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1 NAIST-IS-DD

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3 ,.,,,.,,,,.,.,,.,.,.,,.,,., (FIR), Kalman-Yakubovich- Popov,, (IIR),,, NAIST-IS- DD16116, i

4 .,,.,.,.,.,,.,,, Kalman-Yakubovich-Popov, ii

5 Energy-Efficient Power Assisting Methods for Periodic Motions Kazuyoshi Hatada Abstract In recent years, the research on the power assist technology is wide spreading because of its potential benefit to our society in medical, welfare, industrial and other fields. The simplest assist method would be to apply additional force generated by machines in proportion to the instantaneous value of the force generated by human. Although there seems to be no other choice when the motion is irregular and unpredictable, in case of the periodic motions, as is typical in our persistent tasks, such a strategy must be undesirable since it amplify the pulsation of periodic force and result in unsteady, inefficient dynamical behavior. Thus, in this thesis, the optimal power assist control method for periodic motions from the viewpoint of energy-efficiency is investigated. First we consider the pedaling motion of a human on a bicycle. A dynamical model of the power train of an electric power-assisted bicycle is derived. The optimality condition for the pedaling is examined based on this model. The result implies that to flatten the torque pattern by removing the input pulsation improves the efficiency. This fact justifies the previous work to use a version of the repetitive control to reject the non-dc components of the periodic torque disturbance invoked by the human pedaling action. Another type of repetitive controller synthesis based on the generalized Kalman-Yakubovich-Popov (KYP) Doctoral Dissertation, Department of Information Systems, Graduate School of Information Science, Nara Institute of Science and Technology, NAIST-IS-DD16116, March 15, 213. iii

6 lemma is proposed in the literature. In this framework, the delay component in the standard repetitive controller is replaced by a Finite Impulse Response (FIR) filter and the gain specifications at a set of independent finite frequency ranges are satisfied by using the gkyp lemma. A power assist controller design via yet another synthesis framework is proposed here. The key feature is that the usage of Infinite Impulse Response (IIR) filter with the gkyp lemma to derive reduced-order controllers satisfying the design specifications. The effectiveness of proposed method is demonstrated by comparing the energy efficiency of each method via numerical simulations and experiments. Next we consider the engine speed control of a ship cruising in regular heading waves. Such a ship experiences periodic force disturbances and this is an analogous situation to the bicycle pedaling discussed above. Therefore our design procedure can be applied to this problem to obtain a feedback engine speed controller. The energy efficiency of the proposed method against the case of constant throttle opening is evaluated through numerical simulations. Keywords: periodic motions, power assist control, repetitive control, generalized Kalman- Yakubovich-Popov lemma, engine speed control iv

7 KYP KYP KYP KYP KYP FIR IIR v

8 A. Shur 75 vi

9 (1) (2) (2) FIR K RC (z) IIR F I (z) τ h v vii

10 29 ( ) (PPC) (MRC) (IIRRC) G c (s) Ψ K I (z) viii

11 R C x x, x R(s) s I(s) s s s R n m n m C n m n m I n n A T A A A He(A) A He(A):= A + A H n n tr(a) A A B A, B A B (i, j) A =[a ij ] Ba ij. Ker(A) A Im(A) A L 2 R G 2 : G H 2. G 2 :=. σ max (A) A 1 2π tr(g (jω)g(jω))dω G : G H. G := sup ω R σ max (G(jω)). {c, d} λ, s, z. ix

12 % [1].,.,,,,., [2], [3],., [4] [5], [6], - [7] HAL(Hybrid Assistive Limb)[8].,,. HAL.,,.,., 3. (a) (b) (c),. 1

13 (Hybrid Electric Vehicle: HEV ) (Electric Vehicle: EV ) [9]. (a),, HEV EV, [1]. (b), HEV [11]. (c), [12]. (c),,. 1.2., [13, 14].,,,.,.,.,., [15], 8±1[rpm].,.,,. [16], 2

