word - N a si V s)= i=0 m1 m2 mk(sp 1 sp 2 sp k M V s)= i=1...

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V ( s )= i = 0 N a, s i ( s p 1 ) m 1 ( s p 2 ) m 2 ⋅⋅⋅ ( s p k ) mk = N N ( s ) D M ( s ) (B.1) V ( s )= i = 1 M K i s p i ( B .2 ) K i =[( s p i ) V ( s )]∨ s = p i (B.3) ( s p i ) V ( s )= K 1 ( s p i ) ( s p i ) + ⋅⋅⋅ + K i 1 ( s p i ) s p i 1 + K i + K i + 1 ( s p i ) s p i + 1 + ⋅⋅⋅ V ( s )= ( s + 2 )( s + 4 ) ( s + 1 )( s + 3 )( s + 5 ) K 1 s + 1 + K 2 s + 3 + K 3 s + 5 K 1 =( s + 1 ) ( s + 2 )( s + 4 ) ( s + 1 )( s + 3 )( s + 5 ) s =− 1 = 1 x 3 2 x 4 = 3 8 K 2 =( s + 3 ) ( s + 2 )( s + 4 ) ( s + 1 )( s + 3 )( s + 5 ) s =− 3 = (− 1 ) 1 (− 2 ) 2 = 1 4 K 3 =( s + 5 ) ( s + 2 )( s + 4 ) ( s + 1 )( s + 3 )( s + 5 ) s =− 5 = (− 3 )(− 1 ) (− 4 )(− 2 ) = 3 8 V ( s )= N N ( s ) D M ( s ) = N N ( s ) ( s p 1 ) 3 ( s p 2 )( s p 3 ) ; N < M V ( s )= K 31 ( s p 1 ) 3 + K 21 ( s p 1 ) 2 + K 11 ( s p 1 ) + K 2 ( s p 2 ) + K 3 ( s p 3 )
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V ( s )= i = 0 N a, s i ( s p 1 ) m 1 ( s p 2 ) m 2 ⋅⋅⋅ ( s p k ) mk = N N ( s ) D M ( s ) (B.1) V ( s )= i = 1 M K i s p i ( B .2 ) K i =[( s p i ) V ( s )]∨ s = p i (B.3) ¿ s ¿ ¿ ¿ s V ( s )= K 31 + K 21 ( s p 1 )+ K 11 ¿ ¿ ¿ ¿ d ds [ ( s p 1 ) 3 V ( s ) ] = K 21 + 2 K 11 ( s p 1 ) + [ all remaining terms have ( s p 1 ) in the numerator ] K 11 = 1 2 d 2 d s 2 [( s p 1 ) 3 V ( s )]∨ s = p 1 K ij = 1 ( m j i ) ! d mj 1 d s mj i [( s p j ) mj V ( s )]∨ s = pj ,i = 1,2,... ,m j (B.4) V ( s )= j = 1 k i = 1 mj K ij ( s p j ) i ,k = number of different poles ( B .5 ) V ( s )= 1 ( s 1 ) 2 s 3 p 1 = 1, j = 1, m 1 = 2 p 2 = 0, j = 2, m 2 = 3
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¿ s V ( s )= s 3 ¿ ¿ s ¿ ¿ ¿ ¿ K 21 = 1 ( 2 2 ) ! [( s 1 ) 2 V ( s )]= s 3 s = 1 = 1 K 11 = 1 ( 2 1 ) ! d ds ( s 3 )=− 3 s 4 s = 1 = 3 ¿ s s = 0 = 1 ¿ K 32 = 1 ( 3 3 ) ! [ s 3 V ( s )]= ¿ ¿ s s = 0 = 2 ¿ K 22 = 1 ( 3 2 ) ! d ds [( s 1 ) 2 ]=− 2 ¿ K 12 = 1 ( 3 1 ) ! d 2 d s 2 [( s 1 ) 2 ]= 1 2 [ 6 ( s 1 ) 4 ]= 3 V ( s )= 1 ( s 1 ) 2 + 3 s 1 + 1 s 3 + 2 s 2 + 3 s v ( t )= i = 1 M K i e pit t n 1 ( n 1 ) !
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