2012 September 21, 2012, Rev.2.2

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1 212 September 21, 212, Rev.2.2

2 s

3 s PID

4

5 ( )2 (1) (2) (3)

6 I II 5

7 1 1.1 f(t) =, t < F (s) = f(t)e st dt (1.1) f(t) F (s) = L[f(t)] t t s s t step1 s step2 step3 t step1 s step2 s s step3 t 6

8 1.1: 1.2 L[ ] = G L[ ] (1.2) (1) (2) (3) : 7

9 m x mẍ + kx = f (1.3) ms 2 X + kx = F (1.4) F, X X F = 1 ms 2 + k (1.5) mg 1.3 s s L[x(t) g(t)] = X(s)G(s) (1.6) Y (s) = X(s)G(s) (1.7) s 1.3 s F 2 (s) = G 1 (s)f 1 (s) Y (s) = G 2 (s)f 2 (s) Y (s) = G 2 (s)g 1 (s)f 1 (s) 8

10 L[f 1 (t) + f 2 (t)] = F 1 (s) + F 2 (s) L[f 1 (t) f 2 (t)] = F 1 (s)f 2 (s) L[f 1 (t)f 2 (t)] = F 1 (s)f 2 (s) (1.8) F 1 (s)f 2 (s) s 1.3: 1.4 P 1.4 C 1 C : E E = C 2 R Y Y = C 1 P E 9

11 Y = C 1 P (C 2 R Y ) (1 + C 1 P )Y = C 1 C 2 P R Y R = C 2C 1 P 1 + C 1 P (1.9) PID 1

12 1.5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) PID ( ) 11

13 2 s (SISO) 9% Y (s) = G(s) 1 + G(s)H(s) (2.1) U(s) Y (s) U(s) Y (s) = K 1 s s 1 + K 2 s s K n s s n + K+ s s i + K s s i (2.2)

14 2.1: s 1 s n 1 + G(s)H(s) = (2.1) y(t) y(t) = K 1 e s 1t + K 2 e s 2t + + K n e s nt + K + e s it + K e s it (2.3) s i = σ i + jγ i y i (t) s i K + e sit = K + e (σ i+jγ i )t (2.4) K e s it = K e (σ i jγ i )t (2.5) y(t) [ ] K y(t) = L 1 + s s i + L 1 [ K s + s i = 1 2jω e(σ i+jω i )t 1 2jω e(σ i jω i )t = 1 ω eσ it ejω it e jω it 2j = 1 ω eσ it sin(ω i t) (2.6) ] (1)s 1 s n (2.6) σ < t (2) s 1 s n s 1 s n g(t) = 1 ω e (2.7) t 13

15 (3)s 1 s n (2.6) σ i = g(t) = sin(ω i t) t (4)s 1 s n (2.6) σ > ω i = ω i σ i > σ i = σ i < : jω > 1 + G(s)H(s) = H(s) 14

16 2.3 G(s) = N(s) D(s) (2.8) (D(s) = ) (N(s) = ) U(s) Y (s) G(s) = K(s z 1)(s z 2 ) (s z m ) (s p 1 )(s p 2 ) (s p n ) U(s) = 1 s y(s) (2.9) Y (s) = K s + K 1 + K n (2.1) s p 1 s p n y(t) = K + K 1 e p 1t + K n e p nt (2.11) K K 1 K = [Y (s)s] s= = K( z 1)( z 2 ) ( z m ) ( p 1 )( p 2 ) ( p n ) (2.12) K 1 = [Y (s)(s p 1 )] s=p1 = K(p 1 z 1 )(p 1 z 2 ) (p 1 z m ) (2.13) p 1 (p 1 p 2 ) (p 1 p n ) K 2 K 3 (1) (K i p 1 K i ) (2) p 1 z i K i (3) (4) t = 15

17 2.3 p 1 I R 1. t z1 p1 I R p/z 1. t I p/z p 1 z 1 R 1. t 2.3: ( 1 Y (s) = p 1 z 1 s + z 1 s + p 1 1 s = K s + K 1 s + p 1 (2.14) K = p 1 [ (s + z 1) s z 1 (s + p 1 ) s ] s= = p 1 z 1 = 1 z 1 p 1 (2.15) K 1 = p 1 [ s + z 1 s + p 1 z 1 s + p ] s= p1 = p 1 p 1 + z 1 = p 1 z 1 1 s z 1 p 1 z 1 (2.16) y(t) = 1 + p 1 z 1 e p 1t z 1 (2.17) t = y() = 1 + p 1 z 1 = p 1 z > p ( z 1 z 1 ) y() 1 p > z ( ) y() 1 16

18 2.4 I 1. I 1. p2 p1 R p 2 R 1. t p 1 t I I 1. p 2 p 1 z 1 R p 2 p 1 R I 1. t p 3 I 1. t p 2 z1 p 2 p1 R p 1 R t p 3 t 2.4: ( 1 17

