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Unformatted text preview: Sept. 10, 2009 Linear Systems © Douglas Looze 1 Lecture 2 ECE 604 State Variable Analysis Doug Looze Sept. 10, 2009 Linear Systems © Douglas Looze 2 Announcements Email sent Reading – Ch. 2: p. 28–34 – Ch. 1: p. 10–11 Sept. 10, 2009 Linear Systems © Douglas Looze 3 Last Time Concept of state variable models – Know • State • Future inputs – Determine • Future outputs The state of a system at time t is information at time t that, together with { u ( t ): t ≥ t } and the system model determine uniquely the system behavior for t ≥ t Sept. 10, 2009 Linear Systems © Douglas Looze 4 State Model Assume – Finite dimensional ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , , ; , , d t t t t t dt t t t t = = = x f x u x x y g x u – Nonlinear differential equation for state – Memoryless, nonlinear output equation Sept. 10, 2009 Linear Systems © Douglas Looze 5 Linear System State variable model ( 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. 10, 2009 Linear Systems © Douglas Looze 6 Constant coefficients implies LTI – Coefficients given as matrices LTI for systems with appropriately varying coefficients – But : Assume LTI implies constant coefficients • Can always find such a model ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 d t t t t dt t t t = + = = + x Ax Bu x x y Cx Du Sept. 10, 2009 Linear Systems © Douglas Looze 7 Today Linearization – Obtaining approximate linear models from nonlinear models Examples Sept. 10, 2009 Linear Systems © Douglas Looze 8 Linearization Most system are nonlinear ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , , given (*) , , d t t t t t dt t t t t = = x f x u x y g x u Assume we can find a solution to (*) Desire: approximate solution for small deviations from nominal solution ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 , , given (*) , , d t t t t t dt t t t t = = x f x u x y g x u Sept. 10, 2009 Linear Systems © Douglas Looze 9 Define ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 perturbations (assume small) t t t t t t t t t δ δ δ = = = x x x u u u y y y Assume – ( 29 ( 29 ( 29 , continuous piecewise continuous t t t x y u – f , g have Taylor series expansions with positive convergence radii around nominal solutions for each t . Sept. 10, 2009 Linear Systems © Douglas Looze 10 i th differential equation ( 29 ( 29 ( 29 ( 29 , , i i x t f t t t = x u & – Taylor series: Sept. 10, 2009 Linear Systems © Douglas Looze 11 ( 29 ( 29 ( 29 ( 29 , , i i x t f t t t = x u & δ + x x δ + u u Sept. 10, 2009 Linear Systems © Douglas Looze 12 ( 29 ( 29 ( 29 ( 29 , , i i x t f t t t = x u & Sept. 10, 2009Sept....
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 Spring '10

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