余瀟 黃輝先
摘 要: 針對(duì)神經(jīng)滑??刂葡到y(tǒng)中存在的對(duì)先驗(yàn)數(shù)據(jù)依賴性較強(qiáng)的問(wèn)題,結(jié)合RBF神經(jīng)網(wǎng)絡(luò)的泛化能力和自學(xué)習(xí)能力以及模糊推理算法的強(qiáng)適應(yīng)能力,提出基于模糊RBF神經(jīng)網(wǎng)絡(luò)的永磁同步電機(jī)分?jǐn)?shù)階速度控制系統(tǒng)。模糊推理的引入為神經(jīng)網(wǎng)絡(luò)的不確定性提供了有效的指導(dǎo)作用,同時(shí),分?jǐn)?shù)階微積分算子的引入增加了傳統(tǒng)滑模控制器的自由度,從而對(duì)該控制器進(jìn)行了進(jìn)一步的優(yōu)化。仿真結(jié)果表明,相比RBF神經(jīng)滑模控制器,提出的模糊RBF神經(jīng)分?jǐn)?shù)階滑??刂破骶哂懈玫目刂菩阅堋?/p>
關(guān)鍵詞: 永磁同步電機(jī); 滑模控制器; RBF神經(jīng)網(wǎng)絡(luò); 分?jǐn)?shù)階; 模糊推理; 自由度
中圖分類(lèi)號(hào): TN876?34; TM461 文獻(xiàn)標(biāo)識(shí)碼: A 文章編號(hào): 1004?373X(2018)11?0087?04
Optimization design of fractional?order sliding mode controller
based on fuzzy RBF neural network
YU Xiao, HUANG Huixian
(College of Information Engineering, Xiangtan University, Xiangtan 411105, China)
Abstract: Since sliding mode control based on neural network has the problem of strong dependence on prior information, the generalization ability and self?learning ability of RBF neural network and strong adaptability of fuzzy reasoning algorithm are combined to propose the fuzzy RBF neural network based fractional order speed control system of permanent magnet synchronous motor (PMSM). The introduction of fuzzy reasoning provides an effective guidance for the uncertainty of the neural network, and the introduction of fractional?order calculus operator can increase the degree of freedom of the traditional sliding mode controller, so as to further optimize the controller. The simulation results show that, in comparison with the sliding mode controller based on RBF neural network, the fractional order sliding mode controller based on fuzzy RBF neural network has better control performance.
Keywords: PMSM; sliding mode controller; RBF neural network; fractional order; fuzzy inference; degree of freedom
永磁同步電機(jī)在機(jī)器人、數(shù)控機(jī)床、醫(yī)療設(shè)備等領(lǐng)域內(nèi)得到了廣泛應(yīng)用,但受制于系統(tǒng)中的參數(shù)變化和負(fù)載擾動(dòng)等因素,電機(jī)的轉(zhuǎn)速控制性能受到一定的影響?;?刂萍夹g(shù)對(duì)系統(tǒng)內(nèi)外部干擾所體現(xiàn)出的強(qiáng)魯棒性為電機(jī)的高性能調(diào)速提供了一條有效途徑,但考慮到滑??刂破鞯睦硐胼敵鍪歉哳l切換的開(kāi)關(guān)量,而運(yùn)動(dòng)控制系統(tǒng)中執(zhí)行機(jī)構(gòu)在時(shí)間上的延遲將導(dǎo)致系統(tǒng)狀態(tài)在滑模面上的運(yùn)動(dòng)軌跡不會(huì)準(zhǔn)確發(fā)生在設(shè)定的切換流形面,系統(tǒng)抖振將隨之發(fā)生[1],這無(wú)疑限制了滑??刂萍夹g(shù)的應(yīng)用范圍。
