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Nonlinear Control Lecture # 20 Special nonlinear Forms Nonlinear - PowerPoint PPT Presentation

Nonlinear Control Lecture # 20 Special nonlinear Forms Nonlinear Control Lecture # 20 Special nonlinear Forms Normal Form 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


  1. Nonlinear Control Lecture # 20 Special nonlinear Forms Nonlinear Control Lecture # 20 Special nonlinear Forms

  2. Normal Form 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 def ˙ ∂x [ f ( x ) + g ( x ) u ] = L f h ( x ) + L g h ( x ) u L f h ( x ) = ∂h ∂xf ( x ) is the Lie Derivative of h with respect to f or along f Nonlinear Control Lecture # 20 Special nonlinear Forms

  3. L g L f h ( x ) = ∂ ( L f h ) g ( x ) ∂x f h ( x ) = L f L f h ( x ) = ∂ ( L f h ) L 2 f ( x ) ∂x h ( x ) = ∂ ( L k − 1 h ) f h ( x ) = L f L k − 1 f L k f ( 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 ) [ f ( x ) + g ( x ) u ] = L 2 f h ( x ) + L g L f h ( x ) u ∂x Nonlinear Control Lecture # 20 Special nonlinear Forms

  4. y (2) = L 2 L g L f h ( x ) = 0 ⇒ f h ( x ) y (3) = L 3 f h ( x ) + L g L 2 f h ( x ) u L g L ρ − 1 L g L i − 1 h ( x ) = 0 , i = 1 , 2 , . . . , ρ − 1; h ( x ) � = 0 f f y ( ρ ) = L ρ f h ( x ) + L g L ρ − 1 h ( x ) u f Definition 8.1 The system x = f ( x ) + g ( x ) u, ˙ y = h ( x ) has relative degree ρ , 1 ≤ ρ ≤ n , in R ⊂ D if ∀ x ∈ R L g L i − 1 L g L ρ − 1 h ( x ) = 0 , i = 1 , 2 , . . . , ρ − 1; h ( x ) � = 0 f f Nonlinear Control Lecture # 20 Special nonlinear Forms

  5. Example 8.1 Controlled van der Pol equation x 2 = ε [ − x 1 + x 2 − 1 3 x 3 x 1 = x 2 /ε, ˙ ˙ 2 + u ] , y = x 1 x 2 /ε = − x 1 + x 2 − 1 3 x 3 y = ˙ ˙ x 1 = x 2 /ε, y = ˙ ¨ 2 + u Relative degree two over R 2 x 2 = ε [ − x 1 + x 2 − 1 3 x 3 x 1 = x 2 /ε, ˙ ˙ 2 + u ] , y = x 2 y = ε [ − x 1 + x 2 − 1 3 x 3 Relative degree one over R 2 ˙ 2 + u ] , x 2 = ε [ − x 1 + x 2 − 1 3 x 3 y = 1 2 ( ε 2 x 2 1 + x 2 x 1 = x 2 /ε, ˙ ˙ 2 + u ] , 2 ) y = ε 2 x 1 ˙ x 2 = εx 2 2 − ( ε/ 3) x 4 ˙ x 1 + x 2 ˙ 2 + εx 2 u Relative degree one in { x 2 � = 0 } Nonlinear Control Lecture # 20 Special nonlinear Forms

  6. Example 8.2 (Field-controlled DC motor) x 1 ˙ = d 1 ( − x 1 − x 2 x 3 + V a ) x 2 ˙ = d 2 [ − f e ( x 2 ) + u ] x 3 ˙ = d 3 ( x 1 x 2 − bx 3 ) y = x 3 y = ˙ ˙ x 3 = d 3 ( x 1 x 2 − bx 3 ) y = d 3 ( x 1 ˙ ¨ x 2 + ˙ x 1 x 2 − b ˙ x 3 ) = ( · · · ) + d 2 d 3 x 1 u Relative degree two in { x 1 � = 0 } Nonlinear Control Lecture # 20 Special nonlinear Forms

  7. Example 8.3 H ( s ) = b m s m + b m − 1 s m − 1 + · · · + b 0 s n + a n − 1 s n − 1 + · · · + a 0 x = Ax + Bu, ˙ y = Cx 0 1 0 0  . . . . . .   0  0 0 1 0 . . . . . . 0     . . ... .     . . .     . . .      ...            A = , B =  ... .    .     .        .  . ... . .     . . 0         0 0 0 1     1 − a 0 − a 1 . . . . . . − a m . . . . . . − a n − 1 � � C = b 0 b 1 . . . . . . b m 0 . . . 0 Nonlinear Control Lecture # 20 Special nonlinear Forms

