A Simple and Robust Model Predictive Current Control of PMSM Using Stator Current Predictor and Target-Oriented Cost Function

Model predictive current control of permanent magnet synchronous motor(PMSM), which inherits from the vector control framework, has become a promising control strategy because of its simple structure and fast dynamic response. In order to protect the stator current prediction model from the influence of mismatched parameters, a simple and robust predictive current control with stator current predictor and target-oriented cost function (abbreviated as SCP-TOSF-PCC) is proposed in this paper. The feedback mechanism is introduced into the prediction equation in the discrete domain, and its design principle and stability analysis are also described in detail. In addition, different from the traditional cost function design method which is constructed by the tracking errors of stator current, this paper proposes a direct target-oriented cost function, which takes the average pulsations of <inline-formula> <tex-math notation="LaTeX">${d}$ </tex-math></inline-formula>-axis current and <inline-formula> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula>-axis current as the judge index. The design method of this cost function takes into account the influence of historical values on the voltage vector selection. The experimental results show that compared with the traditional methods, the proposed method has better dynamic, steady-state and robust performance.

and is widely used in the field of power electronics and 23 electrical drive [1], [2], [3], [4], [5], [6]. Model predictive 24 current control (MPCC) strategy of permanent magnet syn-25 chronous motor, which retains the basic principle of vector 26 control, is considered to be a promising method. In [7], for the 27 PMSM drive system of electric vehicle, the author compares 28 the vector control and predictive current control strategies 29 The associate editor coordinating the review of this manuscript and approving it for publication was Fangfei Li . in detail. The experimental results show that predictive cur-30 rent control has faster dynamic performance, smaller stator 31 current harmonics (at the same switching frequency), smaller 32 electromagnetic torque ripple and electromagnetic interfer-33 ence performance. 34 For predictive current control of PMSM, mismatched 35 motor parameters will lead to stator current prediction errors 36 and reduce the control performance of the system [8], [9]. 37 Scholars mainly improve the prediction accuracy of stator 38 current from two aspects. One method is to propose deadbeat 39 predictive current control (DPCC) based on the disturbance 40 observer. The voltage vector reference is calculated based on 41 the deadbeat principle. The voltage vector calculation error 42 caused by mismatch parameters is regarded as a disturbance, 43 and the error is estimated and compensated in realtime based 44 on the disturbance observer. In [10], static-errorless deadbeat 45 predictive current control using a second-order sliding-mode 46 disturbance observer is proposed to avoid to suffer from 47 developed. The calculation method converting torque and 95 flux magnitude into stator flux vector is presented in detail. 96 This method not only cancels the weight factor, but also 97 has excellent control performances. In [18], model predictive 98 flux control for PMSM drives is developed, and then the 99 robustness is analysed and the compensated strategy based on 100 sliding mode observer is utilized to improve the robustness.

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In recent years, scholars have introduced neural networks, 102 intelligent optimization algorithms and other theories into 103 the setting of weight factors. In [19], artificial neural net-104 works(ANN) are trained by simulation data or experimental 105 data through the fitness function, and then ANN are utilized 106 to design the weighing factors for different states of induction 107 machine. In [20], online weighting factor optimization by 108 simplified simulated annealing for induction machine drives 109 are proposed, where during every control period, the weigh-110 ing factor is optimized by intelligent algorithm. This strat-111 egy ensures the optimal factor for every control period,and 112 the limitation of the maximum iteration step prevents the 113 algorithm from falling into a dead loop. In [21], for pre-114 dictive current controller of a six-phase induction machine, 115 a weighting factor design based on particle swarm optimiza-116 tion algorithm is proposed to coordinated torque currents and 117 harmonic currents. It is known that the introduction of intel-118 ligent optimization algorithm provides a better solution to set 119 the weight factors, but it also increases the complexity of the 120 system. In addition to the cost function, the implementation 121 of intelligent optimization algorithm introduces the fitness 122 function. Therefore, the simple and effective setting methods 123 of weight factors has always been the goal of academia.

