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Unformatted text preview: Nonlinear Systems and Control Lecture # 27 Stabilization Partial Feedback Linearization – p. 1/1 1 Consider the nonlinear system ˙ x = f ( x ) + G ( x ) u [ f (0) = 0] Suppose there is a change of variables z = bracketleftBigg η ξ bracketrightBigg = T ( x ) = bracketleftBigg T 1 ( x ) T 2 ( x ) bracketrightBigg defined for all x ∈ D ⊂ R n , that transforms the system into ˙ η = f ( η,ξ ) ˙ ξ = Aξ + Bγ ( x )[ u − α ( x )] ( A,B ) is controllable and γ ( x ) is nonsingular for all x ∈ D – p. 2/1 1 u = α ( x ) + γ − 1 ( x ) v ˙ η = f ( η,ξ ) , ˙ ξ = Aξ + Bv Suppose the origin of ˙ η = f ( η, 0) is asymptotically stable v = − Kξ, where ( A − BK ) is Hurwitz Lemma 13.1: The origin of ˙ η = f ( η,ξ ) , ˙ ξ = ( A − BK ) ξ is asymptotically stable if the origin of ˙ η = f ( η, 0) is asymptotically stable Proof: V ( η,ξ ) = V 1 ( η )+ k radicalbig ξ T Pξ – p. 3/1 1 If the origin of ˙ η = f ( η, 0) is globally asymptotically stable, will the origin of...
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 Spring '08
 CHOI
 Nonlinear system, Stability theory, η, ξ

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