Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Passivity-Based Control x = f ( x, u ) , ˙ y = h ( x ) , f (0 , 0) = 0 V = ∂V u T y ≥ ˙ ∂x f ( x, u ) Theorem 9.1 If the system is (1) passive with a radially unbounded positive definite storage function and (2) zero-state observable, then the origin can be globally stabilized by y T φ ( y ) > 0 ∀ y � = 0 u = − φ ( y ) , φ (0) = 0 , Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Proof V = ∂V ˙ ∂x f ( x, − φ ( y )) ≤ − y T φ ( y ) ≤ 0 ˙ V ( x ( t )) ≡ 0 ⇒ y ( t ) ≡ 0 ⇒ u ( t ) ≡ 0 ⇒ x ( t ) ≡ 0 Apply the invariance principle A given system may be made passive by (1) Choice of output, (2) Feedback, or both Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Choice of Output ∂V x = f ( x ) + G ( x ) u, ˙ ∂x f ( x ) ≤ 0 , ∀ x No output is defined. Choose the output as � T � ∂V def y = h ( x ) = ∂x G ( x ) V = ∂V ∂x f ( x ) + ∂V ˙ ∂x G ( x ) u ≤ y T u Check zero-state observability Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Example 9.14 x 2 = − x 3 x 1 = x 2 , ˙ ˙ 1 + u V ( x ) = 1 4 x 4 1 + 1 2 x 2 2 ˙ V = x 3 1 x 2 − x 2 x 3 With u = 0 1 = 0 Take y = ∂V ∂x G = ∂V = x 2 ∂x 2 Is it zero-state observable? with u = 0 , y ( t ) ≡ 0 ⇒ x ( t ) ≡ 0 u = − (2 k/π ) tan − 1 ( x 2 ) u = − kx 2 or ( k > 0) Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Feedback Passivation Definition The system x = f ( x ) + G ( x ) u, ˙ y = h ( x ) ( ∗ ) is equivalent to a passive system if ∃ u = α ( x ) + β ( x ) v such that x = f ( x ) + G ( x ) α ( x ) + G ( x ) β ( x ) v, ˙ y = h ( x ) is passive Theorem [20] The system (*) is locally equivalent to a passive system (with a positive definite storage function) if it has relative degree one at x = 0 and the zero dynamics have a stable equilibrium point at the origin with a positive definite Lyapunov function Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Example 9.15 ( m -link Robot Manipulator) M ( q )¨ q + C ( q, ˙ q ) ˙ q + D ˙ q + g ( q ) = u M = M T > 0 , ( ˙ M − 2 C ) T = − ( ˙ M − 2 C ) , D = D T ≥ 0 Stabilize the system at q = q r e = q − q r , e = ˙ ˙ q M ( q )¨ e + C ( q, ˙ q )˙ e + D ˙ e + g ( q ) = u ( e = 0 , ˙ e = 0) is not an open-loop equilibrium point ( K p = K T u = g ( q ) − K p e + v, p > 0) M ( q )¨ e + C ( q, ˙ q )˙ e + D ˙ e + K p e = v Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
M ( q )¨ e + C ( q, ˙ q )˙ e + D ˙ e + K p e = v 2 e T K p e V = 1 e T M ( q )˙ e + 1 2 ˙ e T ( ˙ ˙ V = 1 e T D ˙ e T K p e + ˙ e T v + e T K p ˙ e T v 2 ˙ M − 2 C )˙ e − ˙ e − ˙ e ≤ ˙ y = ˙ e Is it zero-state observable? Set v = 0 e ( t ) ≡ 0 ⇒ ¨ ˙ e ( t ) ≡ 0 ⇒ K p e ( t ) ≡ 0 ⇒ e ( t ) ≡ 0 e T φ (˙ v = − φ (˙ e ) , [ φ (0) = 0 , ˙ e ) > 0 , ∀ ˙ e � = 0] u = g ( q ) − K p e − φ (˙ e ) K d = K T Special case: u = g ( q ) − K p e − K d ˙ e, d > 0 Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Passivity-Based Control: Cascade Connection x = f a ( x ) + F ( x, y ) y, ˙ z = f ( z ) + G ( z ) u, ˙ y = h ( z ) f a (0) = 0 , f (0) = 0 , h (0) = 0 ∂V ∂z f ( z ) + ∂V ∂W ∂z G ( z ) u ≤ y T u, ∂x f a ( x ) ≤ 0 U ( x, z ) = W ( x ) + V ( z ) � � T � U ≤ ∂W � ∂W ˙ ∂x F ( x, y ) y + y T u = y T u + ∂x F ( x, y ) � T � ∂W ˙ U ≤ y T v u = − ∂x F ( x, y ) + v ⇒ Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
