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ece604 lecture 13

# ece604 lecture 13 - Lecture 13 ECE 604 State Variable...

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Oct. 22, 2009 Linear Systems © Douglas Looze 1 Lecture 13 ECE 604 State Variable Analysis Doug Looze

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Oct. 22, 2009 Linear Systems © Douglas Looze 2 Announcements PS4 due today PS5 available Due Thursday, October 29 Midterm exam Wednesday, November 4 7–9 P.M. LGRT A203 Open book, notes No electronic devices
Oct. 22, 2009 Linear Systems © Douglas Looze 3 Discrete-time systems: Reachability Last time ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 , 1 , 1 2 , 1 f f f f f f k k k k k k k k k = - - - + R B B B Φ Φ L ( 29 0 rank , f k k n = R ( 29 ( 29 ( 29 ( 29 ( 29 0 1 0 , , 1 , 1 f k T T f f f j k k k k j j j k j - = = + + W B B Φ Φ

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Oct. 22, 2009 Linear Systems © Douglas Looze 4 Discrete-time systems: Observability ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 0 1 1, rank , 1 1, f f f k k k k k k n k k k + + = = - - C C O C Φ Φ M ( 29 ( 29 ( 29 ( 29 ( 29 0 1 0 0 0 , , , f k T T f j k k k j k j j j k - = = M C C Φ Φ
Oct. 22, 2009 Linear Systems © Douglas Looze 5 Time invariant discrete-time systems Reachability 1 rank rank n n - = = R B AB A B L [ ] rank z n z - = 2200 ∈ I A B C Observability 1 rank rank n n - = = C CA O CA M rank z n z - = 2200 ∈ I A C C

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Oct. 22, 2009 Linear Systems © Douglas Looze 6 Today Canonical decomposition Reading Rugh Ch. 13
Oct. 22, 2009 Linear Systems © Douglas Looze 7 Canonical Decomposition We’ll show that every LTI system can be decomposed into (C) and (O) subsystem (C) and not (O) subsystem Not (C) and (O) subsystem Not (C) and not (O) subsystem IO response corresponds to (C) and (O) Homogeneous response corresponds to (O)

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Oct. 22, 2009 Linear Systems © Douglas Looze 8 Procedure Define controllable subspace Find basis Transform Define unobservable subspace Find basis Transform
Oct. 22, 2009

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