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NAIST-IS-MT0851087 2010 3 17

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, Ecological Economical,.,.,,.,,, FIR.,,,.,,,.,,, FIR,, NAIST-IS- MT0851087, 2010 3 17. i

Energy Efficiency of Power Assisting Control Methods for Electric Bicycles Kazuyoshi Hatada Abstract This paper considers energy efficiency of power assisting methods for electric bicycles. In recent years, by a surge to the environmental problem, the use of the bicycle which is an Ecological, Economical vehicle attracts attention again. The electric bicycles which applied power assist technology to a bicycle are spreading rapidly by such a background. Most commercial bikes are using proportional assist to human power. However, it is less efficient since the pulsation of the speed is amplified inevitably. Thus we proposed several methods based on repetitive control in our previous works. In this paper, we compare the energy efficiency of each method through experiments by using newly developed pedaling robot. Keywords: electric bicycle, periodic motion, repetitive control, FIR filter, generalized KYP lemma Master s Thesis, Department of Information Systems, Graduate School of Information Science, Nara Institute of Science and Technology, NAIST-IS-MT0851087, March 17, 2010. ii

1. 1 1.1................................ 1 1.2................................ 2 1.3............................. 3 2. 4 2.1...................... 4 2.2................... 6 3. 8 3.1...................... 8 3.2............................. 9 3.3............................. 10 3.4....................... 13 4. 15 4.1 (PPC)......................... 15 4.2 [13]............ 17 4.2.1 [14]................... 17 4.2.2 (MRC)........ 17 4.2.3..................... 19 4.3 [17]...... 20 4.3.1..................... 20 4.3.2 FIR (FIRRC) 22 5. 27 5.1.............................. 27 5.2............................. 30 iii

6. 31 6.1................................ 31 6.2............................ 32 6.3...................... 32 6.4................................ 34 6.4.1 ( ) 34 6.4.2... 36 7. 40 8. 41 42 43 46 A. 46 iv

1....................... 2 2 ( )............... 5 3................. 5 4 ( ).................. 6 5........................ 8 6....................... 9 7............................. 11 8.................. 11 9............................. 13 10............................. 15 11...................... 16 12.................... 17 13............. 18 14............. 19 15 FIR............................. 21 16 FIR.................... 21 17 FIR......... 22 18 K RC (z)............................. 22 19.............. 28 20 (ω 0 =8.38[rad/sec])................. 29 21 (ω 0 =8.20[rad/sec])................. 29 22............................ 31 23.................... 33 24......................... 33 25 (ω 0 = 8.38[rad/sec])................. 34 26 (ω 0 = 8.20[rad/sec])................. 35 27................................ 36 28 ( )................. 37 v

29 ( )...................... 37 30 (FIR )......... 38 31 (FIR ).............. 38 1 Ψ........................ 25 2......................... 46 3.................... 46 vi

1. 1.1, -.,,, [1],.,,,., [2],, [3], [4],,,., 1990,.,,.,.,,.,, [5][6].,.,, [7],. 1

1.2,,. 1 [8].,. 10.,, 1.,,,.,,,.,, [15], FIR [23]., 2

,.,,,,. 1.3, 2,., 3,,. 4,, FIR. 5. 6,. 7, 8. 3

2.,. 2.1,, [9]. 1993, PAS.,, 2008 [10].,. 2. 3,,.,.,., 4.,,,., 2. 4

図 2 電動アシスト自転車 (チェーン駆動型) 図 3 電動アシスト自転車のアシスト機構 5

4 ( ) 2.2,,,,.,,,,.,..,,,,,., [11].,,.,,. [12] 6

2,,,,,,. 7

3.,,,,. 3.1 5. 5,,,, J, D, r, τ[nm] v[km/h],, Jω 0 + Dω 0 = τ, v = rω 0 (1) v(s) = r τ(s) =: G(s)τ(s) (2) Js + D.,., (MAG-500), ( 6). 8

