Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems - PowerPoint PPT Presentation

Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Input-to-State Stability (ISS) Definition 4.4 The system ˙ x = f ( x, u ) is input-to-state stable if there exist β ∈ KL and γ ∈ K such that for any initial state x ( t 0 ) and any bounded input u ( t ) � � �� � x ( t ) � ≤ max β ( � x ( t 0 ) � , t − t 0 ) , γ sup � u ( τ ) � t 0 ≤ τ ≤ t for all t ≥ t 0 ISS of ˙ x = f ( x, u ) implies BIBS stability � � x ( t ) is ultimately bounded by γ sup t 0 ≤ τ ≤ t � u ( τ ) � lim t →∞ u ( t ) = 0 ⇒ lim t →∞ x ( t ) = 0 The origin of ˙ x = f ( x, 0) is GAS Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Theorem 4.6 Let V ( x ) be a continuously differentiable function α 1 ( � x � ) ≤ V ( x ) ≤ α 2 ( � x � ) ∂V ∂x f ( x, u ) ≤ − W 3 ( x ) , ∀ � x � ≥ ρ ( � u � ) > 0 ∀ x ∈ R n , u ∈ R m , where α 1 , α 2 ∈ K ∞ , ρ ∈ K , and W 3 ( x ) is a continuous positive definite function. Then, the system x = f ( x, u ) is ISS with γ = α − 1 ˙ ◦ α 2 ◦ ρ 1 Proof Let µ = ρ ( sup τ ≥ t 0 � u ( τ ) � ) ; then ∂V ∂x f ( x, u ) ≤ − W 3 ( x ) , ∀ � x � ≥ µ Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Apply Theorem 4.4 β ( � x ( t 0 ) � , t − t 0 ) , α − 1 � � � x ( t ) � ≤ max 1 ( α 2 ( µ )) � � �� � x ( t ) � ≤ max β ( � x ( t 0 ) � , t − t 0 ) , γ sup � u ( τ ) � τ ≥ t 0 Since x ( t ) depends only on u ( τ ) for t 0 ≤ τ ≤ t , the supremum on the right-hand side can be taken over [ t 0 , t ] Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Lemma 4.5 Suppose f ( x, u ) is continuously differentiable and globally Lipschitz in ( x, u ) . If ˙ x = f ( x, 0) has a globally exponentially stable equilibrium point at the origin, then the system x = f ( x, u ) is input-to-state stable ˙ Proof: Apply (the converse Lyapunov) Theorem 3.8 Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Example 4.12 x = − x 3 + u ˙ x = − x 3 is globally asymptotically stable The origin of ˙ V = 1 2 x 2 − x 4 + xu ˙ V = − (1 − θ ) x 4 − θx 4 + xu = � 1 / 3 � | u | − (1 − θ ) x 4 , ≤ ∀ | x | ≥ θ 0 < θ < 1 The system is ISS with γ ( r ) = ( r/θ ) 1 / 3 Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Example 4.13 x = − x − 2 x 3 + (1 + x 2 ) u 2 ˙ x = − x − 2 x 3 is globally exponentially stable The origin of ˙ 2 x 2 V = 1 − x 2 − 2 x 4 + x (1 + x 2 ) u 2 ˙ V = − x 4 − x 2 (1 + x 2 ) + x (1 + x 2 ) u 2 = − x 4 , ∀ | x | ≥ u 2 ≤ The system is ISS with γ ( r ) = r 2 Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Example 4.14 x 2 = − x 3 x 1 = − x 1 + x 2 , ˙ ˙ 1 − x 2 + u x 2 = − x 3 Investigate GAS of x 1 = − x 1 + x 2 , ˙ ˙ 1 − x 2 ˙ V ( x ) = 1 4 x 4 1 + 1 2 x 2 V = − x 4 1 − x 2 ⇒ 2 2 Now u � = 0 ˙ V = − x 4 1 − x 2 2 + x 2 u ≤ − x 4 1 − x 2 2 + | x 2 | | u | ˙ V ≤ − (1 − θ )[ x 4 1 + x 2 2 ] − θx 4 1 − θx 2 2 + | x 2 | | u | (0 < θ < 1) Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

