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# lec04_print - MAE143b Linear Control Theory and...

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MAE143b Linear Control - Theory and Applications Lecture 4, Thursday Aug. 13, 2:00-4:50pm Prof. R.A. de Callafon [email protected] CONTENTS OF THIS LECTURE Dynamic Response ( chap 3, section 7.4 ) Recap time domain solution for TF and SS models Review Laplace or s-domain computation – Laplace of common signals and dynamic operations – Inverse Laplace transform via PFE MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 4, Page 1 Dynamic Response - time domain response SS model Recap - response y ( t ) on the input u ( t ) for a SS model ˙ x ( t ) = Fx ( t ) + Gu ( t ) , with x (0) = x 0 y ( t ) = Hx ( t ) + Du ( t ) is given by the convolution integral : y ( t ) = H e Ft x 0 + H t τ =0 e F ( t τ ) Gu ( τ )d τ + Du ( t ) where F , G , H and D are now state space matrices and e Ft := 1 + F t 1! + F 2 t 2 2! + F 3 t 3 3! + · · · Simplification on case x (0) = 0: y ( t ) = t τ =0 G ( t τ ) u ( τ )d τ + Du ( t ) , where G ( t ) = H e Ft G MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 4, Page 2

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Dynamic Response - time domain response TF model Recap - response y ( t ) on the input u ( t ) for a TF model y ( s ) = G ( s ) u ( s ) is given by the convolution integral : y ( t ) = t τ =0 G ( t τ ) u ( τ )d τ where G ( t ) is given by the inverse Laplace transform: G ( t ) := L 1 { G ( s ) } MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 4, Page 3 Dynamic Response - impulse response of TF and SS model A simple application of the convolution integral is the computa- tion of the impulse response y ( t ) for t > 0: Impulse input at t = 0: u ( t ) = δ ( t ), then for TF model: y ( t ) = G ( t ) δ ( t ) = t τ =0 G ( t τ ) δ ( τ )d τ = G ( t ) and G ( t ) = L 1 { G ( s ) } is directly the impulse response! Impulse input at t = 0: u ( t ) = δ ( t ), then for SS model with initial zero condition x (0) = 0: y ( t ) = H t τ =0 e F ( t τ ) ( τ )d τ = H e Ft G = G ( t ) where e Ft := 1 + F t 1! + F 2 t 2 2! + F 3 t 3 3! + · · · MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 4, Page 4
Dynamic Response - Laplace or s-domain computation In general, for more complicated input signals u ( t ), the convolu- tion integral has to be solved (numerically). An alternative to working in the time domain and solving con- volution integrals, is to compute solutions in the s-domain . Important property of Laplace transformation we saw before – the Laplace transform of a convolution integral: L{ t τ =0 f 1 ( t τ ) f 2 ( τ )d τ } = F 1 ( s ) F 2 ( s ) where F 1 ( s ) := L{ f 1 ( t ) } , F 2 ( s ) := L{ f 2 ( t ) } Laplace transform of convolution L{ f 1 ( t ) f 2 ( t ) } in time domain, becomes multiplication F 1 ( s ) F 2 ( s ) in the s-domain. MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 4, Page 5 Dynamic Response - Laplace or s-domain computation Computation of the response y ( t ) of a system via s-domain (Laplace transform) would then amount to the following steps: 1. Compute Laplace transform u ( s ) = L{ u ( t ) } of input u ( t ) in time domain 2. Compute solution y ( s ) of convolution integral via multiplica- tion in s-domain: y ( s ) = G ( s ) u ( s ) for TF model y ( s ) = ( H ( sI F ) 1 G + D ) u ( s ) = G ( s ) u ( s ) for SS model 3. Compute inverse Laplace transform y ( t ) = L 1 { y ( s ) } of out- put y ( s ) in s-domain MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 4, Page 6

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Dynamic Response - Laplace or s-domain computation
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lec04_print - MAE143b Linear Control Theory and...

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