基于改進(jìn)粒子群優(yōu)化算法的永磁球形電機(jī)驅(qū)動策略研究

周嗣理 李國麗 王群京 鄭常寶 文 彥

基于改進(jìn)粒子群優(yōu)化算法的永磁球形電機(jī)驅(qū)動策略研究

周嗣理1,2李國麗2,3王群京2,4鄭常寶2,3文 彥2,5

（1. 安徽大學(xué)計(jì)算機(jī)科學(xué)與技術(shù)學(xué)院 合肥 230601 2. 安徽大學(xué)高節(jié)能電機(jī)及控制技術(shù)國家地方聯(lián)合實(shí)驗(yàn)室 合肥 230601 3. 安徽大學(xué)電氣與自動化工程學(xué)院 合肥 230601 4. 安徽大學(xué)工業(yè)節(jié)電與電能質(zhì)量控制安徽省級協(xié)同創(chuàng)新中心 合肥 230601 5.安徽大學(xué)互聯(lián)網(wǎng)學(xué)院 合肥 230601）

永磁球形電機(jī)（PMSpM）是一種結(jié)構(gòu)緊湊、可多自由運(yùn)動的單關(guān)節(jié)傳動裝置。該文提出一種適用于PMSpM驅(qū)動策略優(yōu)化的改進(jìn)粒子群優(yōu)化（IPSO）算法，該算法可實(shí)時(shí)計(jì)算PMSpM期望轉(zhuǎn)矩所對應(yīng)的線圈驅(qū)動電流。首先，通過圓環(huán)函數(shù)建立PMSpM轉(zhuǎn)矩解析模型，并構(gòu)建轉(zhuǎn)矩Map圖；然后，在確定種群數(shù)量后為標(biāo)準(zhǔn)粒子群優(yōu)化（PSO）算法引入自適應(yīng)動態(tài)慣性權(quán)重和自適應(yīng)學(xué)習(xí)因子，將所提IPSO算法與PSO算法進(jìn)行仿真對比，仿真結(jié)果表明，在同樣的精度下采用IPSO算法計(jì)算驅(qū)動電流比采用PSO算法有更快的計(jì)算速度；最后，通過PMSpM控制試驗(yàn)進(jìn)一步證明了該仿真結(jié)論的正確性。

永磁球形電機(jī) 改進(jìn)粒子群優(yōu)化 自適應(yīng)動態(tài)慣性權(quán)重 自適應(yīng)學(xué)習(xí)因子 驅(qū)動電流

0 引言

永磁球形電機(jī)（Permanent Magnet Spherical Motor, PMSpM）是一種結(jié)構(gòu)緊湊的單關(guān)節(jié)多自由度電機(jī)[1-2]，有廣泛的應(yīng)用前景[3]。PMSpM的閉環(huán)控制需要計(jì)算驅(qū)動電流，驅(qū)動電流計(jì)算需要建立電磁轉(zhuǎn)矩模型。國內(nèi)外學(xué)者在PMSpM轉(zhuǎn)矩建模領(lǐng)域經(jīng)多年研究提出了很多方法，主要有麥克斯韋張量法[4]、虛位移法[5-6]和洛倫茲力法[7-9]。以上方法的計(jì)算速度都因計(jì)算量大而無法滿足PMSpM實(shí)時(shí)控制的需求。而PMSpM驅(qū)動電流計(jì)算需利用轉(zhuǎn)矩模型逆運(yùn)算，且關(guān)系到控制的實(shí)時(shí)性，國內(nèi)外學(xué)者提出了多種驅(qū)動電流計(jì)算方法。

用解析法[10-11]或有限元法[12-13]計(jì)算一個(gè)線圈和一個(gè)磁極的轉(zhuǎn)矩位置關(guān)系，通過疊加定理計(jì)算轉(zhuǎn)子總轉(zhuǎn)矩，再使用偽逆矩陣求解驅(qū)動電流。該方法解的非唯一性無法支持后續(xù)PMSpM通電策略的優(yōu)化研究。

采用支持向量機(jī)[14]、高斯過程[15]等數(shù)據(jù)驅(qū)動方法將PMSpM作為黑盒，繞過復(fù)雜的三維電磁建模機(jī)理，精度也足夠，但數(shù)據(jù)集采集工作很有挑戰(zhàn)性。