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Aiming at the two problems of robustness and weight 125 factor design method of predictive current control, this paper 126 proposes robust stator current predictor and target-oriented 127 cost function design methods, respectively. The arrangement 128 of this paper is as follows: Section II introduces the tradi-129 tional predictive current control, and then analyzes the robust-130 ness of stator current predictive equation under mismatched 131 parameters. In Section III, the design method and stabil-132 ity analysis of robust stator current predictor are described 133 in detail. Section III also expounds the design principle of 134 target-oriented cost function. Section IV verifies the effec-135 tiveness of the proposed algorithm through experiments. 136 Finally, this paper is summarized in Section V.  Stator current equation (3) can be converted into the rotor 150 magnetic field frame system, which is expressed as (4). 151 v dqs and i dqs are the vectors in the rotor magnetic field frame 152 system. Transformation equation for i s and i dqs is as follows: 153 VOLUME 10, 2022 i dqs = i s e −jθ r , where θ r is the rotor position of PMSM [22]. v dqs = R s i dqs + L s di dqs dt + jL s ω r i dqs + jω r ψ f (4) In the model predictive current control, according to the stator 157 current equation (3) and the first-order Euler discretization 158 equation (5), the predictive equation of stator current can be 159 expressed as equation (6).
The cost function for predictive current control is designed 164 as (7), where i * s (k + 1) denotes the stator current reference.
The block diagram of traditional predictive current control Step I: measure the stator currents i a , i b , and obtain the 171 stator current vector i s at the sampling time k by Clarke 172 transform.

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Step II: based on the stator current prediction equation (6), 174 calculate the stator current vectors i s (k + 1) j at the sampling 175 time k + 1 for seven different voltage vectors.

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Step III: based on the cost function (7), the voltage vector 177 that minimizes the cost function is selected as the optimal one, 178 and applied at the sampling time k + 1. flux linkage ψ f . According to the prediction equation (6), 188 the mismatched PMSM parameters will lead to the prediction 189 errors of stator current vector.

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In order to analyze the influence of mismatched parameters 191 on the prediction errors of stator current vector, a detailed 192 analysis in theory is presented in this paper. The nominal 193 parameters of PMSM are set as R s , L s and ψ f , and the 194 corresponding prediction equation is the equation (6). The 195 mismatch parameters of PMSM are set asR s ,L s andψ f , and 196 the corresponding prediction equation is expressed as (8).
The stator current prediction error is defined The angle prediction error of stator current vector is defined 203 as | ρ| |ρ| , where ρ = i s and ρ = ĩ s − i s . When 204 R s = 2.0R s , the prediction error of stator current vector is 205 presented in Fig.2(a). It can be seen that, from Fig.2(a), The 206 stator current error | i s (k+1)| |i s (k+1)| increases with the increase of 207 electromagnetic torque, while | i s (k+1)| |i s (k+1)| decreases with the 208 increase of the rotor speed. The maximum value of stator 209 current error is 0.019, i.e. 1.9%. In Fig.2(b), the prediction 210 error increases with the increase of rotor speed, while the 211 electromagnetic torque has little effect on it, but the elec-212 tromagnetic torque has a greater effect only under light-load 213 state. Under high-speed and light-load states, the maximum 214 value of | i s (k+1)| |i s (k+1)| is 17.5%. In Fig.2(c), | i s (k+1)| |i s (k+1)| is directly 215 proportional to the speed and torque, and the maximum value 216 of it is 41.2%. WhenL s = 0.5L s andψ f = 0.5ψ f , stator 217 current prediction error is more serious due to the combined 218 action of mismatched inductance and permanent magnet flux, 219 which is illustrated in Fig.2 221 and Fig.2(e). Therefore, it can be concluded that the influ-222 ence of mismatched stator resistance is very small compared 223 withL s andψ f . Fig.2(e) presents the angle prediction error 224 of stator current vector, and its maximum value is 19.3%.

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From the above analysis, it can be seen that the mismatched 226 stator resistance has little impact on the stator current predic-227 tion error, while the mismatched excitation inductance and 228 permanent magnet flux linkage have a great impact on the 229 stator current prediction error. The mismatched parameters 230 have little effect on the angle prediction error. The stator 231 current prediction error is related to electromagnetic torque, 232 speed, rotor position and voltage vector, and its variation law 233 is complex.  The step response method is utilized to achieve the dis-250 crete mathematical equations, and the feedback compen-251 sation mode is proposed to eliminate the prediction error.

