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A Design Methodology of an Optimal Torque Minimizing the Total Loss of an Induction Motor by Means of LMI

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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㟁ືᶵࡢᦆኻࢆ᭱ᑠ࡜ࡍࡿ᭱㐺ࢺࣝࢡࡢ⥺ᙧ⾜ิ୙➼ᘧࢆ⏝࠸ࡓタィἲ

஭ୖࠉ㤾ࠉ࣭ࠉ༡ᒣࠉṇಇࠉ࣭ࠉຍ⸨ࠉ฼ḟ

1. ࡣࡌࡵ࡟

㟁ືᶵࢆ⏝࠸ࡓᦙ㏦㌴ࡸ㟁ື㌴୧࡞࡝ࡣ㸪ῶ㏿᫬࡟

㟁Ẽⓗ࡟ไືຊࢆⓎ⏕ࡉࡏࡿᅇ⏕ࣈ࣮ࣞ࢟ࢆ⏝࠸࡚ᶵ Ეⓗ࢚ࢿࣝࢠ࣮ࢆ㟁ຊ࡟ኚ᥮ࡋ㸪ࡇࢀࢆຍ㏿᫬࡟෌฼

⏝ࡍࡿࡇ࡜࡛ຠ⋡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ࡃ࡜ᅇ⏕୰ࡢࢺࣝࢡࡢ࠿

ࡅ᪉࡟ࡼࡗ࡚ኚ໬ࡍࡿ79)㸬ࡇࡢࡓࡵ㸪ࡼࡾ᭷ຠ࡟࢚

ࢿࣝࢠ࣮ࢆ฼⏝ࡍࡿࡓࡵ࡟ࡣ㸪ࡇࢀࡽࡢ᮲௳ୗ࡟࠾࠸

࡚㸪ᦆኻࡀ᭱ᑠࡍ࡞ࢃࡕᅇ⏕㟁ຊ㔞ࡀ᭱኱࡜࡞ࡿ᭱㐺

࡞ᅇ⏕ࢺࣝࢡࡢ࠿ࡅ᪉ࢆ᳨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ࡋ࠿ࡋ㸪ࡇࡢ᪉ἲ࡛ࡣᅇ

⏕ࢺࣝࢡࡢ኱ࡁࡉ࡟㛵ࡍࡿไ⣙᮲௳ࢆຍ࿡ࡋ࡚࠸࡞࠸

ࡓࡵ㸪ồࡵࡓ᭱㐺ࢺࣝࢡࡢ኱ࡁࡉࡀᶵჾࡢᐃ᱁್ࢆ㉺

࠼࡚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௳ࠕ᫬้t0࠿ࡽt1࡟࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm0

࠿ࡽrm1ࡲ࡛ῶ㏿ࠖ࡟ຍ࠼㸪᪂ࡓ࡞ไ⣙᮲௳ࠕᅇ⏕

ࢺࣝࢡTeࡢ኱ࡁࡉࢆไ㝈್௨ୗ࡟ᢚไࠖ㸪ࡢ཮᪉ࢆ

(2)

Fig. 2. Schematic diagram of an optimal regenerative torque under the constraint condition

of torque amplitudeTezmin.

