NAIST-IS-DD16116 213 3 15
( )
,.,,,.,,,,.,.,,.,.,.,,.,,., (FIR), Kalman-Yakubovich- Popov,, (IIR),,, NAIST-IS- DD16116, 213 3 15. i
.,,.,.,.,.,,.,,, Kalman-Yakubovich-Popov, ii
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
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
1. 1 1.1... 1 1.2... 2 1.3... 3 1.4... 4 2. 5 2.1... 5 2.1.1... 5 2.1.2... 6 2.1.3... 9 2.2 KYP... 11 2.2.1 KYP... 11 2.2.2 KYP. 12 2.2.3 KYP... 13 2.2.4 KYP... 18 3. 2 3.1... 2 3.2... 2 3.3... 22 3.4... 26 3.5... 27 3.6... 3 3.6.1... 31 3.6.2 FIR... 33 3.6.3 IIR... 35 3.7... 39 v
3.7.1... 39 3.7.2... 42 3.7.3... 46 3.7.4... 48 3.8... 51 4. 52 4.1... 52 4.2... 52 4.3... 54 4.3.1... 54 4.3.2... 57 4.4... 59 5. 6 5.1... 6 5.2... 6 5.3... 63 67 68 75 A. Shur 75 vi
1... 5 2... 6 3... 7 4 (1)... 8 5... 9 6... 9 7 (2)... 1 8 (2)... 1 9... 14 1... 19 11... 19 12... 22 13... 23 14... 23 15... 24 16... 25 17... 26 18... 27 19... 3 2... 31 21... 32 22... 32 23 FIR... 33 24 K RC (z)... 34 25 IIR... 35 26 F I (z)... 4 27 τ h v... 41 28... 43 vii
29 ( )... 45 3... 46 31... 47 32... 47 33 (PPC)... 48 34 (MRC)... 49 35 (IIRRC)... 49 36... 5 37... 5 38... 52 39... 55 4 G c (s)... 57 41... 58 42... 58 43... 61 44... 64 45... 65 1 Ψ... 15 2 K I (z)... 41 3... 55 4... 66 viii
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
1. 1.1 27 65 21% [1].,.,,,,., [2], [3],., [4] [5], [6], - [7] HAL(Hybrid Assistive Limb)[8].,,. HAL.,,.,., 3. (a) (b) (c),. 1
(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
, [17]., [18],,,.,,.. [19],, V.,,, [2].,,. 1.3 1.1,., /,,.,.,.,, 2 3 [21], 1993, [22, 23, 24].,,.,. 3
,.,,.,,,.,,.,.,. CO 2 [25, 26, 27, 28, 29],.,,.,,. 1.4. 2, Kalman-Yakubovich-Popov.,. 3.,.,,.,,. 4,,. 5. 4
2. 2.1, [3].,.,,.,,.,,.,. 2.1.1 [31, 32]. 1., r, y, K(s), G(s). 1 2.1 ( ) 1, y r, (KG(s)). 5
,, 1/s., T, T -., [31],., 1 e st 1., [33].. 2.1.2 2., 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
., 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] -5 1-15 -2-25 1-1 1 1 1 1 2 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
. e(s), e(s) =(1 G(s))e st e(s)+d e (s) (1)., D e (s) =(1 e st )r(s)., (1), 2 4. 4 (1) [34],. 2.2 [3] G(s), 1 G(s) < 1 (2), e L 2, L 2..,,.,., 3,. 8
2.1.3 2. T =1, r(t) = sin 2πt +1, G(s) =.9s +.8 s +1., 1 G(s) =.2 (2). 5 6. 2.5 2. 1.5 1..5 -.5 2 4 6 time [s] 8 1 5 2. 1.5 error 1..5 2 4 6 8 1 time [s] 6 9
5 r, y.,.,., G(s) = 7.5s +7.5 s 2 +1s +7.5, 1 G(s) =1 (2).,. 2.5 2. 1.5 1..5 -.5 2 4 6 8 1 time [s] 7 (2) error 2. 1.5 1..5 -.5 2 4 6 time [s] 8 1 8 (2) 1
,,.,, Linear Matrix Inequality: LMI Kalman-Yakubovich-Popov KYP ). 2.2 KYP KYP [35, 36]., KYP.,,., KYP KYP [37]., KYP KYP. 2.2.1 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
(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], 2. 1. (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. 2.2.2 KYP (6) [38]. Λ c Λ d, λ. Λ {c,d} := { λ C σ(λ, Φ {c,d} )=} (8) 12
, Λ c Λ d Φ {c,d} ] ] Φ c = [ 1 1, Φ d := [ 1 1., [ P ] P =Φ c P, 2. 1.. 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. 2.2.3 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
, 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
,,, 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)
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
,., 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
