A Design Methodology of an Optimal Torque Minimizing the Total Loss of an Induction Motor by Means of LMI
Kaoru I
NOUE, Masatoshi M
INAMIYAMA†, and Toshiji K
ATO‡(Received May 9, 2009)
When the motor speed is reduced by a regenerative brake, the mechanical energy of rotation can be converted into the regenerative electric energy. In order to improve the e^ciency of the motor drive systems, the optimal regenerative and acceleration torques of the motor should be studied to maximize the regenerative electric energy and to minimize the required energy for acceleration. This paper proposes a design methodology of the optimal torques for a three phase squirrel-cage induction motor, which minimize its total loss by means of LMI (Linear Matrix Inequality). The derived optimal torque meets the given constrained conditions of the torque amplitude, the operation time period, and the rotating speed range; the regenerative electric energy becomes maximum, and the required electric energy for acceleration becomes minimum. The e&ectiveness of the proposed method will be illustrated by means of both simulations and experiments.
Key words : Induction Motor, Optimal Torque, LMI(Linear Matrix Inequality), Optimization Problem.
࣮࣮࢟࣡ࢻ: ㄏᑟ㟁ືᶵ㸪᭱㐺ࢺࣝࢡ㸪LMI(⥺ᙧ⾜ิ➼ᘧ)㸪᭱㐺ၥ㢟.
ㄏᑟ㟁ືᶵࡢᦆኻࢆ᭱ᑠࡍࡿ᭱㐺ࢺࣝࢡࡢ⥺ᙧ⾜ิ➼ᘧࢆ⏝࠸ࡓタィἲ
ୖࠉ㤾ࠉ࣭ࠉ༡ᒣࠉṇಇࠉ࣭ࠉຍ⸨ࠉḟ
1. ࡣࡌࡵ
㟁ືᶵࢆ⏝࠸ࡓᦙ㏦㌴ࡸ㟁ື㌴୧࡞ࡣ㸪ῶ㏿
㟁ẼⓗไືຊࢆⓎ⏕ࡉࡏࡿᅇ⏕ࣈ࣮ࣞ࢟ࢆ⏝࠸࡚ᶵ Ეⓗ࢚ࢿࣝࢠ࣮ࢆ㟁ຊኚࡋ㸪ࡇࢀࢆຍ㏿
⏝ࡍࡿࡇ࡛ຠ⋡ࡼࡃ㐠⏝࡛ࡁࡿ16)㸬ࡇࡢࡼ࠺࡞㟁
ືᶵࡢ⏝㏵࠾࠸࡚ࡣ㸪ᛴᓧ࡞㏿ᗘᛂ⟅ᛶࡣ࠶ࡲࡾせ ồࡉࢀ࡞࠸ࡀ㸪ᅇ⏕ࡍࡿᮇ㛫ࡸᅇ⏕ᚋࡢᅇ㌿㏿ᗘ࡞
ࡣไ⣙ࡀ࠶ࡿሙྜࡀከ࠸㸬ࡇࡢࡼ࠺࡞᮲௳ୗ࠾ࡅ
ࡿ㟁ືᶵࡢᦆኻࡣ㸪㕲ᦆࢆ㝖ࡃᅇ⏕୰ࡢࢺࣝࢡࡢ
ࡅ᪉ࡼࡗ࡚ኚࡍࡿ79)㸬ࡇࡢࡓࡵ㸪ࡼࡾ᭷ຠ࢚
ࢿࣝࢠ࣮ࢆ⏝ࡍࡿࡓࡵࡣ㸪ࡇࢀࡽࡢ᮲௳ୗ࠾࠸
࡚㸪ᦆኻࡀ᭱ᑠࡍ࡞ࢃࡕᅇ⏕㟁ຊ㔞ࡀ᭱࡞ࡿ᭱㐺
࡞ᅇ⏕ࢺࣝࢡࡢࡅ᪉ࢆ᳨ウࡍࡿᚲせࡀ࠶ࡗࡓ㸬ࡑࡇ
Department of Electrical Engineering, Doshisha University, Kyoto
Telephone: +81-774-65-6296, Fax: +81-774-65-6296, E-mail: kaoinoue@mail.doshisha.ac.jp
† Telephone: +81-774-65-6318, Fax: +81-774-65-6801
‡ Telephone: +81-774-65-6322, Fax: +81-774-65-6812, E-mail: tkato@mail.doshisha.ac.jp
Fig. 1. The objective induction motor drive system.
࡛ⴭ⪅ࡽࡣ㸪3┦ㄏᑟ㟁ືᶵࢆᑐ㇟ࡋ࡚㸪ኚศἲࢆ
⏝࠸࡚ࡇࡢࡼ࠺࡞ไ⣙᮲௳ୗ࠾ࡅࡿ᭱㐺࡞ᅇ⏕ࢺࣝ
ࢡࢆゎᯒⓗᑟฟࡋࡓ10,11)㸬ࡋࡋ㸪ࡇࡢ᪉ἲ࡛ࡣᅇ
⏕ࢺࣝࢡࡢࡁࡉ㛵ࡍࡿไ⣙᮲௳ࢆຍࡋ࡚࠸࡞࠸
ࡓࡵ㸪ồࡵࡓ᭱㐺ࢺࣝࢡࡢࡁࡉࡀᶵჾࡢᐃ᱁್ࢆ㉺
࠼࡚ࡋࡲ࠺ሙྜࡶ࠶ࡾ࠼ࡿ㸬ࡇࡢࡓࡵ㸪ᅇ⏕⤊้
࠾࠸࡚ᡤᮃࡢᅇ㌿㏿ᗘࡀᚓࡽࢀ࡞࠸ྍ⬟ᛶࡀ࠶ࡗࡓ㸬 ࡑࡇ࡛ᮏㄽᩥ࡛ࡣ㸪Fig. 1♧ࡍ㛫᥋ᆺ࣋ࢡࢺࣝไ ᚚࢆ⏝࠸ࡓ୕┦ㄏᑟ㟁ᶵ㥑ືࢩࢫࢸ࣒ࢆᑐ㇟ࡋ㸪ᚑ ᮶ࡢ᮲௳ࠕ้t0ࡽt1࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm0
ࡽrm1ࡲ࡛ῶ㏿ࠖຍ࠼㸪᪂ࡓ࡞ไ⣙᮲௳ࠕᅇ⏕
ࢺࣝࢡTeࡢࡁࡉࢆไ㝈್௨ୗᢚไࠖ㸪ࡢ᪉ࢆ
Fig. 2. Schematic diagram of an optimal regenerative torque under the constraint condition
of torque amplitudeTezmin.