14 , [17]., [18],,,.,,.. [19],, V.,,, [2].,, ,., /,,.,.,.,, 2 3 [21], 1993, [22, 23, 24].,,.,. 3

15 ,.,,.,,,.,,.,.,. CO 2 [25, 26, 27, 28, 29],.,,.,, , Kalman-Yakubovich-Popov.,. 3.,.,,.,,. 4,,. 5. 4

16 2. 2.1, [3].,.,,.,,.,,., [31, 32]. 1., r, y, K(s), G(s) ( ) 1, y r, (KG(s)). 5

17 ,, 1/s., T, T -., [31],., 1 e st 1., [33] , 2 T. R c (s). R c (s) = = e st 1 e st 1 e st 1, R c (jω) = 6 1 e jωt 1

18 ., 1 R c (jω) =, ω. ω = 2πk, k =, 1, 2, T T =1, G(s) =(.9s +.8)/(s +1) (1/(1 + R c G(s))) 3. 3, T 5 magnitude [db] frequency [rad/s] 3., T., 1. 2,, e(s) = r(s) y(s) y(s) = Gu(s) u(s) = e st e(s) 1 e st 7

19 . e(s), e(s) =(1 G(s))e st e(s)+d e (s) (1)., D e (s) =(1 e st )r(s)., (1), (1) [34],. 2.2 [3] G(s), 1 G(s) < 1 (2), e L 2, L 2..,,.,., 3,. 8

20 T =1, r(t) = sin 2πt +1, G(s) =.9s +.8 s +1., 1 G(s) =.2 (2) time [s] error time [s] 6 9

21 5 r, y.,.,., G(s) = 7.5s +7.5 s 2 +1s +7.5, 1 G(s) =1 (2)., time [s] 7 (2) error time [s] (2) 1

22 ,,.,, Linear Matrix Inequality: LMI Kalman-Yakubovich-Popov KYP ). 2.2 KYP KYP [35, 36]., KYP.,,., KYP KYP [37]., KYP KYP KYP, n u n y n p G(jω)=C(jωI A) 1 B + D (3). ( ) ( ), : G (jω)g(jω) <γ 2 I, ω R (4) : G(jω)+G (jω) >, ω R (5). G C n y n u Π H ny +n u σ σ(g, Π) := [ G I nu ] Π [ ] G I nu 11

23 (4) (5) σ(g(jω), Π) <, ω R (6)., (6), [ ] Iny Π=Π br := γ 2 I nu, Π=Π pr := [ I nu I nu ]., KYP [35, 36], (6). 2. n p n p P. [ A B ] [ P ][ A B ] I np P I np +Θ< (7), [ C D ] [ C D ] Θ := Π I nu I nu KYP (6) [38]. Λ c Λ d, λ. Λ {c,d} := { λ C σ(λ, Φ {c,d} )=} (8) 12

24 , Λ c Λ d Φ {c,d} ] ] Φ c = [ 1 1, Φ d := [ 1 1., [ P ] P =Φ c P, σ(g(λ), Π) <, λ Λ {c,d} (9) 2. n p n p P. ] ] [ A B I np (Φ {c,d} P ) [ A B I np +Θ<, KYP KYP, KYP, (LF) (MF) (HF) LMI. Φ, Ψ H 2,. Λ(Φ, Ψ) := { λ C σ(λ, Φ) =, σ(λ, Ψ) } (1), Ψ [ 1 λc ] Ψ c = λ c r 2 λ c 2 13

25 , σ(λ, Ψ c )=r 2 λ λ c 2, λ c, r., Ψ [ ] a + jb Ψ l = a jb 2c, σ(λ, Ψ l )=ar(λ)+bi(λ)+c. (8), (1). 9.,. 9 14

26 ,,, 1. 1 Ψ LF MF HF [ ] [ ] [ 1 1 jϖc 1 jω ϖ 2 l e jθ [ 1 1 2cosϑ l ] [ jϖ c ϖ 1 ϖ 2 e jϑ c e jϑc 2cosϑ ϖl 2 ] [ 1 1 ] 2cosϑ h ], ϖ c,ϑ c,ϑ. ϖ c := ϖ 1 + ϖ 2 2 ϑ c := ϑ 1 + ϑ 2 2 ϑ := ϑ 2 ϑ 1 2, KYP [37]. 2.3 [37, 38] A, B, C, D Π H ny +n u (1) Λ(Φ, Ψ). det(λi A), σ(g(λ), Π), λ Λ(Φ, Ψ) (11), P H np,q H np. [ A B I np ] (Φ P +Ψ Q) 15 [ A B I np ] +Θ (12)