國(guó)內(nèi)外學(xué)者通過(guò)對(duì)抖振削弱方法的研究,獲得了大量成果[2?10]。其中,文獻(xiàn)[2?4]提出高階滑模控制算法,但這種方法較為復(fù)雜,控制器的輸出信號(hào)中存在著與其導(dǎo)數(shù)的耦合,不利于滑??刂坡傻脑O(shè)計(jì);文獻(xiàn)[5?7]依據(jù)干擾觀測(cè)器對(duì)負(fù)載轉(zhuǎn)矩進(jìn)行觀測(cè),并設(shè)計(jì)出一類(lèi)積分型滑??刂破鲗?duì)干擾進(jìn)行抑制,但在這類(lèi)方法作用下,系統(tǒng)的動(dòng)態(tài)性能會(huì)受到一定的影響;文獻(xiàn)[8?10]將智能算法引入滑??刂破鞯膬?yōu)化設(shè)計(jì)過(guò)程中,分別采用RBF神經(jīng)網(wǎng)絡(luò)和模糊推理方法整定滑??刂破鞯拈_(kāi)關(guān)增益,但在這類(lèi)方法作用下的系統(tǒng)中會(huì)存在靜差。
本文綜合考慮模糊推理算法和RBF神經(jīng)網(wǎng)絡(luò)在滑??刂破鲀?yōu)化設(shè)計(jì)過(guò)程中的應(yīng)用,利用模糊推理算法的強(qiáng)適應(yīng)能力調(diào)整RBF神經(jīng)網(wǎng)絡(luò)中的權(quán)值,進(jìn)而利用RBF神經(jīng)網(wǎng)絡(luò)來(lái)訓(xùn)練得出分?jǐn)?shù)階滑模控制器的實(shí)際輸出量,達(dá)到了較好的綜合控制性能。
永磁同步電機(jī)在兩相旋轉(zhuǎn)坐標(biāo)系下的數(shù)學(xué)模型為:
[uq=Rsiq+λq+ωfλdud=Rsid+λd-ωfλqλq=Lqiqλd=Ldid+LmdIdfωf=npωr] (1)
式中:[ud,uq]是兩相旋轉(zhuǎn)d?q坐標(biāo)系下的定子電壓;[id,iq]是定子電流;[λd,λq]是定子磁鏈;[Rs]和[Ld,Lq]是定子電阻和電感;[ωf,ωr]是電機(jī)電角度和給定轉(zhuǎn)速;[Lmd]是定子相電感;[Idf]是等效電流;[np]是磁極對(duì)數(shù)。
電磁轉(zhuǎn)矩方程為:
[Te=Jωr+Bmωr+TL] (2)
式中:[Te,TL]是電磁轉(zhuǎn)矩和負(fù)載力矩;[J]是轉(zhuǎn)動(dòng)慣量;[Bm]是摩擦因子。電磁轉(zhuǎn)矩可描述為:
[Te=32npLmdIdfiq+Ld-Lqiqid] (3)
對(duì)于隱極式永磁同步電機(jī),有[Ld=Lq],此時(shí)式(3)可簡(jiǎn)化為:
[Te=kpiq=32npLmdIdfiq] (4)
將式(4)代入電磁轉(zhuǎn)矩方程(2),可得:
[ωr=-BmJωr+kpJiq-1JTL] (5)
將式(5)轉(zhuǎn)化為狀態(tài)方程,可得:
[x=Ax+Bu+d] (6)
式中:[x=θωT];[A=010-BmJ];[B=0kpJ];[u=iq];[d=0-TLJ]。
將式(6)轉(zhuǎn)換為離散狀態(tài)方程為:
[xk+1=Axk+Buk+d] (7)
式中[xk=θkωkT]。
定義式(6)中狀態(tài)變量的誤差及其變化率為:
[ek=θ?k-θk, dek=ω?k-ωk] (8)
式中:[θ?k]為位置指令;[ω?k]為位置指令變化率;[ek]為位置誤差;[dek]為位置變化率的誤差。根據(jù)式(8)可列出離散誤差狀態(tài)方程:
[xek+1=Axek-Buk+fk+d] (9)
式中:[xek=ekdekT;]
[fk=-θ?k-ω?k+θ?k+1BmJω?k+ω?k+1]。
定義切換函數(shù)為:
[sk=Cxek+Δζxek] (10)
由于:
[sk+1=Cxek+1+Δζxek+1=CAxek+C-Buk+fk+d+Δζxek+1] (11)
式中:[Δζ]為離散域下的分?jǐn)?shù)階微積分算子,可描述為:
[Δζxek+1=xek+1-j=1k+1-1jζjxek+1-j] (12)
式中:
[ζj=diagζjζjζj=1,j=0ζζ-1…ζ-j+1j!,j>0]
根據(jù)式(10)和式(11),可得:
