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2 単純適応制御 SAC サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行当時のものです.

3 A B F E 7 C D URL FAX

4 MRAC MRAC I. Bar-Kana 4 2

5 ii *10 -% SAC 1 SAC SAC SAC SAC SAC PID SAC 10

6 iii 2008

7 ... viii SAC

8 v

9 vi 9 PID PID PID PID PID

10 vii PID PID A 194 B 3.2 φ {i} γ M C D E F

11

12 SAC PID AD/DA

13 adaptive control [1.1] 1950 [1.2] model reference adaptive control MRAC [1.3] [1.4]

14 1.2 3 [1.5] [1.6] n m r [1.7] (E.J. Davison) [1.8] n m + r 1 MRAC

15 SAC almost strictly positive real ASPR [1.9] [1.10] simple adaptive control SAC SAC H. Kaufman 1982 [1.11] [1.12] 2 ASPR J.R. Broussard command generator tracker : CGT [1.13] I. Bar-Kana parallel feedforward compensator : PFC PFC ASPR SAC [1.14] [1.15] SAC [1.16] A.L. Fradkov [1.17] CGT

16 1.4 5 shunt filter SAC PFC PFC [1.18] [1.19] PFC ladder network form PFC PFC PFC [1.20]. SAC [1.21] [1.22] SAC positive real PR strictly positive real SPR - Kalman-Yakubovich SAC ASPR CGT 3 SAC PFC 4 1 SAC SAC ASPR 5 PFC SAC ASPR PFC SAC 7 SAC SAC SAC PFC SAC

17 6 1 9 PFC PID PID 10 PFC 11 PFC SAC PFC SAC SAC de Prony [1.23]

18 SAC 6 SAC 1 SAC SAC ASPR SAC SAC 1 SAC 6.1 SAC 1 SAC [6.1] [6.2] n m ẋ(t) = Ax(t) + Bu(t) 6.1 y(t) = Cx(t) x R n y R m u R m n m m n m n ẋ m (t) = A m x(t) + B m u m (t) 6.2 y m (t) = C m x m (t)

19 ASPR y m (t) CGT u m (t) i u (i) m (t), i = 0, 1,, m SAC u(t) = K(t) z(t) 6.3 z(t) = [e(t) T, x m (t) T, u m (t) T ] T e(t) = y(t) y m (t) K(t) σ- K(t) = K I (t) + K P (t) K I (t) = e(t) z(t) T Γ I σ I (t) K I (t) K P (t) = e(t) z(t) T Γ P e(t) T e(t) σ I (t) = σ e(t) T e(t) + σ 2 Γ I =Γ I T > 0, Γ P =Γ P T > 0, σ 1,σ 2 > σ k = u (i) m (t) 0, i = 0, 1,, m σ I = 0 lim t e(t) = SAC 1

20 SAC 2 ASPR ASPR 2 ASPR PFC SAC ASPR ASPR ẋ(t) = x(t) + u(t) y(t) = x(t) y m (t) = G m (s) [ u m (t) ] [ G m (s) = diag 1 s + 1, 1 s + 1 u m (t) = [ u m1 (t), u m2 (t) ] T ] u m1 (t) 1 u m2 (t) 2 SAC Γ I = diag [ 10 4 I 2, 10 2 ] I 4, ΓP = diag [ 10 3, 10 2, 10 2, 30, 30 ], σ 1 = 0.01, σ 2 = 0.05 ASPR SAC 6.1 a b SAC c d

21 66 6 Outputs y 1 y m1 y 2 y m Time [s] a (y 1 (t), y 2 (t)) (y m1 (t), y m2 (t)) Errors e 1 e Time [s] c (e 1 (t), e 2 (t)) Control inputs Adaptive gain Time [s] b (u 1 (t), u 2 (t)) u 1 u 2 k e1 k e Time [s] d (k e1 (t), k e2 (t)) 6.1 ASPR ASPR PFC ASPR g(t) ẋ(t) = x(t) + u(t) + g(t) y(t) = x(t) 2sin(2πt/5) cos (2πt/7) g(t) = sin (2πt/10) 2cos(2πt/5) 6.7

22 ẋ(t) = x(t) u(t) + g(t) y(t) = x(t) 2sin(2πt/5) cos (2πt/7) g(t) = sin (2πt/10) 2cos(2πt/5) 2sin(2πt/3) 1 1 2s 2 6s + 5 2s 2 6s + 7 G(s) = (s 2 3s + 3)(s 2 3s + 1) s 2 3s + 4 3s 2 9s (s 1)(s 2)(s 3) G(s) = 2 (s 2)(s 5) 2 (s 1)(s 4) 2 (s 4)(s 5) ASPR PFC ASPR 1 F(s) = diag [ 0.08/(s + 5), 0.08/(s + 5) ] F(s) = F 1 (s) + F 2 (s) 6.12

23 68 6 F(s) = diag [ 0.1/(s + 20) 2, 0.01/(s + 20) ] F(s) = diag [ 0.01/(s + 20), 0 ] Γ I = diag [ 10 8 I 2, 10 3 ] I 4, ΓP = diag [ 10 6 I 2, 10 2 ] I 4 σ 1 = 0.01, σ 2 = y y m y 2 y m Time [s] a (y 1 (t), y 2 (t)) (y m1 (t), y m2 (t)) Outputs Control inputs u 1 u Time [s] b (u 1 (t), u 2 (t)) Errors Time [s] c (e 1 (t), e 2 (t)) e 1 e 2 Adaptive gain k e1 k e Time [s] d (k e1 (t), k e2 (t)) 6.2 ASPR 1