27 KYP [37],., (11), [ (λinp A) 1 ] [ ] [ ][ ] B C D C D (λi A) 1 B σ(g(λ), Π) = Π I nu I nu I nu I nu [ (λinp A) 1 ] [ ] B (λinp A) 1 B = Θ (13) I nu I nu. (12), [ (λinp A) 1 ] [ B A B σ(g(λ), Π) I nu I np [ ][ ] A B (λinp A) 1 B ] (Φ P +Ψ Q) I np., (13) [ ][ ] A B (λinp A) 1 B I np I nu = = = I nu [ ] A(λInp A) 1 B + B (λi np A) 1 B [ ] A +(λinp A) (λi np A) 1 B [ λinp I np ] I np (λi np A) 1 B. [ B [(λi np A) 1 ] λinp ] I np ] B [(λi np A) 1 ] [ λinp I np (Φ P ) (Ψ Q) [ λinp I np [ λinp I np ] (λi np A) 1 B ] (λi np A) 1 B (14), λ Λ, P (14)., σ(λ, Ψ), Q [ ] [ ] λinp λinp (Ψ Q) I np 16 I np

28 ,., G(λ). KYP, ẋ(t) = Ax(t)+B 1 w(t)+b 2 u(t) G : z(t) = C 1 x(t)+d 11 w(t)+d 12 u(t) y(t) = C 2 x(t)+d 21 w(t) G, u(t) =K s x(t) (15), K s., x(t), w(t), u(t), z(t), y(t),,,,. (15) C 2 = I, D 21 = G cl : { ẋ(t) = (A + B2 K s )x(t)+b 1 w(t) z(t) = (C 1 + D 12 K s )x(t)+d 11 w(t)., P> K s. P (A + B 2 K s )+(A + B 2 K s ) T P<, LMI., P 1. X, (A + B 2 K s )X + X(A + B 2 K s ) T <, X> K s. LMI, X := K s X LMI, K s X X 1., G cl <γ K s, (7) (15), P K s. 17

29 KYP,., LMI., [ ] Π11 Π 12 Π=, Π 11 H nny, Π 11 (16) Π 21 Π 22, SISO. Π 11 > (9), Π 11 =. (16), (12) Schur [39]. [ Γ(P, Q, C, D) C D] S ] < S [C D R, Π 11 = SR 1 S, rankr =rankπ 11 Γ(P, Q, C, D) := [ A B I np ] [ ] [ ] A B C Π 12 (Φ P +Ψ Q) + I np Π 12 D Π 12 +Π 12D +Π 22., P, Q C, D, Π 11 A, B C, D Π 22 LMI.,. FIR, PID [38]., KYP FIR KYP, FIR KYP, FIR., γ 2., gkypsdp[4]. F (e jθ ).1 (= 4dB) :.4π ω π F (e jθ ) e jdθ <γ : ω.3π 18

30 magnitude [db] normalized frequency 1 5 phase [rad] normalized frequency 11 19

31 3. 3.1,.,,.,.,.., 3.2, 3.3,., 3.4, 3.5., 3.6.,,., 3.7, ,,.,.,,. 2

32 ,..,,, [41].,.,..,.,. DC,.,,.,.,

33 12 3.3,., 6[deg] 12[deg],, 18[deg]. 18[deg],.,. [42]

34 13, 14., [N] force pedaling time [s] 14.,., [15], 23

35 , 13 8 ± 1[rpm] 5,.,,. 14.,.,,,,,. 15,

36 angle [deg] time [s] (a) θ 1 angle [deg] angle [deg] time [s] 4 5 (b) θ time [s] 4 5 (c) θ ,. 25

