2017 3

217 3

1 1 1.1................................... 1 1.2........................ 3 1.2.1............................. 3 1.2.2.............................. 4 1.3................................... 5 2 9 2.1..................................... 9 2.2.......... 9 2.3................... 1 2.3.1.................... 11 2.3.2..................... 12 2.3.3....................... 13 2.4....................................... 14 3 17 3.1..................................... 17 3.2................................... 17 3.2.1.............................. 17 3.2.2.................. 22 3.3.............................. 24 3.4.................. 25 3.4.1 2................ 26 3.4.2........................ 27 3.4.3............................ 3 3.4.4................................. 3 3.5....................................... 33 i

4 39 4.1..................................... 39 4.2....................... 4 4.2.1......................... 4 4.2.2..................... 41 4.2.3........................ 43 4.3................................... 43 4.3.1.......... 43 4.3.2....................... 46 4.3.3................................. 49 4.4....................................... 5 5 53 5.1..................................... 53 5.2 2................. 54 5.3........................ 56 5.4.... 58 5.5....................................... 61 6 63 6.1..................................... 63 6.2............................... 64 6.2.1..................... 64 6.2.2.................... 66 6.3....................... 67 6.3.1..................... 67 6.3.2 1.............. 7 6.3.3 2............. 71 6.3.4 3............ 72 6.4..................... 73 6.4.1.................................. 74 6.4.2 1................... 74 6.4.3 2.................. 76 6.4.4 3................. 78 6.4.5........................... 8 6.5....................................... 8 ii

7 83 7.1.............................. 83 7.2................................ 84 87 88 93 iii

1.1................................... 6 2.1........................... 9 2.2....................... 1 2.3.......................... 12 2.4........................ 13 2.5.......................... 14 3.1................................ 2 3.2 2..................... 26 3.3 FSC FFSC........................... 29 3.4 FSC FFSC.................. 29 3.5 6 % +6 % ( )...................................... 31 3.6 FSC FFSC +6 % 32 3.7 FSC FFSC............................... 34 3.8 FSC FFSC +6 %................................... 34 3.9 FSC FFSC +6 % e........................... 35 3.1 T = 25 C................... 36 3.11 T = 55 C................... 36 3.12 T = 55 C............... 37 4.1........................... 41 4.2............................... 42 4.3............... 43 4.4 (.1 mm )............................. 45 4.5 (1 mm ).............................. 46 4.6 (1 mm )............................. 47 v

4.7 (1 mm )......................... 48 4.8............................ 49 4.9 (1 mm )........................ 51 4.1 (1 mm ).................... 51 4.11 (1 mm )........................ 52 4.12 (1 mm ).................... 52 5.1 2............. 54 5.2.................. 57 5.3..................... 57 5.4............................... 58 5.5..................... 59 5.6....................... 59 5.7 /............. 6 5.8 J........................... 6 6.1.................. 67 6.2............................... 69 6.3............................ 69 6.4...................... 75 6.5 ( ±1µm )..... 76 6.6 1 ( ±1µm ).... 77 6.7 2 ( ±1µm ).... 78 6.8 3 ( 2µm ) 79 6.9...................... 8 vi

2.1...................... 12 2.2..................... 13 2.3...................... 14 vii

1 1.1 196 [1] [2 5] CPU 1996 CPU [6] 2 fθ CO2 UV-YAG 2 CO2 UV-YAG CO2 UV-YAG CO2 UV-YAG 1

/ [7] PTP(Point to Point) CP(Continuous Path) 2 PTP CP PTP 1999 [8] 1.5 4 [9] [1] PI DSP 22 [11 14] [15 17] 1mm 1999 3ms.5ms ±5µm ±3µm [7, 8] 2

[18 2] 1.2 1.2.1 2 2 [15] LMI Linear Matrix Inequality [16, 21 23] [21] [22] DC [16,23] 3

[24 29] [3] [31] [32,33] 2 [19] PSD(Position Sensitive Detector) [34,35] [36] [37] [2] 1.2.2 1.1 (i) (ii) 2 2 (i) PWM 2 ±96V ( ) 24A 4

[15] (LMI) LMI [38] Matlab Compiler [39] LMI 3 (ii) 1.3 7 1.1 1 2 5

1.1: 3 (LMI) LMI 4 5 6

6 3 (i) (ii) (iii) (ii) 7 7

2 2.1 2.2 i u y s 2.1: 9

CPU 2.1 2 f θ CO2 UV-YAG 2 CO2 UV-YAG CO2 UV-YAG 355nm CO2 1.6µ m UV-YAG 2.3 2.2: 1

2.2 DSP D/A R S i re f i V M 2.3.1 i re f y enc 2.3 [Hz/unit] 1 1 Hz/unit 1/(22.2τ) τ 2.3 2 2 µs P enc (s) = P mech (s) e τ d s (2.1), P mech P n P 1, P 2 P mech (s) = P n (s) + P 1 (s) + P 2 (s) (2.2) P n (s) = k e (2.3) s 2 P i (s) = k ei s 2 + 2ζ ei ω ei s + ω 2 ei (i = 1, 2) (2.4) P enc (s) 2.3 2.1 11