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. 2.2.4 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
magnitude [db] -2-4 -6.1.2.3.4.5 normalized frequency 1 5 phase [rad] 1 15 2 25 3.1.2.3.4.5 normalized frequency 11 19
3. 3.1,.,,.,.,.., 3.2, 3.3,., 3.4, 3.5., 3.6.,,., 3.7,. 3.8. 3.2,,.,.,,. 2
,..,,, [41].,.,..,.,. DC,.,,.,., 12. 21
12 3.3,., 6[deg] 12[deg],, 18[deg]. 18[deg],.,. [42]. 13. 22
13, 14., [N] force pedaling 6 5 4 3 2 1 1 2 3 4 5 time [s] 14.,., [15], 23
, 13 8 ± 1[rpm] 5,.,,. 14.,.,,,,,. 15, 16. 15 24
angle [deg] 2 15 1 5 1 2 3 time [s] 4 5 15 (a) θ 1 angle [deg] angle [deg] 1 5 1 2 3 time [s] 4 5 (b) θ 2 1 95 9 85 8 1 2 3 time [s] 4 5 (c) θ 3 16 16,. 25
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
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
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
, 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
, 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
2,. PPC ( ) G c (s) =(1+δ)G o (s)., G o (jω k ) (2). 3.1,.. 3.6.1 T τ h, [45]. 2 2. 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
., 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 2 + + z +1), z =1. (Modified-type Repetitive Control: MRC ) 22. 22 32
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
, 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
, 3.1,. [ω l,ω h ], ω d. KYP,. 3.6.3 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
. 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
. 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
, 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] 2. 1. (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
, 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 3.7 3.7.1 T s.1[s]. (17)... G o (z) = 7.15 1 2 z 9.11 1, M d(z) =2.85 1 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 9.5 1 1 G o M d (z) K F (z) = 3.33z 3.33 z 9.5 1 1 39
. FIR X(z) 4, (25), γ 1 =1.2, γ 3 =.5, ω l =7.96, ω h =8.8, ω d =3.14 1 2 2.3. FIR X k (k =1,, 4). IIRRC, 26 F I (z)., magnitude [db] 1 8 6 4 2-2 -4-6 -8-1 1-1 1 1 1 1 2 frequency [rad/s] 26 F I (z) F I (z) = 2 1 4 (z 2 9.95 1 1 z +2.66 1 3 ) z 2 1.99z +1., (26), 3.1. γ 1 =.93, γ 2 =.82 ω l =7.96, ω h =8.8, ω d =2 1 1 4
, K I (z). K I (z) = K 3 i=1 (z z i) 3 i=1 (z p j) 2. 27., FIRRC IIRRC MRC magnitude [db] 1 MRC 1-2 -3 IIRRC -4-5 FIRRC -6-7 -8-9 -1 1 1 1 1 1 1 2 frequency [rad/s] 27 τ h v. MRC 75, FIRRC 41, IIRRC 5. 2 K I (z) K 1.38 1 4 p 1.49 z 1.39 p 2, p 3.15 ±.33j z 2, z 3.88 ±.9j 41
3.7.2,,. ω 1 =8.38, [15] ω 2 =8.2 2... τ h (t) =1.4 sin ωt +3.8, PPC, 2 δ =.4. 28. 42
velocity deviation [km/h] 1.5 1..5 -.5-1. -1.5 2 PPC MRC, FIRRC, IIRRC 21 22 23 24 25 time [s] (a) Nominal frequency case (ω = ω 1 ) velocity deviation [km/h] 1.5 1..5 -.5-1. -1.5 2 PPC MRC FIRRC, IIRRC 21 22 23 24 25 time [s] (b) Perturbed frequency case (ω = ω 2 ) 28 43
(a) (ω = ω 1 ), (b) (ω = ω 2 )., MRC, FIRRC, IIRRC,., PPC.,,, MRC. FIRRC IIRRC,.,. ( 29). 44
velocity deviation [km/h] 1.5 1..5 -.5-1. -1.5 2 PPC MRC IIRRC 21 22 23 24 25 time [s] (a) Nominal frequency case (ω = ω 1 ) velocity deviation [km/h] 1.5 1..5 -.5-1. -1.5 2 PPC MRC IIRRC 21 22 23 24 25 time [s] (b) Perturbed frequency case (ω = ω 2 ) 29 ( ) 45
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. 3.7.3 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
