Lecture note 10: II Osaka Institute of Technology
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1 Lecture note 10: II Osaka Institute of Technology
2 , MATLAB, G. Goodwin, et al., Control System Design, Prentice Hall, New Jersey, Osaka Institute of Technology II 2
3 ẋ = Ax Bu y = Cx v u B ẋ 1 s A x C y u = Kx v K Osaka Institute of Technology II 3
4 ẋ = Ax B(Kx v) = Ax (BKx Bv) s 1 ẋ A B K u v C x y s 1 ẋ A B K v C x y B Osaka Institute of Technology II 4
5 ẋ = Ax (BKx Bv) = (Ax BKx) Bv s 1 ẋ A B K v C x y B s 1 ẋ A B BK v C x y Osaka Institute of Technology II 5
6 ẋ = (Ax BKx) Bv = (A BK)x Bv v B ẋ 1 s A BK x C y A BK v B ẋ 1 s x C y ABK Osaka Institute of Technology II 6
7 ẋ = (A BK)x Bu y = Cx A BK : K Osaka Institute of Technology II 7
8 (SISO) n ẋ = Ax Bu y = Cx Du A R n n B R n 1 C R 1 n D R Osaka Institute of Technology II 8
9 : 1. (A, B) 2. rank[b AB A n 1 B] = n 3. rank[λi A B] = n λ A (PBH Popov-Belevich-Hautus test) 4. K A BK ( ) 5. SISO Osaka Institute of Technology II 9
10 PBH Osaka Institute of Technology II 10
11 (1 4) (A, B) T R n n x = T ξ ξ = Ãξ Bu à = T 1 AT, B = T 1 B, à = α 0 α 1 α 2 α n 1, B = Osaka Institute of Technology II 11
12 (1 4) u = Kξ Ã B K {λ 1,, λ n } K = det ( si (Ã [ k0,, k n 1 ] ) B K) ( si (Ã ) B K) = s n (α n 1 k n 1 )s n 1 (α 1 k 1 )s (α 0 k 0 ) (s λ 1 ) (s λ n ) = s n a n 1 s n 1 a 1 s a 0 Osaka Institute of Technology II 12
13 (1 4) 2 α n 1 k n 1 = a n 1, α 1 k 1 = a 1, α 0 k 0 = a 0, k i (i = 0,, n 1) Ã B K {λ 1,, λ n } K Osaka Institute of Technology II 13
14 (1 4) K := KT 1 T ( det si (Ã ) B K) = det ( si (T 1 AT T 1 BKT ) ) = det [ T 1 (si (A BK)) T ] = det T 1 det (si (A BK)) det T Ã B K A BK K [ ] K = α n 1 a n 1 α 1 a 1 α 0 a 0 T 1 A BK {λ 1,, λ n } Osaka Institute of Technology II 14
15 (4 3) 4. ( ) 3. (PBH ) ( ) A λ C rank[λi A B] < n w C n w T (λi A) = 0, w T B = 0 K w T (λi A BK) = w T (λi A) w T BK = 0 A λ A BK K λ rank[λi A B] = n Osaka Institute of Technology II 15
16 (3 1) 3. (PBH ) 1. ( ) ( ) T x = T ξ, Ã = T 1 AT, B = T 1 B ξ = Ãξ Bu = Ã11 Ã 12 0 Ã 22 ξ B 1 u 0 Osaka Institute of Technology II 16
17 (3 1) z T Ã22 λ w T := [0 z T ] [λi à B] [ w T λi à B ] [ ] λi = 0 z T Ã11 Ã12 B 1 0 λi Ã22 0 [ ] = 0 λ z T z T à 22 0 = 0 Osaka Institute of Technology II 17
18 (3 1) w T := w T T 1 [ 0 = w T λi à B ] ] = w T λi T 1 AT T 1 B [ = w T T 1 λt AT ] ] B = w T λt AT B w T (λi A)T = 0, w T B = 0 T w T (λi A) = 0, w T B = 0 Osaka Institute of Technology II 18
19 (3 1) [ ] rank λi A B < n 3. Osaka Institute of Technology II 19