19 G = 1 1 Y = R (3.1) Y = D (3.2) E = R Y U = CE Y = P (U + D) 18

20 Y = P (CE + D) = P (CR CY + D) Y = P CR P CY + P D (1 + P C)Y = P CR + P D r d G y r + e + d C - u P y 3.1: Y = G yr R + G yd D P C = 1 + P C R + P 1 + P C D (3.3) U = G ur R + G ud D C = 1 + P C R + P C 1 + P C D (3.4) G yr G yd G ur G ud C = Y = R + D 19

21 1 + P C = SISO G(s) G(s) = N(s) D(s) (3.5) D(s) = 1 + P C 1 + P C = Im Im Im X X X X Re X Re X Re X X X!!"# $! 3.2: 3.1 P = b s + a C = k p Y = G yr R + G yd D G yr = P C 1 + P C = bk p s + (a + bk p ) G yd = P 1 + P C = b s + (a + bk p ) (3.6) (3.7) s = (a + bk p ) < k p > a b 2

22 3.3 H(s) = 1 Y, R 1 1 G(s) = s(s + 2),H(s) = 1 G 1 + GH = 1 s(s + 2) + 1 = 1 (s + 1) 2 2 G(s) = 1 s 2,H(s) = 1 G 1 + GH = 1 s G(s) = s(s 1),H(s) = 1 G 1 + GH = 1 s(s 1) + 1 = 1 s 2 s + 1 1) 1 2 2) ±j 3) s = 1 ± 3j 2 r + - G H y Im Im Im j X X 3 2 X -1 Re -j X Re X 3 2 Re 3.3: 21

23 n a s n + a 1 s n a n = (3.8) [1] [2] [3] 22

24 a s n + a 1 s n a n 1 s + a n = (4.1) 2 [1] a,, a n [2] a s 7 + a 1 s 6 + a 2 s 5 + a 3 s 4 + a 4 s 3 + a 5 s 2 + a 6 s + a = (4.2) s 7 s 6 s 7 a a 2 a 4 a 6 s 6 a 1 a 3 a 5 a 7 s 5 23

25 4.1: b 1 = b 3 = s 4 c 1 = c 3 = a a 2 a 1 a 3 = (a a 3 a 1 a 2 ) (4.3) a 1 a 1 a a 4 a 1 a 5 = (a a 5 a 1 a 4 ) (4.4) a 1 a 1 a 1 a 3 b 1 b 3 = (a 1b 3 a 3 b 1 ) (4.5) b 1 b 1 a 1 a 5 b 1 b 5 = (a 1b 5 b 1 a 5 ) (4.6) b 1 b 1 s (a, a 1, b 1, c 1, d 1, e 1. ) 24

26 1 s 4 + 1s s 2 + 5s + 24 = (4.7) (s + 1)(s + 2)(s + 3)(s + 4) = (4.8) [1] [2] b 1 = b 3 = = = (5 35) 1 ( 24) 1 = 3 (4.9) = 24 (4.1) c 1 = c 3 = (24 15) = = 42 (4.11) = (4.12) 3 25

27 d 1 = = ( 24 42) 42 = 24 (4.13) s 3 + 2s 2 + 3s + 1 = (4.14) s = s =.227 ± b 1 = = (1 6) 2 = 2 (4.15) c 1 = = ( + 2) 2 = 1 (4.16)

28 3 K K > r + - K y 4.2: k Y R = G(s) = (s + 1)(s + 2)(s + 3) k 1 + (s + 1)(s + 2)(s + 3) k = (s + 1)(s + 2)(s + 3) + k (4.17) (s + 1)(s + 2)(s + 3) + k = (s + 1)(s 2 + 5s + 6) + k = s 3 + 5s 2 + 6s + s 2 + 5s k = s 3 + 6s s k (4.18) [1] (s ) 6 + k > [2] k b 1 = 6 = [(6 + k) ] 6 = 6 k 6 (4.19) c 1 = k 6 k 6 6 k 6 27 = 6 + k (4.2)

29 k 6+k s 1 s 6 k 6 > (4.21) 6 + k > (4.22) k > 6+k > 6 k > < k < a s n + a 1 s n a n 1 s + a n = (4.23) 2 [1] a,, a n [2] 1x1 2x2 3x3 28

30 1 = a 1 > a 1 a 3 2 = a a 2 > a 1 a 3 a 5 3 = a a 2 a 4 > (4.24) a 1 a s 3 + 2s 2 + 3s + 1 = (4.25) [1] [2] = 2 > = = = 4 < = 1 3 = = 4 < (4.26) 29

31 5 s 4 + 1s s 2 + 5s + 24 = (4.27) = 1 > = = 35 5 = 3 > = = = = 126 > = = 24 3 >

32 5 G(s) C = K i s P = 1 s 2 + 2ζs + ω 2 r + - C P y 5.1: ( ) 1 + CP s 3 + 2ζωs 2 + ω 2 s + K i = (5.1) 31

33 ω = 6 ζ = 5 2ω 1 s 3 + 5s 2 + 6s + K i = (5.2) 5.2 K i = 1 3.2, 1.5,.2 K i = 2 3., 1.,.5 K i = 9 4.,.46 ± 1.4j K i K i = 9 K i = 31 3 (a) (b) (c) K i = 1 K i = 9 y y y (a) K i =1 time (b) K i =2 time (c) K i =9 time 5.2: 32