‶㊊ࡋࡘࡘᅇ⏕㟁ຊ㔞ࢆ᭱ࡍࡿ᭱㐺ᅇ⏕ࢺࣝࢡࡢ タィ᪉ἲࢆᥦࡍࡿ㸬ලయⓗࡣ㸪LMI(⥺ᙧ⾜ิ
➼ᘧ)ࢆ⏝࠸࡚ᩘ್ⓗゎࡃ᪉ἲࢆᥦࡋ㸪ᚑ᮶ࡢኚ ศἲࡼࡿ᪉ἲࢆไ⣙᮲௳ࡁ࡛ゎᯒⓗゎࡃᨵⰋἲ ࡢ⤖ᯝ12)ẚ㍑᳨࣭ウࢆ⾜࠸㸪ᥦᡭἲࡢ᭷ຠᛶࢆ
☜ㄆࡍࡿ㸬ࡲࡓຍ㏿࠾࠸࡚ࡶྠᵝࡢᡭἲࡀ㐺⏝࡛
ࡁ㸪ไ⣙᮲௳ࠕ้t2ࡽt3࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆ
rm2ࡽrm3ࡲ࡛ຍ㏿ࠖ࠾ࡼࡧࠕຊ⾜ࢺࣝࢡTeࡢ
ࡁࡉࢆไ㝈್௨ୗᢚไࠖࢆ‶㊊ࡋࡘࡘ㸪ຍ㏿ᚲ せ࡞㟁ຊ㔞ࢆ᭱ᑠࡍࡿ᭱㐺ຊ⾜ࢺࣝࢡࢆᑟฟࡍࡿ㸬
௨ୗ㸪2.࡛ࡣࢺࣝࢡไ⣙ࢆ⪃៖ࡍࡿᚲせᛶࢆ㏙㸪 3.࡛ࡣᑐ㇟ࢩࢫࢸ࣒ࡢᐃᘧࢆ⾜࠺㸬4.࡛ࡣLMIࢆ
⏝࠸ࡓ᭱㐺ࢺࣝࢡタィἲࡘ࠸࡚㏙㸪᭱ᚋ5.
࠾࠸࡚㸪ᥦᡭἲࡢ᭷ຠᛶࡘ࠸࡚ࢩ࣑࣮ࣗࣞࢩࣙࣥ
ᐇ㦂ࡢ୧᪉ࡼࡾ᳨ドࡍࡿ㸬
2. ࢺࣝࢡࡢࡁࡉ㛵ࡍࡿไ⣙ࡢᚲせᛶ
ᚑ᮶ࡢኚศἲࢆ⏝࠸ࡓ᭱㐺ᅇ⏕ࢺࣝࢡࡣᅇ⏕㛤ጞ┤
ᚋࡣẚ㍑ⓗࡁࡃ㸪ᅇ㌿ゅ㏿ᗘࡀᑠࡉࡃ࡞ࡿࡘࢀ࡚
ᑠࡉࡃ࡞ࡿ10,11)㸬ࡇࢀࡣᅇ㌿ゅ㏿ᗘࡀࡁ࠸ࡁ
ࡣᅇ㌿ῶ⾶ࡼࡿ࢚ࢿࣝࢠ࣮ᾘ㈝ࡀࡁ࠸ࡓࡵ㸪ࡁ
࡞ࢺࣝࢡࢆࡅ࡚ᅇ㌿ゅ㏿ᗘࢆపୗࡉࡏࡘࡘ㸪Ⓨ㟁㟁 ຊࢆࡁࡃࡋ࡚ᅇ⏕ࡋ࡚࠸ࡿࡇࢆ♧ࡋ࡚࠸ࡿ㸬ࡋ
ࡋ㸪ࢺࣝࢡࡢࡁࡉࡀᶵჾࡢᐃ᱁್ࢆ㉸࠼࡚ࡋࡲ࠺ྍ
⬟ᛶࡀ࠶ࡿ㸬
Fig. 3. Schematic diagram of an optimal acceleration torque under the constraint condition
of torque amplitude Tezmax.
