analyze_LTI_using LT

analyze_LTI_using LT - Analysis of LTI System Using Laplace...

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1 Analysis of LTI System Using Laplace Transform 1. Transfer function. For an LTI system with input x(t) and output y(t), the transfer function is defined as (zero-state response) Y ) ( ) ( ) ( s X s s H = ) ( s X ) ( s H ) ( ) ( ) ( s X s H s Y = where () () x tX s , yt Ys Transfer function H(s) is the LT of the impulse response function h(t): ht H s
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2 Transfer function and Differential equation. Consider a system described by the ODE ) 1 ( ) ( ) 1 ( ) ( b b b y a a a m m n n + + + = + + + ) ( ........ ) ( ) ( ) ( ........ ) ( ) ( 0 1 0 1 t x t x t x t t y t y m m n n Assume that the system is at rest at t=0, and the input x(t)=0, for t<0. Taking LT with the use of the differentiation property => H(s) = 2. Block Diagram Parallel interconnection ) ( 1 s H + Y(s) ) ( 2 s H X(s) + + Series connection ) ( 1 s H Y(s) ) ( 2 s H X(s)
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3 Feedback connection ) ( 1 s H Y(s) X(s) + X 1 (s) ) ( 2 s H - 3. Stability We focus on systems with a rational transfer function. i.e ) )...... ( )( ( ) ( ) ( 2 1 0 N N N n M m m m p s p s p s a s B a s b s H = = = 0 n n s = We also assume M N, and there are no common poles and zeros. A system is said to be stable if Re( p i ) < 0 for i = 1, 2……N where p i ’ are the poles of H(s), i.e. all the poles are located in the open left half plane (LHP). < = 0 ) ( 0 ) ( lim dt t h t h t
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4 A system is marginally stable if its impulse response function h(t) is bounded. Re( p i ) 0 for all non-repeated poles Re( p i ) < 0 for all repeated poles A system is unstable if = ) ( lim t h t Re( p i ) > 0 for at least one poles p i or Re( p i ) = 0 for at least one repeated poles p i For rational system (system with rational transfer function), 0 ) ( lim = t h t is equivalent to absolute integrability of h(t), i.e. < 0 ) ( dt t h . Improper Rational Transfer function: M N. ) )...... ( )( ( ) ( ) ( ) ( 2 1 1 1 N p s p s p s s N s Q s H + = = K k k k s c s Q 0 1 ) ( K 0, M=N+K = When K=1 ) ( ) ( ) ( ) ( ) ( ) ( 1 0 1 s X s A s N s X c s c s Y + + = ........ ) ( ) ( ) ( 0 1 + + = t x c t x c t y 10 () () .
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analyze_LTI_using LT - Analysis of LTI System Using Laplace...

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