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 m +b 1 s m 1 + +b m a 0 s n +a 1 s n 1 + +a n.,. : 3
(3), 1 1, Y(s)/U(s) (U(s) Y(s) u(t) y(t) Laplace ) s,. : 4
(4). G(s) = N(s) D(s) = b 0s m +b 1 s m 1 + +b m a 0 s n +a 1 s n 1 + +a n N(s),D(s),b 0 0, a 0 0., m, n. : 5
(5) denominator, numerator,, D(s), N(s) ( ). G(s), H(s). 1. : 6
(6), G(s) = b 0 a 0 s m + b 1 a 0 s m 1 + + bm a 0 s n + a 1 s n 1 + + a n a 0 a }{{ 0},. : 7
(7) N(s) D(s) ( )1. 1,,.,,. : 8
(8) n, m. n m,., n > m,.,. : 9
(9) (s β 1 ) (s β m ) G(s) G(s) = g 0 (s α 1 ) (s α n ) {α 1,...,α n }. {β 1,...,β m }. g 0 = β 0 /α 0. : 10
(10) G(s) = n i=0 b is n i n i=0 a is n i, (a 0 0, b 0 0). ) n,g(s) = b 0 i=1( b i b 0 a 0 a i s n i + a n 0 i=0 a is n i.,. : 11
(11) G(s), = G 0 + G 1 (s), G 0, G 1 (s),., U(s),G 0 U(s)+ G 1 (s)u(s). Laplace. : 12
(12) G 0 U(s) Laplace G 0 u(t).,. G 1 (s)u(s) Laplace,G 1 (s) Laplace ( g 1 (t) ) u(t). g 1 (t),. : 13
(13) G(s) = g 0 (s β 1 ) (s β m ) (s α 1 ) (s α n ). {α( 1,...,α n }, G(s) = A1 g 0 + + A ) n. s α 1 s α n. : 14
(14) A i (i = 1,...,n), ( n i=1 A i j i j)) (s α ( 1 ). {α 1,...,α n },. : 15
(15) (s β 1 ) (s β m ) G(s) = g 0 (s α 1 ) r 1 (s αk ) r k. ( ri )) A ij (s α i ) j ( k, G s = g 0 i=1 j=1.. :
(16) A ij. L 1 [ Laplace ], t 0 L 1 1 = tk 1 (s α) k (k 1)! exp[αt], [ ] 1 L 1 = tk 1 s k (k 1)!,. : 17
(1) Laplace ( ),, sx(s) = L[x (t)]+x(0) ( 2 ) [ ] t k 1 1 L (k 1)! eαt = (s α) (k, α k 2 ) 3 (L[ ] Laplace ). : 18
(2), ( ). x t, k. dk x x (k) dt k ] L [x (k) ( s) = 0 0 x (k) e st dt = [ x (k 1) e st ] 0 ] x (k 1) e st dt = x (k 1) (0)+sL [x (k 1) : 19
(3) [ f(t) ] 0 = f( ) f(0)., s, x (k). ] ], L [x (k) = sl [x (k 1) x (k 1) (0) = s ( sl[x (k 2) ] x k 2 (0) ) x (k 1) (0) = ], L [x (k) = s k X(s) s k 1 x(0) x (k 1) (0) :
(4). cosβt = eiβt +e iβt, sinβt = eiβt e iβt 2 2i [ ] L cosβt = 1 ( 1 2 s iβ + 1 ) s+iβ [ ] L sinβt = 1 ( 1 2i s iβ 1 ) s+iβ : 21
Laplace. ẏ +y = sint, y(0) = 1 : (sy(s) y(0))+y(s) ( 1 : 1 ) 2i s i 1 s+i : ( (s+1)y(s) = 1+ 1 1 ) 1 2i s i s+i Y(s) = 1 + ( 1 1 ) 1 s+1 2i(s+1) s i s+i 2 3 Laplace. : 22
1 = A1 + B1 (s+1)(s i) s+1 s i A 1,B 1, A 1 = 1(1 i), B 2 1 = 1(1 i) 2 1 = A2 + B2 (s+1)(s+i) s+1 s+i A 2,B 2, A 2 = 1 (1+i), B 2 2 = 1 (1+i) 2, y(t) = e 1 + 1 2i ( 12 (1 i)+ 12 (1+i) ) + 1 2i e t ( 1 2 (1 i)eit 1 ) 2 (1+i)e it : 23
