lect7 - EL 625 Lecture 7 1 EL 625 Lecture 7 Frequency...

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EL 625 Lecture 7 1 EL 625 Lecture 7 Frequency domain analysis of time-invariant systems Theorem : If S is a linear time-invariant continuous-time system, S { e st } = H ( s ) e st Proof : S { e st } = S { e st } e st | {z } H ( t,s ) .e st = H ( t, s ) e st S { e s ( t - τ ) } = H ( t - τ, s ) e s ( t - τ ) = e - S { e st } = e - H ( t, s ) e st = H ( t, s ) = H ( t - τ, s ) for any τ Thus, H ( t, s ) is independent of t . H ( t, s ) = H ( s ) Thus, S { e st } = H ( s ) e st The response of a linear time-invariant continuous-time system to a complex exponential is the same complex exponential with a change of complex magnitude (magnitude and phase).
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EL 625 Lecture 7 2 Laplace transform( L transform) : U ( s ) = Z 0 u ( t ) e - st dt Inverse Laplace transform : 1 2 πj Z c + j c - j U ( s ) e st ds Output, y ( t ) = S { u ( t ) } = 1 2 πj Z c + j c - j U ( s ) S { e st } ds = 1 2 πj Z c + j c - j U ( s ) H ( s ) e st ds Y ( s ) = H ( s ) U ( s ) Differentiation theorem of L transforms : L{ ˙ x ( t ) } = s X ( s ) - x (0) Convolution theorem of L transforms :
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EL 625 Lecture 7 3 If y ( t ) = R 0 x 1 ( t - τ ) x 2 ( τ ) ,then, Y ( s ) = X 1 ( s ) X 2 ( s ) ˙ x ( t ) = A x ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) L{ ˙ x ( t ) } = A L{ x ( t ) } + B L{ u ( t ) } L{ y ( t ) } = C L{ x ( t ) } + D L{ u ( t ) } = s X ( s ) - x (0) = A X ( s ) + B U ( s ) Y ( s ) = C X ( s ) + D U ( s ) Thus, X ( s ) = [ sI - A ] - 1 x (0) + [ sI - A ] - 1 B U ( s ) Y ( s ) = C [ sI - A ] - 1 x (0) + C [ sI - A ] - 1 B + D U ( s ) Using, x ( t ) = φ ( t ) x (0) + Z t 0 φ ( t - τ ) B u ( τ ) y ( t ) = ( t ) x (0) + Z t 0 ( t - τ ) B u ( τ ) + D u ( t ) X ( s ) = L{ φ ( t ) } x (0) + L ( Z t 0 φ ( t - τ ) B u ( τ ) )
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EL 625 Lecture 7 4 = L{ φ ( t ) } x (0) + L{ φ ( t ) } B U ( s ) Y ( s ) = C L{ φ ( t ) } x (0) + [ C L{ φ ( t ) } B + D ] U ( s ) L{ φ ( t ) } = [ sI - A ] - 1 Transfer function matrix, H ( s ) = L{ H ( t ) } = C [ sI - A ] - 1 B + D Example : A = 0 1 - 2 3 ( sI - A ) = s - 1 2 s - 3
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