Nonlinear Control Lecture # 9 State Feedback Stabilization - PowerPoint PPT Presentation

Nonlinear Control Lecture # 9 State Feedback Stabilization Nonlinear Control Lecture # 9 State Feedback Stabilization

Basic Concepts We want to stabilize the system x = f ( x, u ) ˙ at the equilibrium point x = x ss Steady-State Problem: Find steady-state control u ss s.t. 0 = f ( x ss , u ss ) x δ = x − x ss , u δ = u − u ss def x δ = f ( x ss + x δ , u ss + u δ ) ˙ = f δ ( x δ , u δ ) f δ (0 , 0) = 0 u δ = φ ( x δ ) ⇒ u = u ss + φ ( x − x ss ) Nonlinear Control Lecture # 9 State Feedback Stabilization

State Feedback Stabilization: Given x = f ( x, u ) ˙ [ f (0 , 0) = 0] find u = φ ( x ) [ φ (0) = 0] s.t. the origin is an asymptotically stable equilibrium point of x = f ( x, φ ( x )) ˙ f and φ are locally Lipschitz functions Nonlinear Control Lecture # 9 State Feedback Stabilization

Notions of Stabilization x = f ( x, u ) , ˙ u = φ ( x ) Local Stabilization: The origin of ˙ x = f ( x, φ ( x )) is asymptotically stable (e.g., linearization) Regional Stabilization: The origin of ˙ x = f ( x, φ ( x )) is asymptotically stable and a given region G is a subset of the region of attraction (for all x (0) ∈ G , lim t →∞ x ( t ) = 0 ) (e.g., G ⊂ Ω c = { V ( x ) ≤ c } where Ω c is an estimate of the region of attraction) Global Stabilization: The origin of ˙ x = f ( x, φ ( x )) is globally asymptotically stable Nonlinear Control Lecture # 9 State Feedback Stabilization

Semiglobal Stabilization: The origin of ˙ x = f ( x, φ ( x )) is asymptotically stable and φ ( x ) can be designed such that any given compact set (no matter how large) can be included in the region of attraction (Typically u = φ p ( x ) is dependent on a parameter p such that for any compact set G , p can be chosen to ensure that G is a subset of the region of attraction ) What is the difference between global stabilization and semiglobal stabilization? Nonlinear Control Lecture # 9 State Feedback Stabilization

Example 9.1 x = x 2 + u ˙ Linearization: x = u, ˙ u = − kx, k > 0 Closed-loop system: x = − kx + x 2 ˙ Linearization of the closed-loop system yields ˙ x = − kx . Thus, u = − kx achieves local stabilization The region of attraction is { x < k } . Thus, for any set {− a ≤ x ≤ b } with b < k , the control u = − kx achieves regional stabilization Nonlinear Control Lecture # 9 State Feedback Stabilization

The control u = − kx does not achieve global stabilization But it achieves semiglobal stabilization because any compact set {| x | ≤ r } can be included in the region of attraction by choosing k > r The control u = − x 2 − kx achieves global stabilization because it yields the linear closed-loop system ˙ x = − kx whose origin is globally exponentially stable Nonlinear Control Lecture # 9 State Feedback Stabilization

Linearization x = f ( x, u ) ˙ f (0 , 0) = 0 and f is continuously differentiable in a domain D x × D u that contains the origin ( x = 0 , u = 0) ( D x ⊂ R n , D u ⊂ R m ) x = Ax + Bu ˙ � � A = ∂f B = ∂f � � ∂x ( x, u ) ; ∂u ( x, u ) � � � � x =0 ,u =0 x =0 ,u =0 Assume ( A, B ) is stabilizable. Design a matrix K such that ( A − BK ) is Hurwitz u = − Kx Nonlinear Control Lecture # 9 State Feedback Stabilization

