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

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Sept. 17, 2009 Linear Systems © Douglas Looze 1 Lecture 4 ECE 604 State Variable Analysis Doug Looze
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Sept. 17, 2009 Linear Systems © Douglas Looze 2 Announcements PS1 available Due next Thursday
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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
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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
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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
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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
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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 = =
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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
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