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. http://www.morikita.co.jp/books/mid/091961 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.
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MRAC MRAC I. Bar-Kana 4 2
ii *10 -% -100.00-20.00 60.00 8.00 16.00 24.00 32.00 40.00 1 SAC 1 SAC SAC SAC SAC SAC PID SAC 10
iii 2008
... viii 1... 1 1.1 1 1.2 2 1.3 SAC 4 1.4 5 2... 7 2.1 7 2.2-11 2.3 14 2.4 16 2.5 18 3... 27 3.1 1 27 3.2 29 3.3 32 3.4 33 3.5 35 3.6 37 4... 41 4.1 41
v 4.2 46 4.3 49 5 53 5.1 53 5.2 56 5.3 58 6... 63 6.1 63 6.2 65 6.3 70 6.4 73 7... 75 7.1-76 7.2 77 7.3 78 7.4 80 7.5 82 7.6 87 7.7 94 8... 97 8.1 98 8.2 99 8.3 102 8.4 104 8.5 106 8.6 107
vi 9 PID... 112 9.1 PID 113 9.2 116 9.3 PID 117 9.4 9.3 119 9.5 PID 123 9.6 PID 127 10... 131 10.1 1 132 10.2 132 10.3 134 10.4 135 10.5 137 10.6 139 10.7 142 11 1... 144 11.1 145 11.2 146 11.3 1 148 11.4 152 12... 156 12.1 157 12.2 158 12.3 160 12.4 161 13... 165 13.1 165 13.2 166
vii 13.3 168 14... 170 14.1 171 14.2 172 14.3 175 14.4 PID 176 14.5 PID 177 14.6 179 15... 182 15.1 183 15.2 1 185 15.3 187 15.4 188 15.5 191... 194 A 194 B 3.2 φ {i} γ M 1 195 C 7.4 196 D 7.5 200 E 8.1 201 F 203... 204... 211
SAC 1 1.1 PID AD/DA
2 1 1.2 adaptive control [1.1] 1950 [1.2] 30 1980 model reference adaptive control MRAC [1.3] [1.4]
1.2 3 [1.5] [1.6] n m r [1.7] (E.J. Davison) [1.8] n m + r 1 MRAC
4 1 1.3 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
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] 1.4 2 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 3 4 5 6 SAC 7 SAC SAC SAC 8 9 10 11 3 PFC SAC
6 1 9 PFC PID PID 10 PFC 11 PFC 1 12 15 SAC PFC 12 13 14 15 SAC SAC de Prony [1.23]
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) 6.1 6.2
64 6 1 6.1 ASPR 6.1 2 6.1 2.2 y m (t) CGT 3 2.8 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) 1 4.2 3 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 1 1 + e(t) T e(t) + σ 2 Γ I =Γ I T > 0, Γ P =Γ P T > 0, σ 1,σ 2 > 0 6.4 6.4 σ- 4.19 k = 2 6.1 6.1 u (i) m (t) 0, i = 0, 1,, m σ I = 0 lim t e(t) = 0 6.1 1 SAC 1
6.2 65 6.2 SAC 2 ASPR ASPR 2 ASPR PFC SAC 6.2.1 ASPR ASPR 2 1 0 0 0 3 1 0 2 0 0 2 0 ẋ(t) = x(t) + u(t) 0 0 3 0 2 7 0 0 0 4 0 1 1 2 0 1 y(t) = 0 1 1 8 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 ] 6.5 6.6 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 2 6.5 SAC 6.1 a b SAC c d
66 6 Outputs 2 1 0-1 -2 y 1 y m1 y 2 y m2 0 10 20 30 40 Time [s] a (y 1 (t), y 2 (t)) (y m1 (t), y m2 (t)) Errors 0.015 0.010 0.005 0-0.005-0.010 e 1 e 2-0.015 0 10 20 30 40 Time [s] c (e 1 (t), e 2 (t)) Control inputs Adaptive gain 30 20 10 0-10 -20-30 0 10 20 30 40 Time [s] 0-5 -10-15 b (u 1 (t), u 2 (t)) u 1 u 2 k e1 k e2 0 10 20 30 40 Time [s] d (k e1 (t), k e2 (t)) 6.1 ASPR 6.2.2 ASPR PFC ASPR g(t) 1 2 1 1 1 0 0 0 0 2 3 1 0 2 2 ẋ(t) = x(t) + u(t) + g(t) 0 0 0 1 0 0 1 0 2 3 1 3 1 0 0 0 y(t) = 0 0 1 0 x(t) 2sin(2πt/5) cos (2πt/7) g(t) = sin (2πt/10) 2cos(2πt/5) 6.7
6.2 67 2 1 0 0 0 0 1 1 0 2 0 0 0 1 0 ẋ(t) = 0 0 3 0 0 x(t) + 1 0 u(t) + g(t) 0 0 0 4 0 0 1 0 0 0 0 5 1 15 1 2 1 1 0 y(t) = 0 1 0 15 1 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 + 8 6.8 6.9 2 2 (s 1)(s 2)(s 3) G(s) = 2 (s 2)(s 5) 2 (s 1)(s 4) 2 (s 4)(s 5) 6.10 2 6.1 6.6 3 1 2 ASPR PFC ASPR 1 F(s) = diag [ 0.08/(s + 5), 0.08/(s + 5) ] 6.11 2 F(s) = F 1 (s) + F 2 (s) 6.12
