Control ENG HW_Part_36

Control ENG HW_Part_36 - Y (t ) = e r1t e r2t 1 2 τ + + 2...

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Unformatted text preview: Y (t ) = e r1t e r2t 1 2 τ + + 2 r1 (r1 − r2 ) r2 (r2 − r1 ) τ 1 r2t r1t Y (t ) =1 − r − r r1e − r2 e 1 2 [ Y (t ) = 1 − φr1e r t − r2 e r t 2 1 (r1 − r2 ) (b) For inflection point , d 2s d 3s = 0& 2 = 0 dt 2 dt r r (e r2t − e r2t ) ds =− 1 2 r1 − r2 dt r r (r e r2t − r1e r2t ) d 2s =− 1 2 2 =0 r1 − r2 dt 2 = u 2 e r2t = r1e r1t = r2 = e ( r1 −r2 ) ti r1 r ln 2 r = ti = 1 r1 − r2 (c ) ds (t ) dt t = ti r1 r2 r2 =− r1 − r2 r1 = s ' (t i ) ] r r1 − r2 r − 2 r 1 r1 r2 − r r1 r1 r2 r2 =− r1 − r2 r1 r r1 (r1 − r2 ) r2 =− r1 − r2 r1 ds (t ) dt Also ds (t ) dt ds (t ) dt t = ti t = ti r =− t = ti r1 r2 − r r1 r1 r2 (e r2t − e r1t ) =− ( r1 − r2 ) r1 r2 − e r1t (r1 − r2 ) r1 − 1 r 2 = −r1e r1t = − r2 e r2t (d) s (t i ) = 1 − r1e − r2 e r1 − r2 r2t i rt i1 r r s 1 (t i ) 1 − 2 r2 r1 =1+ r1 − r2 r r s 1 (t i ) 1 − 2 r2 r1 = s (t i ) = = 1 + r1 − r2 Now r1 r2 − r r1 r1 r2 − r r1 r = −r1 2 r1 ds (t ) dt t = ti r2 − r1− r2 r1 2 1 r1 − r2 ...
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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 HW_Part_36 - Y (t ) = e r1t e r2t 1 2 τ + + 2...

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