to meet the requirement of i≠r1≠r2≠r3.(3)
Crossover process. The process is designed to generate a test vector and can be expressed as(8)uj,ig+1=vj,ig+1,rand0,1≤crorj=jrandxj,ig,other.In equation (8), cr is the crossover probability, generally taken as a small positive number between 0 and 1. vj,i is the variation vector and xj,i is the target vector, and the cr value determines the proportion of vj,i and xj,i in the test vector. F and cr determine the convergence speed and search capability of the DE algorithm. Small cr and larger F are beneficial to improving the global search capability of the algorithm, whereas larger cr and small F are beneficial to improving the convergence speed and local search capability of the algorithm.(4)
Greedy selection. Calculating the fitness ofuj,ig+1 in the crossover process and making the selection according to the following equation:(9)xj,ig+1=uj,ig+1,fuj,ig+1≤fxj,igxj,ig,other,wheref⋅ represents the fitness function.(5)
Boundary treatment. Individuals out of range are treated according to the following equation:(10)x=xmin,x<xmin,xmax,x>xmax,x,other.


### 2.3. IUPF Algorithm

The UPF algorithm generates an important density function based on the unscented Kalman filter algorithm, which can solve the particle degradation problem under the traditional particle filter framework. The specific implementation steps of the UPF algorithm are as follows:(1)
Algorithm initialization. Settingk = 0, extracting N particles from the prior probability distribution p(x0), that is(11)x0ii=1N∼px0,whereN is a particle number and x0i represent a particle set. Setting ω0i=1/N, meanwhile, setting k = 0.(2)
Importance density sampling. Obtaining each random sampling pointxki and the recommended distribution function Nx¯ki,Pki based on the UKF algorithm and extracting particles from the recommended distribution function, it can be expressed as(12)x^ki∼Nx¯ki,Pki.The particle weights are calculated and normalized.(3)
Resampling a particle set according to the resampling algorithm, obtaining the new particle set expressed asxki,ω˜ki, and setting ω˜ki=1/N.(4)
Outputting the estimation results of the state vector, which is as follows:(13)x^k=∑i=1Nω˜kixki,Pk=∑i=1Nω˜kixki−x^xki−x^T.Wherex^k and pk represent the estimates of the state vector and the estimation error covariance matrix at time k. Setting k = k + 1, return to step 2 and repeat the algorithm.The IUPF algorithm proposed in this paper utilizes the DE algorithm to improve the resampling process of the traditional UPF algorithm, which can solve the problem of traditional UPF algorithm particle diversity reduction. A new optimal particle set was obtained, and its normalized weights were calculated according to the following equation:(14)ω˜ki=ωxki∑i=1Nωxki.In equation (14), k represents the kth iteration and i represents the ith particle.


## 2.1. MPCC Algorithm

### 2.1.1. Prediction Model

In thed-q axis synchronous rotation coordinate system, the PMSM dynamic current equation can be expressed as(1)diddt=−RsLdid+ωeLqLdiq+1Ldud,diqdt=−RsLqiq−ωeLdLqid+1Lquq−ωeLqψf.where ud,q, id,q, and Ld,q represent d-q axis stator voltage, d-q axis stator current, and d-q axis stator inductance, respectively, ωe is rotor electric angular speed, Rs is stator resistance, ψf is permanent magnet flux linkage, and d/dt represents a differential operator.If the current sample period is short enough, the discrete model of equation (1) can be expressed by the Taylor series and expressed in the following equation:(2)diddt=idk+1−idkTs,diqdt=iqk+1−iqkTs.In equation (2), Ts represents the sampling period, id(k), iq(k), and id(k + 1) and iq(k + 1) represents the d-q axis stator current at the moment of (k)Ts and (k + 1)Ts, respectively. Equation (1) is discretized by equation (2) to obtain the discrete current predictive model of PMSM, which is expressed as follows:(3)idpk+1=1−TsRsLdidk+TsudkLd+TsωekLqLdiqk,iqpk+1=1−TsRsLqiqk+TsuqkLq−TsωekLdLqidk−TsωekψfLq.In equation (3), idpk+1 and iqpk+1 represent (k + 1)thd-q axis predictive current, respectively.