37 3.4 17,, 18. θ(t) 17 [rad], τ h (t)[n m], τ m (t)[n m].,,,, J d [kg m 2 ], r b [m].,, D d [N s/m]., τ h (t)+τ m (t) v(t)[m/s] J d θ(t)+dd θ(t) =τh (t)+τ m (t), v = r b θ(t), r b v = (τ h + τ m ) J d s + D d =: G o (s)(τ h + τ m ) (17) 26

38 18., (17) G o (s).,, τ m τ m = C(τ h,v) (18)., (17) (18), G c (s)., v = G c (s)τ h (19). 3.5 G c (s). G c (s),.,. 27

39 3.1, τ h [, ˆω] T, G c (j).,. G c (jω k )= (2) ω k = 2πk ˆωT, k =1, 2, 3,,N, N = T 2π τ h. N τ h = α k e jω kt k= N (21), τ h, α k =ᾱ k., v (19) (21)., 1 T T v = N k= N vdt = 1 T α k G c (jω k )e jω kt T N k= N = α G c (j) α k G c (jω k )e jω kt dt,. E D d v v,. E = T D d v 2 dt { T N 2 = D d α k G c (jω k )e kt} jω dt k= N, f k (t). f k = α k G c (jω k )e jω kt 28

40 , f(t), g(t), f,g = T f(t)g(t)dt., E. N N E = D d f k, f l, T k= N l= N { T (k = l) e jωkt e jωlt dt = (k l), E N E = D d T α k 2 G c (jω k ) 2 (22) k= N.,. N D d T α k 2 G c (jω k ) 2 D d T α 2 G c (j) 2 k= N (2).,,.,.,, (N =) (N =1) 2., v := α G c (j), β := α 1 G c (jω 1 ). E 1 = D d Tv 2., v = v + βe jω1t + βe jω 1t = v +2 β sin(ω 1 t + φ) 29

41 , v 2 β β = v 2. (22) E 2 = D d T ( v 2 +2β 2) = 3 2 D dtv 2 = 3 2 E 1., 5[%]. 3.6 (Pedaling force Proportional Control: PPC ).,,. PPC 19. δ>. 19, DC δ. [43, 44] ( ) 3

42 2,. PPC ( ) G c (s) =(1+δ)G o (s)., G o (jω k ) (2). 3.1, T τ h, [45] K R (s), M d (s). 2 2, τ h v G c (s) = G o (s) 1+R c K R M d G o (s), G c (jω k )=,k =, 1, 2,, (2)., (2) k =,., k = 31

43 ., τ h,.,,. T s T, ζ = T/T s., R c (z) = z ζ 1 z = 1 ζ z ζ 1 (23), 21., (23) 21 z ζ 1=(z 1)(z ζ 1 + z ζ z +1), z =1. (Modified-type Repetitive Control: MRC )

44 3.6.2 FIR.,,., FIR, [46]. FIR, [47]. FIR LMI [46]. FIR (FIR Filter-type Repetitive Control: FIRRC ) [48]., 21 z ζ ξ X(z) = X k z k k=., ξ = ζ, X = X 1 = = X ξ 1 =,X ξ =1,., X(z). FIR X(z), KYP,.,.. FIRRC 23. K F (z) 23 FIR 33

45 , K RC (z). K RC (z) = XL(z) 1 X(z) (24). K RC. 24 K RC (z) FIRRC [46], K F (z) 1+K F M d G o (z) =,., L(z) = 1+K F M d G o (z) K F M d G o (z) [49, 5]., (24) τ h v G c (z) = G o (z) (1 X(z)) 1+K F M d G o (z)., K F (z) 1 X(z) G c (z)., FIR X(z) X k. minimize γ 2 subject to 1 X(e jωt s ) γ 1 1 X(e jωt s ) γ 2,ω [ω l,ω h ] X(e jωt s ) γ 3, ω ω d (25) 34

46 , 3.1,. [ω l,ω h ], ω d. KYP, IIR FIR., FIR IIR [51]., KYP ( ) [52].,.., IIR F I (z)., KYP,., [52] K I (z). IIR (IIR filter-type Repetitive Control: IIRRC ) ( 25). 25 IIR 35