31,,. [54]. 32. 32 47
3.7.4 33-35., PPC (a), (b). MRC (a), (b). IIRRC (a) (b),. velocity deviation [km/h] 1.5 1..5 -.5-1. 1.59-1.5 2 22 24 26 28 21 time [s] velocity deviation [km/h] 1.5 1..5 -.5-1. -1.5 2 1.45 22 24 26 28 21 time [s] (a) Nominal case (b) Perturbed case 33 (PPC) 48
1.5 1.5 velocity deviation [km/h] 1..5 -.5-1..41 velocity deviation [km/h] 1..5 -.5-1. 1.87-1.5 2 22 24 26 28 21 time [s] -1.5 2 22 24 26 28 21 time [s] (a) Nominal case (b) Perturbed case 34 (MRC) 1.5 1.5 velocity deviation [km/h] 1..5 -.5-1..65 velocity deviation [km/h] 1..5 -.5-1..98-1.5 2 22 24 26 28 21 time [s] -1.5 2 22 24 26 28 21 time [s] (a) Nominal case (b) Perturbed case 35 (IIRRC), ( ) 36 37. 49
12 1 distance [m] 8 6 4 PPC 2 MRC IIRRC 2 3 4 5 6 7 time [s] 36.5 voltage drop [V].4.3.2.1 PPC MRC IIRRC 2 3 4 5 6 7 time [s] 37 5
,..,,.,. 36,,. 37 MRC, IIRRC. 3.8,.,.,,. 51
4. 4.1, [28],.,. [26],.,., 3.,.. 4.2,. 4.3, 3,., 4.4. 4.2, 38, [26, 55]. x s. 38 52
, 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
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) 4.3 4.3.1, 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
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 2.47 1 6 [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 177 1 3, v s 2.5[m/s].,,, 37.5[s] (ω =.17[rad/s]). 55
G o (jω ) 2.26 1 6., ω ±5[%] G c (jω) G o (jω ) 4[dB].,. F s (s) = 8s +.56 s 2 +.3 F s (s), 4[dB]. K(s). ±5[%] ω l =.15, ω h =.18. G o (jω ) γ 1 =2 1 8. KYP,., K(s). K(s) = 2.55 14 (s +3.71)(s +.53)(s +.26) (s +.1)(s 2 +15.4s +156.4) X w v s G c (s) 4. 56
12 magnitude [db] 14 16 18 2 22 1 2 1 1 1 frequency [rad/s] 4 G c (s) 4.3.2 41., 41,.,., 42. 41, 42,., (33). d s, f s f s = d s /n 2 s.,. f c f o f c f o =12.3 1 2, 2[%]. 57
4. 3.5 velocity [m/s] 3. 2.5 2. 1.5 1..5 constant engine speed proposed method 8 84 88 92 96 1 time [s] 41 8 7 constant engine speed proposed method engine speed [rpm] 6 5 4 3 2 1 8 84 88 92 96 1 time [s] 42 58
4.4,.,,.,. 59
5. 5.1,.,,,., KYP,.,,.,,,.,,.,.,,. 5.2,.,.,,,, [57]. 43. 6
(a) (b) (c) (d) 43 61
[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 + 4 12 cos 3ωT + 3 4 1 ) } 12, T (T =2π/ω).. D s (v 3 + 3 2 v v 2 a,., 3,. 62 )
,. 8.,,.,,,.,, -. 5.3.. 44., 8.38[rad/sec],.,., B,., 44. 44(a) 44(b) 45., 45, (1 ν (1/ω) µ ). 63
spectral density [V /(rad/s)] 2 1 2 3 2 1 1 2 3 4 5 frequency [rad/s] (a) spectral density [V /(rad/s)] 2 1 2 3 2 1 1 2 3 4 5 frequency [rad/s] (b) A spectral density [V /(rad/s)] 2 1 2 3 2 1 1 2 3 4 5 frequency [rad/s] (c) B spectral density [V /(rad/s)] 2 1 2 3 2 1 1 2 3 4 5 frequency [rad/s] (d) C 44 64
spectral density [V /(rad/s)] 2 1 1 1 1 1 1 2 1 3 1 4 1 1 1 1 2 frequency [rad/s] (a) spectral density [V /(rad/s)] 2 1 1 1 1 1 1 2 1 3 1 4 1 1 1 1 2 frequency [rad/s] (b) A 45, µ =,,., µ =1,, 1/f [59]., ν 65
. A, B, C 4, 1/f., B, µ ν. 4 µ ν.29-1.76 A 1.4 -.5 B.84 -.61 C.98 -.22 [15],., 1/f., 1 (µ =) µ =2., 1/f -1[dB/dec],., 1/f [6, 61].,,,.,,,. [62],.,. 66
,,.,. 1,,..,.,..,.,.,.,.. 1,.,..,,..,.,..,,.,.,. 67