### 2.1.2. Cost Function Design

The three-phase PMSM drive system powered by a two-level inverter has eight voltage vectors. In order to achieve high-performance stator current tracking of PMSM drive systems with the MPCC algorithm, a reasonable cost function should be defined, and the voltage vector with the minimum cost function should be taken as the optimal voltage vector of the PMSM drive system in the next sampling period.In this paper, the expression of the defined cost function is(4)gi=id∗−idpk+12+iq∗−iqpk+12+fidpk+1,iqpk+1.In equation (4), i = 0,1, …, 7; id∗ and iq∗ represent d axis and q axis reference current, respectively, and fidpk+1,iqpk+1 is the nonlinear equation, which is expressed in the following equation:(5)fidpk+1,iqpk+1=∞,idpk+1>idmax,or,iqpk+1>iqmax,0,idpk+1<idmax,and,iqpk+1<iqmax.In equation (5), idmax is the d-axis current limiting value, and iqmax represents the q-axis current limiting value. According to equation (5), if the predictive current generated by the voltage vector exceeds idmax and iqmax, the cost function is infinite, and its corresponding voltage vector will be discarded. If the predictive current is within the allowable range, the value of equation (5) is 0, and then, the cost function of (4) only has the first two terms; the optimal voltage vector with the minimum cost function is calculated, and then, it is applied to the next control cycle of the PMSM drive system.


## 2.1.1. Prediction Model

In thed-q axis synchronous rotation coordinate system, the PMSM dynamic current equation can be expressed as(1)diddt=−RsLdid+ωeLqLdiq+1Ldud,diqdt=−RsLqiq−ωeLdLqid+1Lquq−ωeLqψf.where ud,q, id,q, and Ld,q represent d-q axis stator voltage, d-q axis stator current, and d-q axis stator inductance, respectively, ωe is rotor electric angular speed, Rs is stator resistance, ψf is permanent magnet flux linkage, and d/dt represents a differential operator.If the current sample period is short enough, the discrete model of equation (1) can be expressed by the Taylor series and expressed in the following equation:(2)diddt=idk+1−idkTs,diqdt=iqk+1−iqkTs.In equation (2), Ts represents the sampling period, id(k), iq(k), and id(k + 1) and iq(k + 1) represents the d-q axis stator current at the moment of (k)Ts and (k + 1)Ts, respectively. Equation (1) is discretized by equation (2) to obtain the discrete current predictive model of PMSM, which is expressed as follows:(3)idpk+1=1−TsRsLdidk+TsudkLd+TsωekLqLdiqk,iqpk+1=1−TsRsLqiqk+TsuqkLq−TsωekLdLqidk−TsωekψfLq.In equation (3), idpk+1 and iqpk+1 represent (k + 1)thd-q axis predictive current, respectively.


## 2.1.2. Cost Function Design

The three-phase PMSM drive system powered by a two-level inverter has eight voltage vectors. In order to achieve high-performance stator current tracking of PMSM drive systems with the MPCC algorithm, a reasonable cost function should be defined, and the voltage vector with the minimum cost function should be taken as the optimal voltage vector of the PMSM drive system in the next sampling period.In this paper, the expression of the defined cost function is(4)gi=id∗−idpk+12+iq∗−iqpk+12+fidpk+1,iqpk+1.In equation (4), i = 0,1, …, 7; id∗ and iq∗ represent d axis and q axis reference current, respectively, and fidpk+1,iqpk+1 is the nonlinear equation, which is expressed in the following equation:(5)fidpk+1,iqpk+1=∞,idpk+1>idmax,or,iqpk+1>iqmax,0,idpk+1<idmax,and,iqpk+1<iqmax.In equation (5), idmax is the d-axis current limiting value, and iqmax represents the q-axis current limiting value. According to equation (5), if the predictive current generated by the voltage vector exceeds idmax and iqmax, the cost function is infinite, and its corresponding voltage vector will be discarded. If the predictive current is within the allowable range, the value of equation (5) is 0, and then, the cost function of (4) only has the first two terms; the optimal voltage vector with the minimum cost function is calculated, and then, it is applied to the next control cycle of the PMSM drive system.


## 2.2.  The  Principle of the DE Algorithm

The DE algorithm is a global optimization algorithm, and its implementation process is as follows [17]:(1)
Initializing population. The initialization expression is(6)xj,i0=xj,imin+r,0,1xj,imax−xj,imin,wherexj,imax and xj,imin represent search space upper limit value and lower limit value, respectively, xj,i0 represents jth dimension of the ith individual in the 0th generation, and rand(0,1) represents a random number evenly distributed between (0,1).(2)
The mutation process. The mutation strategy can be expressed as(7)vig+1=xr3g+Fxr1g−xr2g,wherexrig represent rith individual of the gth population; F is the zoom factor, which is generally a small positive number between 0 and 2, and it needs to meet the requirement of i≠r1≠r2≠r3.(3)
Crossover process. The process is designed to generate a test vector and can be expressed as(8)uj,ig+1=vj,ig+1,rand0,1≤crorj=jrandxj,ig,other.In equation (8), cr is the crossover probability, generally taken as a small positive number between 0 and 1. vj,i is the variation vector and xj,i is the target vector, and the cr value determines the proportion of vj,i and xj,i in the test vector. F and cr determine the convergence speed and search capability of the DE algorithm. Small cr and larger F are beneficial to improving the global search capability of the algorithm, whereas larger cr and small F are beneficial to improving the convergence speed and local search capability of the algorithm.(4)
Greedy selection. Calculating the fitness ofuj,ig+1 in the crossover process and making the selection according to the following equation:(9)xj,ig+1=uj,ig+1,fuj,ig+1≤fxj,igxj,ig,other,wheref⋅ represents the fitness function.(5)
Boundary treatment. Individuals out of range are treated according to the following equation:(10)x=xmin,x<xmin,xmax,x>xmax,x,other.


