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Control ENG HW_Part_8

# Control ENG HW_Part_8 - Y(t = i 4e −it − 1 4e −it...

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Unformatted text preview: Y (t ) = i / 4e −it − 1 / 4e −it −1 / 4e it − 1 / 4te it Y (t ) = 1 / 4(ie − it − te −it − ie it − te it ) Y (t ) = 1 / 4(i (Cost − iSin t ) − t (Cost − i Sin t ) − i(Cos t + iSin t ) − t (Cos t + i Sin t ) ) Y (t ) = 1 / 4( 2 Sin t − 2t Cos t ) Y ( t ) = 1 / 2 ( Sin 3.8 f ( s) = = f ( s) = t − t Cos t) 1 s ( s + 1) 2 AB C ++ 2 s s s +1 = A( s + 1) + Bs( s + 1) + Cs 2 = 1 Let s=0 ; A=1 s=1; 2A+B+C=1 s=-1: C=1 B=-1 11 1 f ( s) = 2 + + s s s +1 f (t ) = (t − 1) + e − t PROPERTIES OF TRANSFORMS 4.1 If a forcing function f(t) has the laplace transforms f ( s) = = 1 e − s − e −2 s e −3s + − s s2 s 1 − e −3s e − s − e −2 s + s s2 f (t ) = L−1{ f ( s )} = [u(t ) − u(t − 3)] + [(t − 1)u (t − 1) − (t − 2)u (t − 2)] = u(t ) + (t − 1) u(t − 1) − (t − 2)u(t − 2) − u (t − 3) graph the function f(t) 4.2 Solve the following equation for y(t): t ∫ y (τ ) dτ 0 = dy (t ) y ( 0) = 1 dt Taking Laplace transforms on both sides t dy (t ) L{∫ y (τ ) dt} = L dt 0 1 . y ( s ) = s. y ( s ) − y (0) s ...
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