[ueqk=-CB-1CA-Ixek+Δζxek+1+Cfk+Cd](13)
式中:分?jǐn)?shù)階微積分算子中的階次[ζ]的確定尚無(wú)系統(tǒng)的理論推導(dǎo)方法,本文通過(guò)反復(fù)測(cè)試,確定該值為[ζ=0.14]。據(jù)上所述,得出總的滑??刂坡蔀椋?/p>
[uk=ueqk+unk] (14)
式中[unk]為模糊RBF神經(jīng)網(wǎng)絡(luò)的輸出。
模糊RBF神經(jīng)網(wǎng)絡(luò)中的信號(hào)傳播及各層的功能如下:
1) 輸入層
設(shè)定模糊RBF神經(jīng)網(wǎng)絡(luò)的輸入為:
[xn1=sk,xn2=sk-sk-1] (15)
輸入層的各個(gè)節(jié)點(diǎn)直接與輸入量的各分量相連接,對(duì)該層的每個(gè)節(jié)點(diǎn)i,其輸入輸出關(guān)系表示為:
[f1i=xi] (16)
2) 模糊化層
模糊化層的每個(gè)節(jié)點(diǎn)具有隸屬函數(shù)的功能,采用高斯函數(shù)作為隸屬函數(shù)。則有:
[f2i,j=exp-f1i-cij2bj2] (17)
式中[cij]和[bj]是第i個(gè)輸入變量在第j個(gè)模糊集合高斯函數(shù)上的均值和標(biāo)準(zhǔn)差。
3) 模糊推理層
模糊推理層通過(guò)與模糊化層的連接完成模糊規(guī)則的匹配,每個(gè)節(jié)點(diǎn)的輸出為該節(jié)點(diǎn)所有輸入信號(hào)的乘積,即:
[f3j=j=1Nf2i,j] (18)
4) 輸出層
該層的輸出為該層節(jié)點(diǎn)所有輸入信號(hào)的加權(quán)和,則有:
[un=f4=W?f3=j=1Nwjf3j] (19)
式中[W=w1,w2,…,wNT]為模糊推理層和輸出層之間的權(quán)重向量。
選取模糊RBF神經(jīng)網(wǎng)絡(luò)的學(xué)習(xí)指標(biāo)為:
[Ek=12sk2] (20)
根據(jù)式(9)和式(10),則有:
[?sk?unk=-B] (21)
根據(jù)梯度下降算法,模糊RBF神經(jīng)網(wǎng)絡(luò)的權(quán)值學(xué)習(xí)方法為:
[Δwj=-?Ek?wj=-sk?Ek?unk?unk?wj=skBf3j]
[wjk=wjk-1+ηΔwj+αwjk-1-wjk-2]
[Δbj=-?Ek?bj=-sk?Ek?unk?unk?bj=skBwjf3f1-Cj2b3j]
[bjk=bjk-1+ηΔbj+αbjk-1-bjk-2]
[Δcij=-?Ek?cij=-sk?Ek?unk?unk?cij=skBwjf1-cijb2j]
[cijk=cijk-1+ηΔcij+αcijk-1-cijk-2]
式中:[η]為學(xué)習(xí)速率;[α]為慣性系數(shù)。
本文以Matlab軟件為仿真工具,采用圖1所示的仿真平臺(tái)。永磁同步電機(jī)參數(shù)如下:[Rs=1.5 Ω,Ld=Lq=8.5×]
[10-3 H,np=4,J=2.5×10-3 kg?m2,Bm=0.8×10-3 N?m?s]。
輸入信號(hào)采用正弦信號(hào):[θ?k=0.5sin6πk]。模糊RBF神經(jīng)網(wǎng)絡(luò)結(jié)構(gòu)分別選取為:輸入層2個(gè),模糊化層36個(gè),模糊推理層36個(gè),輸出層1個(gè)。網(wǎng)絡(luò)權(quán)值[W]的初始值選取為[-1,1]的隨機(jī)值。學(xué)習(xí)速率和慣性系數(shù)分別選取為:[η=0.6,α=0.05]。為驗(yàn)證本文所提算法的優(yōu)越性,基于RBF神經(jīng)網(wǎng)絡(luò)優(yōu)化的滑??刂破饔脕?lái)進(jìn)行對(duì)比分析。
本文對(duì)模糊RBF神經(jīng)分?jǐn)?shù)階滑??刂破鞯臉?gòu)建過(guò)程進(jìn)行了詳細(xì)論述,并給出了網(wǎng)絡(luò)結(jié)構(gòu)中各層之間權(quán)重系數(shù)、各節(jié)點(diǎn)的中心向量和基寬向量的更新方法。本文所提算法的優(yōu)越性在與RBF神經(jīng)滑??刂破鞯谋容^中得到了體現(xiàn),仿真驗(yàn)證從轉(zhuǎn)速響應(yīng)、系統(tǒng)狀態(tài)向滑模面收斂的軌跡和控制量三個(gè)方面展開(kāi)。仿真結(jié)果表明,本文提出的算法具有較高的綜合控制性能。
參考文獻(xiàn)
[1] 高為炳.變結(jié)構(gòu)控制的理論及設(shè)計(jì)方法[M].北京:科學(xué)出版社,1996.