6 3.2,., 1 1,.,,,.,,.. 9

3.3, τ, θ,. T s 0.01[sec]. G(s) = 1/J s + D/J =: b s(s + a). G(s), (3) G(z) = b(1 e at s )/a z e at s T s z 1 =: β T s z + α z 1 (4), α, β,., θ(2) θ(1) θ(0) θ(1) T s τ(0) [ ]. =.. α β θ(n) θ(n 1) θ(n 2) θ(n 1) T s τ(n 2) (5)., θ(2) θ(1) θ(0) θ(1) T s τ(0) [ ] b =., A =.. α, x = β θ(n) θ(n 1) θ(n 2) θ(n 1) T s τ(n 2) (6), b Ax 2 x, x = (A T A) 1 A T b G(z) = 0.070 z 0.901,, G(z) T s. G(s) = 7.393 s + 10.460 10 (7) (8)

5.0 4.5 4.0 3.5 torque[nm] 3.0 2.5 2.0 1.5 1.0 0.5 0 5 10 15 20 25 time[sec] 7 angular velocity[rad/sec] 3.5 3.0 2.5 2.0 1.5 1.0 0.5 measured value estimated value 0 5 10 15 20 25 time[sec] 8 11

, 7., 8.,. 12

3.4,,,. without pulsation with pulsation velocity 0 time 9 v D, Dv 2. v = v 0 (const), v = v 0 +v a sin ω 0 t ( 9). W, W 1 = 1 T T 0 Dv 2 dt = D T [v2 0t] T 0 = Dv 2 0 (9) 13

W 2 = 1 T = D T = D T T 0 T 0 Dv 2 dt (v 0 + v a sin ω 0 t) 2 dt [ v0 2 2v 0v a cos ω 0 t + 1 ω 0 2 v2 at v2 a sin 2ω 0 t 4ω 0 = Dv0 2 + 1 2 Dv2 a + 2v 0v a D(1 cos ω 0 T ) v2 a D sin 2ω 0 T (10) ω 0 4ω 0., 1 T = 2π/ω 0, W 1 = Dv 2 0, W 2 = Dv0 2 + (1/2)Dva 2.,.,,., v a = v 0, 33.3[%]. ] T 0 14

4.,,,. 4.1 (PPC),,. 10.,. 10,., 11, E, F, d, r 1, r 2, f 1, f 2, f 1 = F r 1 (11) f 2 = τ r 2 (12) 15

. α, f 2 = αf 1 E, E = r 2 r 1 αf (13).,, r 1 = 0.175[m], r 2 = 0.033[m], E =.. 0.2αF (14)., 2. 11 16

4.2 [13],,., [14],. 4.2.1 [14]. L, N, 12. 12, L, T s N, L = NT s.,, 4.2.2 (MRC),.,, [15].,,. 17

0.,,.. N F 1 (z) F 1 (z) = 1 z N 1 (15)., z = 1 F 1 (z) F 2 (z), F 2 (z) = z 1 z N 1 1 = z N 1 + z N 2 + + z + 1 (16). 13.,.,,. 13 18

4.2.3,., 8.38[rad/sec] 14.,.,,,. 10 4 2.0 1.8 1.6 1.4 power 1.2 1.0 0.8 0.6 0.4 0.2 0-1.5-1.0-0.5 0 0.5 1.0 1.5 angular velocity[rad/sec] 14 19

4.3 [17],., FIR, [18]., KYP [22],. [23],. 4.3.1 [19]. FIR : y(n) = IIR : y(n) = N α k u(n k) (17) k=0 N α k u(n k) + k=0 j=1 M β j y(n j) (18) (17) FIR (Finite Impulse Response Filter)., (18) IIR (Infinite Impulse Response Filter). (17) z, Y (z) = N α k z k U(z) (19) k=0, H(z) = Y (z) N U(z) = α k z k (20)., FIR, FIR 16. k=0 20

15 FIR 16 FIR 21

FIR α k = 1, FIR, α k. 4.3.2 FIR (FIRRC) FIR 17., K RC (z), K o (z)., K RC (z) FIR X(z) = K k=κ X kz k L(z)., K RC (z). 17 FIR 18 K RC (z), K o (z), R KC (z)., 22

