version 0.01 : 2004/04/16 1 2 1.1................. 2 1.2.......................... 3 1.3................. 5 1.4............... 6 1.5.............. 7 2 9 2.1........................ 9 2.2...................... 11 2.3................. 11 2.4............. 12 2.5.................... 13 2.6............... 15 3 17 3.1....................... 17 3.2....................... 18 3.3 1, 2, 5, 10,.................. 18 4 19 4.1....................... 19 1
4.2.................... 20 4.3.................. 21 4.4 ( )............... 22 4.5 ( )...... 24 4.6 ( )........ 25 4.7 ( )..... 26 5 28 5.1 PID........... 28 5.2 PID 3............... 29 5.3 PID................ 32 5.4 PID...................... 34 1 1.1 *1 1.1 r y *1 p.465 2
1.2(b) r u y C P H D 1.1 1.2 1.1 D C 3
D C D C (1.1) f(a 1, a 2, a 3, ) a 1 a 1 a 1 a 1 + a 1 a 1 f *2 f a 1 f a 1 S = f f a 1 a 1 (1.2) S 1 % % S 1 (1.2) a 1 0 S 0 = f f a 1 a 1 = (ln f) (ln a 1 ) (1.3) *2 1kg 4
1.3 1.2(a) r + u y C P r C u y P 1+ PCH H (a) (b) 1.2 r y T T y r = P C 1 + P CH (1.4) P P + P T T 1.2(a) P T = 1 + (P + P )CH T P / S T T P = 1 P 1 + (P + P )CH 1 S 0 = 1 + P CH (1.5) (1.6) (1.7) * 3 *3 5
(1.7) S 0 PCH (1.6) T T = S P P (1.8) S 1 % % 1.4 1.2(a) 1.2(b) 1. 1.2 (a) (b) r y T 2. 1.2(a) P P + P y B y B = P C {1 + (P + P )CH} (1 + P CH) r (1.9) 3. 1.2(b) P P + P P y F y F = P C 1 + P CH r (1.10) (1.9) (1.10) (1.6) y B = S y F (1.11) S S < 1 6
PCH (1.7) S < 1 1.5 1.3 d n d r + C u + P y + n H 1.3 (1.17) 7
1.3 L = P CH T ry y r = P C 1 + L T dy y d = P 1 + L T ny y n = L 1 + L S 0 (1.7) S 0 = 1 1 + L T dy = S 0 P (1.12) (1.13) (1.14) (1.15) (1.16) S 0 + T ny = 1 (1.17) (1.17) (1.13) T dy (1.16) S 0 (1.17) T ny T ny T dy 1.3 H H L (1.14) L (1.15) (1.16) H (1.17) H H (1.17) S 0 T ny (1.17) 8 S 0
1 S 0 2 2.1 2.1 9
(a) (b) (c) 2.1 2.1 2.1 (a) (b) (c) A A A (a) 0 *4 *5 (b) *4 *5, p.79, 10
(a) 2.2 (2.1) (2.2) } d x(t) = Ax(t) + Bu(t) dt (2.1) y(t) = Cx(t) x(0) x(t) x(t) = ε At x(0) + t 0 ε A(t τ) Bu(τ)dτ (2.2) *6 (2.2) (2.1) ε At 2.3 (2.1) u(t) = 0 x(0) lim x(t) = 0 (2.3) t (1981) *6, p.63, (1981) 11
A Re {λ i (A)} < 0 ( i) (2.4) u(t) = 0 (2.2) *7 x(t) = ε At x(0) = T ε λ1t ε λ2t... ε λnt T 1 x(0) (2.5) T A (2.3) 3 x(0) t 0 3 t 0 (2.4) 2.4 A (2.1) 1 sx(s) x(0) = AX(s) + BU(s) *7, p.126, (1992.1), p.63, (1981) 12
