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# Lect_37 - Nonlinear Systems and Control Lecture 37...

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Nonlinear Systems and Control Lecture # 37 Observers Linearization and Extended Kalman Filter (EKF) – p. 1/1

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Linear Observer via Linearization ˙ x = f ( x, u ) , y = h ( x ) 0 = f ( x ss , u ss ) , y ss = h ( x ss ) Linearize about the equilibrium point: ˙ x δ = Ax δ + Bu δ , y δ = Cx δ x δ = x x ss , u δ = u u ss , y δ = y y ss What are A , B , C ? ˙ ˆ x δ = A ˆ x δ + Bu δ + H ( y δ C ˆ x δ ) , ˆ x = x ss + ˆ x δ ( A HC ) is Hurwitz It will work locally for sufficiently small bardbl x δ (0) bardbl , bardbl ˆ x δ (0) bardbl , and bardbl u δ ( t ) bardbl – p. 2/1
Feedback Control: ˙ ˆ x δ = A ˆ x δ + Bu δ + H ( y δ C ˆ x δ ) u δ = K ˆ x δ , u = u ss K ˆ x δ Verify that the closed-loop system has an equilibrium point at x = x ss , ˜ x = 0 and linearization at the equilibrium point yields bracketleftBigg ˙ x δ ˙ ˜ x bracketrightBigg = bracketleftBigg ( A BK ) BK 0 ( A HC ) bracketrightBigg bracketleftBigg x δ ˜ x bracketrightBigg Which theorem would justify this controller locally? – p. 3/1

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Nonlinear Observer via Linearization ˙ x = f ( x, u ) , y = h ( x ) 0 = f ( x ss , u ss ) , y ss = h ( x ss ) ˙ ˆ x = f x, u ) + H [ y h
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