34 5.2 x(t) y(t) Y (s) = G(s)X(s) (5.3) t y time 5.3: 5.4 A i x(t) =A i sin(ωt) (5.4) X(s) =L[A i sin(ωt)] = A iω s 2 + ω 2 = A i ω (s + jω)(s jω) (5.5) G(s) = N(s) D(s) = N(s) (s s 1 )(s s 2 ) (s s n ) (5.6) s i 33

35 x G(s) y 5.4: Y (s) = k 1 (s s 1 ) + k 1 (s s 1 ) + + k n (s s n ) +{ k + (s jω) + k (s + jω) } (5.7) k + (s jω) k (s + jω) y(t) =k 1 e s 1t + + k n e s nt + {k + e jωt + k e jωt } (5.8) G(s) t { } [ A i ω K + = G(s) s 2 + ω ]s=jω (s jω) 2 [ A i ω K = G(s) s 2 + ω ]s= jω (s + jω) 2 y s (t) =k + e jωt + k e jωt (5.9) = G(jω) A iω 2jω = G(jω)A i 2j = G( jω) A iω 2jω = G( jω)a i 2j G(jω) G( jω) G(jω) = G(jω) e jθ G( jω) = G(jω) e jθ θ = G(jω) (5.1) y s (t) y s (t) = G(jω) e jθ A iω 2jω e jωt + G(jω) e jθ A iω 2jω ejωt = G(jω) A i [e j(ωt+θ) e j(ωt+θ) ] 2j = G(jω) A i sin(ωt + θ) = A o sin(ωt + θ) (5.11) 34

36 x(t) = A i sin(ωt) y(t) = A sin(ωt + θ) G(jω) = A o : A i G(jω) = θ : s jω G(jω) 5.3 G(s) G(jω) G(jω) G(jω 1 ) G(jω 2 ) G(jω 3 ) ω i : 35

37 [1] G(s) [2] [1] [2] 36

38 6 6.1 G(jω) [1] G(jω) [2] G(jω) Im 2 G( jω ) ω 2 ω 1 G( jω ) 2 Re 6.1: 37

39 6.2 9deg. ω G(jω) = G(jω) = 1 jωt = 1 ωt ( j) G(jω) = 1 ωt G(jω) = 9deg (6.1) Im -9deg Re 6.2: 6.3 9deg ω G(jω) = G(jω) = jωt G(jω) = ωt G(jω) = 9deg (6.2) 38

40 Im +9deg Re 6.3: 6.4 G(jω) = K 1 + jωt G(jω) = jωt K = K 1 + (ωt ) 2 G(jω) = K (1 + jωt ) = tan 1 ωt (6.3) ω G() = G() = K ω G( ) = tan 1 ωt = 9 G( ) = K 1 + jωt = K KωT j 1 + ω 2 T }{{ ω } 2 T }{{ 2 } x y (6.4) x 2 + y 2 = 1 + ω2 T 2 (1 + ω 2 T 2 ) 2 K2 = (x K 2 )2 K2 4 + y2 = (x K 2 )2 + y 2 = ( K 2 )2 K ω 2 T 2 = Kx 39

41 Im x Re 6.4: 6.5 deg G(jω) = K G(jω) = K G(jω) = deg (6.5) Im k Re 6.5: 6.6 G(jω) = KjωT 1 + jωt { KjωT ω 1 >> ωt = K ω 1 << ωt (6.6) 4

42 1 >> ωt 1 << ωt G(jω) = jωt 1 + jωt K G(jω) = jωt 1 + jωt K = ωt 1 + (ωt ) K 2 G(jω) = jωt (1 + jωt ) = 9deg tan 1 ωt (6.7) ω = G() = 9deg G() = ω = G( ) = deg G( ) = K Im x Re 6.6: 6.7 G(jω) = e jωl G(jω) = e jωl = 1 G(jω) = ωl (6.8) G(jω) = ωl L ω 41

43 Im 1 Re 6.7: 6.8 K G(s) = s(1 + st ) = K s st = G 1(s)G 2 (s) (6.9) G(jω) = K jω(1 + jωt ) (6.1) G(jω) = G 1 (jω) G 2 (jω) (6.11) G(jω) = G 1 (jω) + G 2 (jω) (6.12) ω 1 + jωt 1 G(jω ) K jω (6.13) G(jω ) K ω = (6.14) G(jω ) 9 + = 9deg (6.15) ω ω 1 + jωt jωt G(jω ) 42 K (jω) 2 T = (6.16)

44 G(jω ) K ω 2 T = (6.17) G(jω ) 9 + ( 9) = 18deg (6.18) ω 2 18deg ω G(s) = K s(1 + st ) (6.19) 2 9 n(n = 2) 18 ω Im Re 6.8: 43

45 : (1) 6.1: (2) 44

46 ( 7 ) Y (s) = G 1(s)G 2 (s) 1 + G 1 (s)g 2 (s) R(s) + G 2 (s) D(s) (7.1) 1 + G 1 (s)g 2 (s) r + - G 1 (s) d + + G 2 (s) y 7.1: 7.1 G 1 (s)g 2 (s) 45