采用智能優(yōu)化算法的方法[16-17]，假定轉(zhuǎn)子不動，一個(gè)線圈沿轉(zhuǎn)子表面在三維空間運(yùn)動建立轉(zhuǎn)矩Map圖，再通過智能算法計(jì)算PMSpM的驅(qū)動電流，避免了偽逆矩陣的問題，但往往計(jì)算速度不夠。

本文以文獻(xiàn)[18]所提的臺階式永磁球形電機(jī)為研究對象，計(jì)及電機(jī)控制對算法的實(shí)時(shí)性要求，基于圓環(huán)函數(shù)建立PMSpM轉(zhuǎn)矩解析模型，進(jìn)而構(gòu)建轉(zhuǎn)矩Map圖。線圈驅(qū)動電流可基于該轉(zhuǎn)矩Map圖上快速插值計(jì)算得出對應(yīng)的轉(zhuǎn)矩，避免了解析模型中大量的積分計(jì)算。為進(jìn)一步提升PMSpM驅(qū)動電流的計(jì)算速度，本文提出改進(jìn)粒子群優(yōu)化（Improved Particle Swarm Optimization, IPSO）算法，以線圈電流為粒子，在轉(zhuǎn)矩Map圖上快速尋找到最優(yōu)的驅(qū)動電流，提高了控制的實(shí)時(shí)性。

1 PMSpM結(jié)構(gòu)原理及轉(zhuǎn)矩Map的構(gòu)建

1.1 PMSpM的結(jié)構(gòu)及工作原理

1.1.1 PMSpM結(jié)構(gòu)

本文所研究的PMSpM轉(zhuǎn)子有三層24個(gè)NdFe35的臺階式圓柱永磁體（Permanent Magnet, PM），如圖1a～圖1c所示。轉(zhuǎn)子磁極陣列N、S交替排布，充磁效果如圖1d所示。為避免復(fù)雜的磁耦合因素影響并降低轉(zhuǎn)子的轉(zhuǎn)動慣量，轉(zhuǎn)子本體采用空心的鋁制球形結(jié)構(gòu)，輸出軸從轉(zhuǎn)子頂端接出。圖1e展示了球殼狀定子的剖面圖，24個(gè)集中繞制的圓柱形空心線圈均勻?qū)ΨQ地排布在定子球殼體的兩層上，這兩層所在極角與赤道面的角度差均為22.5°。為避免復(fù)雜的磁耦合問題，滿足輕量化需求，定子殼體采用聚碳酸酯材料。

圖1 PMSpM結(jié)構(gòu)

PMSpM的氣隙長度是1mm。PMSpM總成如圖1f所示，其詳細(xì)尺寸參數(shù)見表1。

表1 PMSpM定轉(zhuǎn)子關(guān)鍵參數(shù)

Tab.1 Key parameters of the PMSpM rotor and stator

1.1.2 PMSpM工作原理

該P(yáng)MSpM可實(shí)現(xiàn)偏轉(zhuǎn)、俯仰和自旋三自由度運(yùn)動。圖1f展示了轉(zhuǎn)子繞s、s和s對應(yīng)的三自由度運(yùn)動模式，其中sss是定子坐標(biāo)系。為便于分析，將所有永磁體和線圈沿方位角方向（赤道方向）展開成圖2所示平面圖。沿方位角方向給各線圈依次通電，因定轉(zhuǎn)子極數(shù)不同形成步進(jìn)角，電機(jī)可實(shí)現(xiàn)自旋運(yùn)動。若在同一方位角下給沿極角方向的兩個(gè)線圈通電，則轉(zhuǎn)子可實(shí)現(xiàn)俯仰或偏轉(zhuǎn)運(yùn)動。

圖2 二維展平的定轉(zhuǎn)子磁極分布圖

1.2 PMSpM轉(zhuǎn)矩Map圖的構(gòu)建

1.2.1 PMSpM轉(zhuǎn)矩解析模型

根據(jù)電磁場電流等效模型理論，圖3所示臺階式圓柱永磁體的上層外部矢量磁位可以表示為

圖3 局部坐標(biāo)系下的第j個(gè)永磁體

將電流密度矢量變換到轉(zhuǎn)子坐標(biāo)系下，根據(jù)洛倫茲力法可得

最終可得轉(zhuǎn)子的總轉(zhuǎn)矩解析模型為

1.2.2 PMSpM轉(zhuǎn)矩Map圖的構(gòu)建

PMSpM轉(zhuǎn)矩模型因計(jì)算量大而無法滿足電機(jī)實(shí)時(shí)控制需求。為此，本文在第1.2.1節(jié)所提轉(zhuǎn)矩解析模型基礎(chǔ)上構(gòu)建轉(zhuǎn)矩Map圖，使計(jì)算量前置。