20 2. 3. ( ) Osaka Institute of Technology II 20
21 ẋ = Ax Bu y = Cx x = T ξ ((Ã11, B 1 ): ) ξ 1 ξ 2 = y = Ã11 Ã 12 0 Ã 22 ξ 1 [ C1 C2 ] ξ 1 ξ 2 B 1 Osaka Institute of Technology II 21 ξ 2 0 u
22 [ ] u = K 1 K 2 ξ 1 v ξ 2 u B 1 ξ 1 A 11 A 12 ξ 1 s 1 C 1 y C 2 v u B 1 K 1 ξ 1 A 11 s 1 A 12 ξ 1 C 1 C 2 y ξ 2 s 1 A 22 ξ 2 K 2 ξ 2 s 1 A 22 ξ 2 Osaka Institute of Technology II 22
23 ξ 1 = Ã11ξ 1 Ã12ξ 2 B 1 (K 1 ξ 1 K 2 ξ 2 v) = Ã11ξ 1 Ã12ξ 2 ( B 1 K 1 ξ 1 B 1 K 2 ξ 2 B 1 v) v u B 1 K 1 ξ 2 ξ 1 A 11 s 1 A 12 s 1 ξ 2 A 22 ξ 1 C 1 C 2 y v B 1 ξ 2 K 1 B 1 B 1 ξ 1 A 11 s 1 A 12 s 1 ξ 2 A 22 ξ 1 C 1 C 2 y K 2 K 2 Osaka Institute of Technology II 23
24 ξ 1 = Ã11ξ 1 Ã12ξ 2 ( B 1 K 1 ξ 1 B 1 K 2 ξ 2 B 1 v) = (Ã11 B 1 K 1 )ξ 1 (Ã12 B 1 K 2 )ξ 2 B 1 v v B 1 B 1 K 1 A 11 ξ 1 ξ s 1 1 C 1 y A 12 C 2 B 1 ξ s 1 2 ξ 2 A 22 v B 1 ξ 2 ξ 1 A B K 11 s 1 A 12B 1K2 C 2 s A 22 ξ 1 ξ 2 C 1 y K 2 Osaka Institute of Technology II 24
25 ξ 1 ξ 2 = Ã11 B 1 K 1 Ã 12 B 1 K 2 0 Ã 22 ξ 1 ξ 2 B 1 0 u y = [ C1 C2 ] ξ 1 ξ 2 χ(s) = det(si Ã11 B 1 K 1 ) det(si Ã22) = 0 Osaka Institute of Technology II 25
26 χ(s) = det(si Ã11 B 1 K 1 ) det(si Ã22) = 0 det(si Ã11 B 1 K 1 ): det(si Ã22): Osaka Institute of Technology II 26
27 : ( ) Ackerman 1 (B R n 1 ) Osaka Institute of Technology II 27
28 Ackerman (Ackerman s formula) : 1 ẋ = Ax Bu, y = Cx Du u = Kx K 1. M c M 1 c k T ( 1 M c ) 2. χ d (s) = s n a n 1 s n 1 a 1 s a 0 3. K = k T (A n a n 1 A n 1 a 1 A a 0 I) Osaka Institute of Technology II 28
29 ẋ = y = [ ] 1 0 x x 1 1 u Osaka Institute of Technology II 29
30 M c M o [ ] rank M c = rank B AB = rank 1 2 = 1 < [ ] rank Mo T = rank C T A T C T = rank 1 1 = Osaka Institute of Technology II 30
31 χ(s) = det si = s 2 s 2 = (s 2)(s 1) Osaka Institute of Technology II 31
32 [ ] 1 G(s) = 1 0 si [ ] = 1 0 s 1 1 (s 2)(s 1) 2 s 1 1 = s 1 (s 2)(s 1) = 1 s 2 s = 1 Osaka Institute of Technology II 32
33 : x = T ξ, T = ξ = y = [ ] 1 1 ξ ξ 1 0 u Osaka Institute of Technology II 33
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微分積分 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 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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1 c Koichi Suga, ISBN
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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I 6 : 1 (1) u(t) y(t) : n m a n i y (i) = b m i u (i) i=0 i=0 t, y (i) y i (u )., a 0 0, b 0 0. : 2 (2), Laplace, (a 0 s n +a 1 s n 1 + +a n )Y(s) = (b 0 s m + b 1 s m 1 + +b m )U(s),, Y(s) U(s) = b 0s