47 G 1 (s)g 2 (s) G(s) F (s) = (7.2) 1 + G(s)H(s) 1 + G(s)H(s) r + - G(s) H(s) y 7.2: 7.3 Im Re 7.3: G(s) H(s) 46

48 G(s) H(s) [1] [2] [3] G(s)H(s) s = σ + jω G(s)H(s) s G(s)H(s) s G(s)H(s) = -1 S = GH 7.4: s s = σ + jω 1 + G(s)H(s) = (7.3) G(σ + jω)h(σ + jω) = 1 (7.4) G(s)H(s) ( 1, j) s G(s)H(s) ( 1, j) s 7.5 s σ + jω GH ( 1, j) 47

49 s G(s)H(s) ( 1, j) 7.5 ( 1, j) Im Im Re (-1, j) Re Im ( ) Im Re (-1, j) Re S GH 7.5: s GH GH G(s) H(s) 48

50 7.3 s ω [1]s s = jω, jω s = jω ω Im s= s=- R Re 7.6: s [2] R G(s)H(s) s = Re jθ (7.5) π 2 < θ < π 2 G(s)H(s) = b s m + b 1 s m b m a s n + a 1 s n a n (7.6) n m (7.5) (7.6) G(s)H(s) b R m e jmθ a R n e jnθ = b R m n e j(m n)θ a (7.7) n m R b if n = m (... R (n m) = R = 1) G(s)H(s) a if n m (... (7.8) R (n m) = R and R ) 49

51 R G(s)H(s) G(s)H(s) = 1 (st 1 + 1)(sT 2 + 1) (7.9) ω = 7.7 ω = Im = j = = Re = =j 7.7: 1 (st 1 +1)(sT 2 +1) [1]ω G(jω)H(jω) [2][1] [3][1][2] ( 1, j) ( 1, j) 1 G(s)H(s) =?? G(s)H(s) = T s (T 1 s + 1)(T 2 s + 1) 7.4 ( ) s = σ + jω G(jω) 5

52 7.8: ω = G(s) s = jω ω = 1 + j [1] 1 [2] 1 [3] 1 51

53 Im (-1,j) Re 7.9: P = (s +.1)(s + 1) C = K s K = 1 K =.8 s = 2 K = K =.8 1 K = 1 P C = s(s +.1)(s + 1) 1 + s(s +.1)(s + 1) = s s 2 +.1s ± j.8 52

54 図 7.1: 簡易化した安定判別例 7.5 ゲイン余裕と位相余裕 ナイキスト線図では 安定限界からどの程度離れているかを判定でき る 図 7.12 は ベクトル軌跡が実軸上の 1 を左に見て通過する安定な システムである この安定な G(jω)H(jω) の位相角が 18deg となるとき 位相交点 交 差 周波数 (ωϕ ) とよぶ その逆数は 1 G(jωϕ )H(jωϕ ) (7.1) となり デシベルで表すと gm = 2 log G(jωϕ )H(jωϕ ) (7.11) となる gm をゲイン余裕とよぶ また G(jω)H(jω) のゲインが 1 となる ω はゲイン交点 交差 周波数 (ωc ) と呼ばれる 原点と ωc の角度は位相余裕とよばれ ϕm = 18 + G(jωc )H(jωc ) となる 53 (7.12)

55 Im Im = = -1+j Re -1+j Re = = 7.11: Im 1/ Re 7.12: 54

56 SISO = = log n = log 1 x 1.1, n 1 x = 1 p 1 = = 1 p 2 = 1 p p = log 1 2(1 2 p) 55

57 p p = log 1 x 1 = 1 p p = log 1 1 = 2 = 1 p p = log = 1 p p = log = 1 p p = log 1 4 = log 1 (2 2) = log log = 1 p 1 p = log 1 5 = log 1 2 = log 1 1 log = = 1 p p = log 1 6 = log 1 (2 3) = log log = 1 p p = log = 1 p p = log 1 8 = log 1 (2 2 2) = 3 log = 1 p p = log 1 9 = log 1 (3 3) = log log = 1 p p = log 1 1 = 1 p=log 1 x X=1 p 1 (1) (1.25)(1.58)(1.99) (2.51)(3.16)(3.98) (5.1)(6.31) (7.94)(1) : log 1 2 log 1 3 log X = 1 X = 1 1 X = 1 2 [Hz] p = log 1 1 = p = log 1 1 = 1 p = log 1 (1 1) = log log 1 1 = 2 Hz GHz 56

58 [Hz] [rad/s] G(jω) [db] G(jω)[deg] G(jω) = G 1 (jω)g 2 (jω) (8.1) G(jω) = G 1 (jω) G 2 (jω) G(jω) = G 1 (jω) + G 2 (jω) (8.2) log a MN = log a M + log a N M log a N = log a M log a N log a M p = p log a M log a M = log b M log b a (8.3) [ 1] G 1 (jω)g 2 (jω) [db] [db] 1 [db] 2 log 1 G(jω) = 2 log 1 G 1 (jω)g 2 (jω) (8.4) = 2 log 1 G 1 (jω) + 2 log 1 G 2 (jω) (8.5) [deg] G(jω) = G 1 (jω) + G 2 (jω) (8.6) [deg] 57