ࡑࡇ࡛ᮏㄽᩥ࡛ࡣ㸪ᚑ᮶ࡢ᮲௳ࠕ้t0ࡽt1࠾
࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm0ࡽrm1ࡲ࡛ῶ㏿ࠖຍ࠼㸪 ᪂ࡓ࡞ไ⣙᮲௳ࠕᅇ⏕ࢺࣝࢡTeࡢࡁࡉࢆไ㝈್௨
ୗᢚไࠖࢆຍ࠼ࡓሙྜࡘ࠸࡚⪃࠼ࡿ㸬⪃࠼᪉ࡢཎ
⌮ᅗࢆFig. 2♧ࡍ㸬1Ⅼ㙐⥺ࡢanalytical solution ࡣ㸪ᚑ᮶ࡢࢺࣝࢡไ⣙ࢆ⪃៖ࡋ࡞࠸᭱㐺ᅇ⏕ࢺࣝࢡࢆ
⾲ࡋ࡚࠸ࡿ㸬ࡇࡢሙྜ㸪Teࡣ้t0㏆࠾࠸࡚ࡑ ࡢไ㝈್ࡢୗ㝈zminࢆୗᅇࡗ࡚࠾ࡾ㸪⌮ㄽ㏻ࡾࡢࢺ
ࣝࢡࢆⓎ⏕࡛ࡁ࡞࠸㸬ࡑࡇ࡛㸪ࡋࡤࡽࡃไ⣙್୍ᐃࢺ
ࣝࢡࢆຍ࠼࡚ῶ㏿ࡉࡏࡿ㸬ࡋࡋ㸪1Ⅼ㙐⥺ไ㝈್
ࡢⅬࡽࡍࡄࡉࡲᚑ᮶ἲᚑࡗ࡚ᅇ⏕ࢆ⾜࠺㸪2
Ⅼ㙐⥺࡛♧ࡍࡼ࠺༑ศᅇ㌿ゅ㏿ᗘࡀῶᑡࡋࡁࡽࡎ㸪
้t1࠾࠸࡚ᡤᮃࡢrm1฿㐩ࡋ࡞࠸㸬
୍᪉㸪ຊ⾜ࡢኚศἲࢆ⏝࠸ࡓ᭱㐺ࢺࣝࢡࡣ㸪ຊ⾜
㛤ጞ┤ᚋࡣᑠࡉ࠸ࡀᅇ㌿ゅ㏿ᗘࡀࡁࡃ࡞ࡿࡘࢀ࡚
ḟ➨ࡁࡃ࡞ࡿ㸬ࡇࡢࡓࡵ㸪ᅇ⏕ࡣ㏫㸪ຊ⾜
ࡢ㏵୰࡛ࢺࣝࢡไ⣙್ࡢୖ㝈zmax㐩ࡋ࡚ࡋࡲ࠺ྍ
⬟ᛶࡀ࠶ࡿ㸬⪃࠼᪉ࡢཎ⌮ᅗࢆFig. 3♧ࡍ㸬ࡇࡢሙ
ྜ㸪ຊ⾜⤊࠾࠸࡚ᡤᮃࡢᅇ㌿ゅ㏿ᗘrm3ࡲ࡛
ࡢ༑ศ࡞ຍ㏿ࡀᚓࡽࢀ࡞࠸㸬
௨ୖࡢࡇࡽ㸪ࢺࣝࢡࡢࡁࡉไ⣙ࡀ࠶ࡿሙྜ
࠾࠸࡚ࡶ㸪ᡤᮃࡢ้࠾࠸࡚ᡤᮃࡢᅇ㌿ゅ㏿ᗘ
฿㐩࡛ࡁ㸪ᦆኻࢆ᭱ᑠࡍࡿ᭱㐺ࢺࣝࢡࢆᑟฟࡍࡿᚲ せࡀ࠶ࡿ㸬
3. ᑐ㇟ࢩࢫࢸ࣒ࡢᐃᘧ
ㄏᑟ㟁ືᶵࡢ☢᮰ࢺࣝࢡࡣ㛫᥋ᆺ࣋ࢡࢺࣝไᚚ13)
ࡼࡗ࡚ไᚚࡉࢀࡿ㸬࡞࠾㸪࣋ࢡࢺࣝไᚚ࡛⏝࠸ࡿྛ
ไᚚࢤࣥࡣ㸪ᴟ㓄⨨ἲࢆ⏝࠸࡚Ỵᐃࡍࡿ14)㸬
3.1≧ែ᪉⛬ᘧ
ㄏᑟ㟁ືᶵࡢᅇ㌿Ꮚࡢ㐠ື᪉⛬ᘧࡣ㸪័ᛶ࣮࣓ࣔࣥ
ࢺࢆJ㸪ῶ⾶ಀᩘࢆ㸪ᅇ㌿ゅ㏿ᗘࢆrm㸪ࢺࣝࢡࢆ
Teࡍࡿ㸪
Jrm(t) +rm(t) =Te(t) (1)
࡛࠼ࡽࢀࡿ㸬ࡓࡔࡋ㸪Teࡣຊ⾜ࢆṇ㸪ᅇ⏕ࢆ㈇
ࡍࡿ㸬rmࡣṇࡢሙྜࡢࡳࡋ㸪ᅇ⏕㏫ᅇ㌿ࡍ
ࡿࡇࡣ࡞࠸ࡍࡿ㸬
ࡇࡇ࡛㸪(1)ᘧࢆࢧࣥࣉࣜࣥࢢ࿘ᮇTs࡛㞳ᩓࡍ
ࡿ㸪㞳ᩓ㛫⣔ࡢ≧ែ᪉⛬ᘧ ( rm[i+ 1] =Arm[i] +BTe[i]
y[i] =crm[i] +dTe[i] (2)
ࢆᚓࡿ㸬ࡓࡔࡋ㸪i= 0,1,2,· · ·࡛࠶ࡾ㸪 A=eJTs, b= 1
J Z Ts
0
eJ+d+, c= 1, d= 0 (3)
࡛࠶ࡿ㸬
3.2ᦆኻ
ㄏᑟ㟁ືᶵࡢᶵᲔⓗᦆኻࡣᅇ㌿ࡢῶ⾶ࡼࡿࡶࡢ࡛
࠶ࡿࡢ࡛㸪
Plm=2rm (4)
࡛࠼ࡽࢀࡿ㸬୍᪉㸪࣋ࢡࢺࣝไᚚࡉࢀࡓㄏᑟ㟁ືᶵ
࡛ࡣ㸪㕲ᦆࢆᚤᑡࡋ࡚↓どࡍࡿሙྜࡀከ࠸ࡢ࡛㸪ࡇ ࡢࡁ㟁ẼⓗᦆኻPleࡣ୍ḟ(ᅛᐃᏊ)㖡ᦆḟ(ᅇ
㌿Ꮚ)㖡ᦆࡢ
Ple=a+bTe2 (5)
࡛࠼ࡽࢀࡿ9)㸬ࡇࡇ࡛㸪a= MRs2*r2㸪b= (RMsL22r + Rr)p2*1r2 ࡛࠶ࡿ㸬ࡲࡓ㸪M㸪Lr㸪Rs㸪Rr㸪p㸪*rࡣ ࡑࢀࡒࢀ㸪┦ࣥࢲࢡࢱࣥࢫ㸪ḟᕳ⥺ࣥࢲࢡࢱ
ࣥࢫ㸪୍ḟᢠ㸪ḟᢠ㸪ᴟᑐᩘ㸪ḟ☢᮰ࡢ㍈ ᡂศࢆ⾲ࡍ㸬ᮏ◊✲࡛ࡣ㸪▐ࢺࣝࢡࡢไᚚ㛫᥋ᆺ
࣋ࢡࢺࣝไᚚࢆ⏝࠸ࡿࡓࡵ*rࡣ୍ᐃ್ไᚚࡉࢀࡿ
ࡢ࡛㸪a㸪bࡣඹᐃᩘ࡛࠶ࡿ㸬
้t0ࡽt1ࡲ࡛ᅇ⏕ࡍࡿࡁࡢᦆኻࡣ㸪
Iloss= Z t1
t0
(Plm+Ple)dt (6)
