Gp4_Chapter 4 - EE2010 Systems and Control (Chapter 4)...

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EE2010 Systems and Control (Chapter 4) 4-1 The impulse response of a LTI system is the system’s output signal when the input signal, u ( t ), is the unit impulse function, ( t ), with zero initial conditions Suppose the transfer function representation of a LTI system is When u ( t ) = ( t ), then U ( s ) = L { ( t )} = 1 and System transfer function, G ( s ) = L {Impulse Response} Impulse Response ) ( ) ( ) ( s U s Y s G   ) ( response, Impulse ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( 1 t h s G t y s U s G s U s G s Y L
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EE2010 Systems and Control (Chapter 4) 4-2 Impulse response of a system, G ( s ), is where p n ( n = 1,…, N ) are the system poles. Since L { ( t )} = 1, input/excitation pole does not exist Steady-state response is determined by input poles. Hence, impulse response does not have a steady-state term For a stable system, all system poles p n ( n = 1,…, N ) lie on the open left half plane. Then, impulse response decays to zero as t             N n t p n t p N t p N N N M n N e A e A e A p s A p s A p s A p s p s p s z s z s z s K s G t h 1 1 2 2 1 1 1 2 1 2 1 1 1 ... ... ... ... ) ( ) ( 1 L L L
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EE2010 Systems and Control (Chapter 4) 4-3 Let f ( t ) and g ( t ) be piecewise continuous functions defined on (0, ∞). The convolution of f ( t ) and g ( t ), denoted by f ( t ) g ( t ), is defined by Convolution is an integral that expresses the amount of overlap of one function f ( t ) as it is shifted over another function g ( t ) Essentially, convolution “blends” one function with another Possible to obtain the output signal of a system by blending two signals? Convolution           0 ) ( ) ( 0 0 t d t g f d g t f t g t f t t
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EE2010 Systems and Control (Chapter 4) 4-4 Laplace Transform of Convolution rule: If L { p ( t )} = P ( s ) and L { q ( t )} = Q ( s ), then From the definition of a system transfer function G ( s ), Y ( s ) = G ( s ) U ( s ), where u ( t ) = L -1 { U ( s )} and y ( t ) = L -1 { Y ( s )} are the input and output signal respectively. Hence, The output signal, y ( t ), of a system may be obtained by convolving the impulse response, h ( t ), with the input/forcing signal, u ( t ).     ) ( ) ( ) ( ) ( or ) ( ) ( ) ( ) ( 1 s Q s P t q t p s Q s P t q t p L L       ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 s G t h t u t h s U s G s Y t y L L L
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EE2010 Systems and Control (Chapter 4) 4-5 Example: Given that and input signal u ( t ) = 1 − U ( t 1) Use convolution to derive the output signal, y ( t ) Impulse response is 1 2 ) ( s s G   t e s s G t h 2 1 2 ) ( ) ( 1 1 L L
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Gp4_Chapter 4 - EE2010 Systems and Control (Chapter 4)...

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