## 2.3. IUPF Algorithm

The UPF algorithm generates an important density function based on the unscented Kalman filter algorithm, which can solve the particle degradation problem under the traditional particle filter framework. The specific implementation steps of the UPF algorithm are as follows:(1)
Algorithm initialization. Settingk = 0, extracting N particles from the prior probability distribution p(x0), that is(11)x0ii=1N∼px0,whereN is a particle number and x0i represent a particle set. Setting ω0i=1/N, meanwhile, setting k = 0.(2)
Importance density sampling. Obtaining each random sampling pointxki and the recommended distribution function Nx¯ki,Pki based on the UKF algorithm and extracting particles from the recommended distribution function, it can be expressed as(12)x^ki∼Nx¯ki,Pki.The particle weights are calculated and normalized.(3)
Resampling a particle set according to the resampling algorithm, obtaining the new particle set expressed asxki,ω˜ki, and setting ω˜ki=1/N.(4)
Outputting the estimation results of the state vector, which is as follows:(13)x^k=∑i=1Nω˜kixki,Pk=∑i=1Nω˜kixki−x^xki−x^T.Wherex^k and pk represent the estimates of the state vector and the estimation error covariance matrix at time k. Setting k = k + 1, return to step 2 and repeat the algorithm.The IUPF algorithm proposed in this paper utilizes the DE algorithm to improve the resampling process of the traditional UPF algorithm, which can solve the problem of traditional UPF algorithm particle diversity reduction. A new optimal particle set was obtained, and its normalized weights were calculated according to the following equation:(14)ω˜ki=ωxki∑i=1Nωxki.In equation (14), k represents the kth iteration and i represents the ith particle.


## 3. State Equation of the Identified System

Compared with state vectors such as the stator current, permanent magnet flux linkage almost causes no change during a control cycle of the PMSM drive system [7, 8, 10]. Combined with the PMSM dynamic current equation shown in (1), the state equation used for the identification of permanent magnet flux linkage is obtained, which is expressed in the following equation:(15)diddt=udLd−RsLdid+ωeLqLdiq,diqdt=uqLq−RsLqiq−ωeLdLqid−ωeψfLq,dψfdt=0,where d/dt represents the differential operator. The state vector, the output vector, and the input vector of the state equation are defined as x, y, andu, respectively, which can be expressed as(16)x=idiqψfT,u=udLduqLqT,y=idiqT.According to (16), the permanent magnet flux linkage ψf is included in the state vector of the identified system, and its online identification can be easily achieved by state vector estimation based on the proposed IUPF algorithm.


## 4. Simulation Study

In this paper, the proposed method was simulated in the MPCC system of PMSM, and the system structure is shown in Figure1. In Figure 1, the controlled motor is an interior PMSM, and its parameters are listed in Table 1; the speed and current sampling period and the simulation step size of the PMSM drive system are all set at 0.1 ms, and the load torque is set at 50 N m. Non-Gaussian random noise with an amplitude of 3 is injected into the measurements, and the reference speed of the PMSM is shown in Figure 2. For balancing the identification stability, identification speed, and identification accuracy of the proposed algorithm, the particle number is set at 50 in the simulation study.Figure 1
The model predictive current control system of PMSM.Table 1
Controlled motor parameters.
ParametersRated valueParametersRated valuePower50 KWd-axis inductance2.517 mHSpeed900 rpmq-axis inductance5.99 mHPole pairs4Permanent magnet flux linkage0.173 WbStator resistance0.0154 ΩRotational inertia0.00625 kg·m2Figure 2
Reference speed and its response of PMSM.The speed response in Figure2 is actual speed, Figure 3 shows the d-axis reference current and its response, and Figure 4 shows the q-axis reference current and its response. According to Figures 2 to 4, when the actual controlled object parameters are adapted to the MPCC algorithm control parameters, the MPCC algorithm can achieve fast tracking of PMSM speed without overshoot and accurate control of d-q axis current.Figure 3
d-axis reference current and its response.Figure 4
q-axis reference current and its response.Figures5 and 6 show the identification results of the permanent magnet flux linkage based on the traditional UPF algorithm and the IUPF algorithm with the same particle number, respectively. Combined with the design value of the permanent magnet flux linkage shown in Table 1, the reference speed shown in Figure 2, and the non-Gaussian random noise injected into the measurements, it can be seen that in the non-Gaussian measurement noise environment and dynamic and steady operation conditions of the PMSM drive system, the accurate identification of permanent magnet flux linkage with smaller identification variance can be realized with the IUPF algorithm, and the IUPF algorithm has better identification performance than the traditional UPF algorithm.Figure 5
The simulation identification result based on the traditional UPF algorithm.Figure 6
The simulation identification result based on the IUPF algorithm.Figure7 shows the particle weight distribution of the traditional UPF algorithm and the IUPF algorithm after 5000 iterations. According to Figure 7, during the implementation of the traditional UPF, the small weight particle number is constantly reduced, and the large weight particles are constantly duplicated, which leads to a significant decrease in particle diversity; then, the identification performance of traditional UPF algorithms which employ particle dispersion to approximate the system state is reduced, whereas the weight of each particle is not equal in the iterative process of the IUPF algorithm. Therefore, it maintains a perfect particle diversity, which is also an important reason why the IUPF algorithm can obtain smaller identification variance and more smooth identification results of permanent magnet flux linkage in the MPCC system for PMSM.Figure 7
The particle weight of the traditional UPF and IUPF algorithm. (a) The particle weight of the traditional UPF algorithm. (b) The particle weight of the IUPF algorithm.
(a)(b)


