ece604 lecture 03

ece604 lecture 03 - Sept 15 2009 Linear Systems © Douglas...

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Unformatted text preview: Sept. 15, 2009 Linear Systems © Douglas Looze 1 Lecture 3 ECE 604 State Variable Analysis Doug Looze Sept. 15, 2009 Linear Systems © Douglas Looze 2 Announce Email sent (2) Office hours – Tomorrow 3–5 pm http://www.ecs.umass.edu/ece604 Sept. 15, 2009 Linear Systems © Douglas Looze 3 Last Time Linearization of state equation ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 [ ] ( 29 ( 29 , , where , , , , t t t t d t t t t t t dt t t t t δ δ δ δ ∂ ∂ ∂ ∂ = = = = = + ≡ ≡ x x u u x x u u x A x B u x f A x u x f B x u u Sept. 15, 2009 Linear Systems © Douglas Looze 4 Last Time (cont.) Linearization of output equation ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 [ ] ( 29 ( 29 , , where , , , , t t t t y t t t t t t t t t δ δ δ ∂ ∂ ∂ ∂ = = = = = + ≡ ≡ x x u u x x u u C x D u g C x u x g D x u u Sept. 15, 2009 Linear Systems © Douglas Looze 5 Today Solutions of state equations – Homogeneous equations • Peano-Baker series • Transition matrix Reading – Ch. 3, p. 40–47 – Ch. 4, p. 61–64 Sept. 15, 2009 Linear Systems © Douglas Looze 6 Linear State Variable Model Does this imply x ( t ) is a state? – Yes, if • There is a solution (existence) • There is only 1 solution (uniqueness) – What can go wrong? ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 d t t t t t t dt t t t t t = + = = + x A x B u x x y C x D u Sept. 15, 2009 Linear Systems © Douglas Looze 7 Problems Existence – Let the differential equation be Assume all matrices are piecewise continuous Sept. 15, 2009 Linear Systems © Douglas Looze 8 Problems Uniqueness – Let the differential equation be Linearity prevents this Sept. 15, 2009 Linear Systems © Douglas Looze 9 Homogeneous Systems Linear system – All matrices defined, continuous...
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ece604 lecture 03 - Sept 15 2009 Linear Systems © Douglas...

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