S(z)(= S o (z)m s (z)), K RC (z) = 0 S o (z), T o (z). [17]. 1 K o (z) ( ) K o (z), K RC (z) = 0 S o (z),.,,.,, S o (e jω 0T s ). 2 L(z) ( ) FIR X S o (z) T o (z) K o (z), L(z) = T o (z) 1. 3 X(z) ( ),. minimize γ p, subject to M s (e jω 0T s ) ϵ, (21) M s (e jω 0T s ) γ p,, ω l ω 0 ω h, (22) X(e jω 0T s ) ζ. (23),,,. KYP [24]. 23

KYP [24] A C n n, B C n m, C C p n, D C p m, Π H n n (28) Λ(Φ, Ψ), det(λi A) 0, λ Λ., σ(g(λ), Π) < 0, λ Λ(Φ, Ψ) (24), P H n, Q > 0 H n. [ A B ] [ ] A B I 0 (Φ P + Ψ Q) I 0 + Θ < 0 (25), Θ := [ C D ] I 0 Π [ C D ] I 0 (26), Π, (27). ] ] Π = Π br := [ Im 0 0 γ 2 I m, Π = Π pr := [ 0 Im I m 0 (27) Φ., Φ c, Φ d ] ] Φ c = [ 0 1 1 0, Φ d = [ 1 0 0 1 (28). (24). Ψ. Ψ 1. 24

1 Ψ Ω Ψ [ ] 1 0 LF ω 0 ϖ l 0 ϖl 2 [ ] 1 jϖc MF ϖ 1 ω 0 ϖ 2 jϖ c ϖ 1 ϖ 2 [ ] 1 0 HF ω 0 ϖ h 0 ϖh 2, ϖ c := (ϖ 1 + ϖ 2 )/2 Θ Ψ [ ] 0 1 LF θ ϑ l 1 2 cos ϑ l [ ] 0 e jϑ c MF ϑ 1 θ ϑ 2 e jϑ c 2 cos ϑ [ ] 0 1 HF ϑ h θ π 1 2 cos ϑ h, ϑ c := (ϑ 1 + ϑ 2 )/2, ϑ := (ϑ 2 ϑ 1 )/2 25

(25),.,, LMI, FIR. (25) Schur complement. [ Γ(P, Q, C, D) C D] S ] < 0 (29) S [C D R, Π 11 = SR 1 S :R > 0 Γ(P, Q, C, D) :=. [ A B I 0 ] [ ] [ ] A B 0 C Π 12 (Φ P +Ψ Q) + I 0 Π 12 D Π 12 + Π 12D + Π 22 (30) 26

5.,, ( 13), FIR ( 17). 5.1, FIR., G(z), F (z), G(z) = 0.078 z 0.901 (31) F (z) = 2.56 (32)., d d = d 1 sin ω 0 t + d 0, (33) d 1 = 1.4, d 0 = 2., T s = 0.01[s]., N = 75, K(z) = 4.589z 4.132 z 0.905 (34)., FIR K = 40, K(z) = 3.33z 3.33 z 0.905 (35), ω h = 8.98, ω l = 7.85 26[dB]. 19., ω 0 8.38[rad/sec]. FIR, 27

gain[db] -10-20 MRC FIRRC -30-40 -50-60 -70-80 7 8 9 angular velocity[rad/sec] 10 19.,, 20, 21. FIR (ω 0 = 8.20[rad/sec]),. 28

velocity deviation[km/h] velocity deviation[km/h] 2.0 1.5 1.0 PPC MRC FIRRC 0.5 0-0.5-1.0-1.5-2.0 170 171 172 173 174 175 176 177 178 179 180 time[sec] 20 (ω 0 =8.38[rad/sec]) 2.0 1.5 1.0 PPC MRC FIRRC 0.5 0-0.5-1.0-1.5-2.0 170 171 172 173 174 175 176 177 178 179 180 time[sec] 21 (ω 0 =8.20[rad/sec]) 29