假設(shè)轉(zhuǎn)子固定，一個(gè)線圈沿方位角和極角依次遍歷整個(gè)轉(zhuǎn)子氣隙球面，利用式（13）計(jì)算出每個(gè)遍歷點(diǎn)的對應(yīng)轉(zhuǎn)矩，可得到、、三個(gè)自由度方向上的轉(zhuǎn)矩Map圖，分別如圖4、圖5和圖6所示。其中，構(gòu)建轉(zhuǎn)矩Map時(shí)轉(zhuǎn)矩解析模型所對應(yīng)的PMSpM幾何參數(shù)見圖1和表1，所選在整個(gè)轉(zhuǎn)子氣隙球面上遍歷的線圈電流設(shè)定為1A。

圖4 PMSpM轉(zhuǎn)矩Map圖（）

圖5 PMSpM轉(zhuǎn)矩Map圖（）

圖6 PMSpM轉(zhuǎn)矩Map圖（）

在PMSpM控制過程中，已知當(dāng)前位置期望轉(zhuǎn)矩，利用智能算法在Map圖上可快速地尋找到最優(yōu)的PMSpM驅(qū)動電流。顯然，所采用算法的收斂速度直接影響PMSpM控制的實(shí)時(shí)性。粒子群優(yōu)化（Particle Swarm Optimization, PSO）算法因?yàn)橛?jì)算量小、收斂速度快而廣泛應(yīng)用于實(shí)時(shí)控制領(lǐng)域[21]。本文以標(biāo)準(zhǔn)PSO算法為基礎(chǔ)，提出改進(jìn)的IPSO算法用于PMSpM驅(qū)動策略研究，進(jìn)一步提升了驅(qū)動電流計(jì)算速度。

2 PMSpM驅(qū)動策略優(yōu)化的IPSO算法

2.1 標(biāo)準(zhǔn)粒子群優(yōu)化算法

早期的粒子群優(yōu)化算法是1995年由美國R. Eberhart和J. Kennedy根據(jù)模仿鳥類覓食行為而提出的。1998年Y. Shi和R. Eberhart又引入慣性權(quán)重以提高粒子的搜索能力，進(jìn)而得到標(biāo)準(zhǔn)PSO算法。標(biāo)準(zhǔn)PSO算法收斂速度快，代碼簡潔高效，近年來在供配電[22-23]、光伏與微電網(wǎng)[24]、參數(shù)辨識[25]、電機(jī)設(shè)計(jì)優(yōu)化[26-27]等領(lǐng)域得到廣泛應(yīng)用。

PSO算法通過式（15）和式（16）對所有粒子的位置和速度進(jìn)行更新[28-29]。

2.2 標(biāo)準(zhǔn)PSO算法的改進(jìn)

2.2.1 慣性權(quán)重的改進(jìn)

2.2.2 學(xué)習(xí)因子改進(jìn)

2.3 PMSpM驅(qū)動策略優(yōu)化的算法流程

圖7 驅(qū)動電流計(jì)算的IPSO算法流程

3 IPSO算法改進(jìn)的仿真對比研究

3.1 仿真環(huán)境

本文在相同仿真條件下將IPSO算法與PSO算法進(jìn)行仿真對比，通過比較改進(jìn)前后算法的收斂速度證明IPSO算法改進(jìn)的有效性。

本文所采用仿真設(shè)備為DELL移動工作站Precision 3541，配備處理器的型號是Intel(R) Core(TM) i7-9750H CPU@2.60GHz (12 CPUs)～ 2.59GHz，運(yùn)行內(nèi)存是8.00G，操作系統(tǒng)是Windows 10，仿真軟件版本為Matlab 2018b。