4 6 Experimental Model Gain [db] 8 1 12 14 1 1 1 9 Phase [deg] 18 27 36 45 1 1 1 Frequency [Hz/unit] 2.3: 2.1: k e 17.5 1 3 k e1 2.56 1 3 ζ e1 3.85 1 3 k e2 17. 1 3 ζ e2 8.41 1 3 ω e2 /ω e1 2.14 2.3.2 i re f y enc 2.4 y mir PSD(Position Sensitive Detector) 2.4 2 2 P mir (s) = k m s 2 2 + k mi s 2 + 2ζ mi ω mi s + ω 2 e τ d s mi i=1 (2.5) 12

4 5 Encoder Mirror Gain[dB] 6 7 8 9 1 11 12 13 14 1 1 1 Frequency[Hz/unit] 2.4: k ω ni ±6% 2.2 2.2: k m 17.5 1 3 k m1 24.4 1 3 ζ m1 3.85 1 3 ω m1 /ω e1 1. k m2 5.68 1 3 ζ m2 8.41 1 3 ω m2 /ω e1 2.14 2.3.3 13

y v, P v (s) = k v s 2 + 2ζ v ω v s + ω 2 v (2.6) ω v ±5% (2.1) (2.6) P enc (s) P v (s) 2.5 (2.6) 2.3 Gain [db] 5 1 15 P enc (s) P v (s) 2 1 1 1 1 1 Phase [deg] 9 18 27 36 1 1 1 1 1 Frequency [Hz/unit] 2.5: 2.3: k v 1.6 1 2 ζ v 13.4 1 3 ω v /ω e1.37 2.4 14

15

3 3.1 2 [4, 41] Final-State Control: FSC 3.2 3.2.1 [15, 31] [42, 43] m 1 x[k + 1] = Ax[k] + Bu[k] (3.1) y[k] = Cx[k] (3.2) x[] x[n] N m u[k] (3.1) k k =, 1,, N 1 x[n] = A N x[] + ΣU (3.3) Σ = [ A N 1 B A N 2 B B ] (3.4) U = [ u[] u[1] u[n 1] ] T (3.5) 17

(A, B) N m (3.4) Σ Σ (3.3) U N m Σ U (3.3) U J = U T QU, Q > (3.6) 2λ J = U T QU + 2λ(X ΣU) X := x[n] A N x[] Q J Q > Q 1 (X ΣU) = (3.8) J U = 2QU 2ΣT λ T = (3.7) U = Q 1 Σ T λ T (3.8) X ΣQ 1 Σ T λ T = (3.9) (A, B) Σ ΣQ 1 Σ T (3.8) (3.1) λ T = (ΣQ 1 Σ T ) 1 X (3.1) U = Q 1 Σ T (ΣQ 1 Σ T ) 1 (x[n] A N x[]) (3.11) Q Q = I J N 1 J = U T U = u 2 [k] 2 3.1 1/(z 1) u c [k] u[k] 2 u c [k] k= 18

3.1 1 1 m P c (s) ẋ c (t) = A c x c (t) + B c u c (t) (3.12) y(t) = C c x c (t) (3.13) P c (s) τ P d [z] x d [k + 1] = A d x d [k] + B d u c [k] (3.14) y[k] = C d x d [k] (3.15) A d := e A cτ B d := τ y(kτ) u c (t) eact B c dt C d := C c x d [k] = x c (kτ) y[k] = u c (kτ + θ) = u c [k], θ < τ (3.16) (A c, B c ) (A d, B d ) P[z] x[k + 1] = Ax[k] + Bu[k] (3.17) y[k] = Cx[k] (3.18) x[k] := [ x T d [k], ut c [k] ] T B d A = A d 1, B = 1, C = [ C d ] (3.19) x d [] = x d [N] = x N u[k] 2 (3.17) (3.6) Q u c u c [] = u c [N] = x[] = (m+1) 1, x[n] = x N (3.2) FSC (Final-State Control) FSC 3.1 u c (t) Û c (ω) Û c (ω) = Nτ 19 u c (t)e jωt dt (3.21)

P[z] P d [z] u[k] 1 z 1 u c [k] H u c (t) P c (s) y(t) τ y[k] 3.1: ω i (i = 1,, l) N 1 l J ω = u 2 [k] + q i Û c (ω i ) 2 (3.22) k= i=1 1 FSC 2 u c (t) q i ω i (3.22) u u c (t) Û c (ω) u c (t) N 1 u c (t) = P i (t)u c [k] (3.23) k= P i (t) := 1, iτ t < (i + 1)τ, t < iτ or t (i + 1)τ (3.24) u c (t) Û c (ω) Nτ Û c (ω) = N 1 = k= kτ u c (t)e jωt dt (3.25) (k+1)τ ωτ 2 sin( = ω u c [k]e jωt dt (3.26) ) N 1 2 e jωτ/2 u c [k]e jωτk (3.27) k= Û c (ω) = Û 1 (ω) Û 2 (ω) (3.28) 2