  8. y = CAx + CBu, ˙ If m = n − 1 , CB = b n − 1 � = 0 ⇒ ρ = 1 CA i − 1 B = 0 , i = 1 , . . . , n − m − 1 , CA n − m − 1 B = b m � = 0 y ( n − m ) = CA n − m x + CA n − m − 1 Bu ⇒ ρ = n − m 1 H ( s ) = N ( s ) N ( s ) Q ( s ) D ( s ) = Q ( s ) N ( s ) + R ( s ) = R ( s ) 1 1 + Q ( s ) N ( s ) u + e y ✲ ✲ 1 ✲ ❦ Q ( s ) − ✻ w R ( s ) ✛ N ( s ) Nonlinear Control Lecture # 20 Special nonlinear Forms

  9. � y, . . . , y ( ρ − 1) � State model of 1 /Q ( s ) : ξ = col y, ˙ ˙ ξ = ( A c + B c λ T ) ξ + B c b m e, y = C c ξ  0 1 0 0  . . .  0  0 0 1 0 . . . 0     . .  ...  . . .     . � � A c = . . , B c = , C c = 1 0 . . . 0 0   .      .    . 0   . 0 1     1 0 . . . . . . 0 0 State model of R ( s ) /N ( s ) η = A 0 η + B 0 y, ˙ w = C 0 η Nonlinear Control Lecture # 20 Special nonlinear Forms

  10. State model of H ( s ) η ˙ = A 0 η + B 0 C c ξ ˙ A c ξ + B c ( λ T ξ − b m C 0 η + b m u ) ξ = y = C c ξ The eigenvalues of A 0 are the zeros of H ( s ) Nonlinear Control Lecture # 20 Special nonlinear Forms

  11. Change of variables: φ 1 ( x )   . . .     φ n − ρ ( x )     φ ( x ) η     def  def z = T ( x ) = − − − = − − − = − − −        h ( x ) ψ ( x ) ξ     . .   .   L ρ − 1 h ( x ) f φ 1 to φ n − ρ are chosen such that T ( x ) is a diffeomorphism on a domain D x ⊂ R When ρ = n, z = T ( x ) = ψ ( x ) = ξ Nonlinear Control Lecture # 20 Special nonlinear Forms

  12. ∂φ η ˙ = ∂x [ f ( x ) + g ( x ) u ] = f 0 ( η, ξ ) + g 0 ( η, ξ ) u ˙ ξ i = ξ i +1 , 1 ≤ i ≤ ρ − 1 ˙ L ρ f h ( x ) + L g L ρ − 1 ξ ρ = h ( x ) u f y = ξ 1 Choose φ ( x ) such that T ( x ) is a diffeomorphism and ∂φ i ∂x g ( x ) = 0 , for 1 ≤ i ≤ n − ρ, ∀ x ∈ D x Always possible (at least locally) η = f 0 ( η, ξ ) ˙ Nonlinear Control Lecture # 20 Special nonlinear Forms

  13. Theorem 8.1 Suppose the system x = f ( x ) + g ( x ) u, ˙ y = h ( x ) has relative degree ρ ( ≤ n ) in R . If ρ = n , then for every x 0 ∈ R , a neighborhood N of x 0 exists such that the map T ( x ) = ψ ( x ) , restricted to N , is a diffeomorphism on N . If ρ < n , then, for every x 0 ∈ R , a neighborhood N of x 0 and smooth functions φ 1 ( x ) , . . . , φ n − ρ ( x ) exist such that ∂φ i ∂x g ( x ) = 0 , for 1 ≤ i ≤ n − ρ � φ ( x ) � is satisfied for all x ∈ N and the map T ( x ) = , ψ ( x ) restricted to N , is a diffeomorphism on N Nonlinear Control Lecture # 20 Special nonlinear Forms

  14. Normal Form: η ˙ = f 0 ( η, ξ ) ˙ ξ i = ξ i +1 , 1 ≤ i ≤ ρ − 1 ˙ L ρ f h ( x ) + L g L ρ − 1 ξ ρ = h ( x ) u f y = ξ 1  0 1 0 . . . 0    0 0 0 1 . . . 0 0     . . ...    .  . . . A c = . . , B c =     .     . .    0  . 0 1     1 0 . . . . . . 0 0 � 1 0 � C c = 0 . . . 0 Nonlinear Control Lecture # 20 Special nonlinear Forms

