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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 callafon@ucsd.edu CONTENTS OF THIS LECTURE Dynamic Response ( chap 3, section 7.4 ) Recapt imedoma inso lut ionfo rTFandSSmode ls 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 )fo raSSmode l ˙ 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 Z t τ =0 e F ( t τ ) Gu ( τ )d τ + ( t ) where F , G , H and D are now state space matrices and e := 1 + F t 1! + F 2 t 2 2! + F 3 t 3 3! + ··· Simpli±cation on case x (0) = 0: y ( t Z t τ =0 G ( t τ ) u ( τ )d τ + ( t ) , where G ( t H e 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 )fo raTFmode l y ( s )= G ( s ) u ( s ) is given by the convolution integral : y ( t Z 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 r t> 0: Impulse input at t =0: u ( t δ ( t ), then for TF model: y ( t ) = G ( t ) δ ( t Z 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 Z t τ =0 e F ( t τ ) ( τ )d τ = H e Ft G = G ( t ) where e := 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{ Z 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 )fo rTFmode l y ( s )=( H ( sI F ) 1 G + D ) u ( s G ( s ) u ( s rSSmode l 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 General remarks on computation of response y ( s
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This note was uploaded on 05/27/2010 for the course MAE 143B taught by Professor Paoc.chau during the Spring '06 term at UCSD.

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lec04_print - MAE143b Linear Control - Theory and...

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