Û 1 (ω) = Û 2 (ω) = ωτ 2 sin( ) 2 (3.29) ω N 1 u c [k]e jωτk (3.3) k= U c = [u c [],..., u c [N 1]] T N 1 Re[Û 2 (ω)] = u c [k] cos(kωτ) = S R U c (3.31) k= N 1 Im[Û 2 (ω)] = u c [k] sin(kωτ) = S I U c (3.32) k= S R (ω) = [cos(), cos(ωτ),, cos ((N 1)ωτ)] (3.33) S I (ω) = [sin(), sin(ωτ),, sin ((N 1)ωτ)] (3.34) Û 2 (ω) 2 = U T c ( S T R S R + S T I S I ) Uc (3.35) U c = Ω z U (3.36) Ω z = 1........... 1 1 (3.37) (3.22) Û c (ω) 2 = Û 1 (ω) 2 U T Ω T z (S T RS R + S T I S I )Ω z U (3.38) J w = U T Q w U (3.39) 21

l Q w = I n + q i Q U (ω i ) (3.4) i=1 Q U (ω) = Û 1 (ω) 2 Ω T z (S T RS R + S T I S I )Ω z (3.41) (3.22) U (3.6) Q Q w FFSC (Frequency-shaped Final-State Control) [43] 3.2.2 x[k] u[k] z[k] = C z x[k] + D z u[k] (3.42) z[k] z max (3.43) z[k] z max (3.3) (3.6) LMI [21] LMI [38] Matlab Compiler [39] LMI U A EQ U = b EQ (3.44) A INEQ U b INEQ (3.45) 22

U J = U T QU, Q > (3.46) A EQ A INEQ b EQ b INEQ (3.45) MATLAB quadprog [44] (3.3) (3.44) (3.6) (3.46) (3.42) k k =, 1,, N 1 z[] = C z x[] + D z u[] z[1] = C z A[] + C z Bu[] + D z u[1] (3.47). z[n 1] = C z A N 1 x[] + + D z u[n 1] Z = Φ z x[] + Ω z U (3.48) Z = [ z[],, z[n 1] ] T C z C z A Φ z =. C z A N 1 D z Ω z = C z B D z........... C z A N 2 B C z B D z (3.49) (3.5) (3.51) (3.43) U (3.45) Ω z U Z max Φ z x[] (3.52) Ω z U Z max + Φ z x[] (3.53) 23

Z max = z max [1,..., 1] T (3.3) (3.52), (3.53) J U 3.3 [45] Û A EQ Û = b EQ (3.54) A INEQ Û b INEQ (3.55) Û J = F T Û (3.56) A EQ A INEQ b EQ b INEQ (3.55) MATLAB linprog [44] (3.3) ˆΣÛ = x[n] A N x[] (3.57) ˆΣ = [ Σ N 1 ] (3.58) Û = [ U T z max ] T (3.59) N 1 = [,..., ] T N 24

(3.52),(3.53) Ω z U Z max Φ z x[] (3.6) Ω z U Z max Φ z x[] (3.61) ˆΩ z Û ˆΦ z (3.62) Ω z 1 N 1 ˆΩ z = (3.63) Ω z 1 N 1 ˆΦ z = Φ z x[] (3.64) Φ z 1 N 1 = [ 1,..., 1] T N F = [,..,, 1] T J J = F T U = z max (3.57) (3.62) J U z max 3.4 3.2 [15] [46] FFSC: Frequency shaped Final-State Control [47] [48] 25

3.2: 2 [47] [46] [48] (1) (2) 1 2 1 2 1 3.4.1 2 3.2 2 FSC Input C P mech e τ d s G r G r G r (s) = P mech (s)e τ d s (3.65) y r e C y G r G r G r e 26

3.4.2 FSC FFSC FSC [48] FFSC FSC [46] FSC FSC FSC (2.2) A n b n P mech [z] = A 1 b 1 A 2 b 2 c n c 1 c 2 (3.66) b n P n [z] = A n c n (3.67) P 1 [z] = A 1 b 1 c 1, P 2[z] = A 2 b 2 c 2 (3.68) P n [z], P 1 [z], P 2 [z] (2.3) (2.4) P n (s), P 1 (s), P 2 (s) τ P n [z], P 1 [z], P 2 [z] x n [k], x 1 [k], x 2 [k] x n [k]= x np, x 1[k]= x 1p, x 2[k]= x 2p (3.69) x nv x np, x nv x 1p, x 1v x 2p, x 2v 2 P mech x d x 1v x 2v x d [k] = [ x T n [k], x T 1 [k], xt 2 [k]] T (3.7) 27

1 z 1 FSC FSC x n [] x n [N] x d [] = x 1 [], x d[n] = x 1 [N] x 2 [] x 2 [N] (3.71) y r m x n [] = x 1 [] = x 2 [] = x n [N] = [r m ] T, x 1 [N] = x 2 [N] = N N = 79 3.3 3.4 3.3 [s/unit] 1 [A/unit] r m = 1 3.4 2 3.4 2 FFSC FFSC FFSC (3.22) ω i 2 ω i ±6% 5 q i 1 1 9 2 ±6% 5 q i 5 1 7 2 q i FFSC 3.3 3.4 2 FSC FFSC 3.3 FSC FFSC 28

2 15 FFSC FSC 1 FSC/FFSC input [A/unit] 5 5 1 15 2.2.4.6.8 1 Time [s/unit] 3.3: FSC FFSC 2 4 Gain [db] 6 8 1 FFSC FSC 12 1 1 1 Frequency [Hz/unit] 3.4: FSC FFSC 29

FFSC q i q i 3.4.3 FSC 3.2 2 FSC FSC N 6% +6% 1% y enc 3.5 [rad/unit] 1 FFSC FFSC y enc 3.5 FFSC FSC 3.4.4 FSC FSC y enc 3.7 ±.8% 1.33 s/unit 1 rad/unit µrad ±.8% 3

Position [rad/unit] 1.4 1.2 1.998 FSC.996.5 1 1.5 2 2.5 3 Position [rad/unit] 1.4 1.2 1.998 FFSC.996.5 1 1.5 2 2.5 3 Time [s/unit] 3.5: 6 % +6 % ( ) 1µrad +6% 3.6 +6% FSC y enc 3.8 3.7 1.33 s/unit 1.94 s/unit FSC FFSC FFSC FSC FFSC y enc 3.7 31