GAO Weibing. Theory and design method of variable structure control [M]. Beijing: Science Press, 1996.
[2] 陳杰,李志平,張國(guó)柱.不確定非線性系統(tǒng)的高階滑??刂破髟O(shè)計(jì)[J].控制理論與應(yīng)用,2010,27(5):563?569.
CHEN Jie, LI Zhiping, ZHANG Guozhu. Design of high order sliding mode controller for uncertain nonlinear systems [J]. Control theory and application, 2010, 27(5): 563?569.
[3] BELTRAN B, AHMED?ALI T, BENBOUZID M. High order sliding mode control of variable speed wind turbines [J]. IEEE transactions on industrial electronics, 2009, 56(9): 3314?3321.
[4] FRANCESCO D, ANTONELLA F. Higher order sliding mode controllers with optimal reaching [J]. IEEE transactions on automatica control, 2009, 54(9): 98?104.
[5] 李政,胡廣大,崔家瑞,等.永磁同步電機(jī)調(diào)速系統(tǒng)的積分型滑模變結(jié)構(gòu)控制[J].中國(guó)電機(jī)工程學(xué)報(bào),2014,34(3):431?437.
LI Zheng, HU Guangda, CUI Jiarui, et al. Integral sliding mode variable structure control of permanent magnet synchronous motor speed control system [J]. Proceedings of the CSEE, 2014, 34(3): 431?437.
[6] HAN H C. LMI?based sliding surface design for integral sliding mode control of mismatched uncertain systems [J]. IEEE transactions on automatic control, 2007, 52(4): 736?742.
[7] 王麗梅,鄭浩,賈啟.永磁同步平面電動(dòng)機(jī)的滑??刂破髟O(shè)計(jì)[J].電機(jī)與控制學(xué)報(bào),2014,18(7):101?106.
WANG Limei, ZHENG Hao, JIA Qi. Design of sliding mode controller for permanent magnet synchronous planar motor [J]. Journal of motor and control, 2014, 18(7): 101?106.
[8] 劉治鋼,王軍政,趙江波.永磁同步電機(jī)神經(jīng)網(wǎng)絡(luò)自適應(yīng)滑??刂破髟O(shè)計(jì)[J].電機(jī)與控制學(xué)報(bào),2009,13(2):290?295.
LIU Zhigang, WANG Junzheng, ZHAO Jiangbo. Design of adaptive sliding mode controller for permanent magnet synchronous motor neural network [J]. Journal of motor and control, 2009, 13(2): 290?295.
[9] HSIAO M Y, Li T H S, LEE J Z, et al. Design of interval type?2 fuzzy sliding?mode controller [J]. Information sciences, 2008, 178(6): 1696?1716.
[10] 逄海萍,江姝妍.伺服系統(tǒng)模糊滑??刂破鞯脑O(shè)計(jì)與仿真[J].系統(tǒng)仿真學(xué)報(bào),2005,17(12):2972?2974.
PANG Haiping, JIANG Shuyan. Design and simulation of fuzzy sliding mode controller for servo systems [J]. Journal of system simulation, 2005, 17(12): 2972?2974.