47 . G c (e jωts ) γ 1, ω [ω l,ω h ] G c (e jωt s ) γ 2, ω [,ω d ] (26),. KYP [52]. G p (λ) K(λ) [ ] [ ] z w = G p (λ), u = K(λ)y y u G p (λ) = K(λ) = A k A B 1 B 2 C 1 D 11 D 12 C 2 D 21 C k., G p (λ), K(λ) n p, n c,, w(t) R nw, u(t) R nu, z(t) R nz, y(t) R ny. w z G wz (λ). G wz (λ) := A c C c B c D c B k D k A + B 2 D k C 2 B 2 C k B 1 + B 2 D k D 21 = B k C 2 A k B k D 21 C 1 + D 21 D k C 2 D 12 C k D 11 + D 12 D k D 21, (12) [ ] [ ] Ac I np +n k (Φ T P +Ψ T Ac I np +n Q) k C c C c [ ] [ ] Bc Bc + Π < (27) D c I nz 36 D c I nz

48 . LMI. 3.1 [53], n := n p + n k, J R (2n+n z) 2n, H C (2n+nz) (nw+nz), L C (2n+nz) n. [ ] I n I2n J :=, H := B c, L := A c D c, P, Q H n, R C n (2n+nz), Φ, Ψ H 2, Π H nw +n z. N, R Ker(R) = Im(N)., 2. I nz 1. (27),. N (J(Φ T P +Ψ T Q)J )N + N (HΠH )N< C c 2. W C n n. J(Φ T P +Ψ T Q)J + HΠH < He(LW R) (28), (27) (28). X, Y, U, V C np np W [ ] X (Inp XY )(V 1 ) W = U UY (V 1 ), M, G, H, L. [ ] M G H L := + [ Y AX ] [ V ][ YB2 Ak ][ B k U ] I nu C k D k C 2 X I ny 37

49 , A C := B D AX + B 2 H A+ B2 LC2 B 1 + B 2 LD21 M YA+ GC 2 YB 1 + GD 21 C 1 X + D 21 H C1 + D 12 LC2 D 11 + D 12 LD21 F := [ ] Inp, Y V F := diag(f, F, I nz ) Z := YX+ VU, [ ] X Inp W := Z Y (29), R RF = F R, (28). 3.2 [52] (11) n p K(λ). 2. X, Y, Z, M, G, H, L, P Hn, Q H n > 1.,. J(Φ T P+Ψ T Q)J + HΠH < He(LR) P := FPF, Q := FQF W H := B, L := A D C I nz 1 Z U, V (29), U, V Z. 38

50 , ω [ω a,ω b ], R = [ I 2np e j(ωa+ω b)/2 I 2np ], Φ P<He LMI. ([ ] A W [ ] ) I 2np I 2np I 2np T s.1[s]. (17)... G o (z) = z , M d(z) = τ h T,.75[s]( ω=8.38[rad/sec]). MRC, ζ = T/T s =75. K R (z),. FIRRC K R (z) = 4.78z 4.3 z G o M d (z) K F (z) = 3.33z 3.33 z

51 . FIR X(z) 4, (25), γ 1 =1.2, γ 3 =.5, ω l =7.96, ω h =8.8, ω d = FIR X k (k =1,, 4). IIRRC, 26 F I (z)., magnitude [db] frequency [rad/s] 26 F I (z) F I (z) = (z z ) z z +1., (26), 3.1. γ 1 =.93, γ 2 =.82 ω l =7.96, ω h =8.8, ω d =

52 , K I (z). K I (z) = K 3 i=1 (z z i) 3 i=1 (z p j) , FIRRC IIRRC MRC magnitude [db] 1 MRC IIRRC -4-5 FIRRC frequency [rad/s] 27 τ h v. MRC 75, FIRRC 41, IIRRC 5. 2 K I (z) K p 1.49 z 1.39 p 2, p 3.15 ±.33j z 2, z 3.88 ±.9j 41