5.2, E. E = (36),, 3.4. d ω 0 = 8.38[rad/sec] 20.,, FIR E P P C, E MRC, E F IRRC, E P P C = 1.15, E MRC = 1.00, E F IRRC = 1.04. E P P C, E MRC 13.06[%], E F IRRC 9.55[%]., d ω 0 = 8.20[rad/sec] ( 21) E P P C = 1.18, E MRC = 1.07, E F IRRC = 1.05. E P P C E MRC 9.41[%], E F IRRC 10.71[%]. ω 0,, FIR. 30

6.,,. 6.1 3. 22.,. drive torque bicycle encoder motor USB-CON controller observer PC 22 T s 0.01[sec].. 31

6.2,. (1) θ, θ = D J θ + 1 J τ (37). x = [θ θ] T,. d dt [ ] [ ] [ ] [ ] θ 0 1 θ 0 = + τ θ 0 D/J θ 1/J (38) [ ] [ ] θ y = 1 0 θ (39),. [ ] [ ] 0 1 0 A =, B =, C = 0 10.460 7.393 [ ] 1 0, D = 0 (C, A),. 6.3.,,,.,.,.,,,. 23, 24. 32

図 23 ペダリングロボットの設計図 図 24 ペダリングロボット 33

velocity deviation[km/h] 6.4. 6.4.1 ( ), (39) d d 1 = 1.6, d 0 = 2, ω 0 = 8.38, 8.20[rad/sec]. 25, 26. 1.2 1.0 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1.0 PPC MRC FIRRC -1.2 170 171 172 173 174 175 176 177 178 179 180 time[sec] 25 (ω 0 = 8.38[rad/sec]) 25, 26,,. FIR, 34

velocity deviation[km/h] 1.2 1.0 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1.0 PPC MRC FIRRC -1.2 170 171 172 173 174 175 176 177 178 179 180 time[sec] 26 (ω 0 = 8.20[rad/sec])., FIR., ω 0 = 8.38[rad/sec], E P P C = 1.08, E MRC = 1.02, E F IRRC = 1.01, E P P C E MRC 5.93[%], E F IRRC 6.39[%]., ω 0 = 8.20[rad/sec] E P P C = 1.08, E MRC = 1.07, E F IRRC = 1.03, E P P C E MRC 1.73[%], E F IRRC 5.44[%].,, FIR.,,,. 35

velocity[km/h] 6.4.2,,,, (39) d 0., FIR,. 2 27.,,, 28, 29,, FIR 30, 31., ω 0 = 8.38[rad/sec], (180[ o ]). 30, FIR. 8.0 7.5 PPC FIRRC 7.0 6.5 6.0 5.5 170 171 172 173 174 175 176 177 178 179 180 time[sec] 27 36

80 70 drive force assist force 60 50 force[n] 40 30 20 10 0 170 171 172 173 174 175 176 177 178 179 180 time[sec] 28 ( ) pedal angle[ ] 200 180 160 140 120 100 80 60 40 20 0 170 171 172 173 174 175 176 177 178 179 180 time[sec] 29 ( ) 37

80 70 drive force assist force 60 50 force[n] 40 30 20 10 0 170 171 172 173 174 175 176 177 178 179 180 time[sec] 30 (FIR ) pedal angle[ ] 200 180 160 140 120 100 80 60 40 20 0 170 171 172 173 174 175 176 177 178 179 180 time[sec] 31 (FIR ) 38

,, E P P C = 1.09, E F IRRC = 1.01, E P P C 7.77[%]. 39

7.,,.,, FIR.,,.,,.,,,., FIR.,, FIR 7.77[%]. 40

8.,. FIR,., [28],,.,,.,.,.,,, I/O.,. 41

,.,,.,.,.,.,,.., ( ),., M1,, 42

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A. category 2 spec electric bike personal computer encoder I/O board software 24 in., DC brushless motor(240w) Win XP, 2.6 GHz, 1.99 GB mem. DC5V, 1000 pulse/rad digital controller U (USBA04) Matlab/Simulink R2006b 3 category spec motor DC brushed motor(150w) gear 1:43 encoder DC5V, 2000 pulse/rad 46