Nonlinear Control Lecture # 15 Input-Output Stability Nonlinear - PowerPoint PPT Presentation

Nonlinear Control Lecture # 15 Input-Output Stability Nonlinear Control Lecture # 15 Input-Output Stability

L 2 Gain Theorem 6.4 Consider the linear time-invariant system x = Ax + Bu, ˙ y = Cx + Du where A is Hurwitz. Let G ( s ) = C ( sI − A ) − 1 B + D The L 2 gain ≤ sup ω ∈ R � G ( jω ) � Actually, L 2 gain = sup ω ∈ R � G ( jω ) � Nonlinear Control Lecture # 15 Input-Output Stability

Proof � ∞ u ( t ) e − jωt dt, U ( jω ) = Y ( jω ) = G ( jω ) U ( jω ) 0 By Parseval’s theorem � ∞ � ∞ 1 � y � 2 y T ( t ) y ( t ) dt = Y ∗ ( jω ) Y ( jω ) dω = L 2 2 π −∞ 0 � ∞ 1 U ∗ ( jω ) G T ( − jω ) G ( jω ) U ( jω ) dω = 2 π −∞ � 2 1 � ∞ � U ∗ ( jω ) U ( jω ) dω ≤ sup � G ( jω ) � 2 π ω ∈ R −∞ � 2 � � u � 2 = sup � G ( jω ) � L 2 ω ∈ R Nonlinear Control Lecture # 15 Input-Output Stability

Lemma 6.1 Consider the time-invariant system x = f ( x, u ) , ˙ y = h ( x, u ) where f is locally Lipschitz and h is continuous for all x ∈ R n and u ∈ R m . Let V ( x ) be a positive semidefinite function such that V = ∂V ∂x f ( x, u ) ≤ k ( γ 2 � u � 2 − � y � 2 ) , ˙ k, γ > 0 Then, for each x (0) ∈ R n , the system is finite-gain L 2 stable and its L 2 gain is less than or equal to γ . In particular � V ( x (0)) � y τ � L 2 ≤ γ � u τ � L 2 + k Nonlinear Control Lecture # 15 Input-Output Stability

Proof � τ � τ � u ( t ) � 2 dt − k � y ( t ) � 2 dt V ( x ( τ )) − V ( x (0)) ≤ kγ 2 0 0 V ( x ) ≥ 0 � τ � τ � u ( t ) � 2 dt + V ( x (0)) � y ( t ) � 2 dt ≤ γ 2 k 0 0 � V ( x (0)) � y τ � L 2 ≤ γ � u τ � L 2 + k Nonlinear Control Lecture # 15 Input-Output Stability

Theorem 6.5 If the system x = f ( x, u ) , ˙ y = h ( x, u ) is output strictly passive with u T y ≥ ˙ V + δy T y, δ > 0 then it is finite-gain L 2 stable and its L 2 gain is less than or equal to 1 /δ Proof ˙ u T y − δy T y V ≤ − 1 2 δ ( u − δy ) T ( u − δy ) + 1 2 δ u T u − δ 2 y T y = � 1 δ δ 2 u T u − y T y � ≤ 2 Nonlinear Control Lecture # 15 Input-Output Stability

Theorem 6.6 Consider the time-invariant system x = f ( x ) + G ( x ) u, ˙ y = h ( x ) f (0) = 0 , h (0) = 0 where f and G are locally Lipschitz and h is continuous over R n . Suppose ∃ γ > 0 and a continuously differentiable, positive semidefinite function V ( x ) that satisfies the Hamilton–Jacobi inequality � T ∂V 1 ∂V � ∂V + 1 ∂x G ( x ) G T ( x ) 2 h T ( x ) h ( x ) ≤ 0 ∂x f ( x ) + 2 γ 2 ∂x ∀ x ∈ R n . Then, for each x (0) ∈ R n , the system is finite-gain L 2 stable and its L 2 gain ≤ γ Nonlinear Control Lecture # 15 Input-Output Stability