59 [ 2]G(s) 1 ω G(s) G(s) 1 G(s) 1 G 1 (jω) G(jω) = 1 G 1 (jω) (8.7) 2 log 1 G(jω) = 2 log 1 G 1 (jω) (8.8) G(jω) = G 1 (jω) (8.9) 1 G 1 (jω) G 1(jω) ω log 1 G(jω) = 1 log 1 G(jω) 2 (8.1) 1V 1V 4dB( ) G(jω) db

60 [ 1] G(jω) = G 1 (jω) G 2 (jω) (8.11) 2 log 1 G(jω) = 2 log 1 G 1 (jω) + 2 log 1 G 2 (jω) (8.12) G i (jω) G i (jω) (2 1 5 [N/m 2 ]) 12 db 11 db 2m 1 db 9 db 8 db 7 db 1m 6 db 5 db 4 db 3 db 2 db 1m 5dB 57[dB] 6[dB] 3[dB]

61 9 (Bode, SISO 9.1 [db] 2 log 1 K 2 log 1 G( jω ) ω [rad/sec] [ ] ω [rad/sec] φ : 6

62 K = 2 G(jω) = 2 log 1 2 = 2.31 = 6[dB] G(jω) = K G(jω) = 2 log 1 K G(jω) = deg (9.1) 9.2 G(jω) = 1 jωt G(jω) = 2 log 1 ωt 1[dec]( 1 2[dB] log 1 ωt = ω = 1/T [db] G(jω) = 9[deg] 9[deg] G(jω) = 1 jωt G(jω) = 2 log 1 1 ωt = 2 log 1 ωt G(jω) = 9deg (9.2) 9.3 ω = 1/T [db] 2[dB/dec] G(jω) = 9[deg] [ 2]G(s) 1 ω G(s) ω 9[deg] G(jω) = jωt G(jω) = 2 log 1 ωt G(jω) = 9deg (9.3) 61

63 [db] 2 log 1 G( jω ) 1 ω [rad/sec] 2 [db/decade] [ ] ω [rad/sec] φ : 9.4 G(jω) = jωt G(jω) = 2 log (ωt ) 2 = 2 log (ωt ) 2 [db] G(jω) = tan 1 ωt [deg] (9.4) ωt << 1 ωt = 1 ωt >> 1 ωt << 1 G(jω) = 2 log (ωt ) 2 [db] ωt = 1 G(jω) = 2 log (ωt ) 2 = 2 log 1 2 = 1 log1 2 = 3[dB] ωt >> 1 G(jω) = 2 log (ωt ) 2 2 log 1 (ωt ) 2 = 2 log 1 ωt [db] ωt << 1 G(jω) [deg] ωt = 1 G(jω) = 45[deg] ωt >> 1 G(jω) 9[deg] 62

64 [db] 2 log 1 G( jω ) 1/T ω [rad/sec] 2 [db/decade] [ ] 1/(5T) 1/T 5/T ω [rad/sec] φ : ω = 1 ( ωt = 1) T ω = 1 1 [db] ω = 2[dB] T T ω = 1 1 [db] ω = 2[dB] T T G(jω) = tan 1 ωt x = log 1 ωt ωt = 1 x ϕ = tan 1 ωt = tan 1 1 x ) ϕ dϕ dt = 1x ln (1 x ) 2 x = ( ω = 1 T ) dϕ dt = 1x ln 1 1 = ln 1 + (1 x ) 2 2 x = 45deg( 9 63

65 ) ϕ = ϕ = π 2 ϕ = ln 1 2 x π 4 π x = (log 1 ωt =) 4 ln 1 2 x = (log 1 ωt =) π 4 ln 1 2 = =.68 = =.68 ωt = 1.68 =.21 ω.2 T = 1 5T ωt = 1.68 = 4.8 ω 5 T 1 5T 5 T 9.5 ω G(jω) = 1 + jωt G(jω) = 2 log 1 + (ωt ) 2 [db] G(jω) = tan 1 ωt [deg] (9.5) 9.6 G(jω) = G(jω) = ωn 2 ωn 2 = ( ω) 2 + 2jω n ωζ + ωn 2 (ωn 2 ω 2 ) + 2jω n ωζ ωn 2 (ωn 2 ω 2 ) 2 + 4ωnω 2 2 ζ = 1 [db] (1 2 ( ωωn ) 2 ) 2 + 4ζ 2 ( ωωn ) 2 ω 2ζ( ) G(jω) = tan 1 2ω n ωζ (ωn 2 ω 2 ) = ω n tan 1 1 ( ω [deg] (9.6) 2 ) ω n 64

66 ω ω n << 1 G(jω) 1 2 log 1 G(jω) = [db] ω ω n >> 1 G(jω) 1 ( ω ω n ) 2 2 log 1 G(jω) = 4 log 1 ω ω n [db] ω 1 = 1 2 log ω 1 G(jω) = 2 log 1 n 2ζ [db] ϕ = tan 1 2ζ( ω ω n ) 1 ( ω ω n ) 2 ω << 1 ϕ = (... = ) ω n ω = 1 ϕ = 9(... = ) n ω >> 1 ϕ = 18 ω n G(jω) = dω 4 [db/dec] ζ ζ 18 9[deg] 18[deg] gain [db] Ω = ω / ω ζ =.2 n ζ =.77 ζ = [ db/decade ] Ω = ω / ω n phase [ ] ζ =.77 9 ζ = 1.5 ζ = : 65