࡞ࡿ㸬ᐃᩘaࡢᐃ✚ศࡣ㸪rm㸪Teࡀࡢࡼ࠺࡞್
࡛࠶ࡗ࡚ࡶ୍ᐃ್࡛࠶ࡿࡢ࡛㸪ࡇࢀࢆ㝖࠸ࡓࡶࡢࢆ
ᦆኻ Ibloss=
Z t1
t0
(rm2 +bTe2)dt (7)
࠾ࡃࡇࡍࡿ㸬࡞࠾㸪้t2ࡽt3ࡲ࡛ຊ⾜ࡍ
ࡿሙྜࡣ㸪ࡑࡢࡼ࠺✚ศ༊㛫ࢆኚ᭦ࡍࢀࡤࡼ࠸㸬
3.3 ᣑ⣔ホ౯㛵ᩘ
ࢺࣝࢡࡢࡁࡉ㛵ࡍࡿไ⣙᮲௳ࢆグ㏙ࡍࡿࡓࡵࡢ ኚᩘz[i]ࢆᑟධࡋ㸪ồࡵࡿ᭱㐺࡞ᅇ⏕ࢺࣝࢡࢆTe[i]
⾲ࡍ㸪ไ⣙᮲௳ࢆຍࡋࡓᣑ⣔ࡣ௨ୗ࡛⾲ࡉࢀ
ࡿ15,16)㸬 T z
Z
rm[i+ 1] =Arm[i] +BTe[i]
y[i] =crm[i] +dTe[i]
z[i] =czrm[i] +dzTe[i]
(8)
ࡇࡇ࡛㸪ไ⣙᮲௳ࡣ௨ୗ࡛グ㏙ࡉࢀࡿࡍࡿ㸬
zmin[i]z[i]zmax[i] (9) ᮏ◊✲࡛ࡢไ⣙᮲௳ࡣࠕᅇ⏕ࢺࣝࢡTeࡢࡁࡉࢆ࠶
ࡿ್௨ୗᢚไࡍࡿࠖࡇ࡛࠶ࡿࡢ࡛㸪cz= 0, dz= 1
ࡍࡿz[i] =Te[i] ࡞ࡿ㸬
ḟ㸪ᚑ᮶ࡢ᮲௳ࠕ้t=t0(i= 0)ࡽt1(i=n)
࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm0 = rm[0]ࡽrm1 = rm[n]ࡲ࡛ῶ㏿ࡉࡏࡿࠖࡓࡵ㸪nࢫࢸࢵࣉࡽ࡞ࡿᅇ
⏕ࢺࣝࢡࡢ┠ᶆ್ಙྕิ࣋ࢡࢺࣝ
Te = [Te[0] ࠉTe[1] ࠉ· · ·ࠉTe[n1] ]T (10)
ᅇ㌿ゅ㏿ᗘࡢ┠ᶆ್ಙྕิ࣋ࢡࢺࣝ
y = [rm[1] ࠉrm[2] ࠉ· · ·ࠉrm[n] ]T (11)
ࢆᐃ⩏ࡍࡿ㸬࡞࠾㸪ຊ⾜ࡢሙྜࡶྠᵝ㸪ࠕ้t = t2(i= 0)ࡽt3(i=n)࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm2= rm[0]ࡽrm3 =rm[n]ࡲ࡛ຍ㏿ࡉࡏࡿࠖㄞࡳ
᭰࠼ࢀࡤࡼ࠸㸬
ࡇࢀࡽࢆ⏝࠸࡚㸪௨ୗࡢホ౯㛵ᩘJࢆᐃ⩏ࡍࡿ㸬 T
z
Z
J =J1+J2+J3
J1=|rm[n] rm[n]|2 J2=
n1
X
i=0
w1|Te[i] |2 J3=
Xn
i=1
w2|rm[i] |2
(12)
ࡇࡢホ౯㛵ᩘࡣᑠࡉࡅࢀࡤᑠࡉ࠸⛬ࡼ࠸㸬J2㸪J3ࡣ ࡑࢀࡒࢀ(7)ᘧ࠾ࡅࡿᶵᲔⓗᦆኻ㸪㟁Ẽⓗᦆኻᑐ ᛂࡋ࡚࠸ࡿ㸬J1ࡣt = t1(i = n)࡛┠ᶆ್฿㐩ࡍ
ࡿ࠸࠺ᣊ᮰᮲௳ࢆグ㏙ࡋ࡚࠾ࡾ㸪J2㸪J3ࡢ㔜ࡳw1㸪 w2(>0)ࢆᑠࡉࡃࡿࡇࡼࡗ࡚┠ᶆ್ࡢࡎࢀࢆ
ᑠࡉࡃࡍࡿࡇࡀ࡛ࡁࡿ㸬ࡲࡓw1㸪w2ࡣ㸪J2J3ࡢ
࢚ࢿࣝࢠ࣮ࡢḟඖࢆྜࢃࡏࡿࡓࡵ㸪ᐃᩘࢆ⏝࠸࡚
௨ୗࡢࡼ࠺Ỵᐃࡍࡿ㸬 ( w1=w·b
w2=w· (13)
4. ᭱㐺ࢺࣝࢡࡢᑟฟ
4.1 LMI᭱㐺ၥ㢟
ホ౯㛵ᩘJࡣᑠࡉࡅࢀࡤᑠࡉ࠸㸪ᦆኻࡀᑠࡉ ࡃ㸪ᅇ⏕ࡢሙྜࡣ้t=t1(ຊ⾜ࡢሙྜࡣt3)࠾࠸
࡚┠ᶆ್rm[n] ᐇ㝿ࡢᅇ㌿ゅ㏿ᗘrm[n]ࡀ㏆࡙
ࡃ㸬ࡑࡇ࡛㸪LMIࢆ⏝࠸࡚Jࡢ᭱ᑠၥ㢟ࢆ⪃࠼ࡿ㸬
ࡲࡎ㸪௨ୗ♧ࡍ#ࢆᐃ⩏ࡍࡿ㸬
= [AࠉA2ࠉ· · ·ࠉAn]T (14)
# = m NN NN Nt
B 0 · · · 0
AB B . .. ...
... . .. . .. 0 An1B An2B · · · B