[1] 24 1 1 [2] T. Miyoshi, K. Suzuki and K. Terashima: Development of Five-Degree-of- Freedom Wire Suspension Power-Assisted System Using Linear Cylinders; in Proc. 211 IEEE International Conference on Robotics and Automation, pp. 342-348 (211) [3],,,, Vol. 76, No. 767, pp. 178-1787 (21) [4],, : 3, 1A2-B26 (21) [5] T. Ishida, T. Kiyama, K. Osuka, G. Shirogauchi, R. Oya and H. Fujimoto: Movement analysis of power-assistive machinery with high strengthamplification; in Proc. SICE Annual Conference 21(SICE 21), pp. 222-225 (21) [6],,,,, Vol. 45, No. 12, pp. 638-645 (29) [7], -, Vol. 38, No. 3, pp. 316-323 (22) [8] K. Suzuki, G. Mito, H. Kawamoto, Y. Hasegawa and Y. Sankai: Intentionbased walking support for paraplegia patients with robot suit HAL; Advanced Robotics, Vol. 21, No. 12, pp. 1441-1469 (27) 68
[9],,,,, D, Vol. 116, No. 3, pp. 233-244 (1996) [1],, 211, pp. 85-88 (211) [11], (28) [12], 22, IIC-1-19, pp. 23-28 (21) [13] / /, Vol. 49, No. 1, pp. 393-398 (25) [14] K. Hirata: On Internal Stabilizing Mechanism of Passive Dynamic Walking; SICE Journal of Control, Measurement, and System Integration, Vol. 4, No. 1, pp. 29-36 (211) [15] D. J. Sanderson and A. T. Amoroso: The influence of seat height on the mechanical function of the triceps surae muscles during steady-rate cycling; Journal of Electromyography and Kinesiology, Vol. 19, No. 6, pp. 465-471 (29) [16] A. P. Seyranian and A. O. Belyakov: How to twirl a hula hoop; American Journal of Physics, Vo. 79, No. 7, pp. 712-715 (211) [17] J. Nishizaki, S. Nakaura and M. Sampei: Modeling and Control of Hula- Hoop System; in Proc. Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pp. 4125-413 (29) [18],, 56, pp. 625-626 (212) 69
[19] R. Sepulchre and M. Gerard: Stabilization of periodic orbits in a wedge billiard; in Proc. 42th IEEE Conference on Decision and Control, pp. 1568-1573 (23) [2] R. Ronsse, P. Lefevre and R. Sepulchre: Sensorless Stabilization of Bounce Juggling; IEEE Transactions on Robotics, Vol. 22, No. 1, pp. 147-159 (26) [21], 21 1 16, be on Saturday [22],,,, PAS211 YAMAHA MOTOR TECHNICAL REVIEW, No. 47, pp. 4-6 (211) [23],,, Vol. 69, No. 68, pp. 195-2 (23) [24],,, 54, pp. 182-185 (211) [25], Vol. 76, No. 1, pp. 37-4 (21) [26],, 55, pp. 291-292 (211) [27],,, 23 (211) [28],,, -, Vol. 44, No. 1, pp. 92-94 (29) 7
[29] J. A. Heine, R. N. Schane and J. M. Bielefeld: Minimizing Fuel Consumption in Ocean Vessels; IEEE Transactions on Industrial Electronics and Control Instrumentation, Vol. IECI-2, No. 2, pp. 44-46 (1973) [3],,,, (1989) [31] B. A. Francis and W. M. Wonham: The Internal Model Principle for Linear Multivariable Regulators; Applied Mathematics & Optimization, Vol. 2, No. 2, pp. 17-194 (1975) [32] (22) [33] Y. Yamamoto and S. Hara: Relationships Between Internal and External Stability for Infinite-Dimensional Systems with Applications to a Servo Problem; IEEE Transactions on Automatic Control, Vol. 33, No. 11, pp. 144-152 (1988) [34] S. Skogestad and I. Postlethwaite: Multivariable Feedback Control Analysis and Design Second Edition; WILEY (23) [35] A. Rantzer: On the Kalman-Yakubovich-Popov lemma; Systems & Control Letters, Vol. 28, No. 1, pp. 7-1 (1996) [36] LMI, (212) [37] T. Iwasaki and S. Hara: Generalized KYP Lemma: Unified Frequency Domain Inequalities With Design Applications; IEEE Transactions on Automatic Control, Vol. 5, No. 1, pp. 41-59 (25) [38] KYP, Vol. 44, No. 8, pp. 534-539 (25) [39] LMI, (1997) 71