‶㊊ࡋࡘࡘᅇ⏕㟁ຊ㔞ࢆ᭱኱໬ࡍࡿ᭱㐺ᅇ⏕ࢺࣝࢡࡢ タィ᪉ἲࢆᥦ᱌ࡍࡿ㸬ලయⓗ࡟ࡣ㸪LMI(⥺ᙧ⾜ิ୙

➼ᘧ)ࢆ⏝࠸࡚ᩘ್ⓗ࡟ゎࡃ᪉ἲࢆᥦ᱌ࡋ㸪ᚑ᮶ࡢኚ ศἲ࡟ࡼࡿ᪉ἲࢆไ⣙᮲௳௜ࡁ࡛ゎᯒⓗ࡟ゎࡃᨵⰋἲ ࡢ⤖ᯝ12)࡜ẚ㍑᳨࣭ウࢆ⾜࠸㸪ᥦ᱌ᡭἲࡢ᭷ຠᛶࢆ

☜ㄆࡍࡿ㸬ࡲࡓຍ㏿᫬࡟࠾࠸࡚ࡶྠᵝࡢᡭἲࡀ㐺⏝࡛

ࡁ㸪ไ⣙᮲௳ࠕ᫬้t2࠿ࡽt3࡟࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆ

rm2࠿ࡽrm3ࡲ࡛ຍ㏿ࠖ࠾ࡼࡧࠕຊ⾜ࢺࣝࢡTe

኱ࡁࡉࢆไ㝈್௨ୗ࡟ᢚไࠖࢆ‶㊊ࡋࡘࡘ㸪ຍ㏿࡟ᚲ せ࡞㟁ຊ㔞ࢆ᭱ᑠ໬ࡍࡿ᭱㐺ຊ⾜ࢺࣝࢡࢆᑟฟࡍࡿ㸬

௨ୗ㸪2.࡛ࡣࢺࣝࢡไ⣙ࢆ⪃៖ࡍࡿᚲせᛶࢆ㏙࡭㸪 3.࡛ࡣᑐ㇟ࢩࢫࢸ࣒ࡢᐃᘧ໬ࢆ⾜࠺㸬4.࡛ࡣLMIࢆ

⏝࠸ࡓ᭱㐺ࢺࣝࢡタィἲ࡟ࡘ࠸࡚㏙࡭㸪᭱ᚋ࡟5.࡟

࠾࠸࡚㸪ᥦ᱌ᡭἲࡢ᭷ຠᛶ࡟ࡘ࠸࡚ࢩ࣑࣮ࣗࣞࢩࣙࣥ

࡜ᐇ㦂ࡢ୧᪉࡟ࡼࡾ᳨ドࡍࡿ㸬

2. ࢺࣝࢡࡢ኱ࡁࡉ࡟㛵ࡍࡿไ⣙ࡢᚲせᛶ

ᚑ᮶ࡢኚศἲࢆ⏝࠸ࡓ᭱㐺ᅇ⏕ࢺࣝࢡࡣᅇ⏕㛤ጞ┤

ᚋࡣẚ㍑ⓗ኱ࡁࡃ㸪ᅇ㌿ゅ㏿ᗘࡀᑠࡉࡃ࡞ࡿ࡟ࡘࢀ࡚

ᑠࡉࡃ࡞ࡿ10,11)㸬ࡇࢀࡣᅇ㌿ゅ㏿ᗘࡀ኱ࡁ࠸࡜ࡁ࡟

ࡣᅇ㌿ῶ⾶࡟ࡼࡿ࢚ࢿࣝࢠ࣮ᾘ㈝ࡀ኱ࡁ࠸ࡓࡵ㸪኱ࡁ

࡞ࢺࣝࢡࢆ࠿ࡅ࡚ᅇ㌿ゅ㏿ᗘࢆపୗࡉࡏࡘࡘ㸪Ⓨ㟁㟁 ຊࢆ኱ࡁࡃࡋ࡚ᅇ⏕ࡋ࡚࠸ࡿࡇ࡜ࢆ♧ࡋ࡚࠸ࡿ㸬ࡋ࠿

ࡋ㸪ࢺࣝࢡࡢ኱ࡁࡉࡀᶵჾࡢᐃ᱁್ࢆ㉸࠼࡚ࡋࡲ࠺ྍ

⬟ᛶࡀ࠶ࡿ㸬

Fig. 3. Schematic diagram of an optimal acceleration torque under the constraint condition

of torque amplitude Tezmax.