u UU UU U{
(15)
,#ࢆ⏝࠸ࡿ㸪y ࡣḟᘧࡢࡼ࠺᭩ࡁ┤ࡏࡿ㸬
y =rm[0] +#Te (16)
ࡉࡽInࢆnḟ༢⾜ิࡋ࡚W1=w1In㸪W2= w2Inࡋ㸪y0Myࢆ
y0=rm[n] Anrm[0] (17) My= [An1BࠉAn2Bࠉ· · ·ࠉB] (18)
ࡍࡿ㸪ホ౯㛵ᩘJࡣ௨ୗ࡞ࡿ㸬
J= (rm[n] rm[n])T(rm[n] rm[n]) +TeTW1Te +y TW2y
= (yT0y0+rm[0]TTW2rm[0]) (y0TMyrm[0]TTW2#)Te TeT(MyTy0#TW2rm[0])
+TeT(MyTMy+W1+#TW2#)Te (19) Table 1. Parameters setup.
Rs () 2.63 Rr() 2.42 Ls (H) 0.177 Lr (H) 0.173
M (H) 0.167 p 2
w 104 *r(Wb) 0.5 J (kg·m2) 0.0073 (N·s) 0.0036
zmin (Nm) -3 zmax (Nm) 3
ࡇࡇ࡛㸪ホ౯㛵ᩘJࡢୖ⏺ࡢ୍ࡘࢆ ࡍࡿ㸪
(y0Ty0+rm[0]TTw2rm[0]) + (y0TMyrm[0]TTW2#)Te +TeT(MyTy0#TW2rm[0])
TeT(MyTMy+W1+#TW2#)Te >0 (20)
࡞ࡿ㸬ࡉࡽ㸪MyTMy+W1+#TW2# >0ࡼࡾ㸪 Schur Complement17)ࢆ⏝࠸ࡿ㸪௨ୗࢆᚓࡿ㸬
A< B (21)
ణࡋ㸪
A=
"
11 12
T12 22
# , B=
"
0 0 0 1
#
(22) T
z
Z
11=(MyTMy+W1+#TW2#)1 12=Te
22= (y0Ty0+rm[0]TTw2rm[0]) ࠉࠉࠉ(yT0Myrm[0]TTW2#)Te ࠉࠉࠉTeT(MyTy0#TW2rm[0])
(23)
࡛࠶ࡿ㸬
ࡉࡽ㸪(9)ᘧࢆ᭩ࡁ┤ࡍ㸪 ( zminTe
Te zmax (24)
࡞ࡿ㸬
(21)ᘧ(24)ᘧࡣࡑࢀࡒࢀLMI࡞ࡗ࡚࠾ࡾ㸪ࡇ
ࢀࡽࢆ‶ࡓࡋࡘࡘ ࢆ᭱ᑠࡍࡿᅇ⏕ࢺࣝࢡTe ࢆồ
ࡵࡿ࠸࠺LMI᭱㐺ၥ㢟ᖐ╔ࡉࢀࡿ㸬 4.2 LMI᭱㐺ၥ㢟ࡢゎἲ
ࡇࡢ LMI ᭱㐺ၥ㢟ࢆゎࡃࡓࡵ㸪Mathworks
♫ Matlab ࡢ Robust Control Toolbox ෆ ࠶ ࡿ GEVP(Generalized eigenvalue minimization under LMI constraints)ࣝࢦࣜࢬ࣒ࢆ⏝࠸ࡿ㸬ࡓࡔࡋ㸪ࡇ ࡢࢶ࣮ࣝࡣṇᐃ್ၥ㢟ࢆᑐ㇟ࡋ࡚࠾ࡾ㸪Bࡀ༙ṇᐃ
࡛࠶ࡿࡓࡵ┤᥋ⓗࡣゎࡅ࡞࠸㸬ࡑࡢࡓࡵ᪂ࡓ࡞ኚᩘ
0< ࢆ⏝࠸㸪(21)ᘧࢆ௨ୗኚᙧࡍࡿࡇࡼࡾ㸪
ྠࢶ࣮ࣝࢆ⏝࠸࡚ゎࡃࡇࡀ࡛ࡁࡿ㸬
A<
"
0 0 0 0
#
(25)
5. ᭷ຠᛶࡢ᳨ド
5.1 ᑟฟࡋࡓ᭱㐺ࢺࣝࢡ
Fig. 4. Regenerative torques derived by LMI and variational method without the constraint condition
of torque amplitude.
