ece604 lecture 10

# ece604 lecture 10 - Lecture 10 ECE 604 State Variable...

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Oct. 8, 2009 Linear Systems © Douglas Looze 1 Lecture 10 ECE 604 State Variable Analysis Doug Looze

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Oct. 8, 2009 Linear Systems © Douglas Looze 2 Announcements PS4 available Due 2 weeks from today October 22 No class next Tuesday
Oct. 8, 2009 Linear Systems © Douglas Looze 3 Controllability Define ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 1 1 j j j t t d t t t t dt - - = = - + K B K A K K Then ( 29 ( 29 ( 29 ( 29 , , j j j t t σ ∂σ = K B Φ Φ

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Oct. 8, 2009 Linear Systems © Douglas Looze 4 Theorem: Suppose that, for some t , B ( t ) is q times differentiable and A ( t ) is ( q –1) times differentiable. Then the linear system is controllable over any interval containing t if ( 29 ( 29 0 q rank t t n = K K L Note: ( 29 ( 29 ( 29 , j j j t t t σ ∂σ = = K B Φ
Oct. 8, 2009 Linear Systems © Douglas Looze 5 Observability Define ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 1 1 j j j t t d t t t t dt - - = = + L C L L A L ( 29 ( 29 ( 29 , j j j t t t t t σ = = L C Φ Theorem: Suppose that, for some t , C ( t ) is q times differentiable and A ( t ) is ( q –1) time differentiable. Then the linear system is observable over any interval containing t if ( 29 ( 29 0 q t rank n t = L L M

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Oct. 8, 2009 Linear Systems © Douglas Looze 6 Today Basis change Duality Time-invariant systems Reading Rugh, Ch. 9 p. 144–146, 149
Oct. 8, 2009 Linear Systems © Douglas Looze 7 Change of Basis (C) and (O) unaffected 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 ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 , , , f t T T f t t t t t t t t t dt = W B B Φ Φ ( 29 ( 29 ( 29 ( 29 ( 29 0 0 0 0 , , , f t T T f t t t t t t t t t dt = M C C Φ Φ

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Oct. 8, 2009 Linear Systems © Douglas Looze 8 Basis change
Oct. 8, 2009 Linear Systems © Douglas Looze 9 Same for (O) I I

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Oct. 8, 2009
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ece604 lecture 10 - Lecture 10 ECE 604 State Variable...

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