# Lect_22 - Nonlinear Systems and Control Lecture 22 Normal...

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Nonlinear Systems and Control Lecture # 22 Normal Form – p. 1/1 7

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Relative Degree ˙ x = f ( x ) + g ( x ) u, y = h ( x ) where f , g , and h are sufficiently smooth in a domain D f : D R n and g : D R n are called vector fields on D ˙ y = ∂h ∂x [ f ( x ) + g ( x ) u ] def = L f h ( x ) + L g h ( x ) u L f h ( x ) = ∂h ∂x f ( x ) is the Lie Derivative of h with respect to f or along f – p. 2/1 7
L g L f h ( x ) = ( L f h ) ∂x g ( x ) L 2 f h ( x ) = L f L f h ( x ) = ( L f h ) ∂x f ( x ) L k f h ( x ) = L f L k 1 f h ( x ) = ( L k 1 f h ) ∂x f ( x ) L 0 f h ( x ) = h ( x ) ˙ y = L f h ( x ) + L g h ( x ) u L g h ( x ) = 0 ˙ y = L f h ( x ) y (2) = ( L f h ) ∂x [ f ( x ) + g ( x ) u ] = L 2 f h ( x ) + L g L f h ( x ) u – p. 3/1 7

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L g L f h ( x ) = 0 y (2) = L 2 f h ( x ) y (3) = L 3 f h ( x ) + L g L 2 f h ( x ) u L g L i 1 f h ( x ) = 0 , i = 1 , 2 ,...,ρ 1; L g L ρ 1 f h ( x ) n = 0 y ( ρ ) = L ρ f h ( x ) + L g L ρ 1 f h ( x ) u Definition: The system ˙ x = f ( x ) + g ( x ) u, y = h ( x ) has relative degree ρ , 1 ρ n , in D 0 D if x D 0 L g L i 1 f h ( x ) = 0 , i = 1 , 2 ,...,ρ 1; L g L ρ 1 f h ( x ) n = 0 – p. 4/1 7
Example ˙ x 1 = x 2 , ˙ x 2 = x 1 + ε (1 x 2 1 ) x 2 + u, y = x 1 , ε > 0 ˙ y = ˙ x 1 = x 2 ¨ y = ˙ x 2 = x 1 + ε

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Lect_22 - Nonlinear Systems and Control Lecture 22 Normal...

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