  15. η ˙ = f 0 ( η, ξ ) ˙ f h ( x ) + L g L ρ − 1 L ρ � � ξ = A c ξ + B c h ( x ) u f y = C c ξ ˜ γ ( η, ξ ) = L g L ρ − 1 ψ ( η, ξ ) = L ρ � � f h ( x ) x = T − 1 ( z ) , ˜ h ( x ) � � f x = T − 1 ( z ) ξ = A c ξ + B c [ ˜ ˙ ψ ( η, ξ ) + ˜ γ ( η, ξ ) u ] If x ∗ is an open-loop equilibrium point at which y = 0 ; i.e., f ( x ∗ ) = 0 and h ( x ∗ ) = 0 , then ψ ( x ∗ ) = 0 . Take φ ( x ∗ ) = 0 so that z = 0 is an open-loop equilibrium point. Nonlinear Control Lecture # 20 Special nonlinear Forms

  16. Zero Dynamics η ˙ = f 0 ( η, ξ ) ˙ L ρ f h ( x ) + L g L ρ − 1 � � ξ = A c ξ + B c h ( x ) u f y = C c ξ L ρ f h ( x ( t )) y ( t ) ≡ 0 ⇒ ξ ( t ) ≡ 0 ⇒ u ( t ) ≡ − L g L ρ − 1 h ( x ( t )) f ⇒ ˙ η = f 0 ( η, 0) Definition The equation ˙ η = f 0 ( η, 0) is called the zero dynamics of the system. The system is said to be minimum phase if the zero dynamics have an asymptotically stable equilibrium point in the domain of interest (at the origin if T (0) = 0 ) Nonlinear Control Lecture # 20 Special nonlinear Forms

  17. Z ∗ = { x ∈ R | h ( x ) = L f h ( x ) = · · · = L ρ − 1 h ( x ) = 0 } f y ( t ) ≡ 0 ⇒ x ( t ) ∈ Z ∗ � L ρ f h ( x ) � def ⇒ u = u ∗ ( x ) = − � L g L ρ − 1 h ( x ) � f � x ∈ Z ∗ The restricted motion of the system is described by � L ρ � f h ( x ) def x = f ∗ ( x ) ˙ = f ( x ) − g ( x ) L g L ρ − 1 h ( x ) f x ∈ Z ∗ Nonlinear Control Lecture # 20 Special nonlinear Forms

  18. Example 8.4 x 2 = ε [ − x 1 + x 2 − 1 3 x 3 x 1 = x 2 /ε, ˙ ˙ 2 + u ] , y = x 2 x 2 = ε [ − x 1 + x 2 − 1 3 x 3 y = ˙ ˙ 2 + u ] ⇒ ρ = 1 The system is in the normal form with η = x 1 and ξ = x 2 y ( t ) ≡ 0 ⇒ x 2 ( t ) ≡ 0 ⇒ ˙ x 1 = 0 Non-minimum phase Nonlinear Control Lecture # 20 Special nonlinear Forms

  19. Example 8.5 x 1 = − x 1 + 2 + x 2 3 ˙ u, ˙ x 2 = x 3 , ˙ x 3 = x 1 x 3 + u, y = x 2 1 + x 2 3 y = ˙ ˙ x 2 = x 3 y = ˙ ¨ x 3 = x 1 x 3 + u ⇒ ρ = 2 Z ∗ = { x 2 = x 3 = 0 } u = u ∗ ( x ) = 0 ⇒ x 1 = − x 1 ˙ Minimum phase Nonlinear Control Lecture # 20 Special nonlinear Forms

  20. Find φ ( x ) such that   2+ x 2 3 ∂φ 1+ x 2 � � 3 ∂φ ∂φ ∂φ φ (0) = 0 , ∂xg ( x ) = ∂x 1 , ∂x 2 ,  = 0  0  ∂x 3  1 and � φ ( x ) � T T ( x ) = x 2 x 3 is a diffeomorphism · 2 + x 2 ∂φ + ∂φ 3 = 0 1 + x 2 ∂x 1 ∂x 3 3 φ ( x ) = x 1 − x 3 − tan − 1 x 3 Nonlinear Control Lecture # 20 Special nonlinear Forms

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