X(s) x(0) = 0 (2.1) 2 Y (s) = C(sI A) 1 BU(s) U(s) Y (s) G(s) G(s) = C(sI A) 1 B = Cadj(sI A)B det(si A) (2.6) A =0 (2.6) 3 =0 det(si A) = 0 (2.7) s A A (2.4) 2.5 G(s) 1 1 G(s) = N(s) D(s) = β ns n + β n 1 s n 1 + β 1 s 1 + β 0 s n + α n 1 s n 1 + α 1 s 1 + α 0 3 0 λ i z i 13
λ i G(s) ε λit z i ε z it { K1 Y (s) = G(s)U(s) = + K 2 + + K } n U(s) s λ 1 s λ 2 s λ n U(s) = 1 y(t) = K 1 ε λ 1t + K 2 ε λ 2t + + K n ε λ nt u(t) = ε zit U(s) = 1 G(s) s z i Y (s) = N(s) D(s) 1 = s z i (s z i)ñ(s) D(s) 1 = Ñ(s) s z i D(s) ε zit 14
G(s) = N(s) D(s) -K Y (s) U(s) = N(s) D(s) + KN(s) 2.6 2.2(a) ω : 0 G(jω)H(jω) 2.2(b) (-1+0j) *8 1 + G(s)H(s) = 0 ω : 0 (-1+0j) ( 1 + 0j) k *8 pp.68-73 (1997.9) 15
2.2(a) 0 A A A B B B 180 2.2(b) 180 ω 1, ω 2 180 2.2(a) B A 1 A 2 A 180 A 1 1 180 1 1 (-1+0j) 180 1 16
(-1+0j) Im + A B G( jω ) H( jω ) ω 2-1 ω 1 Re (a) (b) 図 2.2 2.2 3 3.1 17
3.2 d x(t) = Ax(t) + Bu(t) dt y(t) = Cx(t) } (3.1) x(t) u(t) u(t) = F x(t) (3.2) (3.1) 1 (3.2) d x(t) = (A BF )x(t) (3.3) dt (A BF ) A *9 3.3 1, 2, 5, 10,... 1, 2, 5, 10,... 1, 2, 5, 10,... *9, p.116-118, (1981) 18
2 1, 2, 4, 8, 16, 32,... 10 10 1 *10 1, 3, 10,... 1, 10, 100,... 10 10 1/10 1/10 10 10bit 2 10 = 1024 A/D 0.1% 8bit A/D D/A 12bit =0.025% 4 4.1 1964 D. G. Luenberger *10 3 19
1960 R. E. Kalman 4.2 d x(t) = Ax(t) + Bu(t) : A (n n), B(n m) (4.1) dt y(t) = Cx(t) : C(l n) (4.2) x(t) A, B, C u(t), y(t) (4.1) x(0) = x 0 (initial values) (4.3) x(t) = ε At x 0 + t 0 ε A(t τ) Bu(τ)dτ (4.4) x 0 x(t) x 0 20
4.3 y(t) x(t) (4.2) t (4.1) y(t) = Cx(t) ẏ(t) = Cẋ(t) = CAx(t) + CBu(t) ÿ(t) = CAẋ(t) + CB u(t) = CA 2 x(t) + CABu(t) + CB u(t). y (n 1) (t) = CA n 1 x(t) + CA n 2 x(t) + + CBu n 2 (t) Y (t) = Ox(t) + T U(t) (4.5) y(t) ẏ(t) Y (t) = ÿ(t). y (n 1) (t) O = C CA CA 2.. CA n 1 (4.6) (4.7) 21
U(t) = u(t) u(t) ü(t).. u (n 1) (t) 0 0 0 0 CB 0 0 0 T = CAB CB 0 0..... CA n 2 B CA n 3 B CB 0 (4.8) (4.9) x(t) (4.5) O T x(t) x(t) = (O T O) 1 O T (Y (t) T U(t)) (4.10) (O T O) 1 (4.7) O n l ranko = n (4.11) (4.10) x(t) x 0 (4.11) (4.1) (4.2) 0 t u(t) y(t) x 0 (4.6) 4.4 ( ) (4.1) (4.1) 22
4.1 u(t) x(t) (4.1) d dt ˆx(t) = ˆx(t) + ˆBu(t) (4.12) ˆx(0) = ˆx 0 (4.13)  = A ˆB = B (4.1) (4.12) ˆx(t) x(t) (4.13)  = A ˆB = B (4.12) (4.1) d (ˆx(t) x(t)) = A(ˆx(t) x(t)) (4.14) dt e(t) e(t) ˆx(t) x(t) (4.15) (4.14) d e(t) = Ae(t) (4.16) dt (4.16) e(t) = ε At e(0) (4.17) 23
(4.17) (4.15) x(t) = ˆx(t) ε At (ˆx(0) x(0)) (4.18) A t ε At 0 2 ˆx(t) x(t) A 4.1 K = 0 u(t) x(t) y(t) B + C A K + - Plant Observer Bɵ + x(t) ɵ Cɵ y(t) ɵ Aɵ 4.1 4.5 ( ) u(t) y(t) y(t) 24