53 3.7.2,,. ω 1 =8.38, [15] ω 2 = τ h (t) =1.4 sin ωt +3.8, PPC, 2 δ =

54 velocity deviation [km/h] PPC MRC, FIRRC, IIRRC time [s] (a) Nominal frequency case (ω = ω 1 ) velocity deviation [km/h] PPC MRC FIRRC, IIRRC time [s] (b) Perturbed frequency case (ω = ω 2 ) 28 43

55 (a) (ω = ω 1 ), (b) (ω = ω 2 )., MRC, FIRRC, IIRRC,., PPC.,,, MRC. FIRRC IIRRC,.,. ( 29). 44

56 velocity deviation [km/h] PPC MRC IIRRC time [s] (a) Nominal frequency case (ω = ω 1 ) velocity deviation [km/h] PPC MRC IIRRC time [s] (b) Perturbed frequency case (ω = ω 2 ) 29 ( ) 45

57 リミッタを挿入したことで, MRC, IIRRC ともに変動除去性能は低下しているが, 速度偏差の大小関係においてはリミッタなしの場合と同様の傾向を示している. なお, 前述の結果より FIRRC と IIRRC がほぼ同等の性能を有することが確認できたので, FIRRC の結果を省略している実験系の構成人間が長時間にわたって一定のペダリング動作を続けることは難しく, また実験ごとの再現性を確保することも困難である. そこで本研究では実験時のペダリングトルクを正確に発生させるためにモータ駆動されるペダリング機器図 3 を製作した. 図 3 ペダリング機器走行中の負荷は競技用のトレーニング機器図 31 を装着することで再現した. 46

58 31,,. [54]

59 , PPC (a), (b). MRC (a), (b). IIRRC (a) (b),. velocity deviation [km/h] time [s] velocity deviation [km/h] time [s] (a) Nominal case (b) Perturbed case 33 (PPC) 48

60 velocity deviation [km/h] velocity deviation [km/h] time [s] time [s] (a) Nominal case (b) Perturbed case 34 (MRC) velocity deviation [km/h] velocity deviation [km/h] time [s] time [s] (a) Nominal case (b) Perturbed case 35 (IIRRC), ( )

61 12 1 distance [m] PPC 2 MRC IIRRC time [s] 36.5 voltage drop [V] PPC MRC IIRRC time [s] 37 5

62 ,..,,.,. 36,,. 37 MRC, IIRRC. 3.8,.,.,,. 51

63 4. 4.1, [28],.,. [26],.,., 3.,.. 4.2,. 4.3, 3,., , 38, [26, 55]. x s

64 , λ s [m], O s G s ξ G [m], v s [m/s], c s [m/s]. v s = 1 t c, t v s (τ)dτ ξ G =(c s + v s )t (3)., X w, ρ, g, ξ w, k w, d s, b s, s s, S(x S ), X w (ξ G )= ρgξ w k w e k ds 2 bs s s S(x s )W (ξ G,x S )dx s (31)., W (ξ G,x s ) = sin 2π(ξ G + x s )/λ s (32)., (32) (3), W (ξ G,x s ) = sin 2π (c S + v s ) t cos 2π x s λ s, (31),., λ s +cos2π (c s + v s ) X w (ξ G )=α sin ω t + β cos ω t λ s t sin 2π x s λ s α = ρgξ w k w e k ds 2 β = ρgξ w k w e k d s 2 bs s s bs s s S(x s )cos2π x s λ s dx s, S(x s ) sin 2π x s λ s dx s,., ω =2π c s + v s λ s X w = γ s sin(ω t + δ) 53

65 γ s = α 2 + β 2,δ=tan 1 α β., X w 2π/ω., n s, t p, D p, K T (v s )., T h (v s,n s ). T h (v s,n s )=(1 t p )ρd 4 pk T (v s )n 2 s, n s h h n 2 s (33) [56], k T h h T h (v s,h) kk T (v s )h. M s, R s (v s ), (31) (33),. M s dv dt + R s(v s )=γ s sin(ω t + δ)+kk T (v s )h k = kk T ( v s ),R s (v s )= R s v s,. M s dv s dt + R s v s (t) =γ s sin(ω t + δ)+ kh (34) , G o (s) { } v s = G o (s) γ s sin(ω t + δ)+ kh (35). (35) 3.6 IIRRC 39., 39 r s., X w 54