68 6 F(s) = diag [ 0.1/(s + 20) 2, 0.01/(s + 20) ] F(s) = diag [ 0.01/(s + 20), 0 ] 6.13 6.14 4 1 2 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 = 0.05 6.2 6.3 6.4 6.2 1 6.3 1 6.4 2 2 1 0 y 1 0-1 y m1-50 -2 y 2 y m2-100 0 10 20 30 40-150 Time [s] a (y 1 (t), y 2 (t)) (y m1 (t), y m2 (t)) Outputs Control inputs 150 100 50 u 1 u 2 0 10 20 30 40 Time [s] b (u 1 (t), u 2 (t)) Errors 0.015 0.010 0.005 0-0.005-0.010-0.015 0 10 20 30 40 Time [s] c (e 1 (t), e 2 (t)) e 1 e 2 Adaptive gain 0-1000 -2000-3000 k e1 k e2-4000 0 10 20 30 40 Time [s] d (k e1 (t), k e2 (t)) 6.2 ASPR 1
SAC 15 least squares sufficiently rich
15.1 183 [15.1] [15.2] de Prony 15.1 n 1 n λ 1 λ n n y(t) = α i e λit, t 0 15.1 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 1 15.3 15.1 15.3 y 0 = α 1 + + α n y 1 = α 1 x 1 + + α n x n. 15.4
184 15 y k = α 1 x 1 k + + α n x n k. y 2n 1 = α 1 x 1 2n 1 + + 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 ) = 0 15.5 a n x n + a n 1 x n 1 + + a 1 x 1 + a 0 = 0, a n = 1 15.6 15.6 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+1 + + 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 15.6 15.8 a 0 y k + a 1 y k+1 + + a n y k+n = 0 15.9 k + 1, k + 2, a 0 y j + a 1 y j+1 + + a n y j+n = 0, 15.10 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 1... 15.11 15.12 N T N λ i [15.4] 15.6 n x i
15.2 1 185 15.4 15.10 15.12 α i X T X 15.11 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... 15.13 15.14 15.4 n 15.2 1 15.1 15.2 PID PID 1 (3 ) [15.4] 15.2.1 3 K T L 3 K G(s) = 1 + Ts e Ls 15.15 K T L ŷ(t) = αe λ(t τ) + γ, y(t) = ŷ(t) γ = αe λt, α = αe λτ t τ t τ 15.16 γ τ λ >0 y k τ λ γ
186 15 1 λ α e λt = x y k = αx k, k = k 0, k 0 + 1, 15.17 k 0 T τ>0 k 0 a 0 + a 1 x = 0, a 1 = 1 15.18 a 0 15.10 a 0 y k + y k+1 = 0, k = k 0, k 0 + 1,, k 0 + k 1, k 1 1 15.19 y k ỹ k 15.11 a 0 15.17 λ = 1 T log( a 0) 15.20 15.16 15.17 α 2 α τ ŷ(τ) = 0 15.16 α = γ, τ = 1 λ log α α 15.21 3 3 15.15 T = 1 λ, L = τ, K = γ 15.22 15.2.2 Model A : G(s) = 1 (s + 1) 8 [15.2] 1 3 1 G(s) = 3.1055s + 1 e 5.3879s k 0 ỹ(k) k 0 ỹ(k) 40 k 0 a 0
15.3 187 40 100 IAE integral of absolute value of error [15.6] 15.1 40 100 Response 1.2 1.0 0.8 0.6 0.4 Model A 70% 100% 40% 100% 10% 20% 0.2 0 0 5 10 15 20 25 Times [s] 15.1 3 15.3 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 =1 15.23 15.24 0 t T 2 u(t) 1, 0 t T 1 u(t) = 15.25 0, T 1 t T 2 T 1 T 2
188 15 15.23 0 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 ) 15.26 0 0 λ i =1 i =1 i t = T 1 n α i y(t 1 ) = (1 e λ it 1 ) 15.27 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 1 15.28 α i = y(t 1 ) b i 15.29 y(t) n y(t) = α i e λ i(t T 1 ) 15.30 i =1 λ i t = T 1 α i α i = α i λ i e λ 15.31 it 1 1 15.1 T 1 t T 2 (α i,λ i ) 15.23 15.4 15.1 high n AIC AIC low n [15.6]
15.4 189 15.4.1 high n high n IAE ŷ(t) y(t) IAE = 1 N y(kt) ŷ(kt) 15.32 N k=1 T N high n n high n = n :min 1 N y(kt) ŷ n n (kt) N 15.33 k =1 high n, [15.7] 15.4.2 15.4.1 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
1 44 1 94, 120, 185 1 18, 112 1 16, 41 3 185 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- 48 172, 188, 189 156 101 99 82 3 42 4, 14 49 114 50 90 28, 29, 54 129 146 146 131 36 33 84-5, 11 3 118 ASPR 78 78 97, 98 98 5, 8 99 39 9, 22, 194 138 22 8, 194 8 39 36 4, 16 158 156 8 39 22 14, 78 22 11, 194 182 165 183 98 98 11 185 5 DSAC 95 182 34 102 97 SMC 179 SMC 132 102 98 98 102 36 156 8 156
212 195 131 5, 8 47 I 112 42 115 103, 145 9, 22, 194 8, 194 22 SAC 63, 64 18, 112 SPR 19 ASPR 22-19 CGT 25 22 PFC 33 4 50, 136 13 158 189 PID 126 PID 125 PID 123 3 64 42 147 2 42 87 38 83 133, 139 89 124 121 49 114 48 144 156 119, 120 105 76 123 97 PFC 107 D 112 115 35 P 112 115 102 147 PFC 38 29 170 170 171, 183 8, 194 136 4, 28 59 70, 117 195 185 185 2 158 8 PFC 32, 55 CGT 78 SAC 75 75 PFC 88-76 ASPR 77 SPR 76 PR 76 80 26 99 103 3, 46 SAC 49 SAC 70 48
2008 JCLS Printed in Japan ISBN978-4-627-91961-7