Table 1♧ࡍ᮲௳ࡢୗ࡛㸪ᥦࡍࡿᡭἲࢆ⏝࠸࡚
ᅇ⏕ࡢ᭱㐺ࢺࣝࢡࡢᑟฟࡇࢀࢆ⏝࠸ࡓࢩ࣑࣮ࣗࣞ
ࢩࣙࣥࢆ⾜ࡗࡓ㸬㛫ᅇ㌿ゅ㏿ᗘ㛵ࡍࡿ᮲௳ࡣ㸪 t0 = 0.3 (s)㸪t1 = 0.7 (s)㸪rm0 = 167.5 (rad/s)㸪 rm1 = 52.3 (rad/s)ࡋࡓ㸬ࡲࡓࢺࣝࢡࡢࡁࡉ
㛵ࡍࡿ᮲௳ࡣ㸪ࡲࡎ㸪zminࢆ༑ศᑠࡉࡃ㸪zmaxࢆ
༑ศࡁࡃタᐃࡍࡿࡇ࡛㸪ไ⣙᮲௳ࢆ⪃៖ࡋ࡞࠸
ᚑ᮶ࡢኚศἲ10,11)ྠ୍ࡢ⤖ᯝ࡞ࡿࢆㄪࡓ㸬 ᅇ⏕㟁ຊ㔞ࡀ᭱࡞ࡿࡼ࠺࡞ᅇ⏕ࢺࣝࢡࢆኚศἲ࡛
ồࡵࡓ⤖ᯝ㸪ᮏሗ࿌࡛ᥦࡋࡓLMIࢆ⏝࠸࡚ᑟฟ ࡋࡓᅇ⏕ࢺࣝࢡࢆẚ㍑ᅗࢆFig. 4♧ࡍ㸬Teࡣຊ⾜
ࢆṇࡋࡓㄏᑟ㟁ືᶵࡢࢺࣝࢡ㸪rmࡣㄏᑟ㟁ືᶵࡢ ᅇ㌿ゅ㏿ᗘࢆ⾲ࡋ࡚࠸ࡿ㸬ࡇࢀࡼࡿ㸪LMIኚศ ἲ࡛ồࡵࡓᅇ⏕ࢺࣝࢡࡢ㌶㐨ࡣ୍⮴ࡋ࡚࠸ࡿࡇࡀぢ
࡚ྲྀࢀࡿ㸬ࡇࡢࡇࡽ㸪ᮏ◊✲࡛ᥦࡋࡓLMIࢆ
⏝࠸࡚ᑟฟࡋࡓᅇ⏕ࢺࣝࢡࡣ㸪ᦆኻࢆ᭱ᑠࡋࡑࡢ⤖
ᯝ㸪ᅇ⏕㟁ຊ㔞ࢆ࡛᭱ࡁࡿࡇࡀ☜ㄆ࡛ࡁࡿ㸬 ḟ㸪ࢺࣝࢡࡢࡁࡉไ㝈ࢆຍ࠼ࡓሙྜࡢ⤖ᯝࢆ
Fig. 5㸪Fig. 6♧ࡍ㸬ຊ⾜ࡢFig. 6࡛ࡣ㸪0.4 (s)ࡢ 㛫rm2= 52.3 (rad/s)ࡽrm3 = 167.5 (rad/s)
ࡲ࡛ຍ㏿ࡉࡏࡿࡇࡋࡓ㸬ࡲࡓ㸪ᅇ⏕࣭ຊ⾜ࡢ୧᪉ ࡢሙྜ࠾࠸࡚㸪ࢺࣝࢡࡢࡁࡉࡢไ㝈್ࢆ3 (Nm)
ࡋࡓ㸬ࡇࡇ࡛ࡣ㸪ኚศἲࢆ⏝࠸࡚ࢺࣝࢡࡢࡁࡉ
㛵ࡍࡿไ⣙᮲௳ࢆ⪃៖࡛ࡁࡿᨵⰋἲ12)LMIࢆ⏝࠸
ࡓᮏᡭἲࡀྠ୍ࡢ⤖ᯝ࡞ࡿࢆㄪࡓ㸬୧ᅗࡼࡾ㸪
ᩥ⊩12)࡛ᥦࡉࢀ࡚࠸ࡿゎᯒⓗᡭἲࡢᨵⰋἲⰋዲ
୍⮴ࡋ࡚࠸ࡿࡇࡀ☜ㄆ࡛ࡁࡿ㸬ࡲࡓ㸪ᅇ㌿ゅ㏿ᗘ ࡣᅇ⏕ࡢሙྜ࠾࠸࡚ࡶຊ⾜ࡢሙྜ࠾࠸࡚ࡶ㸪ᡤᮃ
Fig. 5. Regenerative torques derived by LMI and variational method under the constraint condition
of torque amplitude|Te|3 (Nm).
