# Hw6_f09 - ME 132 Fall 2009 Solutions to Homework 6 1 There are several different ways to find state-space realizations of single linear ordi nary

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: ME 132 Fall 2009 Solutions to Homework 6 1. There are several different ways to find state-space realizations of single, linear, ordi- nary, differential equations. One approach is listed below. (a) We can write the transfer function form of the ODE as: ( s 4 + 4 s 3 + 10 s 2 + 7 s- 2) Y ( s ) = ( s 3 + 5 s 2- 2 s- 1) U ( s ) Y ( s ) = s 3 + 5 s 2- 2 s- 1 s 4 + 4 s 3 + 10 s 2 + 7 s- 2 U ( s ) Define X ( s ) as X ( s ) := 1 s 4 + 4 s 3 + 10 s 2 + 7 s- 2 U ( s ) so Y ( s ) = ( s 3 + 5 s 2- 2 s- 1) X ( s ). Now, define x 1 := x , x 2 := ˙ x 1 = ¨ x , x 3 := ˙ x 2 = x (3) , and x 4 := ˙ x 3 = x (4) , and we get the state space representation of the differential equation: ˙ x 1 ˙ x 2 ˙ x 3 ˙ x 4 = 1 1 1 2- 7- 10- 4 x 1 x 2 x 3 x 4 + 1 u y = bracketleftbig- 1- 2 5 1 bracketrightbig x 1 x 2 x 3 x 4 (b) We can almost repeat the same procedure as above, except that we have to make the transfer function from u to y strictly proper. If we left the transfer function as is, then the above procedure would not work. The result is a non-zero D matrix, as we’ll see below. The transfer function can be made strictly proper by simply polynomial division: Y ( s ) = 3 s 3- 2 s + 1 s 3 + 5 s 2- 4 s U ( s ) = parenleftbigg- 15 s 2 + 10 s + 1 s 3 + 5 s 2- 4 s + 3 parenrightbigg U ( s ) From here, we can see that the D matrix is D = 3. Now, the procedure the same as before, and the state-space representation is...
View Full Document

## This note was uploaded on 10/20/2010 for the course ME 132 taught by Professor Tomizuka during the Spring '08 term at University of California, Berkeley.

### Page1 / 6

Hw6_f09 - ME 132 Fall 2009 Solutions to Homework 6 1 There are several different ways to find state-space realizations of single linear ordi nary

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

View Full Document
Ask a homework question - tutors are online