{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ece604 lecture 04

# ece604 lecture 04 - Lecture 4 ECE 604 State Variable...

This preview shows pages 1–9. Sign up to view the full content.

Sept. 17, 2009 Linear Systems © Douglas Looze 1 Lecture 4 ECE 604 State Variable Analysis Doug Looze

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

View Full Document
Sept. 17, 2009 Linear Systems © Douglas Looze 2 Announcements PS1 available Due next Thursday
Sept. 17, 2009 Linear Systems © Douglas Looze 3 Last Time Homogeneous systems State equation ( 29 ( 29 ( 29 ( 29 0 0 d t t t t dt = = x A x x x Assume A ( t ) defined, piecewise continuous for all t Existence Successive approximations Weierstrasse M-test Vector and induced norms

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

View Full Document
Sept. 17, 2009 Linear Systems © Douglas Looze 4 Uniqueness Assumed 2 solutions Difference (error) Showed difference must be zero Gronwell-Bellman lemma Vector and induced norms ( 29 ( 29 , a b t t x x ( 29 ( 29 ( 29 a b t t t = - z x x
Sept. 17, 2009 Linear Systems © Douglas Looze 5 Properties of transition matrices Same differential equation Invertible Explicit inversion formula Composition/semi-group rule Fundamental matrices Same matrix equation – Initial condition X 0 (invertible) ( 29 ( 29 ( 29 1 0 0 , t t t t - = X X Φ ( 29 ( 29 ( 29 ( 29 0 0 d t t t t dt = = X A X X X

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

View Full Document
Sept. 17, 2009 Linear Systems © Douglas Looze 6 Operator Norms Any vector norm can induce a norm on the space of operators ( 29 Let and be Banach spaces with norms and • . Let be a linear operator L X Y X Y : L X Y The induced norm on L is 0 sup L L Y x X x x
Sept. 17, 2009 Linear Systems © Douglas Looze 7 Example ( 29 ( 29 ( 29 ( 29 1 cos 0 0 t t t t t = = x A x A & Transition matrix starting at t = 0: ( 29 ( 29 ( 29 ( 29 ,0 ,0 0,0 d t t t dt = = A I Φ Φ Φ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 11 12 11 12 21 22 21 22 11 12 21 22 ,0 ,0 ,0 ,0 1 cos ,0 ,0 ,0 ,0 0 0 0,0 0,0 1 0 0,0 0,0 0 1 t t t t t d t t t t dt φ φ φ φ φ φ φ φ φ φ φ φ = =

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

View Full Document
Sept. 17, 2009 Linear Systems © Douglas Looze 8 Transition matrix starting at t = 0: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 11 12 11 12 21 22 21 22 11 12 21 22 ,0 ,0 ,0 ,0 1 cos ,0 ,0 ,0 ,0 0 0 0,0 0,0 1 0 0,0 0,0 0 1 t t t t t d t t t t dt φ φ φ φ φ φ φ φ φ φ φ φ = = ( 29 ( 29 ( 29 ( 29 ( 29 11 11 21 11 ,0 ,0 cos ,0 0,0 1 t t t t φ φ φ φ = + = & ( 29 ( 29 ( 29 ( 29 ( 29 12 12 22 12 ,0 ,0 cos ,0 0,0 0 t t t t φ φ φ φ = + = & ( 29 ( 29
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern