Control ENG HW_Part_35

Control ENG - Thus tan = And A New = Thus Y(t = 2 A11 = 11 1 2 B A(1 2 2 2 2 A(1 2 2 2 2(using 2 2 A11 B11 = ANew Sin(t proved 8.9 If a second

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Thus 2 11 11 ) ( 1 2 tan ωτ ωτξ φ - - = = B A And, 2 2 2 ) 2 ( ) ) ( 1 (( ξωυ + - = A A New (using New A B A = + 2 2 11 11 Thus, ) ( ) 2 ( ) ) ( 1 (( ) ( 2 2 2 ω + + - = t Sin A t Y proved 8.9) If a second- order system is over damped, it is more difficult to determine the parameters τ ξ experimentally. One method for determining the parameters from a step response has been suggested by R.c Olderboung and H.Sartarius (The dynamics of Automatic controls,ASME,P7.8,1948),as described below. (a) Show that the unit step response for the over damped case may be written in the form. 2 1 2 1 1 2 1 ) ( r r e r e r t s t r t r - - = Where r 1 and r 2 are the roots of 0 1 2 2 2 = + + s s ξτ (b) Show that s(t) has an inflection point at ) ( ) / ln( 1 2 1 2 r r r r t i - = © Show that the slope of the step response at the inflection point ) ( ) ( 1 i t t t s dt s d i = - Where, i t r t r i e r e r t s 2 1 2 1 1 ) ( - = - =
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) ( 1 2 1 2 1 1 r r r r r r - - = (d) Show that the value of step response at the inflection point is ) ( 1 ) ( 1 2 1 2 1 1 i i t s r r r r t s + = and that hence 2 1 1 1 1 ) ( ) ( 1 r r t s t s i i - - = - (e) on a typical sketch
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This note was uploaded on 11/13/2011 for the course COP 4355 taught by Professor Koslov during the Spring '10 term at University of Florida.

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Control ENG - Thus tan = And A New = Thus Y(t = 2 A11 = 11 1 2 B A(1 2 2 2 2 A(1 2 2 2 2(using 2 2 A11 B11 = ANew Sin(t proved 8.9 If a second

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