[4] Z. Liu and L. Vandenberghe: http://www.ee.ucla.edu/zhang/software/gkypsdp [41], E, Vol. 121, No. 11, pp. 599-66 (21) [42] D. G. Wilson: Bicycling Science third edition, The MIT Press [43] [44] [45] 26, (27) [46] G. Pipeleers, B. Demeulenaere, J. De Schutter and J.Swevers: Generalized repetitive control: Better performance with less memory; in Proc. 1th IEEE International Workshop on Advanced Motion Control, pp. 14-19 (28) [47],,,, Vol. 29, No. 11, pp. 1311-1319 (1993) [48] FIR 28, (29) [49] M. Steinbuch: Repetitivecontrol for systems with uncertain period-time; Automatica, Vol. 38, No. 12, pp. 213-219 (22) [5] M. Tomizuka: Zero phase error tracking algorithm for digital control; Journal of Dynamic Systems, Measurement, and Control, Vol. 19, No. 1, pp. 65-68 (1987) [51],,, (1993) 72
[52] T. Iwasaki and S. Hara: Dynamic Output Feedback Synthesis with Frequency Domain Specifications; 16th IFAC World Congress (25) [53] T. Iwasaki and R. E. Skelton: All Controllers for the General H Control Problem: LMI Existence Conditions and State Space Formulas; Automatica, Vol. 3, No. 8, pp. 137-1317 (1994) [54],, (1979) [55], No. 152, pp. 192-21 (1983) [56] O. Bondarenko, M. Kashiwagi, S. Naito: Dynamics of Diesel Engine in the Framework of Ship Propulsion Plant;, No. 8, pp. 335-338 (29) [57] - -, (27) [58] F. Celentano, G. Cortili, P. E. Prampero and P. Cerretelli: Mechanical aspects of rowing; Journal of Applied Physiology, Vol. 36, No. 6, pp. 642-647 (1974) [59],, 1/f 51, pp. 64-67 (28) [6], 1/f, D-II, Vol. J8-D-II, No. 1, pp. 2831-284 (1997) [61] B. Kaulakys, J. Ruseckas, V. Gontis and M. Alaburda: Nonlinear stochastic models of 1/f noise and power-law distributions; Physica A, Vol. 365, pp. 217-221 (26) 73
[62],, - : 2 12 (SI211), pp. 1825-1828 (211) 74
A. Shur A.1 [36, 39] [ A11 A 12 A = A T 12 A 22 ], 3. 1. A> 2. A 22 > A 11 A 12 A 1 22 A T 12 > 3. A 11 > A 22 A T 12P 1 11 A 12 > 75
1.,,,, Vol. 25, No. 2, pp. 28-38 (212) 2.,, Vol. 49, No. 2, pp. 313-315 (213) ( ) 1. Kazuyoshi Hatada and Kentaro Hirata: Energy-Efficient Power Assist Control for Periodic Motions; in Proc. SICE Annual Conference 21(SICE 21), pp. 24-29 (21) 2. Kazuyoshi Hatada and Kentaro Hirata: Energy-Efficient Power Assisting Methods for Periodic Motions and its Experimental Verification; in the 212 IEEE International Conference on Industrial Technology(ICIT212), pp. 869-874 (212) ( ) 1., 1 (SI29), pp. 33-36 (29) 2., 55 (SCI 11), pp. 147-148 (211) 3., 12, P162 (212) 76
4.,, 56 (SCI 12), pp. 623-624 (212) 1. Kiminao Kogiso, Makoto Noguchi, Kazuyoshi Hatada, Naoki Kida, Naofumi Hirade and Kenji Sugimoto: Experimental Validation of Switching Strategy for Tracking Control with Collision Avoidance in Non-Cooperative Situation Using Toy Model Cars; SICE Journal of Control, Measurement, and System Integration, Vol. 3, No. 4, pp. 229-236 (21) ( ) 1. Kentaro Hirata, Mayumi Tomida and Kazuyoshi Hatada: Gain Scheduling Control Experiment of Balancing Transformer Robot using LEGO Mindstorms; in International Conference on Design and Modeling in Science, Education, and Technology(DeMset211), DM42NY (211) ( ) 1. Kiminao Kogiso, Makoto Noguchi, Kazuyoshi Hatada, Naoki Kida and Naofumi Hirade; Experimental Validation of Tracking Control with Collision Avoidance Using Rc Model Cars; SIAM Conference on Control and Its Applications(CT9), PP(29) 1. 29,,, 21 3 77
2. SI29,,, 21 3 78