3.2 IPSO算法參數(shù)改進(jìn)的仿真對比

3.2.1 種群數(shù)量分析與仿真對比

圖8 PSO算法種群數(shù)量仿真對比

表2 不同種群數(shù)量下PSO算法收斂性能對比

Tab.2 PSO performance comparison for different popsize

3.2.2 自適應(yīng)動態(tài)慣性權(quán)重改進(jìn)的仿真對比

圖9 慣性權(quán)重改進(jìn)仿真對比

可以看出，在同樣的收斂精度下，PSO算法配備改進(jìn)的自適應(yīng)動態(tài)慣性權(quán)重能有效提高運(yùn)行效率，在第50代左右就能完成收斂。而當(dāng)算法配備傳統(tǒng)慣性權(quán)重時(shí)需要在近200代才能徹底收斂。通過表3對比可以看出，改進(jìn)為自適應(yīng)動態(tài)慣性權(quán)重后，PSO算法平均運(yùn)行時(shí)間只有改進(jìn)前的22.3%，收斂速度從800ms級降低到200ms級，證明了采用自適應(yīng)動態(tài)慣性權(quán)重的有效性。

表3 慣性權(quán)重改進(jìn)前后收斂性能對比

Tab.3 Inertia weight improvement impact comparison

3.2.3 自適應(yīng)學(xué)習(xí)因子改進(jìn)的仿真對比

由圖10a可以發(fā)現(xiàn)，PSO算法在僅配備自適應(yīng)動態(tài)慣性權(quán)重時(shí)能在50代左右穩(wěn)定收斂。如果進(jìn)一步引入自適應(yīng)學(xué)習(xí)因子，算法可以在40代以內(nèi)穩(wěn)定收斂，如圖10b所示。學(xué)習(xí)因子改進(jìn)前后收斂性能對比見表4。

圖10 學(xué)習(xí)因子改進(jìn)仿真對比

表4 學(xué)習(xí)因子改進(jìn)前后收斂性能對比

Tab.4 Learning factors improvement impact comparison

從表4可以發(fā)現(xiàn)，算法改進(jìn)前平均運(yùn)行時(shí)間約為0.159s，而改進(jìn)后算法平均運(yùn)行時(shí)間縮短到約0.128s，速度提升了近20%。結(jié)果表明，自適應(yīng)學(xué)習(xí)因子的改進(jìn)對PSO算法收斂性能也有明顯的提升。

3.2.4 IPSO算法與標(biāo)準(zhǔn)PSO算法的仿真對比

改進(jìn)前的標(biāo)準(zhǔn)PSO算法仿真結(jié)果如圖9a所示，對應(yīng)的平均運(yùn)行時(shí)間約為0.711s。圖10b展示了改進(jìn)后IPSO算法的仿真結(jié)果，對應(yīng)的平均運(yùn)行時(shí)間約為0.128s。對比兩圖可發(fā)現(xiàn)，標(biāo)準(zhǔn)PSO和IPSO算法的收斂精度均滿足應(yīng)用需求。但在同樣的仿真條件下，IPSO算法運(yùn)行速度遠(yuǎn)高于PSO算法，IPSO算法的收斂曲線也更密集。對比結(jié)果表明改進(jìn)的IPSO算法對PMSpM驅(qū)動策略優(yōu)化問題不僅能快速得出最優(yōu)值，算法魯棒性也足夠好，值得進(jìn)一步挖掘其用于PMSpM實(shí)時(shí)控制的潛力。

4 IPSO算法的PMSpM控制試驗(yàn)驗(yàn)證

為驗(yàn)證采用IPSO算法計(jì)算PMSpM驅(qū)動電流在電機(jī)實(shí)時(shí)控制中應(yīng)用的可行性，本文設(shè)計(jì)了一個(gè)PMSpM閉環(huán)控制試驗(yàn)，并在試驗(yàn)中與采用標(biāo)準(zhǔn)PSO算法實(shí)時(shí)計(jì)算驅(qū)動電流的工況進(jìn)行了比較分析。

4.1 PMSpM控制系統(tǒng)的設(shè)計(jì)

為簡化閉環(huán)驗(yàn)證試驗(yàn)設(shè)計(jì)，論文采用比例積分微分（Proportional Integral Differential, PID）控制策略，并忽略PMSpM動力學(xué)模型中的不確定因素，PMSpM的動力學(xué)方程為

設(shè)計(jì)PID控制器，則PMSpM控制系統(tǒng)結(jié)構(gòu)如圖11所示，其中為控制增益矩陣[34]。

4.2 PMSpM控制試驗(yàn)平臺

該試驗(yàn)平臺由PMSpM樣機(jī)、電機(jī)控制器、上位機(jī)、直流穩(wěn)壓電源、微電機(jī)系統(tǒng)（Microelectro Mechanical System, MEMS）無線位置傳感器（MPU6050）和轉(zhuǎn)子初始位置標(biāo)定架總成構(gòu)成，如圖12所示。