## 5. Experimental Study

In order to realize the experimental study of the IUPF algorithm proposed in this paper, an experimental platform of the MPCC system for PMSM was developed. The experimental motor is a face-mounted PMSM, whose parameters are as follows: rated voltage is 380 V, the permanent magnet flux linkage is designed to be 0.1278 Wb, stator resistance is 0.28 Ω, stator inductance is 1.273 mH, pole pairs is 4, the moment of inertia is 0.00214 kg·m2.Taking both the steady state and dynamic state conditions of the MPCC system for PMSM to experimentally study the identification performance of the proposed IUPF algorithm, the discrete period of the IUPF algorithm and the control period of the PMSM drive system are all set at 0.1 ms, load torque is set at 3 N·m, and the particle number is also set at 50. The measured speed waveform from 1800 rpm to 900 rpm is shown in Figures8(a) and 8(b), and they show the measured current waveform when the speed drops to 900 rpm.Figure 8
The measured rotor speed dynamic and stator current of PMSM. (a) The measured rotor speed waveform. (b) The measured stator current waveform.
(a)(b)The experimental identification result of the traditional UPF algorithm and the IUPF algorithm is shown in Figures9 and 10, respectively. From Figures 9 and 10, it can be seen that the proposed IUPF algorithm can accurately identify permanent magnet flux linkage under both dynamic and steady operating conditions of the MPCC system for PMSM, and the identification result has a smaller variance than that of the traditional UPF algorithm. The experimental results are consistent with those of simulation.Figure 9
The experimental identification result based on the traditional UPF algorithm.Figure 10
The experimental identification result based on the IUPF algorithm.The particle weight distribution of the traditional UPF algorithm and the IUPF algorithm after a certain number of iterations is shown in Figure11. According to Figure 11, with the execution of the UPF algorithm, its particle diversity decreases significantly, while the IUPF algorithm can always maintain the diversity of particle weight during its iteration, which is very beneficial to improving the identification performance of the proposed IUPF algorithm. The experimental results are also consistent with the simulation results.Figure 11
The particle weight of the traditional UPF and IUPF algorithm. (a) The particle weight of the traditional UPF algorithm. (b) The particle weight of the IUPF algorithm.
(a)(b)


## 6. Conclusions

Aiming at the MPCC system of PMSM, a novel identification method based on the IUPF algorithm for permanent magnet flux linkage is proposed in this paper. Simulation research and experimental verification of the proposed method are carried out, and the research results provide a basis for realizing high-performance control of the MPCC system for PMSM under the condition of permanent magnet flux linkage variation, and the following conclusions are obtained.(1)
When the actual controlled object parameters are adapted to the MPCC algorithm control parameters, the MPCC algorithm can achieve fast tracking speed of PMSM without overshoot and accurate control of d-axis current and q-axis current, and the PMSM drive system with the MPCC algorithm has perfect speed and current control performance.(2)
The IUPF algorithm can always keep the diversity of particle weights during its iteration, which effectively solves the problem of the particle diversity decrease with the traditional UPF algorithm.(3)
The IUPF algorithm can achieve accurate identification of permanent magnet flux linkage under non-Gaussian measurement noise and steady state and dynamic state operation conditions of the MPCC system for PMSM, and the identification result has smaller variance. The proposed IUP algorithm is suitable for both interior PMSM and surface mounted PMSM.



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