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Unformatted text preview: Sept. 17, 2009 Linear Systems Douglas Looze 1 Lecture 4 ECE 604 State Variable Analysis Doug Looze 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 d t t t t dt = = x A x x x Assume A ( t ) defined, piecewise continuous for all t Existence Successive approximations Weierstrasse Mtest Vector and induced norms Sept. 17, 2009 Linear Systems Douglas Looze 4 Uniqueness Assumed 2 solutions Difference (error) Showed difference must be zero GronwellBellman 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/semigroup rule Fundamental matrices Same matrix equation Initial condition X (invertible) ( 29 ( 29 ( 29 1 , t t t t = X X ( 29 ( 29 ( 29 ( 29 d t t t t dt = = X A X X X 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 sup L L Y x X x x Sept. 17, 2009 Linear Systems Douglas Looze 7 Example ( 29 ( 29 ( 29 ( 29 1 cos 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 ( 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 1 0,0 0,0 1 t t t t t d t t t t dt = = 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 1 0,0 0,0 1 t t t t t d t t t t dt =...
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 Spring '10

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