ࡑࡇ࡛ᮏㄽᩥ࡛ࡣ㸪ᚑ᮶ࡢ᮲௳ࠕ᫬้t0࠿ࡽt1࡟࠾

࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm0࠿ࡽrm1ࡲ࡛ῶ㏿ࠖ࡟ຍ࠼㸪 ᪂ࡓ࡞ไ⣙᮲௳ࠕᅇ⏕ࢺࣝࢡTeࡢ኱ࡁࡉࢆไ㝈್௨

ୗ࡟ᢚไࠖࢆຍ࠼ࡓሙྜ࡟ࡘ࠸࡚⪃࠼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Teࡣ᫬้t0௜㏆࡟࠾࠸࡚ࡑ ࡢไ㝈್ࡢୗ㝈zminࢆୗᅇࡗ࡚࠾ࡾ㸪⌮ㄽ㏻ࡾࡢࢺ

ࣝࢡࢆⓎ⏕࡛ࡁ࡞࠸㸬ࡑࡇ࡛㸪ࡋࡤࡽࡃไ⣙್୍ᐃࢺ

ࣝࢡࢆຍ࠼࡚ῶ㏿ࡉࡏࡿ㸬ࡋ࠿ࡋ㸪1Ⅼ㙐⥺࡜ไ㝈್

ࡢ஺Ⅼ࠿ࡽࡍࡄࡉࡲᚑ᮶ἲ࡟ᚑࡗ࡚ᅇ⏕ࢆ⾜࠺࡜㸪2

Ⅼ㙐⥺࡛♧ࡍࡼ࠺࡟༑ศ࡟ᅇ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㏿ᗘࡀ኱ࡁࡃ࡞ࡿ࡟ࡘࢀ࡚

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

௨ୖࡢࡇ࡜࠿ࡽ㸪ࢺࣝࢡࡢ኱ࡁࡉ࡟ไ⣙ࡀ࠶ࡿሙྜ

࡟࠾࠸࡚ࡶ㸪ᡤᮃࡢ᫬้࡟࠾࠸࡚ᡤᮃࡢᅇ㌿ゅ㏿ᗘ࡟

฿㐩࡛ࡁ㸪ᦆኻࢆ᭱ᑠ࡜ࡍࡿ᭱㐺ࢺࣝࢡࢆᑟฟࡍࡿᚲ せࡀ࠶ࡿ㸬

3. ᑐ㇟ࢩࢫࢸ࣒ࡢᐃᘧ໬

ㄏᑟ㟁ືᶵࡢ☢᮰࡜ࢺࣝࢡࡣ㛫᥋ᆺ࣋ࢡࢺࣝไᚚ13)

࡟ࡼࡗ࡚ไᚚࡉࢀࡿ㸬࡞࠾㸪࣋ࢡࢺࣝไᚚ࡛⏝࠸ࡿྛ

(3)

ไᚚࢤ࢖ࣥࡣ㸪ᴟ㓄⨨ἲࢆ⏝࠸࡚Ỵᐃࡍࡿ14)

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

Te࡜ࡍࡿ࡜㸪

Jrm(t) +rm(t) =Te(t) (1)

࡛୚࠼ࡽࢀࡿ㸬ࡓࡔࡋ㸪Teࡣຊ⾜᫬ࢆṇ㸪ᅇ⏕᫬ࢆ㈇

࡜ࡍࡿ㸬rmࡣṇࡢሙྜࡢࡳ࡜ࡋ㸪ᅇ⏕᫬࡟㏫ᅇ㌿ࡍ

ࡿࡇ࡜ࡣ࡞࠸࡜ࡍࡿ㸬

ࡇࡇ࡛㸪(1)ᘧࢆࢧࣥࣉࣜࣥࢢ࿘ᮇTs࡛㞳ᩓ໬ࡍ

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

3.2඲ᦆኻ

ㄏᑟ㟁ືᶵࡢᶵᲔⓗᦆኻࡣᅇ㌿ࡢῶ⾶࡟ࡼࡿࡶࡢ࡛

࠶ࡿࡢ࡛㸪

Plm=2rm (4)