2 4 Gain [db] 6 8 1 FFSC FSC 12 1 1 1 Frequency [Hz/unit] 3.6: FSC FFSC +6 % +6% FFSC 3.6 FFSC 3.8 3.6 FSC FFSC 3.8 +6% e 5[s/unit] 3.9 FFSC FSC 1[Hz/unit] FFSC FSC 1.39 s/unit 1.44 s/unit FFSC T T = 25 C T = 55 C 2 32

T = 25 C T = 55 C 2.5% 2 5 % FSC FFSC T = 25 C 3.1 T = 55 C 3.11 FSC T = 55 C FFSC T = 55 C FSC 1.57 s/unit 2.91 s/unit FFSC 1.39 s/unit 1.53 s/unit T = 55 C e 5[s/unit] 3.12 FFSC FSC 1[Hz/unit] FFSC FSC FFSC IC FFSC 3.5 33

1.4 1.3 FFSC FSC 1.2 Position [rad/unit] 1.1 1.999.998.997.996.5 1 1.5 2 2.5 3 Time [s/unit] 3.7: FSC FFSC 1.4 1.3 FFSC FSC 1.2 Position [rad/unit] 1.1 1.999.998.997.996.5 1 1.5 2 2.5 3 Time [s/unit] 3.8: FSC FFSC +6 % 34

3.5 4 x 16 FFSC FSC 3 Power spectral density 2.5 2 1.5 1.5.5.6.7.8.9 1 1.1 1.2 1.3 1.4 1.5 Frequency [Hz/unit] 3.9: FSC FFSC +6 % e 35

1.4 1.3 FFSC FSC 1.2 Position [rad/unit] 1.1 1.999.998.997.996.5 1 1.5 2 2.5 3 Time [s/unit] 3.1: T = 25 C 1.4 1.3 FFSC FSC 1.2 Position [rad/unit] 1.1 1.999.998.997.996.5 1 1.5 2 2.5 3 Time [s/unit] 3.11: T = 55 C 36

3.5 4 x 16 FFSC FSC 3 Power spectral density 2.5 2 1.5 1.5.5.6.7.8.9 1 1.1 1.2 1.3 1.4 1.5 Frequency [Hz/unit] 3.12: T = 55 C 37

4 4.1 CPU 5 µm.1mm 1mm.1mm 1mm.1mm 1mm.1mm.5mm 3.4 PWM 2 [24 29] 39

LMI Linear Matrix Inequality [16, 21 23] [21] [22] DC [16,23] 3.4 [15] 4.2 4.2.1 i(t) i re f (t) 4.1 P c (s) K t J x p (t) x v (t) R L K e K(s) V M (t) V p K V p 4

P c (s) i re f (t) V M (t) 1 K t x v (t) 1 K(s) i(t) R+sL Js ±V P 1 s x p (t) K e 4.1: [21] i(t) i re f (t) i re f (t) 3.1 4.1 V M (t) 1. K 2. K K 4.2.2 i re f (t) i(t) 1 4.2 u c (t) = i(t) = i re f (t) 3.4 i re f (t) = i(t) V M (t) i(t) x v (t) V M (t) = Ri(t) + L di(t) dt + K e x v (t) (4.1) V M (t) V p i(t) i re f (t) i(t) = i re f (t) 4.2 41

P[z] P d [z] u[k] 1 z 1 u c [k] H u c (t) K t Js x v (t) P c (s) 1 s x p (t) τ x p [k] 4.2: (4.1) V M (t) z vol [k] V M (τk) z vol [k] = Ru c [k] + L u c[k + 1] u c [k] τ + K e x v [k] (4.2) x v [k] = x v (τk) 4.2 u[k] u c [k] (4.2) z vol [k] = Ru c [k] + L τ u[k] + K ex v [k] (4.3) z vol [k] u c [k] x v [k] u[k] 4.3 P c (s) x c (t) [ x p (t), x v (t) ] T Pd [z] x d [k] x c (τk) P[z] x[k] x[k] = x d[k] u c [k] = x x p [k] c(τk) u c [k] = x v [k] (4.4) u c [k] z vol [k] z vol [k] = C vol x[k] + D vol u[k] (4.5) C vol = [ K e R ], D vol = L τ (4.5) (3.42) z vol [k] V p 4.2 (4.5) K z vol [k] V M (t) 42

P[z] P d [z] u[k] 1 z 1 u c [k] H u c (t) K t Js P c (s) x v (t) 1 s x p (t) τ x p [k] τ L τ R K e x v [k] z vol [k] 4.3: 4.2.3 x[k] z cur [k] z vel [k] z cur [k] = C cur x[k] + D cur u[k] (4.6) z vel [k] = C vel x[k] + D vel u[k] (4.7) C cur = [ 1 ], D cur = (4.8) C vel = [ 1 ], D vel = (4.9) (3.42) z[k] (3.43) z max z[k] = [ z vol [k] z cur [k] z vel [k] ] T z max = [ z max vol z max z [k] z max cur z max vel ] T 4.3 4.3.1 43

