Nonlinear Control Lecture # 8 Special nonlinear Forms Nonlinear - PowerPoint PPT Presentation

Nonlinear Control Lecture # 8 Special nonlinear Forms Nonlinear Control Lecture # 8 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 → 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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

Example 8.1 (Read) Controlled van der Pol equation 3 x 3 x 2 = ε [ − x 1 + x 2 − 1 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 # 8 Special nonlinear Forms

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 one in { x 1 � = 0 } Nonlinear Control Lecture # 8 Special nonlinear Forms

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 1 0     − a 0 − a 1 . . . . . . − a m . . . . . . − a n − 1 1 � � C b 0 b 1 . . . . . . b m 0 . . . 0 = Nonlinear Control Lecture # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

∂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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

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 # 8 Special nonlinear Forms

x 1 − x 3 − tan − 1 x 3     1 0 ⋆ ∂T  , T ( x ) = x 2 ∂x = 0 1 0    x 3 0 0 1 T ( x ) is a global diffeomorphism 1 + 2 + ξ 2 � � � η + ξ 2 + tan − 1 ξ 2 2 � η ˙ = − ξ 2 1 + ξ 2 2 ˙ ξ 1 = ξ 2 η + ξ 2 + tan − 1 ξ 2 ˙ � � ξ 2 = ξ 2 + u y = ξ 1 Nonlinear Control Lecture # 8 Special nonlinear Forms

Controller Form Definition A nonlinear system is in the controller form if x = Ax + B [ ψ ( x ) + γ ( x ) u ] ˙ where ( A, B ) is controllable and γ ( x ) is a nonsingular matrix for all x in the domain of interest u = γ − 1 ( x )[ − ψ ( x ) + v ] ⇒ x = Ax + Bv ˙ Any system that can be represented in the controller form is said to be feedback linearizable Nonlinear Control Lecture # 8 Special nonlinear Forms

Example 8.7 ( m -link robot) M ( q )¨ q + C ( q, ˙ q ) ˙ q + D ˙ q + g ( q ) = u q is an m -dimensional vector of joint positions and M ( q ) is a nonsingular inertial matrix � 0 � 0 � q � � � I m x = , A = , B = q ˙ 0 0 I m ψ = − M − 1 ( C ˙ γ = M − 1 q + D ˙ q + g ) , Nonlinear Control Lecture # 8 Special nonlinear Forms