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ece604 lecture 06

# ece604 lecture 06 - Lecture 6 ECE 604 State Variable...

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Sept. 24, 2009 Linear Systems © Douglas Looze 1 Lecture 6 ECE 604 State Variable Analysis Doug Looze

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Sept. 24, 2009 Linear Systems © Douglas Looze 2 Announcements PS1 due Solutions to be posted on website Password: linear PS2 available Due next Thursday
Sept. 24, 2009 Linear Systems © Douglas Looze 3 Last Time Discrete-time systems Linearization Transition matrix Properties Exercises

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Sept. 24, 2009 Linear Systems © Douglas Looze 4 Today Solution to linear system Inhomogeneous differential equation Variation of constants formula Reading Ch. 3, p. 47–50 Ch. 5, p. 78–81
Sept. 24, 2009 Linear Systems © Douglas Looze 5 Forced Linear Systems Examine 3 methods Superposition Direct integration Verification of formula ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 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

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Sept. 24, 2009 Linear Systems © Douglas Looze 6 Superposition
Sept. 24, 2009 Linear Systems © Douglas Looze 7 Response to single weighted impulse ( 29 ( 29 0 For 0, response to is t σ δ σ = - x u ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 , t t t t t + < = = x B u B u Φ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 d t t t t t dt t t t t t = + = + x A x B u y C x D u

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Sept. 24, 2009 Linear Systems © Douglas Looze 8 Non-zero initial conditions Again apply superposition By superposition (add & integrate) ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 , , t t t t t t d σ = + x x B u Φ Φ Variation of constants formula ( 29 ( 29 ( 29 ( 29 , t t x B u = Φ
Sept. 24, 2009 Linear Systems © Douglas Looze 9 Direct integration Multiply by inverse of transition matrix ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 d t t t t t dt t t t t t = + = + x A x B u y C x D u ( 29 , t σ Φ ( 29 , t Φ ( 29 , t Φ I ( 29 , t τ Φ ( 29 0 , t t Φ ( 29 ( 29 ( 29 , , d t t t dt = - A Φ Φ

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Sept. 24, 2009 Linear Systems © Douglas Looze 10 Verify formula Initial conditions: General formula: ( 29 ( 29 ( 29 ( 29 ( 29 0 , , 0 0 t t t t t t d τ = + x x B u Φ Φ
Sept. 24, 2009 Linear Systems © Douglas Looze 11 Liebnitz rule ( 29 ( 29 ( 29 { } ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , , , , g x f x g x f x dg x df x h x d h x g x h x f x x dx dx h x d x σ = - + ( 29 ( 29 ( 29 ( 29 ( 29 0 , , 0 0 t t t t t t d τ = + x x B u Φ Φ ( 29 ( 29 , 0 0 t t t A x Φ I

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Sept. 24, 2009 Linear Systems © Douglas Looze 12 Variation of Constants Solution Inhomogeneous linear system ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 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 Solution ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 0 0 , , , , t t t t t t t t d t t t t t t d t t σ = + = + + x x B u y C x C B u D u Φ Φ Φ Φ
Sept. 24, 2009 Linear Systems © Douglas Looze 13 State Transformation

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ece604 lecture 06 - Lecture 6 ECE 604 State Variable...

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