(3.42) 4.2 P mech P mech (3.7) z vol [k] z cur [k] z vel [k] z vol [k] = C vol T x[k] + D vol u[k] z cur [k] = C cur T x[k] + D cur u[k] z vel [k] = C vel T x[k] + D vel u[k] 1 1 1 T = 1 1 1 1 12 A 5 m/s 1/2 36 V 72 V 3.1 mm 1 mm 1 mm Matlab R21a Optimization Toolbox.1 mm 1 mm 1 mm.1 mm 1 mm 1 mm N 37 73 351 (3.22) ω i q i 3.4 ±6% 5 ω i q i 1 1 9 2 ±6% 5 q i 5 1 7 44

15 1 No Voltage Constraint Voltage Constraint 72 V Voltage Constraint 36 V Voltage [V] 5 5 1 15 Current [A] 1 1 6 Velocity [m/s] 4 2 5 1 15 2 25 3 35 4 45 5 Time [sample] 4.4: (.1 mm ) 4.4 4.6 z vol [k] z cur [k] z vel [k] 4.6 1mm 351sample 4.7 1mm 36V 72V 36V 72V 76V 45

15 1 No Voltage Constraint Voltage Constraint 72 V Voltage Constraint 36 V Voltage [V] 5 5 1 15 Current [A] 1 1 6 Velocity [m/s] 4 2 1 2 3 4 5 6 7 8 9 Time [sample] 4.5: (1 mm ) 4.3.2 46

15 1 No Voltage Constraint Voltage Constraint 72 V Voltage Constraint 36 V Voltage [V] 5 5 1 15 Current [A] 1 1 6 Velocity [m/s] 4 2 5 1 15 2 25 3 35 Time [sample] 4.6: (1 mm ) P = 1 Nτ Nτ i(t)(v p V M (t)) dt V M (t) V p ±V power 2V power V p 4.8 47

Voltage [V] 8 6 4 2 2 4 6 8 No Voltage Constraint Voltage [V] 8 6 4 2 2 4 6 8 Voltage Constraint 72 V Voltage [V] 8 6 4 2 2 4 Voltage Constraint 36 V 6 8 3 31 32 33 34 35 36 37 Time [sample] 4.7: (1 mm ) 146 V 146 V 16 V V p 36 V 72 V 52 V 88 V 162 V 4.8.1mm 1mm 1mm 48

Electricity Consumption [W] 12 1 8 6 4 2.1 mm step 1 mm step 1 mm step 2 4 6 8 1 12 Constraint Voltage [V] 14 16 18 4.8: 4.3.3 4.3.1 V p 162 V 76 V 72 V 1 mm 4.9 4.1 ±3 µm 4.1 V p 72 V 36 V 72 V V p 16 V 52 V, 88 V, 162 V 1 mm 4.11 4.12 162 V 72 V 88 V 49

36 V 52 V 36 V 52 V 4.1 72 V.1 mm 1 mm 4.4 5

1 8 V p = 162 V V p = 76 V V p = 72 V Positon [µm] 6 4 2 5 1 15 2 Time [sample] 1 5 4.9: (1 mm ) V p = 162 V V p = 76 V V p = 72 V Positon [µm] 5 1 5 1 15 2 Time [sample] 4.1: (1 mm ) 51

1 8 No Voltage Constraint at V p = 162 V Voltage Constraint 72 V at V p = 88 V Voltage Constraint 36 V at V p = 52 V Positon [µm] 6 4 2 5 1 15 2 Time [sample] 1 4.11: (1 mm ) 5 Positon [µm] 5 No Voltage Constraint at V p = 162 V Voltage Constraint 72 V at V = 88 V p Voltage Constraint 36 V at V p = 52 V 1 5 1 15 2 Time [sample] 4.12: (1 mm ) 52

5 5.1 3.4 / [3,31] [45] [3,31,45] 2 2 2 53

5.2 2 5.1 2 P d [z] P mir [z] y mir [k] u d [k] P nenc [z] r - C[z] y c P enc [z] y enc [k] 5.1: 2 P nenc [z] P enc [z] P mir [z] P nenc (s) P enc (s) P mir (s) τ C[z] u d r u d P nenc r P enc C P mir u d y c y mir 2 u d y enc y mir P nenc [z] x n [k + 1] = A n x n [k] + B n u d [k] (5.1) y n [k] = C n x n [k] (5.2) 54

P enc [z] x e [k + 1] = A e x e [k] + B e (u d [k] + y c [k]) (5.3) y enc [k] = C e x e [k] (5.4) P mir [z] x m [k + 1] = A m x m [k] + B m (u d [k] + y c [k]) (5.5) y mir [k] = C m x m [k] (5.6) C x c [k + 1] = A c x c [k] + B c (y n [k] y e [k]) (5.7) y c [k] = C c x c [k] + D c (y n [k] y r [k]) (5.8) 5.1 2 x d [k + 1] = A d x d [k] + B d u d [k] (5.9) y[k] = C d x d [k] (5.1) y[k] = y enc[k] y mir [k], x d[k] = x n [k] x c [k] x e [k] x m [k] (5.11) A d = A n B c C n A c B c C e B e D c C n B e C c A e B e D c C e B m D c C n B m C c B m D c C e A m (5.12) B d = B n B e B m, C d = C e C m (5.13) 55

5.3 y enc y enc y mir y mir y mir 1 2 ±6% 1 { 6% 5%... %... +5% +6%} 13 2 13 169 5.2 5.3 5m/s 36V 18A x[k] u[k] (3.42) (3.45) y enc y mir 5.4 5.4 ϵ k N ϵ 1 µm k 79 1mm (3.44) Q (3.45) (3.46) J y enc y enc y mir 5.5 y enc 1 µm y mir 5 µm y enc y mir 5.6 y mir 1 µm 56