24

25 SAC 15 least squares sufficiently rich

26 [15.1] [15.2] de Prony 15.1 n 1 n λ 1 λ n n y(t) = α i e λit, t i =1 (α i,λ i ) exponential analysis method [15.3] n α i G(s) = 15.2 s + λ i =1 i 15.1 y(t) T 2n y(0), y(t),, y{(2n 1)T} e λ it = x i, y j = y ( jt) i = 1,, n, j = 0, 1,, 2n y 0 = α α n y 1 = α 1 x α n x n. 15.4

27 y k = α 1 x 1 k + + α n x n k. y 2n 1 = α 1 x 1 2n α n x n 2n 1 2n α i, x i, i = 1,, n 2n [15.3] x 1,, x n (x x 1 )(x x 2 ) (x x n ) = a n x n + a n 1 x n a 1 x 1 + a 0 = 0, a n = a 0,, a n 1 (15.6) n x i λ i = 1 T log x i, i = 1,, n 15.7 α i 15.4 y 2n 15.4 k a 0 k + 1 a 1 k + 2,, k + n a 2,, a n n a 0 y k + a 1 y k a n y k+n = α i x k i (a 0 + a 1 x i + + a n x n i ) i =1 x i a 0 y k + a 1 y k a n y k+n = k + 1, k + 2, a 0 y j + a 1 y j a n y j+n = 0, j = k, k + 1,, k + n, a i β = (N T N) 1 N T y β = a 0. a n 1, y = y k+n. y k+2n. y k y k+1 y k+n 1..., N = y k+n 1 y k+n y k+2n N T N λ i [15.4] 15.6 n x i

28 α i X T X α = (X T X) 1 X T y α = α 1. α n, y = y k. y k+n. k k k x 1 x 2 x n..., X = k+n k+n k+n x 1 x 2 x n n PID PID 1 (3 ) [15.4] K T L 3 K G(s) = 1 + Ts e Ls K T L ŷ(t) = αe λ(t τ) + γ, y(t) = ŷ(t) γ = αe λt, α = αe λτ t τ t τ γ τ λ >0 y k τ λ γ

29 λ α e λt = x y k = αx k, k = k 0, k 0 + 1, k 0 T τ>0 k 0 a 0 + a 1 x = 0, a 1 = a a 0 y k + y k+1 = 0, k = k 0, k 0 + 1,, k 0 + k 1, k y k ỹ k a λ = 1 T log( a 0) α 2 α τ ŷ(τ) = α = γ, τ = 1 λ log α α T = 1 λ, L = τ, K = γ Model A : G(s) = 1 (s + 1) 8 [15.2] G(s) = s + 1 e s k 0 ỹ(k) k 0 ỹ(k) 40 k 0 a 0

30 IAE integral of absolute value of error [15.6] Response Model A 70% 100% 40% 100% 10% 20% Times [s] n 1 G(s) g(t) n α i G(s) = s + λ i =1 i n g(t) = α i e λ it i = t T 2 u(t) 1, 0 t T 1 u(t) = , T 1 t T 2 T 1 T 2

31 t T 1 t t n n y(t) = g(t τ) u(τ) dτ = α i e λi(t τ) α i dτ = (1 e λit ) λ i =1 i =1 i t = T 1 n α i y(t 1 ) = (1 e λ it 1 ) λ i i =1 T 1 t T 2 y(t 1 ) n y(t) = y(t 1 ) b i e λ i(t T 1 ) i =1 t = T α i = y(t 1 ) b i y(t) n y(t) = α i e λ i(t T 1 ) i =1 λ i t = T 1 α i α i = α i λ i e λ it T 1 t T 2 (α i,λ i ) high n AIC AIC low n [15.6]

32 high n high n IAE ŷ(t) y(t) IAE = 1 N y(kt) ŷ(kt) N k=1 T N high n n high n = n :min 1 N y(kt) ŷ n n (kt) N k =1 high n, [15.7] AIC AIC [15.8] [15.6] AIC Γ=α N g i g 2 T log N T g 2 + 4n i + β N G i G 2 ω log N ω G 2 + 4n i

33

34 , 120, , , AIC 172, 189 ASPR 4, 14 CGT 4, 16 CHR 129 DI 102 DSAC 82, 86 DSAC 80 FP 102 H 36 IAE 187 IMC 129 LQ 160 m 24 OFEP 102 OFP 102 OFSP 102 PFC 4 PID 115 PID 112 PR 5, 8 SAC 4 SAC 41 SAC 44 SAC 42 SMC 131 SPR 5, 8 VSS 131 Z-N 129 σ , 188, , , 29, , ASPR , , , 22, , , , , DSAC SMC 179 SMC

35 , 8 47 I , 145 9, 22, 194 8, SAC 63, 64 18, 112 SPR 19 ASPR CGT PFC , PID 126 PID 125 PID , , PFC 107 D P PFC , 183 8, , , PFC 32, 55 CGT 78 SAC PFC ASPR 77 SPR 76 PR , 46 SAC 49 SAC 70 48

36 2008 JCLS Printed in Japan ISBN

2

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