solhwset1

# solhwset1 - Solutions to Homework Set 1 - MAE143B, Summer...

This preview shows pages 1–3. Sign up to view the full content.

Solutions to Homework Set 1 - MAE143B, Summer II, 2009 1.1 First we try to solve for y 2 , as this is part of the feedback. Substitution of the ±rst connection equation u 2 = y c into the dynamic equation (in Laplace form) y 2 ( s )= Q 21 ( s ) u 1 ( s )+ Q 22 ( s ) u 2 ( s ) yields y 2 ( s Q 21 ( s ) u 1 ( s Q 22 ( s ) y c ( s ) and substitution of y c ( s C ( s ) u c ( s ) now gives y 2 ( s Q 21 ( s ) u 1 ( s Q 22 ( s ) C ( s ) u c ( s ) We now use the second connection equation u c = y 2 to ±nd y 2 ( s Q 21 ( s ) u 1 ( s Q 22 ( s ) C ( s ) y 2 ( s ) allowing us to solve for y 2 ( s ): y 2 ( s )=[1 - Q 22 ( s ) C ( s )] - 1 Q 21 ( s ) u 1 ( s ) We can now simply use the other dynamic equation y 1 ( s Q 11 ( s ) u 1 ( s Q 12 ( s ) u 2 ( s ) and the fact that u 2 ( s y c ( s C ( s ) u c ( s C ( s ) y 2 ( s ) to solve for y 1 ( s )asnow y 1 ( s Q 11 ( s ) u 1 ( s Q 12 ( s ) C ( s ) y 2 ( s ) making y 1 ( s Q 11 ( s ) u 1 ( s Q 12 ( s ) C ( s )[1 - Q 22 ( s ) C ( s )] - 1 Q 21 ( s ) u 1 ( s ) and thus y 1 ( s T u 1 y 1 ( s ) u 1 ( s ) , where T u 1 y 1 ( s Q 11 ( s Q 12 ( s ) C ( s )[1 - Q 22 ( s ) C ( s )] - 1 Q 21 ( s ) 1

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

View Full Document
Q - - C ± - u 1 y 1 u 2 y 2 Figure 1: Connection of two dynamical systems described by a Linear Fractional Tranformation 1.2 A parallel connection of Q 11 and C means T u 1 y 1 ( s )= Q 11 ( s )+ C ( s ) From the answer above T u 1 y 1 ( s Q 11 ( s Q 12 ( s ) C ( s )[1 - Q 22 ( s ) C ( s )] - 1 Q 21 ( s ) we see this is obtained if Q 12 ( s )=1 ,Q 21 ( s )=1and Q
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 05/27/2010 for the course MAE MAT143B taught by Professor Linearcontrol during the Spring '10 term at UCSD.

### Page1 / 5

solhwset1 - Solutions to Homework Set 1 - MAE143B, Summer...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online