4 Gain [db] 6 8 1 12 Phase [deg] 14 1 1 9 18 27 36 45 54 63 1 1 1 1 Frequency [Hz/unit] 図 5.2: エンコーダ応答の変動モデルの周波数特性 4 Gain [db] 6 8 1 12 Phase [deg] 14 1 1 9 18 27 36 45 54 63 1 1 1 1 Frequency [Hz/unit] 図 5.3: ミラー応答の変動モデルの周波数特性 57

5.4: 5.4 N 5m/s 36V 18A 1mm k 59 79 N k N = k k + 4 k + 8 2 ϵ ϵ 5.7 5.7 k N k 58

y enc error [µ m] 5 4 3 2 1 1 2 3 4 5 2 4 6 8 1 12 14 16 y mir error [µ m] 5 4 3 2 1 1 2 3 4 5 2 4 6 8 1 12 14 16 Time [step] 5.5: y enc error [µ m] 5 4 3 2 1 1 2 3 4 5 2 4 6 8 1 12 14 16 y mir error [µ m] 5 4 3 2 1 1 2 3 4 5 2 4 6 8 1 12 14 16 Time [step] 5.6: 59

Overshoot / Undershoot [ µ m] 2 1.8 1.6 1.4 1.2 1.8.6.4 w/o Final state cond. N=k* N=k*+4 N=k*+8.2 55 6 65 7 75 8 Time [step] 5.7: / 1 1 w/o Final state cond. N=k* N=k*+4 N=k*+8 Performance Index J 1 1 1 1 2 55 6 65 7 75 8 Time [step] 5.8: J 1 µm k 1 µm 1mm k 59 79 (3.6) J 5.8 5.8 k J N k J 6

5.5 61

6 6.1 [34,35] [36] [37] [2] 3 C % 3.4 [3] [49] 63

2 6.2 3.2 2 6.2.1 m ( m N) z[m] Z m (3.47) Z m = z[m] = Σ m P m U + C z A m x (6.1) 64

Σ m = [ C z A m 1 B, C z AB, C z B, D z ] P m = [ I m O m (N m) ] (6.2) (6.3) U A eq,m U = B eq,m (6.4) A eq,m = Σ m P m, B eq,m = Z m C z A m x (6.4) U E(A, B, C z, D z, x, Z m, m) (6.1) Z m = z[m] U U E(A, B, C z, D z, x, Z m, m) (1) x N x N U U E(A, B, I,, x, x N, N) (6.5) k = m 1,..., m s Z mi = z[m i ], i = 1,..., s (6.6) U (6.6) U U E(A, B, C z, D z, x, Z mi, m i ) (6.7) i=1,...,s (6.7) A eq,m1 B eq,m1 A eq,m2. A eq,ms U = B eq,m2. B eq,ms (6.8) 65

6.2.2 m m N z[m] Z max,m (6.9) (6.9) (6.1) U (6.9) U A ineq,m U B ineq,m (6.1) Σ m P m A ineq,m = (6.11) Σ m P m B ineq,m = Z max,m C z A m x (6.12) Z max,m + C z A m x I(A, B, C z, D z, x, Z max,m, m) (6.13) U I(A, B, C z, D z, x, Z max,m, m) (6.14) (6.14) (3.52) U U I(A, B, C z, D z, x, Z max,mi, m i ) (6.15) i=1,...,n (6.15) k (6.15) A ineq,m1 B ineq,m1 A ineq,m2. A ineq,mn U B ineq,m2. B ineq,mn (6.16) (6.5) (6.15) J U U 66

z s [k] P enc [z] u[k] 1 z 1 u c [k] H P enc (s) τ y enc [k] P v1 [z] 1 z 1 u c [k] H P v1 (s) τ y v1 [k] P vi [z]. 1 z 1 u c [k] H P vi (s) τ y vi [k] P vn [z]. 1 z 1 u c [k] H P vn (s) τ y vn [k] 6.1: 6.3 6.3.1 6.1 P s [z] P s (s) τ 1/(z 1) u c [k] u[k] u c [k] [21, 43] y s [k] z s [k] P s [z] 67

z s [k] x s [k + 1] = A s x s [k] + B s u[k] (6.17) y s [k] = C s x s [k] (6.18) z s [k] = C sz x s [k] + D sz u[k] (6.19) x s [k] = [ x T n [k] x T 1 [k] xt 2 [k] u c[k] ] T (6.2) x n [k] = x np x nv, x 1[k] = x 1p x 1v, x 2[k] = x 2p x 2v (6.21) x p, x v n, 1, 2 6.1 P vi (s) (i = 1,..., n) n P vi [z] (i = 1,..., n) P vi (s) τ P s [z] y vi [k] P vi [z] x vi [k + 1] = A vi x vi [k] + B vi u[k] (6.22) y vi [k] = C vi x vi [k] (6.23) x vi [k] = [ x vpi [k] x vvi [k] u c [k] ] T (6.24) x vpi, x vvi P vi [z](i = 1,..., n) 1 n P v [z] x v [k + 1] = A v x v [k] + B v u[k] (6.25) z v [k] = [ y v1 [k] y vn [k] ] T (6.26) = C v x v [k] (6.27) A v = block diag(a v1,..., A vn ) B v = [ ] B T v 1,..., B T T v n C v = block diag(c v1,..., C vn ) x v = [ ] xv T 1,..., xv T T n 68

