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Lect_23 - Nonlinear Systems and Control Lecture 23...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 23 Controller Form – p. 1/1 8 Definition: A nonlinear system is in the controller form if ˙ x = Ax + Bγ ( x )[ u − α ( x )] where ( A,B ) is controllable and γ ( x ) is a nonsingular u = α ( x ) + γ − 1 ( x ) v ⇒ ˙ x = Ax + Bv The n-dimensional single-input (SI) system ˙ x = f ( x ) + g ( x ) u can be transformed into the controller form if ∃ h ( x ) s.t. ˙ x = f ( x ) + g ( x ) u, y = h ( x ) has relative degree n . Why? – p. 2/1 8 Transform the system into the normal form ˙ z = A c z + B c γ ( z )[ u − α ( z )] , y = C c z On the other hand, if there is a change of variables ζ = S ( x ) that transforms the SI system ˙ x = f ( x ) + g ( x ) u into the controller form ˙ ζ = Aζ + Bγ ( ζ )[ u − α ( ζ )] then there is a function h ( x ) such that the system ˙ x = f ( x ) + g ( x ) u, y = h ( x ) has relative degree n . Why? – p. 3/1 8 For any controllable pair ( A,B ) , we can find a nonsingular matrix M that transforms ( A,B ) into a controllable canonical form: MAM − 1 = A c + B c λ T , MB = B c z = Mζ = MS ( x ) def = T ( x ) ˙ z = A c z + B c γ ( · )[ u − α ( · )] h ( x ) = T 1 ( x ) – p. 4/1 8 In summary, the n-dimensional SI system ˙ x = f ( x ) + g ( x ) u is transformable into the controller form if and only if ∃ h ( x ) such that ˙ x = f ( x ) + g ( x ) u, y = h ( x ) has relative degree n Search for a smooth function h ( x ) such that L g L i − 1 f h ( x ) = 0 , i = 1 , 2 ,...,n − 1 , and L g L...
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