ch08.pdf - 1234567898 8.1 a For step response M s D s 1 s 1...

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8-1 ±²³´µ¶·¸¹¸ 8.1 a) For step response, input is M t u = ) ( , s M s U = ) ( ) ( 1 1 ) ( s U s s K s Y D D c a + ατ + τ = = + ατ + τ ) 1 ( 1 s s s M K D D c = ) ( s Y a ) 1 ( 1 + ατ + + ατ τ s s M K s M K D c D D c Taking inverse Laplace transform ) 1 ( ) ( ) /( ) /( D D t c t c a e M K e M K t y ατ - ατ - - + α = As α 0 M K dt e t M K t y c t t c a D + α δ = = ατ - 0 ) /( ) ( ) ( M K t M K t y c D c a + τ δ = ) ( ) ( Ideal response, ) ( ) ( ) ( s U s G s Y i i = = + τ s s M K D c 1 = K c M τ D + s M K c M K t M K t y c D c i + δ τ = ) ( ) ( Hence ) ( ) ( t y t y i a as 0 α For ramp response, input is Mt t u = ) ( , 2 ) ( s M s U = Solution Manual for Process Dynamics and Control, 2 nd edition, Copyright © 2004 by Dale E. Seborg, Thomas F. Edgar and Duncan A. Mellichamp.
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8-2 ) ( 1 1 ) ( s U s s K s Y D D c a + ατ + τ = = + ατ + τ ) 1 ( 1 2 s s s M K D D c = ) ( s Y a ) 1 ( ) 1 ( 2 + ατ + + ατ τ s s M K s s M K D c D D c + ατ ατ + + ατ - + + ατ ατ - τ = 1 ) ( 1 1 1 2 2 s s s M K s s M K D D D c D D D c Taking inverse Laplace transform [ ] [ ] ) 1 ( 1 ) ( ) /( ) /( - ατ + + - τ = ατ - ατ - D D t D c t D c a e t M K e M K t y As α 0 Mt K M K t y c D c a + τ = ) ( Ideal response, = ) ( s Y i + τ 2 1 s s M K D c = 2 s M K s M K c D c + τ Mt K M K t y c D c i + = τ ) ( Hence ) ( ) ( t y t y i a as 0 α b) It may be difficult to obtain an accurate estimate of the derivative for use in the ideal transfer function. c) Yes. The ideal transfer function amplifies the noise in the measurement by taking its derivative. The approximate transfer function reduces this amplification by filtering the measurement. 8.2 a) 1 1 ) ( ) ( 1 2 1 2 1 2 1 1 + τ + τ + = + + τ = s K s K K K s K s E s P + τ + + τ + = 1 1 ) ( 1 2 1 1 2 2 1 s s K K K K K
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8-3 b) K c = K 1 + K 2 K 2 = K c - K 1 D ατ = τ 1 2 1 2 2 1 1 2 K K K K K K D D + ατ = + τ = τ or 2 1 2 1 K K K + α = α = + 2 2 1 K K K ) 1 ( 2 2 2 1 - α = - α = K K K K Substituting, 1 1 1 ) 1 ( ) 1 ( ) 1 )( ( K K K K K c c - α - - α = - α - = Then, c K K α - α = 1 1 c) If K c = 3 , τ D = 2 , α = 0.1 then, 27 3 1 . 0
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