Y dir. 1st 2nd... P th hole Origin (r 1,) (r 2,) (r p,) X dir. X dir. 6.2: r p r 2 k 1 r 1 N 1 N 2 N p 6.3: Time 6.2 r i (i = 1, 2,..., p) X 6.3 r i N i N i + k i T on T on = {N 1,..., N 1 + k 1, N 2,..., N 2 + k 2,..., N p,..., N p + k p } (6.28) 69

6.3.2 1 r p 6.2 FSC k = k = N = N p + k p u[k] FSC U (a) P s [z] x x N U E f in = E(A s, B s, I, O, x, x N, N) P v [z] x v x vn U E v f in = E(A v, B v, I, O, x v, x vn, N) (b) r i r i N i N i + k i y s [k] r i U E re f = E(A s, B s, C s, O, x, r i, k) i=1,...,p k T on N i N i + k i y v [k] U E v re f = E(A v, B v, C v, O, x v, O, k) i=1,...,p k T on (c) z s [k] z s [k] Zmax s k N I zs = I(A s, B s, C sz, D sz, x, Zmax, s k) (6.29) k=,...,n FSC U E f in E v f in E re f E v re f I z s (6.3) U (3.6) 1 (6.3) 3.2 FSC 7

6.3.3 2 r p 6.2 FSC k = k = N = N p + k p u[k] FSC U (a) x x N U E f in = E(A s, B s, I, O, x, x N, N) (b) r i r i N i N i + k i y s [k] r i U E re f = E(A s, B s, C s, O, x, r i, k) i=1,...,p k T on (c) z s [k] z s [k] Zmax s k N I zs = I(A s, B s, C sz, D sz, x, Zmax, s k) (6.31) k=,...,n (d) 6.1 P vi u[k] y vi [k] k T on y vi [k] z v max, i = 1,..., n (6.32) U (6.32) z v [k] Zmax v (6.33) Zmax v = [1, 1,..., 1] T z v max R n (6.34) R n n (6.32) U I zv = I(A v, B v, C v,, x v, Zmax, v k) (6.35) k T on 71

x v x v [k] FSC U E f in E re f I zs I zv (6.36) U (3.6) 2 6.3.4 3 2 2 n y vi [k] i = 1,..., n δy i, j [k] = y vi [k] y v j [k] ( < i < j n) (6.37) k T on z v max δy i, j [k] z v max, k T on (6.38) U (6.38) z v [k] = δy 1,2 [k] δy 1,3 [k]. δy 2,3 [k] δy 2,4 [k]. δy n 2,n 1 [k] δy n 1,n [k] = Tz v [k] (6.39) 72

1 1 1... T = 1 1.... 1 1 1 1 z v [k] Z v max, k T on (6.4) Z v max = [1, 1,..., 1]z v max R n (6.41) n = n C 2 (6.42) U (6.4) U I zv = I(A v, B v, TC v,, x v, Zmax, v k) k T on (6.43) FSC U E f in E re f I zs I zv (6.44) U (3.6) 3 6.4 6.2 X 1 mm 5 1 16 sample µs ±5% 2.5% 5 73

6.4.1 1 mm FSC z s [k] 144 V 18 A 5 m/s (3.6) Q 1mm x x N z s [k] Z s max N 3.1 FSC N = 56 ±.1µm ±1µm 6.4(a) ±4µm ±5µm ±1µm u c y s z v 6.5 y s z v 4684 sample.37 Hz/unit 6 sample 6 sample 56 sample 6.4.2 1 1 U 5 mm x x v x N x vn 1 mm z s [k] Zmax s N 6.3.2 FSC 74

Processing time [sample] 1 5 1 4 1 3 conventional method proposal method 1 proposal method 2 proposal method 3 1 2 1 1 1 1 1 Processing accuracy of holes [µm] (a) 45 proposal method 2 proposal method 3 Processing time [sample] 4 35 1 1 1 1 1 Processing accuracy of holes [µm] (b) 6.4: 75

uc[a] ys[mm] zv[µm] Laser 2 2 1 2 3 4 5 1 2 3 4 5 5 1 2 3 4 on off 1 2 3 4 Time [sample] 6.5: ( ±1µm ) N = 295 6.6 u c y s z v 6.6 1 295 sample ±1µm 4684 sample 38% 6.4(a) 1 1 6.4(a) ±1µm 6.4.3 2 2 U U r i N i N i 1 mm r 1 = 1 1 3, r i+1 r i = 1 1 3 (i = 1,..., p 1) 16 sample k i = 15 i = 1,..., p 76

1 uc[a] ys[mm] 1 5 1 15 2 25 5 5 1 15 2 25.1 zv[µm] Laser.1 5 1 15 2 25 on off 5 1 15 2 25 Time [sample] 6.6: 1 ( ±1µm ) i). 1 x r 1 = 1 [mm] N = N 1 +k 1 x v z v max (6.36) N 1 1 N 1 ii). 2 1 N 1 2 N = N 2 +k 2 r 1 = 1 1 3 r 2 = 2 1 3 1 z v max (6.36) N 2 1 N 2 iii). 5 6.7 ±1µm u c y s z v 6.7 2 382 sample 4684 sample 92% 77

uc[a] ys[mm] zv[µm] Laser 2 2 5 1 15 2 25 3 35 5 5 1 15 2 25 3 35 5 5 5 1 15 2 25 3 35 on off 5 1 15 2 25 3 35 Time [sample] 6.7: 2 ( ±1µm ) ±.1µm ±1µm 6.4 2 6.4(a) 2 6.4.4 3 3 2.5 3 2 N i i = 1,..., p 2 (6.44) z v max 6.8 z v max = 2 [µm] ±1µm 78