67 9.7 G(jω) = [db] G(jω) = e jωl G(jω) = 2 log 1 = [db] G(jω) = ωl[deg] (9.7) [db] 2 log 1 G( jω ) ω [rad/sec] [ ] ω [rad/sec] φ : 9.8 G(jω) G(jω) = G 1 (jω)g 2 (jω)g 3 (jω) G(jω) = 2 log G 1 (jω) G 2 (jω) G 3 (jω) = 2 log G 1 (jω) + 2 log G 2 (jω) + [db] G(jω) = G 1 (jω) + G 2 (jω) + G 3 (jω) 66

68 G(jω) = 1 jω( jω) 1 jω 1[rad/s] 1.125jω + 1 1/.125 = 8[rad/s] 1 jω 9 1 1/5T = 1/(5.125) =.125jω /T = 5/.125 = 4 9[deg] 9.6 G(j ) G 1 G 2 G(j ) G 2 G : (1) 67

69 2 G(jω) = 6(1 + 2jω) (1 + 5jω)(1 +.4jω) G 1 (s) = s, G 2(s) = s, G 3(s) = 1 + 2s, G 4 (s) = 6 G 1 (jω) 1/5 =.2 2[dB/dec] G 2 (jω) 1/.4 = 2.5 2[dB/dec], G 3 (jω) 1/2 =.5 2[dB/dec] G 4 (jω) G 1 (jω) 1/5T = 1/(5 5) =.4 [deg] 5/T = 5/5 = 1 9[deg] G 2 (jω) 1/5T = 1/(5.4) =.5 [deg] 5/T = 5/.4 = [deg] G 3 (jω) 1/5T = 1/(5 2) =.1 [deg] 5/T = 5/2 = 2.5 9[deg] G 4 (jω) [deg] G(j ) 2 G 3 G G 1 G 2 9 G(j ) G G 4 G 2-9 G 1 9.7: (2) 68

70 9deg 18deg 9deg 2[dB/dec] H(s) = G(s) = 9.9 r + - G(s) y H(s) 9.8: Im Re 9.9: [1] 18[deg] [2]

71 Im 1/ Re 9.1: [1] 18[deg] 1 [2] 18[deg] [deg] db > db < -18 > -18 < 9.11: 1 [db] 1 [db] 18[deg] 18[deg] 1 ( ) 1 [db] 18[deg] 7

72 G(s) [1] 18[deg] 1 [2] [db] 18[deg] 18[deg] 1 1 [db] 18[deg] r y 36 71

73 [1] [2] [1] [2] [3] OS 72

74 Im OS Re 1.1: Time 1.2: [1]ISE( integral of squared error) I = e(t) 2 dt (1.1) [2]ITSE( integral of time multiplied by squared error) I = te(t) 2 dt (1.2) 73

75 ISE ITSE E(s) E(s) = R(s) Y (s) Y (s) = G 1 (s)g 2 (s)e(s) G 2 (s)d(s) (1.3) G 1 (s)g 2 (s) E(s) = R(s) (G 1 (s)g 2 (s)e(s) G 2 (s)d(s)) (1 + G 1 (s)g 2 (s))e(s) = R(s) G 2 (s)d(s) E(s) = G 1 (s)g 2 (s) R(s) G 2 (s) 1 + G 1 (s)g 2 (s) D(s) (1.4) r + - G 1 (s) d + + G 2 (s) y 1.3: e( ) = lim t e(t) = lim s se(s) (1.5) D(s) = 74

76 1.2.1 r(t) = r R(s) = r s e( ) = 1 r lim s s 1 + G 1 (s)g 2 (s) s = r 1 + K p lim G 1 (s)g 2 (s) = K p s K G 1 (s)g 2 (s) = s n (T s + 1) G 1(s)G 2 (s) 1/s 1/s K p = e( ) = K G 1 (s)g 2 (s) = s (T s + 1) K G 1 (s)g 2 (s) = s 1 (T s + 1) K G 1 (s)g 2 (s) = s 2 (T s + 1) K K p = lim s T s + 1 = K, e( ) = r 1 + K K p = lim s K p = lim s K =, e( ) = s(t s + 1) K =, e( ) = s 2 (T s + 1) r(t) = v t R(s) = v s 2 v e( ) = 1 v lim s 1 + G 1 (s)g 2 (s) s = lim s 2 s s + sg 1 (s)g 2 (s) = v K v lim sg 1 (s)g 2 (s) = K v s K G 1 (s)g 2 (s) = s n (T s + 1) G 1(s)G 2 (s) 1/s 2 1/s 2 K G 1 (s)g 2 (s) = s (T s + 1) K G 1 (s)g 2 (s) = s 1 (T s + 1) K G 1 (s)g 2 (s) = s 2 (T s + 1) K v = lim s s K v = lim s s K v = lim s s K T s + 1 =, e( ) = v = K v K s(t s + 1) = K, e( ) = v = v K v K K =, e( ) = s 2 (T s + 1) 75

