ece604 lecture 08

# ece604 lecture 08 - Lecture 8 ECE 604 State Variable...

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Oct. 1, 2009 Linear Systems © Douglas Looze 1 Lecture 8 ECE 604 State Variable Analysis Doug Looze

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Oct. 1, 2009 Linear Systems © Douglas Looze 2 Announcements PS2 due PS3 available Due next Thursday
Oct. 1, 2009 Linear Systems © Douglas Looze 3 Last Time Modal decomposition ( 29 1 0 t t e t e - = A x P P x Λ 142 43 Assume full set of e-vectors { } 1 , , n i i i λ = v v Av v Define diagonalizing transformation Note: = AP P Λ

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Oct. 1, 2009 Linear Systems © Douglas Looze 4 Modal decomposition for inhomogeneous systems ( 29 ( 29 ( 29 ( 29 ( 29 0 1 i n t t H i i i t e d λ σ - = = y Cv w B u E-value of A Right e-vector Left e-vector ( 29 ( 29 ( 29 ( 29 1 1 n H i i i i s s s = = - y Cv w B u ( 29 ( 29 ( 29 1 1 n H i i i i s s = = - G Cv w B Partial Fraction
Oct. 1, 2009 Linear Systems © Douglas Looze 5 Today Discrete-time system solutions Controllability and observability Reading Ch. 20–21 Ch. 9, p. 142–144, 148–149

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Oct. 1, 2009 Linear Systems © Douglas Looze 6 Forced Linear Discrete-Time Systems ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 1 k k k k k k k k k k k + = + = = + x A x B u x x y C x D u Iterate ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 1 k k k k + = + x A x B u ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 0 0 0 0 2 1 1 1 1 k k k k k k k k + = + + + + + + x A A x A B u B u
Oct. 1, 2009 Linear Systems © Douglas Looze 7 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 1 k k k k k k k k k k k + = + = = + x A x B u x x y C x D u Iterate (cont.) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 0 0 0 0 2 1 1 1 1 k k k k k k k k + = + + + + + + x A A x A B u B u ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 1 2 1 2 1 1 2 2 k k k k k k k k k k k k k + = + + + + + + + + + + + + x A A A x A A B u A B u B u

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Oct. 1, 2009 Linear Systems © Douglas Looze 8 Solution ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 1 0 0 1 0 0 , , 1 , , 1 k j k k j k k k k k j j j k k k k k k j k k k k - = - = = + + = + + + x x B u y C x C B u D u Φ Φ Φ Φ Transition matrix ( 29 ( 29 ( 29 ( 29 0 0 0 0 1 2 , k k k k k k k k k - - = = A A A I Φ L ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 1 2 1 2 1 1 2 2 k k k k k k k k k k k k k + = + + + + + + + + + + + + x A A A x A A B u A B u B u
Oct. 1, 2009 Linear Systems © Douglas Looze 9 State Transformation Assume that basis for the state is changed ( 29 ( 29 ( 29 k k k = x P z P ( k ) is assumed to be invertible for all k Transformed state model ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 k k k k k + + = P z A B u ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 k k k k k k k k k k k k k k - - + = + + + = + z P A P z P B u y C P z D u ( 29 ( 29 k k P z

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Oct. 1, 2009 Linear Systems © Douglas Looze 10 Characterization of Equivalence Theorem: Two systems with states x ( k ) and z ( k ) are equivalent if and only if there exists a nonsingular state transformation P ( k ) such that ( 29 ( 29 ( 29 k k k = x P z
Oct. 1, 2009 Linear Systems © Douglas Looze 11 Time-Invariant Systems Solution

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## ece604 lecture 08 - Lecture 8 ECE 604 State Variable...

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