{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# Lect_18 - Nonlinear Systems and Control Lecture 18...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 10

Lect_18 - Nonlinear Systems and Control Lecture 18...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online