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And then, the stator current predictor is developed, which 253 is expressed as (11). v sum (k + 1) represents the total com-254 pensation item, which is mainly composed of two parts. 255 one is the cross coupling part, that is, ω r L q i qs (k + 1) − 256 jω r L d i ds (k + 1) − jω r ψ f , the other is current compensation,
It is known that at k sampling instant, stator cur-266 rent i s (k + 1) cannot be measured, and only the actual stator 267 current i s (k) can be measured by stator current sensor. The 268 actual stator current i s (k + 1) is estimated by Langrange 269 extrapolation method (13) in this paper. And then, the pre-270 diction error of stator current at k + 1 sampling instant can be 271 computed by the equation (14).
According to the control theory of linear discrete sys-275 tem, the transfer function of stator current predictor can be 276 expressed as (15),  where M 1 , M 2 , E 1 , E 2 and E 3 are presented as follows: where the equation (16) is the cost function in stator refer-298 ence frame system and equation (17) is the cost function in 299 dq-axis (17) system. In (16), λ β is the weight factor. Because 300 i α and i β belong to the same dimension, it is usually set as 301 λ β = 1 for simplicity. However, λ β = 1 is not the optimal 302 value, which has been studied in detail in [23].
The above cost functions only are designed by stator cur-306 rent error at k +2 sampling instant, and does not consider sta-307 tor current errors at the past sampling instants (i.e. sampling 308 instant k + 1, k, k − 1, etc.). It is known that, in essence, the 309 cost function is utilized to select the optimal voltage vector 310 to achieve the minimum stator current ripple and the fastest 311 dynamic response.

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In this paper, the target-oriented cost function is proposed 313 for the first time, which directly takes the stator current ripple 314 as the evaluation index and cancels the setting of weight 315 factor. For d-axis stator current, the current ripple RMSE dj 316 is defined as (18), where N represents the calculated number 317 of stator current i dj (k + 2 − i). In this paper, N = 9, which 318 means that i d (k + 2) , . . . , i d (k − 7) are involved in the cal-319 culation of d-axis stator current ripple. Similarly, the current 320 ripple RMSE qj is defined as (19).

RMSE dj
Therefore, the stator current vector ripple is defined as 326 equation (20), and the targeted-oriented cost function is 327 designed as equation (21). The block and flow diagrams of the proposed algorithm are 333 illustrated in Fig.5 and Fig.6. The implementation process of 334 the proposed algorithm is mainly divided into the following 335 steps:

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Step I: measure stator current i s (k) and rotor speed ω r .

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Step II: Based on i dqs (k), v opt (k) and stator current pre-338 dictor, stator current i dqs (k + 1) at k sample instant are esti-339 mated, in order to eliminate the delay of the digital system, 340 where v opt (k) is the optimal vector computed at last sampling 341 period. 342 i dqs (k + 2) is predicted by the proposed stator current pre-344 dictor.

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Step V: The optimal voltage vector v opt (k + 1) is obtained 348 by targeted-oriented cost function (21), and applied at k + 1 349 sample instant.  Table 1.     load torque occurring suddenly for the proposed SCP-TOSF-392 PCC are illustrated in Fig.11. From Fig.11, it can be seen that 393 the proposed SCP-TOSF-PCC has excellent speed recovery 394 ability. The load torque 5.5 N · m is exerted at 4.54 s, and the 395 electromagnetic torque immediately becomes 5.5 N · m. 396 The robustness experiments of T-PCC and SCP-TOSF-397 PCC are illustrated in Fig.12 amd Fig.13. The robustness 398 test method is to use mismatched motor parameters in the 399 control algorithms executed by DSP controller. In 0-4.0 s, 400 the T-PCC algorithm is executed in DSP controller, and in 401 4.0-8.0 s, the proposed SCP-TOSF-PCC algorithm is exe-402 cuted in DSP controller. For L * s = 2.0L s , where L * s denotes 403 the mismatched parameter and L s is nominal parameter, the 404 experimental result is presented in Fig.12. It can be seen that 405 the proposed SCP-TOSF-PCC has better robustness than the 406 traditional T-PCC. For T-PCC of PMSM, the ripple of stator 407 flux magnitude is 0.025 Wb and that of SCP-TOSF-PCC is 408 only 0.015 Wb. The ripple of electromagnetic torque for the 409 proposed SCP-TOSF-PCC is 2.5 N · m, which is 28.6% lower 410 than that of T-PCC.  For ψ * f = 0.5ψ f , Fig.13 shows the results of experiments. the stator current predictor proposed in this paper can over-432 come the prediction error caused by mismatch parameters 433 by introducing a simple feedback mechanism. For the design 434 of cost function, this paper proposes a targeted-oriented cost 435 function design method, which directly takes the ripple of d-436 axis stator current and q-axis stator current as the evaluation 437 index, which has the advantages of simplicity and directness. 438 Experimental results show that the proposed algorithm has 439 excellent dynamic performance, steady-state performance 440 and speed recovery performance. For the mismatched param-441 eters, the proposed SCP-TOSF-PCC algorithm has better 442 robustness than the traditional methods.