Analysis and Robustness Improvement of Finite-Control-Set Model Predictive Current Control for IPMSM With Model Parameter Mismatches

Analysis and robustness improvement of FCS-MPCC for IPMSM with model parameter mismatches are studied in this paper. The prediction error of the current in synchronous rotation coordinate is analyzed and it is divided into two categories according to whether it is related to the selected optimal voltage vector. A robustness improvement method by extracting the information of both kinds of prediction errors in the last sampling period is proposed. The simulation and experimental results demonstrate that the proposed method can effectively improve the ability to resist multiple model parameter mismatches.

of FCS-MPC may be deteriorated with model parameter 27 mismatches [9] and it has been one of the main barriers to 28 its widespread application. 29 To address the problem, some methods have been stud-30 ied. The model parameter mismatches can be viewed as 31 one of the disturbances of the system, and the extended 32 state observer [10] has been designed to compensate for 33 model parameter mismatches. In addition, the sliding-model 34 observers (SMO) have also been studied to enhance the 35 robustness [11], [12], [13], [14], [15]. A multistep error track-36 ing based continuous model predictive control with a SMO 37 differentiator is studied in [11] to improve the robustness. 38 A robust predictive speed control for PMSM using integral 39 SMO is proposed in [12]. Robust MPCC based equivalent 40 input disturbance approach for PMSM drive is studied in [13]. 41 The SMO is introduced in the non-cascade predictive control 42 to estimate and compensate the disturbance caused by the 43 uncertain parameters [14]. A continuous integral-type termi-44 nal SMO has been studied to deal with the mismatched distur-45 bance [ [27], [28]. A robust model reference 85 adaptive system estimator incorporating online parameter 86 identification algorithm for parallel predictive torque con-87 trol of induction motor is studied in [26]. The impact of 88 parameters mismatch on the FCS-MPCC performance of a 89 five-phase induction motor drive is studied in [27]. Influence 90 of covariance-based methods in the performance of predictive 91 controller for five-phase induction motor is studied in [28] to 92 improve the robustness. The models of the motors studied in 93 [26], [27], and [28] are different from IPMSMs, and accord-94 ingly, applying the methods in [26], [27], and [28] to IPMSMs 95 needs further studies. 96 In this paper, analysis and robustness improvement of is related to the selected optimal voltage vector. Then, 103 a parameter mismatch compensation method by calculating 104 both kinds of prediction errors according to the error infor-105 mation in last sampling period is proposed. The main contri-106 bution of this article is that the prediction errors with model 107 parameter mismatches considering the model of IPMSM 108 is studied and a compensation scheme by calculating the 109 prediction errors with a new and simple method is pro-110 posed. The proposed method can deal with multiple param-111 eter mismatches including the stator resistance, magnetic 112 flux linkage, inductances in both d and q axes. The control 113 performances of the proposed method including the steady-114 state errors of current tracking, current harmonics, and torque 115 ripples can ensure almost the same as the FCS-MPCC without 116 parameter mismatches. The effectiveness of the proposed 117 method is verified by the simulation and experimental results. 118 The IPMSM control system based on 2-level voltage source 123 inverter (2L-VSI) is shown in Figure 1 (a). There are 8 switch-124 ing states for 2L-VSI, which generate 8 different voltage 125 vectors (VVs) (V 0 , V 1 , V 2 . . . V 7 ) as shown in Figure 1 (b). 126 The positions of V 0 and V 7 are coincidence and they are 127 defined as zero VVs (ZVVs), and the others are defined as 128 non-zero VVs (NZVVs). The current predictive equations of IPMSM at the end of 130 kth sampling period are given in (1) and (2)  respectively.

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In order to compensate the time delay in the actuation, two-144 step prediction is usually used and the corresponding current 145 predictive equations are given in (3) and (4).
The cost function is designed as (5) to realize the target of then the VV that minimizes the cost function is selected.  The actual currents in d and q axes at the end of kth sam-168 pling period are defined as i d (k + 1) and i q (k + 1) and they where ' ' represents the prediction error, C d and C q can be 178 calculated as (8), M d and M q can be calculated as (9).
As can be seen from (6)-(9), the prediction errors have 182 already appeared in i p d (k + 1) and i p q (k + 1). The currents 183 in d and q axes are predicted twice in a whole process to 184 compensate the system delay. The prediction errors appeared 185 in i p d (k+2) and i p q (k+2) are caused by not only the deviation of 186 the motor parameters but also the prediction errors in i p d (k +1) 187 and i p q (k + 1), which may further increase the prediction 188 errors.