Proof ∂V ∂x f ( x ) + ∂V ∂x G ( x ) u = 2 � � T � � ∂V − 1 � u − 1 + ∂V � � 2 γ 2 γ 2 G T ( x ) ∂x f ( x ) � � ∂x � � � � T 1 ∂V � ∂V + 1 ∂x G ( x ) G T ( x ) 2 γ 2 � u � 2 + 2 γ 2 ∂x V ≤ 1 2 γ 2 � u � 2 − 1 ˙ 2 � y � 2 Nonlinear Control Lecture # 15 Input-Output Stability

Example 6.8 x 2 = − ax 3 x 1 = x 2 , ˙ ˙ 1 − kx 2 + u, y = x 2 , a, k > 0 V ( x ) = a 4 x 4 1 + 1 2 x 2 2 ˙ ax 3 1 x 2 + x 2 ( − ax 3 V = 1 − kx 2 + u ) 2 + x 2 u = − ky 2 + yu − kx 2 = The system is finite-gain L 2 stable and its L 2 gain is less than or equal to 1 /k Nonlinear Control Lecture # 15 Input-Output Stability

Example 6.9 x = Ax + Bu, ˙ y = Cx Suppose there is P = P T ≥ 0 that satisfies the Riccati equation PA + A T P + 1 γ 2 PBB T P + C T C = 0 for some γ > 0 . Verify that V ( x ) = 1 2 x T Px satisfies the Hamilton-Jacobi equation The system is finite-gain L 2 stable and its L 2 gain is less than or equal to γ Nonlinear Control Lecture # 15 Input-Output Stability

Local Versions Lemma 6.2 Suppose V ( x ) satisfies V = ∂V ∂x f ( x, u ) ≤ k ( γ 2 � u � 2 − � y � 2 ) , ˙ k, γ > 0 for x ∈ D ⊂ R n and u ∈ D u ⊂ R m , where D and D u are domains that contain x = 0 and u = 0 , respectively. Suppose further that x = 0 is an asymptotically stable equilibrium point of ˙ x = f ( x, 0) . Then, there is r > 0 such that for each x (0) with � x (0) � ≤ r , the system x = f ( x, u ) , ˙ y = h ( x, u ) is small-signal finite-gain L 2 stable with L 2 gain less than or equal to γ Nonlinear Control Lecture # 15 Input-Output Stability

Theorem 6.7 Consider the system x = f ( x, u ) , ˙ y = h ( x, u ) Assume u T y ≥ ˙ V + δy T y, δ > 0 is satisfied for V ( x ) ≥ 0 in some neighborhood of ( x = 0 , u = 0) and the origin is an asymptotically stable equilibrium point of ˙ x = f ( x, 0) . Then, the system is small-signal finite-gain L 2 stable and its L 2 gain is less than or equal to 1 /δ Nonlinear Control Lecture # 15 Input-Output Stability

Theorem 6.8 Consider the system x = f ( x ) + G ( x ) u, ˙ y = h ( x ) Assume � T � ∂V ∂V 1 ∂V + 1 ∂x G ( x ) G T ( x ) 2 h T ( x ) h ( x ) ≤ 0 ∂x f ( x ) + 2 γ 2 ∂x is satisfied for V ( x ) ≥ 0 in some neighborhood of ( x = 0 , u = 0) and the origin is an asymptotically stable equilibrium point of ˙ x = f ( x ) . Then, the system is small-signal finite-gain L 2 stable and its L 2 gain is less than or equal to γ Nonlinear Control Lecture # 15 Input-Output Stability

Example 6.10 x 2 = − a ( x 1 − 1 3 x 3 x 1 = x 2 , ˙ ˙ 1 ) − kx 2 + u, y = x 2 , a, k > 0 √ � 1 � 1 − 1 + 1 2 x 2 12 x 4 2 x 2 V ( x ) = a 2 ≥ 0 for | x 1 | ≤ 6 1 2 + x 2 u = − ky 2 + yu ˙ V = − kx 2 ˙ V = − kx 2 u = 0 ⇒ 2 ≤ 0 √ x 2 ( t ) ≡ 0 ⇒ x 1 ( t )[3 − x 2 1 ( t )] ≡ 0 ⇒ x 1 ( t ) ≡ 0 for | x 1 | < 3 By the invariance principle, the origin is asymptotically stable when u = 0 . By Theorem 6.7, the system is small-signal finite-gain L 2 stable and its L 2 gain is ≤ 1 /k Nonlinear Control Lecture # 15 Input-Output Stability