77 1.2.3 r(t) = 1 2 a t 2 R(s) = a s 3 e( ) = lim s G 1 (s)g 2 (s) a s 2 3 s = lim s a lim s s 2 G 1 (s)g 2 (s) = K a s 2 + s 2 G 1 (s)g 2 (s) = a K G 1 (s)g 2 (s) = s n (T s + 1) G 1(s)G 2 (s) 1/s 3 1/s 3 K a K G 1 (s)g 2 (s) = s (T s + 1) K G 1 (s)g 2 (s) = s 1 (T s + 1) K G 1 (s)g 2 (s) = s 2 (T s + 1) K a = lim s s 2 K a = lim s s 2 K a = lim s s 2 K =, e( ) = T s + 1 K =, e( ) = s(t s + 1) K s 2 (T s + 1) = K, e( ) = a = a K a K L 1 s 1 s 2 1 s 3 G 1 (s)g 2 (s)( K G 1 (s)g 2 (s) = s n (T s + 1) 76

78 1.4: 1.5: 1.6: G 1 (s)g 2 (s) G 1 (s)g 2 (s) = K(1 + b 1 s + b 2 s 2 ) s n (1 + a 1 s + a 2 s 2 + ) (1.6) 1 s n G 2 (s) (n = 1) 1 s 1 (n = 2) 1 s 2 n = r = r r = v t r = 1a 2 t 2 (n=) r 1 + K 1 (n=1) v K 2 (n=2) a K 9[deg] n = 1( ) n = 2 77

79 G 1 = K p G 2 = 1 s 2 K p 5% K p r + - G 1 (s) d + + G 2 (s) y 1.7: G yr G yd G ur G ud Y = G yr R + G yd D = G 2G 1 G 2 R + D (1.7) 1 + G 2 G G 2 G 1 U = G ur R + G ud D = G G 2 G 1 R + G 2G G 2 G 1 D (1.8) 1 + G 2 G 1 = 1 + K p 1 s 2 = s 2 + K p = s = 2 K p K p > 2 = E = G 2 G 1 R = 1 R = K p s 2 s 2 s 2 + K p R (1.9) r r s e( ) = lim s 2 r s s s 2 + K p s = 5% ( 2)r 2 + K p (1.1) 2r.5 r (1.11) K p 2 K p 2 > K p > 42 78

80 1.3 root-locus technique,, 1.8 K r + K - s( s +1) y 1.8: 1 + K s(s + 1) = s 2 + s + K = s = 1 2 ± 1 4 K K K = s = 1 < K < 1 4 s = 1 2 ± 1 4 K K = 1 4 K > 1 4 s = 1 2 s = 1 2 ± j K

81 1 K s(s + 1) 1.9 Im K=1 K= (-1) K=.25 (.5) K= Re K=1 1.9: K = 1 K = K 1 2 K = 1 4 K K = K 4 K s(s + 1) K K [1] K [2]K < 1 4 K >

82 K 1 1. s(s + a)(s + b) 1 2. s(s + a)(s + b)(s + c) 1 s(s + a)(s + b) K 1 s(s + a)(s + b)(s + c) Im Im -b -a Re -c -b -a Re 1.1: 1.11: 81

83 1 s(s + 1) K(s + b) s(s + a) b Im -b -a Re 1.12: Im 82

84 G 2 (s) G 1 (s) r + - G 1 (s) d + + G 2 (s) y 11.1: [Objective] = [Controller] [Plant] (11.1) [Plant] [Controller] [Objective] [Plant] [Controller] [Plant] [Plant] [Controller] [Controller] PID 83

85 11.1 PID PID e(t) u p (t) =K p e(t) (11.2) I u i (t) = 1 t e(t)dt (11.3) T i D de(t)/dt u d (t) =T d de(t) dt (11.4) PID u(t) =K p (e(t) + 1 T i t e(t)dt + T d de(t) dt ) (11.5) U(s) =K p (1 + 1 T i s + T ds)e(s) 11.2 PID E(s) I P U(s) D 11.2: PID 84

86 11.2 G 2 (s) = K T s + 1 (11.6) G 1 (s) = K p (1 + 1 st i ) (11.7) PI PI (e(t) = r(t) y(t)) G cl (s) = G 1 (s)g 2 (s) 1 + G 1 (s)g 2 (s) = K p K T s + K pk T T i s K pk s + K (11.8) pk T T T i K p T i s K pk s + K pk = (11.9) T T T i 2 s 2 + 2ζω n s + ω 2 n = (11.1) T i K p ω 2 n = K pk T T i 2ζω n = 1 + K pk T K p = 2ζω nt 1 K T i = K pk T ω 2 n 85 = 2ζω nt 1 ω 2 nt

87 ζ ω n PI K p T i T K 11.3 PID Ziegler and Nichols Ziegler and Nichols ZN ZN [1]T I = T D = K p [2] K p [3] T c [4]K p T c PID K p T i T d P.5K c PI.45K c T c 1.2 PID.6K c T c 2 T c 8 PID PID 9% PID 86

