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1 NAIST-IS-MT

2 ( )

3 , Ecological Economical,.,.,,.,,, FIR.,,,.,,,.,,, FIR,, NAIST-IS- MT , i

4 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-MT , March 17, ii

5 (PPC) [13] [14] (MRC) [17] FIR (FIRRC) iii

6 ( ) A. 46 iv

7 ( ) ( ) FIR FIR FIR K RC (z) (ω 0 =8.38[rad/sec]) (ω 0 =8.20[rad/sec]) (ω 0 = 8.38[rad/sec]) (ω 0 = 8.20[rad/sec]) ( ) v

8 29 ( ) (FIR ) (FIR ) Ψ vi

9 1. 1.1, -.,,, [1],.,,,., [2],, [3], [4],,,., 1990,.,,.,.,,.,, [5][6].,.,, [7],. 1

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

11 ,.,,,,. 1.3, 2,., 3,,. 4,, FIR. 5. 6,. 7, 8. 3

12 2.,. 2.1,, [9]. 1993, PAS.,, 2008 [10].,. 2. 3,,.,.,., 4.,,,., 2. 4

13 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

14 4 ( ) 2.2,,,,.,,,,.,..,,,,,., [11].,,.,,. [12] 6

15 2,,,,,,. 7

16 3.,,,, ,,,, 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

17 6 3.2,., 1 1,.,,,.,,.. 9

18 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) = z 0.901,, G(z) T s. G(s) = s (7) (8)

19 torque[nm] time[sec] 7 angular velocity[rad/sec] measured value estimated value time[sec] 8 11

20 , 7., 8.,. 12

21 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

22 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 = Dv 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

23 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

24 . 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).,

25 4.2 [13],,., [14], [14]. L, N, , L, T s N, L = NT s.,, (MRC),.,, [15].,,. 17

26 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 z + 1 (16). 13.,.,,

27 4.2.3,., 8.38[rad/sec] 14.,.,,, power angular velocity[rad/sec] 14 19

28 4.3 [17],., FIR, [18]., KYP [22],. [23], [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

29 15 FIR 16 FIR 21

30 FIR α k = 1, FIR, α k 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

31 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

32 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 = [ , Φ d = [ (28). (24). Ψ. Ψ 1. 24

33 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

34 (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

35 5.,, ( 13), FIR ( 17). 5.1, FIR., G(z), F (z), G(z) = z (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 z (34)., FIR K = 40, K(z) = 3.33z 3.33 z (35), ω h = 8.98, ω l = [dB]. 19., ω [rad/sec]. FIR, 27

36 gain[db] MRC FIRRC angular velocity[rad/sec] ,, 20, 21. FIR (ω 0 = 8.20[rad/sec]),. 28

37 velocity deviation[km/h] velocity deviation[km/h] PPC MRC FIRRC time[sec] 20 (ω 0 =8.38[rad/sec]) PPC MRC FIRRC time[sec] 21 (ω 0 =8.20[rad/sec]) 29

38 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 = 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 = E P P C E MRC 9.41[%], E F IRRC 10.71[%]. ω 0,, FIR. 30

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

40 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),. [ ] [ ] A =, B =, C = [ ] 1 0, D = 0 (C, A),. 6.3.,,,.,.,.,,,. 23,

41 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

42 velocity deviation[km/h] ( ), (39) d d 1 = 1.6, d 0 = 2, ω 0 = 8.38, 8.20[rad/sec]. 25, PPC MRC FIRRC time[sec] 25 (ω 0 = 8.38[rad/sec]) 25, 26,,. FIR, 34

43 velocity deviation[km/h] PPC MRC FIRRC 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

44 velocity[km/h] 6.4.2,,,, (39) d 0., FIR, ,,, 28, 29,, FIR 30, 31., ω 0 = 8.38[rad/sec], (180[ o ]). 30, FIR PPC FIRRC time[sec] 27 36

45 80 70 drive force assist force force[n] time[sec] 28 ( ) pedal angle[ ] time[sec] 29 ( ) 37

46 80 70 drive force assist force force[n] time[sec] 30 (FIR ) pedal angle[ ] time[sec] 31 (FIR ) 38

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

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

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

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

51 [1],, 10, pp (2009) [2],,, 2, 10, pp (2009) [3],,,,,,, Vol. 45, No. 12, pp (2009) [4],,,,,, Vol. 43, No. 3, pp (2007) [5],, - Compass Walking,,,, 17-12, pp (2004) [6],, 8, (2008) [7],,,, (c ), 69, 680, pp (2003) [8] D. G. Wilson, Bicycling Science, 3rd Edition, The MIT Press (2004) [9] [10], , be on Saturday [11],,, Vol. 90, No. 11, pp (2007) 43

52 [12], (2008) [13],, 2006,, (2007) [14],,,,, (1988) [15],,,, 7, pp (2006) [16] B.A.Francis and W.N.Wonham, The Internal Model Principle for Linear Multivariable Regulators, Appl. Math. & Opt., 2-2, pp (1975) [17], FIR, 2008,, (2009) [18] Goele Pipeleers, Generalized Repetitive Control: Better Performance with Less Memory, AMC 08, pp (2008) [19],,,, (1993) [20] M.Steinbuch, Repetitive control for systems with uncertain period-time, Automatica, 38(12), pp (2002) [21] M.Tomizuka, Zero phase error tracking algorithm for digital control.transactions of the ASME: Journal of Dynamic Systems, Measurement and Control, 109(1), pp (1987) [22] T.Iwasaki and S.Hara, Generalized KYP lemma:unified Characterization of Frequency Domain Inequalities with Applications to System Design (2003),Technical reports, (2003) [23],, FIR, SICE, pp (2009) 44

53 [24], KYP,, 45-8, pp (2005),Technical reports, (2003) [25] Z. Liu and L. Vandenberghe, Low-rank structure in semidefiniteprograms derived from the kyp lemma, Reprint submitted to the 46th Conference on Decision and Control (2007) [26],,, / /, 50-12, pp (2006) [27] D.Siokata, S.Hara and T,Iwasaki, From Nyquist/Bode to GKYP design: Design algorithms with CACSD tools, SICE Annual Conference, pp , (2004) [28],,,,,,, 10, pp (2009) 45

54 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

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