Closed-loop system: x = f ( x, − Kx ) ˙ Linearization: � ∂f ∂x ( x, − Kx ) + ∂f � x ˙ = ∂u ( x, − Kx ) ( − K ) x x =0 = ( A − BK ) x Since ( A − BK ) is Hurwitz, the origin is an exponentially stable equilibrium point of the closed-loop system Nonlinear Control Lecture # 9 State Feedback Stabilization

Feedback Linearization Consider the nonlinear system x = f ( x ) + G ( x ) u ˙ x ∈ R n , u ∈ R m f (0) = 0 , Suppose there is a change of variables z = T ( x ) , defined for all x ∈ D ⊂ R n , that transforms the system into the controller form z = Az + B [ ψ ( x ) + γ ( x ) u ] ˙ where ( A, B ) is controllable and γ ( x ) is nonsingular for all x ∈ D u = γ − 1 ( x )[ − ψ ( x ) + v ] ⇒ z = Az + Bv ˙ Nonlinear Control Lecture # 9 State Feedback Stabilization

v = − Kz Design K such that ( A − BK ) is Hurwitz The origin z = 0 of the closed-loop system z = ( A − BK ) z ˙ is globally exponentially stable u = γ − 1 ( x )[ − ψ ( x ) − KT ( x )] Closed-loop system in the x -coordinates: def x = f ( x ) + G ( x ) γ − 1 ( x )[ − ψ ( x ) − KT ( x )] ˙ = f c ( x ) Nonlinear Control Lecture # 9 State Feedback Stabilization

What can we say about the stability of x = 0 as an equilibrium point of ˙ x = f c ( x ) ? z = T ( x ) ⇒ ∂T ∂x ( x ) f c ( x ) = ( A − BK ) T ( x ) ∂f c J = ∂T ∂x (0) = J − 1 ( A − BK ) J, ∂x (0) (nonsingular) The origin of ˙ x = f c ( x ) is exponentially stable Is x = 0 globally asymptotically stable? In general No It is globally asymptotically stable if T ( x ) is a global diffeomorphism Nonlinear Control Lecture # 9 State Feedback Stabilization

What information do we need to implement the control u = γ − 1 ( x )[ − ψ ( x ) − KT ( x )] ? What is the effect of uncertainty in ψ , γ , and T ? Let ˆ γ ( x ) , and ˆ ψ ( x ) , ˆ T ( x ) be nominal models of ψ ( x ) , γ ( x ) , and T ( x ) γ − 1 ( x )[ − ˆ ψ ( x ) − K ˆ u = ˆ T ( x )] Closed-loop system: z = ( A − BK ) z + B ∆( z ) ˙ Nonlinear Control Lecture # 9 State Feedback Stabilization

z = ( A − BK ) z + B ∆( z ) ˙ ( ∗ ) V ( z ) = z T Pz, P ( A − BK ) + ( A − BK ) T P = − I Lemma 9.1 Suppose (*) is defined in D z ⊂ R n If � ∆( z ) � ≤ k � z � ∀ z ∈ D z , k < 1 / (2 � PB � ) , then the origin of (*) is exponentially stable. It is globally exponentially stable if D z = R n If � ∆( z ) � ≤ k � z � + δ ∀ z ∈ D z and B r ⊂ D z , then there exist positive constants c 1 and c 2 such that if δ < c 1 r and z (0) ∈ { z T Pz ≤ λ min ( P ) r 2 } , � z ( t ) � will be ultimately bounded by δc 2 . If D z = R n , � z ( t ) � will be globally ultimately bounded by δc 2 for any δ > 0 Nonlinear Control Lecture # 9 State Feedback Stabilization