The system x ˙ = f a ( x ) + F ( x, y ) y � T � ∂W z ˙ = f ( z ) − G ( z ) ∂x F ( x, y ) + G ( z ) v y = h ( z ) with input v and output y is passive with storage function U [ φ (0) = 0 , y T φ ( y ) > 0 ∀ y � = 0] v = − φ ( y ) , U ≤ ∂W ˙ ∂x f a ( x ) − y T φ ( y ) ≤ 0 , ˙ U = 0 ⇒ x = 0& y = 0 ⇒ u = 0 ˙ ZSO of driving system: U ( t ) ≡ 0 ⇒ z ( t ) ≡ 0 Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Theorem 9.2 Suppose the system z = f ( z ) + G ( z ) u, ˙ y = h ( z ) is zero-state observable and passive with a radially unbounded, positive definite storage function; the origin of ˙ x = f a ( x ) is globally asymptotically stable and W ( x ) is a radially unbounded, positive definite Lyapunov function � T � ∂W Then, u = − ∂x F ( x, y ) − φ ( y ) , globally stabilizes the origin ( x = 0 , z = 0) Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Example 9.16 (see Examples 9.7 and 9.12) x = − x + x 2 z, ˙ z = u ˙ With y = z as the output, the system takes the form of the cascade connection z = u, ˙ y = z 2 z 2 and zero-state observable is passive with V ( z ) = 1 2 x 2 ⇒ ˙ W ( x ) = 1 W = − x 2 x = − x, ˙ u = − x 3 − kz, k > 0 Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Robust State Feedback Stabilization Definition 10.1 The system x = f ( x, u ) + δ ( t, x, u ) ˙ is said to be practically stabilizable by the feedback control u = φ ( x ) if for every b > 0 , the control can be designed such that the solutions are ultimately bounded by b ; that is � x ( t ) � ≤ b, ∀ t ≥ T, for some T > 0 local practical stabilization regional practical stabilization global practical stabilization semiglobal practical stabilization Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Sliding Mode Control Example 10.1 x 1 = x 2 ˙ x 2 = h ( x ) + g ( x ) u, ˙ g ( x ) ≥ g 0 > 0 Sliding Manifold (Surface): s = ax 1 + x 2 = 0 s ( t ) ≡ 0 ⇒ x 1 = − ax 1 ˙ a > 0 ⇒ t →∞ x 1 ( t ) = 0 lim How can we bring the trajectory to the manifold s = 0 ? How can we maintain it there? Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
s = a ˙ ˙ x 1 + ˙ x 2 = ax 2 + h ( x ) + g ( x ) u Suppose � � ax 2 + h ( x ) � � � ≤ ̺ ( x ) � � g ( x ) � V = 1 2 s 2 ˙ V = s ˙ s = s [ ax 2 + h ( x )] + g ( x ) su ≤ g ( x ) | s | ̺ ( x ) + g ( x ) su β ( x ) ≥ ̺ ( x ) + β 0 , β 0 > 0 s > 0 , u = − β ( x ) ˙ V ≤ g ( x ) | s | ̺ ( x ) − g ( x ) β ( x ) | s | ≤ − g ( x ) β 0 | s | Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
s < 0 , u = β ( x ) ˙ V ≤ g ( x ) | s | ̺ ( x ) + g ( x ) su = g ( x ) | s | ̺ ( x ) − g ( x ) β ( x ) | s | ˙ V ≤ g ( x ) | s | ̺ ( x ) − g ( x )( ̺ ( x ) + β 0 ) | s | = − g ( x ) β 0 | s | � 1 , s > 0 sgn( s ) = − 1 , s < 0 ˙ u = − β ( x ) sgn( s ) ⇒ V ≤ − g ( x ) β 0 | s | ≤ − g 0 β 0 | s | √ ˙ V ≤ − g 0 β 0 2 V √ dV 1 � � √ √ ≤ − g 0 β 0 2 dt ⇒ V ( s ( t )) ≤ V ( s (0)) − g 0 β 0 2 t V Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
| s ( t ) | ≤ | s (0) | − g 0 β 0 t s ( t ) reaches zero in finite time The trajectory cannot leave the surface s = 0 s=0 Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Pendulum Equation θ + sin θ + b ˙ ¨ θ = cu 0 ≤ b ≤ 0 . 2 , 0 . 5 ≤ c ≤ 2 Stabilize the pendulum at θ = π/ 2 x 1 = θ − π 2 , x 2 = ˙ θ ⇒ x 1 = x 2 , ˙ x 2 = − cos x 1 − bx 2 + cu ˙ s = x 1 + x 2 , s = x 2 − cos x 1 − bx 2 + cu ˙ � � � � x 2 − cos x 1 − bx 2 (1 − b ) x 2 − cos x 1 � � � � � = � ≤ 2( | x 2 | + 1) � � � � c c � � u = − (2 . 5 + 2 | x 2 | ) sgn( s ) Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
2 0 1.5 −0.5 θ 1 s −1 0.5 −1.5 0 −2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Time Time 5 5 4 Filtered u 3 u 0 2 1 0 −5 −1 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 Time Time b = 0 . 01 , c = 0 . 5 , θ (0) = ˙ θ (0) = 0 Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
Estimate the region of attraction when � ax 2 + h ( x ) � � � ∀ x ∈ D ⊂ R 2 � ≤ ̺ ( x ) , � � g ( x ) � x 1 = − ax 1 + s ˙ s = ax 2 + h ( x ) − g ( x ) β ( x )sgn( s ) ˙ s ˙ s ≤ − g 0 β 0 | s | ⇒ {| s | ≤ c } is positively invariant ˙ V 0 = 1 2 x 2 x 1 = − ax 2 1 + x 1 s ≤ − ax 2 1 ⇒ V 0 = x 1 ˙ 1 + | x 1 | c ∀ | s | ≤ c and | x 1 | ≥ c ˙ V 0 ≤ 0 , a | x 1 | ≤ c � � Ω = a, | s | ≤ c ⊂ D is positively invariant Nonlinear Control Lecture # 10 State Feedback Stabilization and Robust State Feedback Stabilization
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