ࡢᅇ㌿ゅ㏿ᗘ฿㐩࡛ࡁ࡚࠸ࡿࡇࡀ☜ㄆ࡛ࡁࡿ㸬
5.2 ᐇ㦂ࡼࡿ᭷ຠᛶࡢ☜ㄆ
Fig. 7ᥦᡭἲࢆ⏝࠸࡚ᅇ⏕࣭ຊ⾜ࢆ㐃⥆ࡋ࡚⾜ࡗ
ࡓᐇ㦂⤖ᯝࢆ♧ࡍ㸬ไᚚ⨨ࡋ࡚㸪࣐࢙࢘ᢏ◊〇 ࡢPE-Expertϩࢩࢫࢸ࣒ࢆ⏝࠸ࡓ㸬ᅗ୰ࡢVDC2ࡣ㸪
Fig. 1࠾ࡅࡿDCࣜࣥࢡ㟁ᅽࢆ⾲ࡋ࡚࠸ࡿ㸬ᅇ⏕⤊
࣭ຊ⾜⤊ࡶᡤᮃࡢᅇ㌿ゅ㏿ᗘࡀᚓࡽࢀ࡚
࠸ࡿࡇࡀ☜ㄆ࡛ࡁࡿ㸬ࡲࡓ㸪ᅇ⏕ࡣ㟁ຊࡀDC
ࣜࣥࢡࡢ࢟ࣕࣃࢩࢱC2࠼ࡽࢀVDC2ࡀୖ᪼ࡋຊ
⾜ࡣῶᑡࡋ࡚࠸ࡿࡇࡽ㸪ᅇ⏕ࡋ࠼ࡽࢀࡓ㟁 ຊࡀຊ⾜⏝ࡉࢀࡓࡇࡀ☜ㄆ࡛ࡁࡿ㸬௨ୖࡢࡇ
ࡽ㸪ᥦࡋࡓᡭἲࡣᐇ㦂࠾࠸࡚ࡶไ⣙᮲௳ࢆ‶㊊
ࡍࡿࡼ࠺ᅇ⏕࣭ຊ⾜࡛ࡁࡿࡇࡀ᫂ࡽ࡞ࡗࡓ㸬
6. ࠾ࢃࡾ
ᮏㄽᩥ࡛ࡣ㸪ㄏᑟ㟁ືᶵࡢ㟁ẼⓗᦆኻᶵᲔⓗᦆኻ
╔┠ࡋ㸪ࡑࢀࡽࡢᦆኻࢆ᭱ᑠࡍࡿࡼ࠺࡞᭱㐺ࢺ
ࣝࢡLMIࢆ⏝࠸࡚ᩘ್ⓗᑟฟࡍࡿ᪉ἲࢆᥦࡋࡓ㸬 ࢺࣝࢡࡢࡁࡉ㛵ࡍࡿไ⣙᮲௳ࢆ⦆ࡸࡋࡓሙྜ㸪 ኚศἲࡽᑟฟࡋࡓᅇ⏕ࢺࣝࢡ୍⮴ࡋࡓ㸬ࡲࡓ㸪ࢺ
ࣝࢡࡢࡁࡉ㛵ࡍࡿไ⣙᮲௳ࢆཝࡋࡃࡋࡓሙྜ࡛ࡶ㸪 ኚศἲࢆ⏝࠸࡚ࢺࣝࢡࡢࡁࡉ㛵ࡍࡿไ⣙᮲௳ࢆ⪃
៖࡛ࡁࡿᨵⰋἲ࡛ᚓࡓ⤖ᯝ୍⮴ࡋࡓ㸬ࡲࡓ㸪ᐇ㦂
࠾࠸࡚ࡶࡑࡢ᭷ຠᛶࢆ☜ㄆࡋࡓ㸬ᮏᡭἲࢆ⏝࠸ࡿࡇ
࡛㸪ᅇ⏕㛤ጞ⤊ࡢᅇ㌿ゅ㏿ᗘᅇ⏕㛫㸪ᅇ
⏕ࢺࣝࢡࡢࡁࡉࡢไ⣙್ࢆᣦᐃࡍࡿࡔࡅ࡛㸪ࡑࡢ㛫
࡛ࡢᦆኻࡀ᭱ᑠ࡞ࡿ᭱㐺ࢺࣝࢡࡀᑟฟ࡛ࡁࡿ㸬
Fig. 6. Acceleration torques derived by LMI and variational method under the constraint condition
of torque amplitude|Te|3 (Nm).
LMIࢆ⏝࠸ࡓᮏᡭἲࡢ≉ᚩࡣ㸪(8)ᘧ㸪(9)ᘧࡀࡁ
ࢃࡵ୍࡚⯡ⓗ࡞ᙧ࡛グ㏙ࡉࢀ࡚࠾ࡾ㸪ᮏㄽᩥ࡛♧ࡋࡓ ไ⣙᮲௳௨እࡢᵝࠎ࡞᮲௳ᑐࡋ࡚ࡶᰂ㌾ᑐᛂࡀྍ
⬟࡞Ⅼࡀ࠶ࡿ㸬ᚋࡣ㸪ᵝࠎ࡞⏝㏵ࡑࢀࡒࢀᑐࡋ࡚
㐺ࡋࡓไ⣙᮲௳ୗ࠾ࡅࡿ᭱㐺ࢺࣝࢡࡢᑟฟࢆ⾜࠸㸪 ࡑࡢ᭷ຠᛶࢆ᳨ドࡋ࡚⾜ࡃணᐃࢆࡋ࡚࠸ࡿ㸬
ཧࠉ⪃ࠉᩥࠉ⊩
1) Yee-Pien Yang and Tsung-Hsien Hu, “A New En- ergy Management System of Directly-Driven Elec- tric Vehicle with Electronic Gearshift and Regener- ative Braking”, American Control Conference 2007 (ACC ’07), pp.4419-4424 (2007).