88 [s] 11.3 K c = 4.5 T = 5[s] PID K p =.6K c = 2.7 T i = T c /2 = 2.5 T d = T c /8 =.6 y 5 y 5 1 time 5 1 time 11.3: 11.4 PID 11.4 (C 1 C 2 ) PID U(s) = K p (1 + 1 T i s + T ds)e(s) (11.11) 87

89 r(t) C P y(t) + 1 C : G c (s) = 1 + st st 1 (11.12) T 1 < T T 2 1 T 1 Gain[dB] 2dB / dec Phase[deg] 1/T 2 1/T 1 1/T 2 1/T : 2[dB/dec] 11.6 ω g > ω p G c (jω) ω p T 1 T 2 ω p ω p ω p G c(jω)g p (jω) ω g < ω p 88

90 Gain[dB] < Gain[dB] > [deg] [deg] < > 11.6: Gain[dB] -2dB / dec 1/T 1 1/T 2 Phase[deg] 1/T 1 1/T : 89

91 G c (s) = 1 + st st 1 (11.13) T 2 < T 1 1 T 1 1 T 2 2[dB/dec] STEP1 1/T 2 ω g 1dec ω p STEP2 STEP1 STEP2 ω g ω g ω p Gain[dB] Gain[dB] [deg] [deg] : PID 9 PID PID 5 9

92 ( ) ( ) ( ) ( ) ( ) ( ) Bode ( ) ( ) PID PID 91

93 12.2 s i = σ i + jγ i y i (t) g(t) = 1 ω e σ it sin(ω i t) (12.1) : ω i = ω i σ i > σ i = σ i < 92

94 s 4 + 1s s 2 + 5s + 24 = (12.2) 2 [1] a,, a n [2] b 1 b (5 35) b 1 = = = 3 (12.3) ( 24) b 3 = = = 24 (12.4) (a, a 1, b 1, c 1, d 1, e 1. ) 93

95 12.4 A i x(t) =A i sin(ωt) (12.5) X(s) =L[A i sin(ωt)] = A iω s 2 + ω 2 = A i ω (s + jω)(s jω) (12.6) (s i ) (t ) y s (t) = k + e jωt + k e jωt (12.7) K + = K = [ ] A i ω G(s) (s jω) s 2 + jω2 [ ] A i ω G(s) (s + jω) s 2 + jω2 s=jω s= jω = G(jω) A iω 2jω = G(jω)A i 2j = G( jω) A iω 2jω = G( jω)a i 2j G(jω) G( jω) y s (t) G(jω) = G(jω) e jθ G( jω) = G(jω) e jθ θ = G(jω) (12.8) y s (t) = G(jω) e jθ A iω 2jω e jωt + G(jω) e jθ A iω 2jω ejωt = G(jω) A i sin(ωt + θ) = A o sin(ωt + θ) (12.9) 94

96 12.5 ナイキストの安定判別法とボード線図 ナイキストの安定判別を用いると 閉ループ伝達関数 G(s) において s = jω とおき ω = で動かすとき 次のようにまとめられる [1] ベクトル軌跡が 1 を 左 にみて通過する場合は安定 [2] ベクトル軌跡が 1 を通過する場合は安定限界 [3] ベクトル軌跡が 1 を 右 にみて通過する場合は不安定 図 12.2 に簡易化した事例を示す 図 12.2: 簡易化した安定判別例 95

97 (Bode) G(jω) G(jω) = 1 jω( jω) 12.3 [1] 1 1[rad/s] jω 1 9 jω [2] 1 1/.125 = 8[rad/s].125jω /5T = 1/(5.125) = 1.6 5/T =.125jω + 1 5/.125 = 4 9[deg] G(j ) G 1 G 2 G(j ) G 2 G : ) 96

98 12.6 r(t) = r R(s) = r s e( ) = 1 r lim s s 1 + G 1 (s)g 2 (s) s = r 1 + K p lim G 1 (s)g 2 (s) = K p s K G 1 (s)g 2 (s) = 1/s s n (T s + 1) 1/s K G 1 (s)g 2 (s) = s (T s + 1) K G 1 (s)g 2 (s) = s 1 (T s + 1) K G 1 (s)g 2 (s) = s 2 (T s + 1) K K p = lim s T s + 1 = K, e( ) = r 1 + K p K p = lim s K p = lim s K =, e( ) = s(t s + 1) K =, e( ) = s 2 (T s + 1) 12.4 r + - G 1 (s) d + + G 2 (s) y 12.4: E = G 2 G 1 R (12.1) lim f(t) = lim sf (s) t s (12.11) 97

99 G 2 (s) = K T s + 1 (12.12) PI G 1 (s) = K p (1 + 1 st i ) (12.13) G cl (s) = G 1 (s)g 2 (s) 1 + G 1 (s)g 2 (s) = K p K T s + K pk T T i s K pk s + K (12.14) pk T T T i K p T i s K pk s + K pk = (12.15) T T T i K p K i 2 s 2 + 2ζω n s + ω 2 n = (12.16) ζ ω n

100 First version March nd Version March PPT 2.1 Version May Version September

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