圖12 PMSpM控制試驗(yàn)平臺

4.3 IPSO算法的PMSpM控制試驗(yàn)驗(yàn)證

PMSpM閉環(huán)控制自旋圖如圖13所示，其中藍(lán)色實(shí)線表示采用IPSO算法時(shí)PMSpM自旋運(yùn)動閉環(huán)運(yùn)動30°的轉(zhuǎn)子軌跡?？梢钥闯觯琍MSpM閉環(huán)自旋運(yùn)動能夠成功運(yùn)行。此時(shí)轉(zhuǎn)子輸出軸頂端在定子坐標(biāo)系、、三個(gè)坐標(biāo)軸方向上的空間運(yùn)動位移誤差曲線如圖14a所示，可以發(fā)現(xiàn)，該試驗(yàn)自旋運(yùn)動空間位移誤差幅值在、、三個(gè)坐標(biāo)軸方向上均在可接受的范圍內(nèi)，并且從誤差波形可發(fā)現(xiàn)閉環(huán)控制下的轉(zhuǎn)子運(yùn)自旋動空間位移誤差是可控的。該試驗(yàn)采用IPSO算法后PMSpM自旋運(yùn)動30°的軟件執(zhí)行時(shí)間約為2.57s。

圖13 PMSpM閉環(huán)控制自旋圖

圖14 PMSpM自旋運(yùn)動誤差曲線

為提高閉環(huán)試驗(yàn)的可比性，本文在同樣的試驗(yàn)條件和運(yùn)動工況下采用標(biāo)準(zhǔn)PSO算法進(jìn)行閉環(huán)控制試驗(yàn)。圖13中的紅色點(diǎn)畫線表明采用標(biāo)準(zhǔn)PSO算法時(shí)PMSpM自旋閉環(huán)運(yùn)動30°同樣可以成功運(yùn)行，但試驗(yàn)所需的軟件執(zhí)行時(shí)間約為15.63s，比采用IPSO算法時(shí)的軟件執(zhí)行時(shí)間約長6倍，證明了前面仿真結(jié)果的正確性。采用標(biāo)準(zhǔn)PSO算法時(shí)的PMSpM閉環(huán)自旋運(yùn)動所對應(yīng)的轉(zhuǎn)子空間位移誤差曲線如圖14b所示。

試驗(yàn)結(jié)果表明，在同樣的運(yùn)動工況下，所提IPSO算法用于PMSpM實(shí)時(shí)驅(qū)動電流計(jì)算比采用標(biāo)準(zhǔn)PSO算法具有更高的電機(jī)驅(qū)動電流計(jì)算速度，證明了仿真結(jié)果的正確性。

5 結(jié)論

本文提出了一種適用于PMSpM驅(qū)動策略優(yōu)化的IPSO算法?；趫A環(huán)函數(shù)建立PMSpM轉(zhuǎn)矩解析模型并構(gòu)建轉(zhuǎn)矩Map圖，IPSO算法通過轉(zhuǎn)矩Map圖插值計(jì)算可快速地尋找到最優(yōu)的PMSpM驅(qū)動電流。在研究確定PMSpM驅(qū)動策略優(yōu)化問題的粒子群種群數(shù)量后，本文在標(biāo)準(zhǔn)PSO算法的基礎(chǔ)上重點(diǎn)研究了慣性權(quán)重和學(xué)習(xí)因子在PMSpM驅(qū)動策略應(yīng)用中的改進(jìn)，仿真和試驗(yàn)結(jié)果表明：

1）采用自適應(yīng)動態(tài)慣性權(quán)重的PSO算法平均運(yùn)行速度是采用慣性權(quán)重PSO算法的近5.5倍，繼續(xù)改進(jìn)學(xué)習(xí)因子后，算法的平均運(yùn)行速度又可提升約20%。

2）仿真對比IPSO算法和PSO算法可發(fā)現(xiàn)，在同樣精度下，采用IPSO算法計(jì)算驅(qū)動電流比采用標(biāo)準(zhǔn)PSO算法時(shí)有更高的計(jì)算速度。

