problemset1_Soln

problemset1_Soln - The University of Texas at Austin...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic Control—Fall 2009 Solutions to Problem Set One C. Caramanis September 9, 2009. 1. Left to the students. 2. The Taylor expansions of cos( αt ), sin( αt ), e iαt and e- iαt are: cos( αt ) = 1- α 2 t 2 2! + α 4 t 4 4!- ... sin( αt ) = αt- α 3 t 3 3! + α 5 t 5 5!- ... e iαt = 1- α 2 t 2 2! + α 4 t 4 4!- ... + i αt- α 3 t 3 3! + α 5 t 5 5!- ... e- iαt = 1- α 2 t 2 2! + α 4 t 4 4!- ...- i αt- α 3 t 3 3! + α 5 t 5 5!- ... , from which one can see that we have the identities: cos( αt ) = e iαt + e- iαt 2 sin( αt ) = e iαt- e- iαt 2 i , and also the desired identity: e iαt = cos( αt ) + i sin( αt ) . Now consider some a ∈ C , and write a = β + iα . Then we have: e at = e βt + iαt = e βt · e iαt = e βt (cos( αt ) + i sin( αt )) . It is clear from this expression that the magnitude of e at depends only on β = Re ( a )....
View Full Document

This note was uploaded on 10/26/2009 for the course EE 362K taught by Professor Friedrich during the Fall '08 term at University of Texas.

Page1 / 3

problemset1_Soln - The University of Texas at Austin...

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

View Full Document
Ask a homework question - tutors are online