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The prediction errors in i p d (k + 2) and i p q (k + 2) are By combining (6)-(12), the prediction errors in i p d (k + 2) 198 and i p q (k + 2) can be expressed as According to (13) and (14), it can be seen that there are i.e., i d (k + 1) = 0 and i q (k + 1) = 0, (13) and (14) can 221 be rewritten as The form of (16) and (17) is the same as that of (6) and (7). At the kth instant, the current prediction errors in d and q 233 axes are (18) and (19), u d (k −1) and u q (k −1) are known optimal 237 voltage vectors at the (k − 1)th instant. When the optimal 238 voltage vectors are zero VVs, C d and C q are obtained by the 239 difference between the predicted current and actual current as 240 Since the sampling period is quite short, it can be con-243 sidered that the currents in d and q axes remain unchanged 244 between adjacent sampling periods. In the meantime, the 245 motor parameters are also approximately invariant and the 246 rotor speed keeps stable. Hence, if the optimal voltage vectors 247 are non-zero VVs, C d and C q can be regarded as equal to the 248 C d and C q in the last sampling period, respectively.

249
When the optimal voltage vectors are non-zero VVs, M d 250 and M q can be obtained by (22) and (23) as When the optimal voltage vectors are zero VVs, M d and M q 254 can be viewed as equal to the M d and M q in the last sampling 255 period with assuming the motor parameters invariant between 256 adjacent sampling periods.

257
Since u d (k) and u q (k) are optimal voltage vectors deter-258 mined at last sampling period, the current prediction errors at 259 the (k + 1)th instant can be figured out by (6) and (7). Then 260 the currents after compensation at the (k + 1)th instant are 261 expressed as where 'm' represents the modified value after compensation. 268 The modified currents are used for the second prediction 269 while the parameter mismatch errors are compensated in the 270 same way. The modified equations in the second prediction 271 are expressed as In the end, the modified cost function is given by The flowchart of the proposed robustness improvement 280 method is shown in Figure 3.  The simulation results with the stator resistance mismatch are 307 shown in Figure 4 and 5, where the reference speed and load 308 torque are 750r/min and 40 N.m, respectively. In Figure 4, R ∧ is equal to R in the interval of 1-2s and 310 the mismatch happens in the interval of 2-3s where R ∧ is 311 equal to 2R. According to the curves of i d , i q , and the error 312 between i d, i q and their references as shown Figure 4 (a)-(c), 313 the stator resistance mismatch has few effects on the control 314 performance of i d , i q . The total harmonic distortion (THD) of 315 the phase currents as shown in Figure 4 (d) only increases 316 about 0.02%. In addition, the curves of e d_RMS , e q_RMS, 317 and THD of phase current with different ratios of the sta-318 tor resistance mismatch are shown in Figure 5, and it is 319 shown that e d_RMS , e q_RMS, and THD of phase current only 320 change slightly with R ∧ /R. Accordingly, the effects of the 321 stator resistance mismatch on the steady performance of the 322 conventional FCS-MPCC are not obviously. The simulation results with the permanent magnet flux 324 linkage mismatch are shown in Figure 6 and 7, where the 325 reference speed and load torque are the same as Figure 3. 326 In Figure 6, ψf is equal to ψ f in the interval of 1-2s and the 327 mismatch happens in the interval of 2-3s where ψf is equal 328 VOLUME 10, 2022 to 2ψ f . According to the curves of i d , i q , and the error between i d, i q and their references as shown Figure 6    interval of 1-2s and the mismatch happens in the interval of 353 2-3s where Lq is equal to 1.8L q . According to the curves 354 of i d , i q , and the error between i d, i q and their references as 355 shown Figure 9 (a)-(c), the L q mismatch can increase the 356 tracking errors between i d , i q and their references. The total 357 harmonic distortion (THD) of the phase currents as shown in 358 Figure 9 (d) increases from 9.37% to 11.69%.

359
Due to the difference between L d and L q of IPMSM and 360 the coupling between the model in d and q-axis, the induc-361 tance mismatch is much more complex. Figure 10  Simulation results of the speed are shown in Figure 11.

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According to the enlargement in Figure 11, the biggest value 385 of the speed fluctuation exceeds 10r/min for the conven-386 tional FCS-MPCC with multiple parameter mismatches, and 387 it has been reduced to 2.8r/min with the proposed robustness 388 improvement method. Simulation results of i d and i q are shown in Figure 13 and 395 Figure 14, respectively. By comparing the curves before and 396 after 0.4s, it can be found that the fluctuations of i d and i q are 397 large for the conventional FCS-MPCC with multiple parame-398 ter mismatches, and they have been successfully reduced with 399 the proposed robustness improvement method. . .