2) ཎ⣧ኵ,⸨㑥ኵ, ㎷㍤⏕,⚄ཎㄔ,“ㄏᑟ㟁ືᶵࡢ࢚ࢿ
ࣝࢠ࣮ᅇ⏕ࢩࢫࢸ࣒”, ᖹᡂ12ᖺ㟁ẼᏛᅜ, 4-113 (2000).
3) ᕝཱྀΎ, “㕲㐨࠾ࡅࡿ࢚ࢿࣝࢠ࣮㛵㐃ᢏ⾡ࠉࣁࣈ
ࣜࢵࢻ㌴୧⏝ࣇ࣮ࣛ࣍ࣝᘧ㟁⨨”, RRR, 61 [9], 8-11 (2004).
4) S. R. Cikanek and K. E. Bailey, “Regenerative Brak- ing System For A Hybrid Electric Vehicle”, Ameri- can Control Conference 2002 (ACC ’02), 3129-3134 (2002).
5) 㔝ᮧᘯ,᳃ᮌᏹ⮳, “ไᚚ㟁ὶ※ࢆ⏝࠸ࡓᅇ⏕ࣈ࣮ࣞ࢟
ࢩࢫࢸ࣒ࡢᇶ♏◊✲”, ᖹᡂ14ᖺ㟁ẼᏛᅜ, 4-217 (2002).
6) ᶓ㇂ᒎ, ᐩᶔோኵ, ᩘཎᑑᏹ, “㟁ືࣁࣈࣜࢵࢻ⮬
㌿㌴CY-SJ”, SANYO TECHNICALREVIEW, 35 [1], 106-114 (2003).
7) ୖ㤾,ᑠ᪉ྖ,ຍ⸨ḟ, “ㄏᑟ㟁ືᶵࡢ㟁ຊᅇ⏕ࢩ ࢫࢸ࣒ࡢ᳨ウ”,ᖹᡂ17ᖺ㟁ẼᏛ⏘ᴗᛂ⏝㒊㛛, 1-50, I-229-I-232 (2005).
Fig. 7. Experimental results under the constraint condition of torque amplitude|Te|3 (Nm).
8) ୖ㤾,ᑠ᪉ྖ,ຍ⸨ḟ, “ㄏᑟ㟁ືᶵࡢ㟁ຊᅇ⏕
࠾ࡅࡿᅇ⏕ࢺࣝࢡࡢ᳨ウ”,ᖹᡂ18ᖺ㟁ẼᏛ⏘ᴗ ᛂ⏝㒊㛛, 1-30, I-267-I-270 (2006).
9) K. Matsuse, T. Yoshizumi, S. Katsuta, and S.
Taniguchi, “High-Response Flux Control of Direct- Field-Oriented Induction Motor with High E^- ciency Taking Core Loss into Account”, IEEE Transactions on Industry Applications, 35 [1], 62- 69 (1999).
10) K. Inoue㸪K. Ogata㸪and T. Kato, “A Study on an Optimal Torque for Power Regeneration of an Induction Motor”, Proceedings of the 38th IEEE Power Electronics Specialists Conference, 2108-2112 (2007).
11) ୖ㤾, ᑠ᪉ྖ,ຍ⸨ḟ, “ኚศἲࡼࡿ᭱㐺ࢺࣝ
ࢡࢆ⏝࠸ࡓㄏᑟ㟁ືᶵࡢ㧗ຠ⋡㟁ຊᅇ⏕࣭㥑ືἲ”,㟁 ẼᏛㄽᩥㄅD, 128[9], 1098-1105 (2008).
12) ༡ᒣṇಇ,ୖ㤾,ຍ⸨ḟ, “ㄏᑟ㟁ືᶵ࠾ࡅࡿᵝࠎ
࡞ไ⣙᮲௳ࢆ⪃៖ࡋࡓ᭱㐺ࢺࣝࢡࡢᑟฟ”, 㟁ẼᏛ
◊✲㈨ᩱ ༙ᑟయ㟁ຊኚ◊✲, SPC-09-31, 61-66 (2009).
13) ᮡᮏⱥᙪ㸪ᑠᒣṇே㸪⋢ఙ୕, “ACࢧ࣮࣎ࢩࢫࢸ࣒
ࡢ⌮ㄽタィࡢᐇ㝿”㸪⥲ྜ㟁Ꮚฟ∧♫,ᮾி(1990).
14) ୖ㤾,ᯇᮏ๛,ᑠ᪉ྖ,ຍ⸨ḟ, “ᅇ⏕㟁ຊ㈓ⶶ
⨨ࢆ⏝࠸ࡓㄏᑟ㟁ືᶵࡢ┬࢚ࢿࣝࢠ࣮㥑ືἲ”,ྠᚿ
♫Ꮫ⌮ᕤᏛ◊✲ሗ࿌,48[1], 42-49 (2007).
15) ᮡỤಇ㸪ᒣᮏᾈஅ, “≧ែ࠾ࡼࡧධຊࡢไ⣙ࢆ⪃៖ࡋ ࡓ㛢࣮ࣝࣉ⣔ࡢ┠ᶆ್⏕ᡂ”,ィ ⮬ືไᚚᏛㄽᩥ㞟, 37[9], 849-855 (2001).
16) ᮡỤಇ㸪㕥ᮌᐂぢ, “ไ⣙ࢆ᭷ࡍࡿࢩࢫࢸ࣒ᑐࡍࡿ
࿘ᮇ┠ᶆ್ࡢసᡂ”,ࢩࢫࢸ࣒ไᚚሗᏛㄽᩥㄅ,17 [8], 313-320 (2004).
17) ᒾᓮᚭஓ, “LMIไᚚ”,ᇽ㸪ᮾி(1997).