ece604 lecture 11

# ece604 lecture 11 - Lecture 11 ECE 604 State Variable...

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Oct. 15, 2009 Linear Systems © Douglas Looze 1 Lecture 11 ECE 604 State Variable Analysis Doug Looze

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Oct. 15, 2009 Linear Systems © Douglas Looze 2 Announcements PS4 available Due next week October 22
Oct. 15, 2009 Linear Systems © Douglas Looze 3 Basis change Duality Began LTI systems Controllability Matrix ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 d t t t dt t t t = + = = + x Ax Bu x x y Cx Du Controllable if and only if Last Time

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Oct. 15, 2009 Linear Systems © Douglas Looze 4 Observable if and only if Observaability Matrix
Oct. 15, 2009 Linear Systems © Douglas Looze 5 Modal controllability Distinct eigenvalues System { } { } 1 1 , , , , λ n n v v ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 LS1 d t t t dt t t t = + = = + x Ax Bu x x y Cx Du

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Oct. 15, 2009 Linear Systems © Douglas Looze 6 Define:
Oct. 15, 2009 Linear Systems © Douglas Looze 7 Let ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 LS2 d t t t dt t t t = + = = + z z WBu z Wx y CVz Du Λ Theorem: A LTI system with distinct e- values is (C) if and only if WB has no zero rows. It is (O) if and only if CV has no zero columns.

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Oct. 15, 2009 Linear Systems © Douglas Looze 8 Today Modal controllabilty and observability Repeated eigenvalues Hautus tests Reading Rugh Ch. 9 p. 144–146, 149 Ch. 13, p. 221, 232
Oct. 15, 2009 Linear Systems © Douglas Looze 9 Theorem: Consider the LTI system (LS1) and let x ( t ) = W –1 z ( t ) where W = V

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ece604 lecture 11 - Lecture 11 ECE 604 State Variable...

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