࡛୚࠼ࡽࢀࡿ㸬୍᪉㸪࣋ࢡࢺࣝไᚚࡉࢀࡓㄏᑟ㟁ືᶵ

࡛ࡣ㸪㕲ᦆࢆᚤᑡ࡜ࡋ࡚↓どࡍࡿሙྜࡀከ࠸ࡢ࡛㸪ࡇ ࡢ࡜ࡁ㟁ẼⓗᦆኻPleࡣ୍ḟ(ᅛᐃᏊ)㖡ᦆ࡜஧ḟ(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㸪୍ḟ᢬ᢠ㸪஧ḟ᢬ᢠ㸪ᴟᑐᩘ㸪஧ḟ☢᮰ࡢ㍈ ᡂศࢆ⾲ࡍ㸬ᮏ◊✲࡛ࡣ㸪▐᫬ࢺࣝࢡࡢไᚚ࡟㛫᥋ᆺ

࣋ࢡࢺࣝไᚚࢆ⏝࠸ࡿࡓࡵ*rࡣ୍ᐃ್࡟ไᚚࡉࢀࡿ

ࡢ࡛㸪a㸪bࡣඹ࡟ᐃᩘ࡛࠶ࡿ㸬

᫬้t0࠿ࡽt1ࡲ࡛ᅇ⏕ࡍࡿ࡜ࡁࡢ඲ᦆኻࡣ㸪

Iloss= Z t1

t0

(Plm+Ple)dt (6)

࡜࡞ࡿ㸬ᐃᩘaࡢᐃ✚ศࡣ㸪rm㸪Teࡀ࡝ࡢࡼ࠺࡞್

࡛࠶ࡗ࡚ࡶ୍ᐃ್࡛࠶ࡿࡢ࡛㸪ࡇࢀࢆ㝖࠸ࡓࡶࡢࢆ඲

ᦆኻ Ibloss=

Z t1

t0

(rm2 +bTe2)dt (7)

࡜࠾ࡃࡇ࡜࡜ࡍࡿ㸬࡞࠾㸪᫬้t2࠿ࡽt3ࡲ࡛ຊ⾜ࡍ

ࡿሙྜࡣ㸪ࡑࡢࡼ࠺࡟✚ศ༊㛫ࢆኚ᭦ࡍࢀࡤࡼ࠸㸬

3.3 ᣑ኱⣔࡜ホ౯㛵ᩘ

ࢺࣝࢡࡢ኱ࡁࡉ࡟㛵ࡍࡿไ⣙᮲௳ࢆグ㏙ࡍࡿࡓࡵࡢ ኚᩘz[i]ࢆᑟධࡋ㸪ồࡵࡿ᭱㐺࡞ᅇ⏕ࢺࣝࢡࢆTe[i]

࡜⾲ࡍ࡜㸪ไ⣙᮲௳ࢆຍ࿡ࡋࡓᣑ኱⣔ࡣ௨ୗ࡛⾲ࡉࢀ

15,16)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㏙ࡉࢀࡿ࡜ࡍࡿ㸬

zmin[i]z[i]zmax[i] (9) ᮏ◊✲࡛ࡢไ⣙᮲௳ࡣࠕᅇ⏕ࢺࣝࢡTeࡢ኱ࡁࡉࢆ࠶

ࡿ್௨ୗ࡟ᢚไࡍࡿࠖࡇ࡜࡛࠶ࡿࡢ࡛㸪cz= 0, dz= 1

࡜ࡍࡿ࡜z[i] =Te[i] ࡜࡞ࡿ㸬

ḟ࡟㸪ᚑ᮶ࡢ᮲௳ࠕ᫬้t=t0(i= 0)࠿ࡽt1(i=n)

࡟࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm0 = rm[0]࠿ࡽrm1 = rm[n]ࡲ࡛ῶ㏿ࡉࡏࡿࠖࡓࡵ㸪nࢫࢸࢵࣉ࠿ࡽ࡞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ࠕ᫬้t = t2(i= 0)࠿ࡽt3(i=n)࡟࠾࠸࡚ᅇ㌿ゅ㏿ᗘࢆrm2= rm[0]࠿ࡽrm3 =rm[n]ࡲ࡛ຍ㏿ࡉࡏࡿࠖ࡜ㄞࡳ

᭰࠼ࢀࡤࡼ࠸㸬

ࡇࢀࡽࢆ⏝࠸࡚㸪௨ୗࡢホ౯㛵ᩘJࢆᐃ⩏ࡍࡿ㸬 T€

€€

€€

€€

€€ z

€€

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€€

€€

€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ⓗᦆኻ࡟ᑐ 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)ࢆᑠࡉࡃ࡜ࡿࡇ࡜࡟ࡼࡗ࡚┠ᶆ್࡜ࡢࡎࢀࢆ

ᑠࡉࡃࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬ࡲࡓw1㸪w2ࡣ㸪J2࡜J3

(4)

࢚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⏕ࡢሙྜࡣ᫬้t=t1(ຊ⾜ࡢሙྜࡣt3)࡟࠾࠸

࡚┠ᶆ್rm[n] ࡬ᐇ㝿ࡢᅇ㌿ゅ㏿ᗘrm[n]ࡀ㏆࡙

ࡃ㸬ࡑࡇ࡛㸪LMIࢆ⏝࠸࡚Jࡢ᭱ᑠ໬ၥ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)

ࡉࡽ࡟Inࢆnḟ༢఩⾜ิ࡜ࡋ࡚W1=w1In㸪W2= w2In࡜ࡋ㸪y0࡜My

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€

€€

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€z

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€€

€€ 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 ࢆ᭱ᑠ໬ࡍࡿᅇ⏕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)ᘧࢆ௨ୗ࡟ኚᙧࡍࡿࡇ࡜࡟ࡼࡾ㸪

ྠࢶ࣮ࣝࢆ⏝࠸࡚ゎࡃࡇ࡜ࡀ࡛ࡁࡿ㸬

A<

"

0 0 0 0

#

(25)

5. ᭷ຠᛶࡢ᳨ド

5.1 ᑟฟࡋࡓ᭱㐺ࢺࣝࢡ

(5)

Fig. 4. Regenerative torques derived by LMI and variational method without the constraint condition

of torque amplitude.

Table 1࡟♧ࡍ᮲௳ࡢୗ࡛㸪ᥦ᱌ࡍࡿᡭἲࢆ⏝࠸࡚

ᅇ⏕᫬ࡢ᭱㐺ࢺࣝࢡࡢᑟฟ࡜ࡇࢀࢆ⏝࠸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ࡲࡓࢺࣝࢡࡢ኱ࡁࡉ࡟