Example 9.4 (Pendulum Equation) x 1 = x 2 , ˙ x 2 = − sin( x 1 + δ 1 ) − bx 2 + cu ˙ � 1 � u = [sin( x 1 + δ 1 ) − ( k 1 x 1 + k 2 x 2 )] c � � 0 1 A − BK = − k 1 − ( k 2 + b ) � 1 � u = [sin( x 1 + δ 1 ) − ( k 1 x 1 + k 2 x 2 )] c ˆ x 1 = x 2 , ˙ x 2 = − k 1 x 1 − ( k 2 + b ) x 2 + ∆( x ) ˙ � c − ˆ c � ∆( x ) = [sin( x 1 + δ 1 ) − ( k 1 x 1 + k 2 x 2 )] ˆ c Nonlinear Control Lecture # 9 State Feedback Stabilization

| ∆( x ) | ≤ k � x � + δ, ∀ x � � � � � � c − ˆ c c − ˆ c � � � � � k 2 1 + k 2 k = 1 + , δ = � | sin δ 1 | � � � � 2 c ˆ ˆ c � � � � p 11 � p 12 P ( A − BK ) + ( A − BK ) T P = − I, P = p 12 p 22 1 k < ⇒ GUB � p 2 12 + p 2 2 22 1 sin δ 1 = 0 & k < ⇒ GES � p 2 12 + p 2 2 22 Nonlinear Control Lecture # 9 State Feedback Stabilization

Is feedback linearization a good idea? Example 9.5 x = ax − bx 3 + u, ˙ a, b > 0 u = − ( k + a ) x + bx 3 , k > 0 , ⇒ x = − kx ˙ − bx 3 is a damping term. Why cancel it? x = − kx − bx 3 u = − ( k + a ) x, k > 0 , ⇒ ˙ Which design is better? Nonlinear Control Lecture # 9 State Feedback Stabilization

Example 9.6 x 1 = x 2 , ˙ x 2 = − h ( x 1 ) + u ˙ h (0) = 0 and x 1 h ( x 1 ) > 0 , ∀ x 1 � = 0 Feedback Linearization: u = h ( x 1 ) − ( k 1 x 1 + k 2 x 2 ) With y = x 2 , the system is passive with � x 1 h ( z ) dz + 1 2 x 2 V = 2 0 ˙ V = h ( x 1 ) ˙ x 1 + x 2 ˙ x 2 = yu Nonlinear Control Lecture # 9 State Feedback Stabilization

The control u = − σ ( x 2 ) , σ (0) = 0 , x 2 σ ( x 2 ) > 0 ∀ x 2 � = 0 creates a feedback connection of two passive systems with storage function V ˙ V = − x 2 σ ( x 2 ) x 2 ( t ) ≡ 0 ⇒ ˙ x 2 ( t ) ≡ 0 ⇒ h ( x 1 ( t )) ≡ 0 ⇒ x 1 ( t ) ≡ 0 Asymptotic stability of the origin follows from the invariance principle Which design is better? Nonlinear Control Lecture # 9 State Feedback Stabilization

The control u = − σ ( x 2 ) has two advantages: It does not use a model of h The flexibility in choosing σ can be used to reduce | u | However, u = − σ ( x 2 ) cannot arbitrarily assign the rate of decay of x ( t ) . Linearization of the closed-loop system at the origin yields the characteristic equation s 2 + σ ′ (0) s + h ′ (0) = 0 One of the two roots cannot be moved to the left of � Re[ s ] = − h ′ (0) Nonlinear Control Lecture # 9 State Feedback Stabilization

Partial Feedback Linearization Consider the nonlinear system x = f ( x ) + G ( x ) u ˙ [ f (0) = 0] Suppose there is a change of variables � η � T 1 ( x ) � � z = = T ( x ) = ξ T 2 ( x ) defined for all x ∈ D ⊂ R n , that transforms the system into ˙ η = f 0 ( η, ξ ) , ˙ ξ = Aξ + B [ ψ ( x ) + γ ( x ) u ] ( A, B ) is controllable and γ ( x ) is nonsingular for all x ∈ D Nonlinear Control Lecture # 9 State Feedback Stabilization

u = γ − 1 ( x )[ − ψ ( x ) + v ] ˙ η = f 0 ( η, ξ ) , ˙ ξ = Aξ + Bv v = − Kξ, where ( A − BK ) is Hurwitz Nonlinear Control Lecture # 9 State Feedback Stabilization