− θx 2 2 + | x 2 | | u | ≤ 0 for | x 2 | ≥ | u | /θ and has a maximum value of u 2 / (4 θ ) for | x 2 | < | u | /θ 2 ≥ u 2 1 ≥ | u | x 2 or x 2 − θx 4 1 − θx 2 ⇒ 2 + | x 2 | | u | ≤ 0 2 θ θ 2 2 θ + u 2 � x � 2 ≥ | u | − θx 4 1 − θx 2 ⇒ 2 + | x 2 | | u | ≤ 0 θ 2 � 2 θ + r 2 r ρ ( r ) = θ 2 ˙ V ≤ − (1 − θ )[ x 4 1 + x 2 2 ] , ∀ � x � ≥ ρ ( | u | ) The system is ISS Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Lemma 4.6 η = f 1 ( η, ξ ) and ˙ If the systems ˙ ξ = f 2 ( ξ, u ) are input-to-state stable, then the cascade connection ˙ η = f 1 ( η, ξ ) , ˙ ξ = f 2 ( ξ, u ) is input-to-state stable. Consequently, If ˙ η = f 1 ( η, ξ ) is input-to-state stable and the origin of ˙ ξ = f 2 ( ξ ) is globally asymptotically stable, then the origin of the cascade connection ˙ η = f 1 ( η, ξ ) , ˙ ξ = f 2 ( ξ ) is globally asymptotically stable Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Example 4.15 x 1 = − x 1 + x 2 ˙ 2 , x 2 = − x 2 + u ˙ x 1 = − x 1 + x 2 The system ˙ 2 is input-to-state stable, as seen from Theorem 4.6 with V ( x 1 ) = 1 2 x 2 1 ˙ V = − x 2 1 + x 1 x 2 2 ≤ − (1 − θ ) x 2 1 , for | x 1 | ≥ x 2 2 /θ, 0 < θ < 1 The linear system ˙ x 2 = − x 2 + u is input-to-state stable by Lemma 4.5 The cascade connection is input-to-state stable Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Definition 4.5 Let X ⊂ R n and U ⊂ R m be bounded sets containing their respective origins as interior points. The system ˙ x = f ( x, u ) is regionally input-to-state stable with respect to X × U if there exist β ∈ KL and γ ∈ K such that for any initial state x ( t 0 ) ∈ X and any input u with u ( t ) ∈ U for all t ≥ t 0 , the solution x ( t ) belongs to X for all t ≥ t 0 and satisfies � � �� � x ( t ) � ≤ max β ( � x ( t 0 ) � , t − t 0 ) , γ sup � u ( τ ) � t 0 ≤ τ ≤ t The system ˙ x = f ( x, u ) is locally input-to-state stable if it is regionally input-to-state stable with respect to some neighborhood of the origin ( x = 0 , u = 0) Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Theorem 4.7 Suppose f ( x, u ) is locally Lipschitz in ( x, u ) for all x ∈ B r and u ∈ B λ . Let V ( x ) be a continuously differentiable function that satisfies α 1 ( � x � ) ≤ V ( x ) ≤ α 2 ( � x � ) ∂V ∂x f ( x, u ) ≤ − W 3 ( x ) , ∀ � x � ≥ ρ ( � u � ) > 0 for all x ∈ B r and u ∈ B λ , where α 1 , α 2 , ρ ∈ K and W 3 ( x ) is a continuous positive definite function. Suppose α 1 ( r ) > α 2 ( ρ ( λ )) and let Ω = { V ( x ) ≤ α 1 ( r ) } . Then, the system ˙ x = f ( x, u ) is regionally input-to-state stable with respect to Ω × B λ and γ = α − 1 ◦ α 2 ◦ ρ 1 Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems

Local input-to-state stability of ˙ x = f ( x, u ) is equivalent to asymptotic stability of the origin of ˙ x = f ( x, 0) Lemma 4.7 Suppose f ( x, u ) is locally Lipschitz in ( x, u ) in some neighborhood of ( x = 0 , u = 0) . Then, the system x = f ( x, u ) is locally input-to-state stable if and only if the ˙ unforced system ˙ x = f ( x, 0) has an asymptotically stable equilibrium point at the origin The proof uses (converse Lyapunov) Theorem 3.9 Nonlinear Control Lecture # 11 Time Varying and Perturbed Systems