(4.12) y(t) ŷ(t) = C ˆx(t) d dt ˆx(t) = ˆx(t) + ˆBû(t) + K(y(t) C ˆx(t)) (4.19) 4.1  = A ˆB = B (4.19) (4.1) (4.2) (4.15) d e(t) = (A KC)e(t) (4.20) dt (4.16) K K A KC C, A A KC C, A 4.6 ( ) n m y(t) y(t) n m x(t) [ ] y(t) (m) thorder ˆx(t) = (4.21) ẑ(t) (n m) thorder (4.1) [ ] [ ] [ ] [ ] d y(t) A11 A = 12 y(t) B1 + u(t) (4.22) dt ẑ(t) A 21 A 22 ẑ(t) B 2 25
2 z(t) 1 dtẑ(t) d = A 22ẑ(t) + A 21 y(t) + B 2 u(t) estimation + K (ẏ(t) A 11 y(t) A 12 ẑ(t) B 1 u(t)) correction (4.23) ẏ(t) d dtẑ(t) = (A 22 KA 12 )ẑ(t)+(a 21 KA 11 )y(t)+(b 2 KB 1 )u(t)+kẏ(t) (4.24) 4.2 ẏ(t) K y(t) A21 KA11 +. z(t) ɵ + z(t) ɵ u(t) B2 KB1 A22 KA12 4.2 4.7 ( ) 26
(C, A, B) (Ĉ, Â, ˆB) (4.1) u(t) = F ˆx(t) + Gv(t) (4.25) (4.1) (4.19) d x(t) = Ax(t) + BF ˆx(t) + BGv(t) (4.26) dt d ˆx(t) = KCx(t) + (A KC)ˆx(t) + BF ˆx(t) + BGv(t) (4.27) dt (4.26) (4.15) ˆx(t) d x(t) = (A + BF )x(t) + BF e(t) + BGv(t) (4.28) dt (4.27) (4.26) (4.15) d e(t) = (A KC)e(t) (4.29) dt (4.28) (4.29) [ ] [ ] [ ] [ ] d x(t) A + BF BF x(t) BG = + v(t) (4.30) dt e(t) 0 A KC e(t) 0 A + BF A KC 27
A + BF (4.28) F A KC (4.29) K (C, A,B) (Ĉ, Â, ˆB) (4.30) 5 5.1 PID LQ LQG 28
H PID PID 1922 *11 *12 1960 H *13 PID 5.1 PID 84% *14 5.2 PID 3 5.1 PID PID G C (s) 1 G C (s) = K P + K I s + K Ds = K P (1 + 1 T I s + T Ds) K I K P /T I, K D K P T D (5.1a) (5.1b) *11 PID p.7, 6 (1992.7) *12 *13 *14 29 10 pp.953-958, (1990) 29
(5.1a) 2 (Proportional) (Integral) (Differential) PID (5.1a) 2 (5.1a) 3 r + e 1 u y + K P + G P ( s) T I s d T s D GC ( s ) 5.1 PID (K P ) e (K I, K P /T I ) e 0 (K D, K P T D ) e 3 5.1 e u : u(s) = K P (1 + 1 T I s + T Ds)e(s) (5.2) u e : e(s) r(s) y(s) = r(s) G P (s)u(s) (5.3) (5.2) (5.3) 3 e(s) (5.2) 2 e(s) u(s) (5.3) e(s) (5.2) e(s) 0 e(s) 30
(5.2) 3 u(s) u(s) (5.3) 3 e(s) PID 3 1 2 P PI PID 3 *15 5.2 G P (s) G P (s) = ε Ds 1 (1 + T 1 s)(1 + T 2 s) T 2 /D (5.4) 100 10 T 1 =T 2 つまり 二 重 根 の 付 近 4 PID 5 PD 1 0.1 0.1 1 I 1 10 2 PI 100 3 P T 1 /D 定 常 偏 差 をと 取 る くらいしかできない D が 小 さいので,ゲイン を 大 きくしても 大 丈 夫 5.2 PID *15, p.149, (1976), p.131, (1997) 31
5.3 PID PID *16 Ziegler Nichols P T I T D 0 K P K P K C T C 5.1 1 K P T I T D P 0.5K C PI 0.45K C 0.83T C PID 0.6K C 0.5T C 0.125T C K P *16 PID 2 (1992), pp.135-137, (1997) 32
5.3 R L 5.2 y 1 R 0 L t 5.3 2 K P T I T D P 1/RL PI 0.9/RL 3.33L PID 1.2/RL 2L 0.5L 25% 33
5.4 PID 5.1 PID D P D PD D PID PI-D PD PID I-PD D PD 2 PID PID 3 4 5 6 PID 34