problem6_sol - EE221A Problem Set 6 Solutions - Fall 2011...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EE221A Problem Set 6 Solutions - Fall 2011 Problem 1. Linear systems. a) Call this dynamical system L = ( U , , Y ,s,r ) , where U = R n i , = R n , Y = R n o . So clearly U , , Y are all linear spaces over the same field ( R ). We also have the response map ( t,t ,x ,u ) = y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) and the state transition function s ( t,t ,x ,u ) = x ( t ) = ( t,t ) x + t t ( t, ) B ( ) u ( ) d We need to check the linearity of the response map; we have that, t t ,t R + : ( t,t 1 x 1 + 2 x 2 , 1 u 1 + 2 u 2 ) = C ( t ) ( t,t )( 1 x 1 + 2 x 2 ) + t t ( t, ) B ( )( 1 u 1 ( ) + 2 u 2 ( )) d + D ( t )( 1 u 1 ( ) + 2 u 2 ( )) = 1 C ( t )( t,t ) x 1 + t t ( t, ) B ( ) u 1 ( ) d + D ( t ) u 1 ( t ) + 2 C ( t )( t,t ) x 1 + t t ( t, ) B ( ) u 2 ( ) d + D ( t ) u 2 ( t ) = 1 ( t,t ,x 1 ,u 1 ) + 2 ( t,t ,x 2 ,u 2 ) b) Using the definition of time-invariance for dynamical systems, check: ( t 1 + ,t + ,x ,T u ) = Cx ( t 1 + ) + Du (( t 1 + )- ) = C e A ( t 1 + - ( t + )) x + t 1 + t + e A ( t 1 + - ) Bu ( - ) d + Du ( t 1 ) = Ce A ( t 1- t ) x + t 1 t e A ( t 1- s ) Bu ( s ) ds + Du ( t 1 ) = ( t 1 ,t ,x ,u ) Problem 2. A linear time-invariant system....
View Full Document

This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

Page1 / 5

problem6_sol - EE221A Problem Set 6 Solutions - Fall 2011...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online