2 zv [µm] zv[µm] ys[mm] uc[a] 2 5 1 15 2 25 3 35 5 5 1 15 2 25 3 35 5 5 5 1 15 2 25 3 35 5 5 1 15 2 25 3 35 5 ry[µm] Laser 5 5 1 15 2 25 3 35 on off 5 1 15 2 25 3 35 Time [sample] 6.8: 3 ( 2µm ) 2 z v max = 1 [µm] u c y s z v z v r y r ( y ) /2 r y [k] max (y v i [k]) min (y v i [k]) 1 i n 1 i n 6.8 z v z v z v 2 2 3 368 sample 2 382 sample 4% 6.4 3 3 2 79

1 4 Processing time [sample] 1 3 conventional method proposal method 1 proposal method 2 proposal method 3 1 2 1 1 1 1 1 Hole pitch [mm] 6.9: 6.4.5 ±1µm.1mm 3mm 6.9 1.4mm 1.5mm 2.5mm 3mm 2 3.1mm 2mm 2 3 6.5 FSC 1 2 3 1 2 8

3 81

7 7.1 2 3 4 6 4 83

5 6 3 7.2 fθ 84

85

87

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[26] Y. Peng, D. Vrancic and R. Hanus Anti-windup, bumpless, and conditioned transfer techniques for PID controllers, IEEE Control Systems, 16, 4, 48 57 (1996) [27] D 119 1 88 96 (1999) [28] 35 6 754 761 (1999) [29] 2 D 121 7 792 798 (21) [3],,,, : D, 129, 7, 714 724 (29) [31],,, : LMI 2 D, 131, 1, 93 11 (211) [32],,, : 2 D, 133, 3, 29-299 (213) [33],,, : D, 134, 2, 22 228 (214) [34] 22-6255 [35] 214-52294 [36] 25-5561 [37] 22-296533 [38] / 55 5 159 164 (211) 91

[39] MATLAB Compiler User s Guide, MathWorks (216) [4] N.C. Singer and W.P. Seering: Preshaping Command Inputs to Reduce System Vibration, Trans. of ASME, Journal of Dynamic Systems, Measurement, and Control, 112, 1 76 82 (199) [41] L. Van den Broeck, G. Pipeleers, J. De Caigny, B. Demeulenaere, J. Swevers and J. De Schutter: A linear programming approach to design robust input shaping, 1th IEEE International Workshop on Advanced Motion Control, 8 85 (28) [42] T.Totani and H.Nishimura: Final-State Control Using Compensation Input, Trans. of SICE, 3, 3, 253 26 (1994) [43] (27) [44] MATLAB Optimization Toolbox User s Guide, MathWorks (216) [45] U.Boettcher, R.A. de Callafon and F.E. Talke: Multiobjective Time Domain Input Shaping for Closed-Loop Discrete-Time Systems, IFAC, 2 25 (21) [46],, : D, 125, 5, 524 529 (25) [47] M.Hirata and Y.Hasegawa: Vibration minimized trajectory design for information devices, SICE-ICASE International Joing Conference 26, 2777 278 (26) [48] D 128 6 718 725 (28) [49], : C, 69, 682, 155 1555 (23) 92

[1],, : D 129 9 938 944 29 [2] D 133 1 1 19 213 [3] D 136 11 851 86 216 [4] S.Ueda, Y.Kuroki and M.Hirata: Trajectory Design of Galvano Scanner Considering Voltage Constraint of Current Amplifier, Electrical Engineering in Japan 198 1 54 64 217 [2] Selected Paper [1] S.Ueda, Y.Kuroki and M.Hirata: Trajectory Design Method for Voltage Saturation of a Current Amplifier, ASME 212 5th Annual Dynamic Systems and Control Conference joint with the JSME 212 11th Motion and Vibration Conference, 172 177 Fort Lauderdale 212 9 [2] H.Sekine, S.Ueda, M.Suzuki and M.Hirata: System identification of a galvano scanner using input-output data obtained from positioning control, European Control Conference 215 1297 132 Linz 215 7 [1],,, 4, 57 6 211 9 93

[2],,, IIC-11-156, 61 66 211 9 [3],,, IIC-11-156, 67 7 211 9 [4],,,, 55, 212 11 [5],, PWM, 56, 213 11 [6],,,,, MEC- 14-155 45 5 214 9 [7],,, PWM, MEC-14-156 51 56 214 9 ( ) [1], 215-8859, 27 5 7 [2], 573662, 27 5 1 [3], 5726259, 27 4 1 [4], 5393598, 25 1 25 ( ) [1] Computer-readable storage medium, generating method, generating apparatus, driving apparatus, processing apparatus, lithography apparatus, and method of manufacturing article, 2869121, 216 9 21 94

( ) [1] Mirror angular-positioning apparatus and processing apparatus, 2111148747.8, 214 7 3 ( ) [1] Mirror angular-positioning apparatus and processing apparatus, 1-14264, 214 5 26 ( ) [1] Mirror angular-positioning apparatus and processing apparatus, 4695, 215 1 11 95