Lecture note 10: II Osaka Institute of Technology
, 2002. MATLAB, 1998. 1991. G. Goodwin, et al., Control System Design, Prentice Hall, New Jersey, 2001. Osaka Institute of Technology II 2
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
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
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
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
ẋ = (A BK)x Bu y = Cx A BK : K Osaka Institute of Technology II 7
(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
: 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
2. 1. 5. 3. 4. PBH Osaka Institute of Technology II 10
(1 4) (A, B) T R n n x = T ξ ξ = Ãξ Bu à = T 1 AT, B = T 1 B, 0 1 0 0 0 0 1 0 à =..... 0 0 0 1 α 0 α 1 α 2 α n 1, B = 0 0. 0 1 Osaka Institute of Technology II 11
(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
(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
(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
(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
(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
(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
(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
(3 1) [ ] rank λi A B < n 3. Osaka Institute of Technology II 19
2. 3. ( ) Osaka Institute of Technology II 20
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
[ ] 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
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
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 1 1 1 A 22 ξ 1 ξ 2 C 1 y K 2 Osaka Institute of Technology II 24
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
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
: ( ) Ackerman 1 (B R n 1 ) Osaka Institute of Technology II 27
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
ẋ = y = 1 1 2 0 [ ] 1 0 x x 1 1 u Osaka Institute of Technology II 29
M c M o [ ] rank M c = rank B AB = rank 1 2 = 1 < 2 1 2 [ ] rank Mo T = rank C T A T C T = rank 1 1 = 2 0 1 Osaka Institute of Technology II 30
χ(s) = det si 1 1 2 0 = s 2 s 2 = (s 2)(s 1) Osaka Institute of Technology II 31
[ ] 1 G(s) = 1 0 si 1 1 1 2 0 1 1 [ ] = 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
: x = T ξ, T = 1 1 1 1 ξ = y = 2 1 0 1 [ ] 1 1 ξ ξ 1 0 u Osaka Institute of Technology II 33