3）閉環(huán)控制試驗(yàn)表明，在同樣的運(yùn)動工況下，采用IPSO算法應(yīng)用于PMSpM驅(qū)動電流計(jì)算比采用標(biāo)準(zhǔn)PSO算法軟件執(zhí)行時(shí)間更短，證明了仿真結(jié)論的正確性。IPSO算法在PMSpM實(shí)時(shí)驅(qū)動策略上的應(yīng)用潛力值得進(jìn)一步研究挖掘。

本文所提IPSO算法方法同樣也適用于其他復(fù)雜特種電機(jī)驅(qū)動電流的計(jì)算。

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Improved Particle Swarm Optimization Algorithm Based Driving Strategy Research for Permanent Magnet Spherical Motor

Zhou Sili1,2Li Guoli2,3Wang Qunjing2,4Zheng Changbao2,3Wen Yan2,5

（1. School of Computer Science and Technology Anhui University Hefei 230601 China 2. National Engineering Laboratory of Energy-Saving Motor & Control Technology Anhui University Hefei 230601 China 3. School of Electrical Engineering and Automation Anhui University Hefei 230601 China 4. Anhui Collaborative Innovation Center of Industrial Energy-Saving and Power Quality Control Anhui University Hefei 230601 China 5. School of Internet Anhui University Hefei 230601 China）

A permanent magnet spherical motor (PMSpM) is a compact transmission apparatus that is capable of motion in multiple degrees of freedom. To achieve the close loop control of the PMSpM, the driving current of the stator coils needs to be calculated, and the analytic torque model needs to be built in advance. However, if the geometry of the permanent magnet (PM) is a non-circumferential symmetric one, the pseudo-inverse matrix technique is not applicable. Thus, the research on the fast driving strategy of the universal reverse torque model is an essential prerequisite for the PMSpM close-loop control.

This paper takes the PMSpM with the stepped cylindrical PM as the research object. Firstly, this paper proposes new analytical torque models using the toroidal expansion method. To avoid repeating integrations in magnetic and torque analytic calculation, this paper builds torque maps by moving one 1A energized electromagnetic coil on the overall spherical surface of the airgap along the azimuth angle direction and polar angle direction. Secondly, the classical particle swarm optimization algorithm (PSO) is introduced to build the reverse torque model. The current of the stator electromagnetic coils is considered as the particle swarm, and the desired torques are set as optimization targets. Thus, we can use the reverse torque model to calculate the driving current of the stator electromagnetic coils from the torque maps. Thirdly, this paper proposes an improved particle swarm optimization (IPSO) algorithm for the PMSpM driving strategy optimization, which can be used for calculating the real-time driving current for the desired torques of the PMSpM. After the determination of the population size of the PSO algorithm, the adaptive dynamic inertia weight and adaptive learning factors are introduced for IPSO.

The following conclusions can be drawn from the simulation analysis: ① The driving current calculation speed of the PSO algorithm with adaptive dynamic inertia weight is 5.5 times faster than the classical PSO algorithm; ② The comparison result between the classical PSO algorithm and IPSO algorithm indicates that IPSO has a better convergence rate than PSO on the premise of ensuring the accuracy of convergence. ③ The PMSpM control experimental result shows that the proposed IPSO algorithm is effective in the PMSpM driving strategy, and the PMSpM driving current calculation speed of the proposed IPSO algorithm is significantly faster than using the classical PSO algorithm. In addition, the proposed IPSO algorithm is also applicable for the driving current calculation of other complex special motors.

Permanent magnet spherical motor, improved particle swarm optimization, adaptive dynamic inertia weight, adaptive learning factors, driving current

10.19595/j.cnki.1000-6753.tces.210841

TM351; TP18

周嗣理 男，1984年生，博士，講師，研究方向?yàn)殡姍C(jī)設(shè)計(jì)優(yōu)化、電機(jī)控制及相關(guān)算法和新能源汽車電驅(qū)動系統(tǒng)等。E-mail：szhou551@gmail.com

王群京 男，1960年生，教授，博士生導(dǎo)師，研究方向?yàn)殡姍C(jī)、電機(jī)控制、新能源汽車電驅(qū)動系統(tǒng)和機(jī)器人技術(shù)等。E-mail：wangqunjing@ahu.edu.cn（通信作者）

國家自然科學(xué)基金（51637001）和安徽省自然基金（2008085ME156）資助項(xiàng)目。

2021-06-14

2021-10-07

（編輯 赫蕾）