66 v s G c (s) 39 G c (s) = G o (s) 1+KF kg o (s).,. G c (jω) γ 1, ω [ω l,ω h ] [ω l,ω h ]. 3., 3 MCR(Maximum Continuous Rating)., λ s 3[m], ξ w 2.4[m], c s 3 mass [kg] length depth breadth propeller diameter MCR of the engine speed 15 [m] 13.5 [m] 27.2 [m] 5.4 [m] 165 [rpm] 5.5[m/s]., R s , v s 2.5[m/s].,,, 37.5[s] (ω =.17[rad/s]). 55

67 G o (jω ) , ω ±5[%] G c (jω) G o (jω ) 4[dB].,. F s (s) = 8s +.56 s F s (s), 4[dB]. K(s). ±5[%] ω l =.15, ω h =.18. G o (jω ) γ 1 = KYP,., K(s). K(s) = (s +3.71)(s +.53)(s +.26) (s +.1)(s s ) X w v s G c (s) 4. 56

68 12 magnitude [db] frequency [rad/s] 4 G c (s) , 41,.,., , 42,., (33). d s, f s f s = d s /n 2 s.,. f c f o f c f o = , 2[%]. 57

69 velocity [m/s] constant engine speed proposed method time [s] constant engine speed proposed method engine speed [rpm] time [s] 42 58

70 4.4,.,,.,. 59

71 5. 5.1,.,,,., KYP,.,,.,,,.,,.,.,,. 5.2,.,.,,,, [57]

72 (a) (b) (c) (d) 43 61

73 [58], v D s v α s (D s = 4.7 ± 1.,α s =2.95 ±.49).,,., 3.5., 2, 3., 3., v = v + v a sin ωt., T. 1 T T D s v 3 dt = D s T = D s T = D s T = D s T T T T (v + v a sin ωt) 3 dt (v 3 +3vv 2 a sin ωt +3v va 2 sin 2 ωt + va 3 sin 3 ωt)dt {v 3 +3v 2v a sin ωt +3v 2v ( 12 a 12 ) cos 2ωt ( 3 +v 3 4 sin ωt 1 ) } sin 3ωt dt 4 {v 3T 3v 2v a cos ωt +3v 2v ( 12 a T 14 ) sin 2ωT +v 3 ( 3 1 cos ωt cos 3ωT ) } 12, T (T =2π/ω).. D s (v v v 2 a,., 3,. 62 )

74 ,. 8.,,.,,,.,, , 8.38[rad/sec],.,., B,., (a) 44(b) 45., 45, (1 ν (1/ω) µ ). 63

75 spectral density [V /(rad/s)] frequency [rad/s] (a) spectral density [V /(rad/s)] frequency [rad/s] (b) A spectral density [V /(rad/s)] frequency [rad/s] (c) B spectral density [V /(rad/s)] frequency [rad/s] (d) C 44 64

76 spectral density [V /(rad/s)] frequency [rad/s] (a) spectral density [V /(rad/s)] frequency [rad/s] (b) A 45, µ =,,., µ =1,, 1/f [59]., ν 65

77 . A, B, C 4, 1/f., B, µ ν. 4 µ ν A B C [15],., 1/f., 1 (µ =) µ =2., 1/f -1[dB/dec],., 1/f [6, 61].,,,.,,,. [62],.,. 66

78 ,,.,. 1,,..,.,..,.,.,.,.. 1,.,..,,..,.,..,,.,.,. 67

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( ) NAIST-IS-MT0851087 2010 3 17 ( ) , Ecological Economical,.,.,,.,,, FIR.,,,.,,,.,,, FIR,, NAIST-IS- MT0851087, 2010 3 17. i Energy Efficiency of Power Assisting Control Methods for Electric Bicycles Kazuyoshi