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Lect_18 - Nonlinear Systems and Control Lecture 18...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 18 Boundedness & Ultimate Boundedness – p. 1/1 8 Definition: The solutions of ˙ x = f ( t,x ) are uniformly bounded if ∃ c > and for every < a < c, ∃ β = β ( a ) > such that bardbl x ( t ) bardbl ≤ a ⇒ bardbl x ( t ) bardbl ≤ β, ∀ t ≥ t ≥ uniformly ultimately bounded with ultimate bound b if ∃ b and c and for every < a < c, ∃ T = T ( a,b ) ≥ such that bardbl x ( t ) bardbl ≤ a ⇒ bardbl x ( t ) bardbl ≤ b, ∀ t ≥ t + T “Globally” if a can be arbitrarily large Drop “uniformly” if ˙ x = f ( x ) – p. 2/1 8 Lyapunov Analysis: Let V ( x ) be a cont. diff. positive definite function and suppose that the sets Ω c = { V ( x ) ≤ c } , Ω ε = { V ( x ) ≤ ε } , Λ = { ε ≤ V ( x ) ≤ c } are compact for some c > ε > Ω ε c Ω Λ – p. 3/1 8 Suppose ˙ V ( t,x ) = ∂V ∂x f ( t,x ) ≤ − W 3 ( x ) , ∀ x ∈ Λ , ∀ t ≥ W 3 ( x ) is continuous and positive definite Ω c and Ω ε are positively invariant...
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Lect_18 - Nonlinear Systems and Control Lecture 18...

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