㛵ࡍࡿ᮲௳ࡣ㸪ࡲࡎ㸪zminࢆ༑ศ࡟ᑠࡉࡃ㸪zmax

༑ศ࡟኱ࡁࡃタᐃࡍࡿࡇ࡜࡛㸪ไ⣙᮲௳ࢆ⪃៖ࡋ࡞࠸

ᚑ᮶ࡢኚศἲ10,11)࡜ྠ୍ࡢ⤖ᯝ࡜࡞ࡿ࠿ࢆㄪ࡭ࡓ㸬 ᅇ⏕㟁ຊ㔞ࡀ᭱኱࡜࡞ࡿࡼ࠺࡞ᅇ⏕ࢺࣝࢡࢆኚศἲ࡛

ồࡵࡓ⤖ᯝ࡜㸪ᮏሗ࿌࡛ᥦ᱌ࡋࡓLMIࢆ⏝࠸࡚ᑟฟ ࡋࡓᅇ⏕ࢺࣝࢡࢆẚ㍑ᅗࢆFig. 4࡟♧ࡍ㸬Teࡣຊ⾜᫬

ࢆṇ࡜ࡋࡓㄏᑟ㟁ືᶵࡢࢺࣝࢡ㸪rmࡣㄏᑟ㟁ືᶵࡢ ᅇ㌿ゅ㏿ᗘࢆ⾲ࡋ࡚࠸ࡿ㸬ࡇࢀ࡟ࡼࡿ࡜㸪LMI࡜ኚศ ἲ࡛ồࡵࡓᅇ⏕ࢺࣝࢡࡢ㌶㐨ࡣ୍⮴ࡋ࡚࠸ࡿࡇ࡜ࡀぢ

࡚ྲྀࢀࡿ㸬ࡇࡢࡇ࡜࠿ࡽ㸪ᮏ◊✲࡛ᥦ᱌ࡋࡓLMIࢆ

⏝࠸࡚ᑟฟࡋࡓᅇ⏕ࢺࣝࢡࡣ㸪ᦆኻࢆ᭱ᑠ໬ࡋࡑࡢ⤖

ᯝ㸪ᅇ⏕㟁ຊ㔞ࢆ᭱኱໬࡛ࡁࡿࡇ࡜ࡀ☜ㄆ࡛ࡁࡿ㸬 ḟ࡟㸪ࢺࣝࢡࡢ኱ࡁࡉ࡟ไ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㏿ࡉࡏࡿࡇ࡜࡜ࡋࡓ㸬ࡲࡓ㸪ᅇ⏕࣭ຊ⾜ࡢ୧᪉ ࡢሙྜ࡟࠾࠸࡚㸪ࢺࣝࢡࡢ኱ࡁࡉࡢไ㝈್ࢆ3 (Nm)

࡜ࡋࡓ㸬ࡇࡇ࡛ࡣ㸪ኚศἲࢆ⏝࠸࡚ࢺࣝࢡࡢ኱ࡁࡉ࡟

㛵ࡍࡿไ⣙᮲௳ࢆ⪃៖࡛ࡁࡿᨵⰋἲ12)࡜LMIࢆ⏝࠸

ࡓᮏᡭἲࡀྠ୍ࡢ⤖ᯝ࡜࡞ࡿ࠿ࢆㄪ࡭ࡓ㸬୧ᅗࡼࡾ㸪

ᩥ⊩12)࡛ᥦ᱌ࡉࢀ࡚࠸ࡿゎᯒⓗᡭἲࡢᨵⰋἲ࡜Ⰻዲ

࡟୍⮴ࡋ࡚࠸ࡿࡇ࡜ࡀ☜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࠸ࡿࡇ࡜ࡀ☜ㄆ࡛ࡁࡿ㸬

5.2 ᐇ㦂࡟ࡼࡿ᭷ຠᛶࡢ☜ㄆ

Fig. 7࡟ᥦ᱌ᡭἲࢆ⏝࠸࡚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⤊஢᫬࡜ࡶ࡟ᡤᮃࡢᅇ㌿ゅ㏿ᗘࡀᚓࡽࢀ࡚