400
Simulation results of the phase current (i a ) are shown 401 in Figure 15. The THD of i a is 18.57% for the conven-402 tional FCS-MPCC with multiple parameter mismatches and 403 it has been reduced to 11.37% with the proposed robustness 404 improvement method.

405
The THD values of i a with various sample period 406 (T s ) are shown in Figure 16 where the load torque in 407 Figure 16 Figure 18-21, 437 R ∧ is equal to 2R, Ld is equal to 0.5L d , Lq is equal to 1.2L q , 438 and ψf is equal to 1.25ψ f , while, in the process of y, R ∧ is 439 equal to 0.5R, Ld is equal to 2L d , Lq is equal to 0.5L q , and ψf 440 is equal to 0.4ψ f . In Figure 18-21, the speed reference and 441 load torque are set as 750r/min and 80 N.m respectively.  Figure 19 (d) for the proposed FCS-MPCC are almost 471 the same as Figure 19 (a) for the conventional FCS-MPCC 472 without parameter mismatches.

473
As shown in Figure 20 (a), the THD value of i a for the 474 conventional MPCC without parameter mismatches is 4.87%, 475 and it increases to 8.54% and 8.94% in the parameter mis-476 matches process of x and y, respectively. The harmonic 477 contents in the phase current increase substantially with the 478 conventional MPCC in the process of parameter mismatches. 479 As shown in Figure 20  (a) R ∧ is equal to 2R, Ld is equal to 0.5L d , Lq is equal to 1.2L q , and ψf is equal to 1.25ψ f (b) R ∧ is equal to 0.5R, Ld is equal to 2L d , Lq is equal to 0.5L q , and ψf is equal to 0.4ψ f . is set as 0. In the experiment, the reference of the speed is set 537 as 750 r/min and the load torque is about 50N.m.

538
The experimental results for both the conventional and pro-539 posed FCS-MPCC with parameter mismatches are given in 540 Figure 24-26, where the parameters adopted in the prediction 541 satisfy R^= 3R 0 , Ld = 0.4L d0 , Lq = 4L q0 , and ψf = 2ψ f0 . 542 Curves of i d and i q of the conventional and proposed FCS-543 MPCC are shown in Figure 24 and Figure    effectively promoted as shown in Figure 26(b) and the THD 570 has been reduced to 15.92%.

571
Accordingly, the above analysis indicates that the pro-572 posed robustness improvement method is effective and it can 573 achieve satisfactory performance in the case where multiple 574 parameters of IPMSM are mismatched. Compared with the 575 conventional FCS-MPCC for IPMSM, the proposed method 576 increases some computational complexity. The implemen-577 tation time of the proposed increases about 1.7 µs, which 578 is only 2.83% of the sampling period. Accordingly, the 579 increased computational complexity has little impact with the 580 adopted TMSF28377D control board.

581
The current transient response results with the proposed 582 method is shown in Figure 27. The speed keeps 750 r/min 583   Figure 24 with Figure 29, the steady-state error of 597 i d can be mitigated with both the proposed method and the 598 method in [19]. The root-mean square values of the ripples 599 of i d with the proposed method and the method in [19] are 600 9.28 and 19.32A. Accordingly, the proposed method can 601 realize better performances in the aspect of current tracking 602 compared with the previously studied method in [19].

604
In this paper, the prediction error caused by model parameter 605 mismatches in the two-step prediction considering the time-606 delay compensation is analyzed and a robustness improve-607 ment method is proposed to depress the model parameter 608 sensitivity. The conclusion is given as follows: 609 1) The prediction errors with parameter mismatches of the 610 stator resistance, magnetic flux linkage, inductances in both 611 d and q axes are studied. The mismatches of stator resistance 612 and magnetic flux linkage mainly cause the steady-state error 613 of the current tracking, while the mismatches of inductances 614 cause both the steady-state error and the increase of the 615 current ripples. 616 2) The parameter mismatch compensation method by cal-617 culating the prediction errors which have been divided into 618 two kinds can mitigate the steady-state error of the current 619 tracking and reduce the current harmonics caused by param-620 eter mismatches.

621
3) The simulation and experimental results indicate that 622 the proposed method can deal with multiple parameter 623 mismatches and the control performances of the proposed 624 method including the steady-state errors of current tracking, 625 current harmonics, and torque ripples can ensure almost the 626 same as the FCS-MPCC without parameter mismatches. The 627 torque ripples of the proposed method can be reduced by 628 more than 54.4% and 16.3% compared with the conventional 629 FCS-MPCC in the parameter mismatches process of x and 630 y, respectively.