, 1 i = 2e iπ 4, 1 + i = 2e iπ 4, y(t) = 3 2 e 1 + 1 2i ( e i(t π 4 ) e i(t π)) 4 2 = 3 2 e 1 + 1 2 sin(t π 4 ) : 24
(1),..,. : 25
(2),,,,. : 26
U(s) Y(s) U(s) 1 Y(s) K s 2 +s+1 1 Y(s) = KU(s) Y(s) = s 2 +s+1 U(s) : 27
(4)., 1 2. 1 2 : 28
(5),.,. : 29
(6) x + + x+y x + - x-y y y x x x. : 30
(7)., Σ,.. : 31
(8),.,.. : 32
(9),.. : 33
G 1 (s) G 2 (s) = G 1 (s)g 2 (s) G 1 (s) G 2 (s) + + = G 1 (s)+g 2 (s) : 34
R(s) + U(s) Y(s) R(s) G(s) G(s) = 1-G(s)H(s) + H(s) Y(s) Y(s) = G(s)U(s), U(s) = R(s)+H(s)Y(s) Y(s) = G(s)R(s)+G(s)H(s)Y(s) Y(s) = G(s) 1 G(s)H(s) R(s) : 35
(12) s ( ) : 36
( ) U(s) + + Y(s) Y(s) = U(s)+Y(s) U(s) = 0, Y(s) : ; : 37
( ) U(s) + + 0.5 0.5 Y(s) = U(s) 2/3 Y(s) 0.5(U(s)+0.5Y(s)) = Y(s), 0.5U(s) = 0.75Y(s), = 2 3 U(s) : 38
(1),,.. (branch) (node) 2., ( ) ( ).,. : 39
(2).,.,.,.,. : 40
+ + G(s) H(s) G(s) H(s) : 41
(1) n A i j n 1. ( 1) i+j A (i,j), a ij, ã ij. : 42
(2) (i,j) ã ji A, adja. Cramer, A, A 1. A 1 = 1 deta adja : 43
(1),. 1 1 : ẋ = Ax+Bu, y = Cx+Du., x R n, u R, y R, A R n n, B R n 1, C R 1 n, D R. u y. : 44
(2) x, u, y,, A, B, C. A, B, C,. L[ ] Laplace. : 45
(3) L[x(t)] = X(s), L[u(t)] = U(s), L[y(t)] = Y(s). Laplace, sx(s) = AX(s)+BU(s)., (si A)X(s) = BU(s). I n. : 46
(4) P(s) = det(si A) s,, (si A),. (si A) 1 = 1 adj(si A) P(s) : 47
(5) (si A)X(s) = BU(s), X(s) = (si A) 1 BU(s). Y(s) = CX(s)+DU(s), Y(s) = ( C(sI A) 1 B +D ) U(s).,. : 48
(6) (C(sI A) 1 B +D) ( ) 1,., A ( ),. : 49
(1)., : ẋ = Ax+Bu, y = Cx+Du., x R n, u R m, y R p, A R n n, B R n m, C R p n, D R p m. : 50
(2) Laplace, sx(s) = AX(s) + BU(s). si A 1 1, : X(s) = (si A) 1 BU(s) : Y (s) = ( C(sI A) 1 B +D ) U(s) : 51
(3) G(s) = C(sI A) 1 B +D.,G(s) m p, s. G(s). : 52
(4) ( ), : 53
Markov (1) ( (si A) = s I A ) s. 1 1 r = 1+r +r2 + ( r < 1) ( ) : 54
Markov (2) s ( )) 1 (si A) 1 = s = 1 s ( I A s (I + ) As A2 + s + 2 = I s + A s 2 + A2 s 3 + : 55
Markov (3) G(s) = C(sI A) 1 B +D (si A) 1 G(s) = D+ CB s + CAB s 2 + CA2 B s 3 + : 56
Markov (4) G(s) s D, CB, CAB, CA 2 B, Markov. Markov, ( II ) : 57
vs (1), ( ),. : 58
vs (2) ( ), ( ).,. : 59
vs (3),,,. : 60
vs (4), ( ),.,,. : 61
vs (5) ( ),.,.,,. : 62
vs (6),,,. : 63
W. S. Levine (ed.), The Control Handbook, 2/e, CRC Press, 2011, ( ),,, 2015,, 1993, H,, 2000 J. Mikusinsuki (, ), ( ), 15,, 1982 : 64