࠸ࡿࡇ࡜ࡀ☜ㄆ࡛ࡁࡿ㸬ࡲࡓ㸪ᅇ⏕᫬࡟ࡣ㟁ຊࡀDC

ࣜࣥࢡࡢ࢟ࣕࣃࢩࢱC2࡟⵳࠼ࡽࢀVDC2ࡀୖ᪼ࡋຊ

⾜᫬࡟ࡣῶᑡࡋ࡚࠸ࡿࡇ࡜࠿ࡽ㸪ᅇ⏕ࡋ⵳࠼ࡽࢀࡓ㟁 ຊࡀຊ⾜࡟౑⏝ࡉࢀࡓࡇ࡜ࡀ☜ㄆ࡛ࡁࡿ㸬௨ୖࡢࡇ࡜

࠿ࡽ㸪ᥦ᱌ࡋࡓᡭἲࡣᐇ㦂࡟࠾࠸࡚ࡶไ⣙᮲௳ࢆ‶㊊

ࡍࡿࡼ࠺࡟ᅇ⏕࣭ຊ⾜࡛ࡁࡿࡇ࡜ࡀ᫂ࡽ࠿࡜࡞ࡗࡓ㸬

6. ࠾ࢃࡾ࡟

ᮏㄽᩥ࡛ࡣ㸪ㄏᑟ㟁ືᶵࡢ㟁Ẽⓗᦆኻ࡜ᶵᲔⓗᦆኻ

࡟╔┠ࡋ㸪ࡑࢀࡽࡢ඲ᦆኻࢆ᭱ᑠ໬ࡍࡿࡼ࠺࡞᭱㐺ࢺ

ࣝࢡLMIࢆ⏝࠸࡚ᩘ್ⓗ࡟ᑟฟࡍࡿ᪉ἲࢆᥦ᱌ࡋࡓ㸬 ࢺࣝࢡࡢ኱ࡁࡉ࡟㛵ࡍࡿไ⣙᮲௳ࢆ⦆ࡸ࠿࡟ࡋࡓሙྜ㸪 ኚศἲ࠿ࡽᑟฟࡋࡓᅇ⏕ࢺࣝࢡ࡜୍⮴ࡋࡓ㸬ࡲࡓ㸪ࢺ

ࣝࢡࡢ኱ࡁࡉ࡟㛵ࡍࡿไ⣙᮲௳ࢆཝࡋࡃࡋࡓሙྜ࡛ࡶ㸪 ኚศἲࢆ⏝࠸࡚ࢺࣝࢡࡢ኱ࡁࡉ࡟㛵ࡍࡿไ⣙᮲௳ࢆ⪃

៖࡛ࡁࡿᨵⰋἲ࡛ᚓࡓ⤖ᯝ࡜୍⮴ࡋࡓ㸬ࡲࡓ㸪ᐇ㦂࡟

࠾࠸࡚ࡶࡑࡢ᭷ຠᛶࢆ☜ㄆࡋࡓ㸬ᮏᡭἲࢆ⏝࠸ࡿࡇ࡜

࡛㸪ᅇ⏕㛤ጞ᫬࡜⤊஢᫬ࡢᅇ㌿ゅ㏿ᗘ࡜ᅇ⏕᫬㛫㸪ᅇ

⏕ࢺࣝࢡࡢ኱ࡁࡉࡢไ⣙್ࢆᣦᐃࡍࡿࡔࡅ࡛㸪ࡑࡢ㛫

࡛ࡢ඲ᦆኻࡀ᭱ᑠ࡜࡞ࡿ᭱㐺ࢺࣝࢡࡀᑟฟ࡛ࡁࡿ㸬

(6)

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࡞⏝㏵ࡑࢀࡒࢀ࡟ᑐࡋ࡚

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ẼᏛ఍඲ᅜ኱఍, 4-113 (2000).

3) ᕝཱྀΎ, “㕲㐨࡟࠾ࡅࡿ࢚ࢿࣝࢠ࣮㛵㐃ᢏ⾡ࠉࣁ࢖ࣈ

ࣜࢵࢻ㌴୧⏝ࣇࣛ࢖࣍࢖࣮ࣝᘧ⵳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ẼᏛ఍඲ᅜ኱఍, 4-217 (2002).

6) ᶓ㇂࿴ᒎ, ᐩᶔோኵ, ᩘཎᑑᏹ, “㟁ືࣁ࢖ࣈࣜࢵࢻ⮬

㌿㌴CY-SJ”, SANYO TECHNICALREVIEW, 35 [1], 106-114 (2003).

7) ஭ୖ㤾,ᑠ᪉೺ྖ,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,ᑠ᪉೺ྖ,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. InoueK. Ogataand 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, ᑠ᪉೺ྖ,ຍ⸨฼ḟ, “ኚศἲ࡟ࡼࡿ᭱㐺ࢺࣝ

ࢡࢆ⏝࠸ࡓㄏᑟ㟁ືᶵࡢ㧗ຠ⋡㟁ຊᅇ⏕࣭㥑ືἲ”, ẼᏛ఍ㄽᩥㄅD, 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࿴๛,ᑠ᪉೺ྖ,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 ⮬ືไᚚᏛ఍ㄽᩥ㞟, 37[9], 849-855 (2001).

16) ᮡỤಇ἞㸪㕥ᮌᐂぢ, “ไ⣙ࢆ᭷ࡍࡿࢩࢫࢸ࣒࡟ᑐࡍࡿ

࿘ᮇ┠ᶆ್ࡢసᡂ”,ࢩࢫࢸ࣒ไᚚ᝟ሗᏛ఍ㄽᩥㄅ,17 [8], 313-320 (2004).

17) ᒾᓮᚭஓ, “LMI࡜ไᚚ”,᫛᫭ᇽ㸪ᮾி(1997).

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