Module 4 p4 - ECE 514 Nonlinear & Adaptive Control Module...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
1 ECE 514 Nonlinear & Adaptive Control Module 4-Part 4 LYAPUNOV STABILITY
Background image of page 1

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

View Full DocumentRight Arrow Icon
2 Topics ± Converse Theorems ± Boundedness & ultimate boundedness ± Input-to-state stability Material covered in Chapter 4 of Khalil, sections 4.8 and 4.9.
Background image of page 2
3 Converse Theorems i) if stable ii) uniformly asymtotically (exponentially) stable See Theorems 4.14, 4.15, 4.16, and 4.17 in Khalil V V
Background image of page 3

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

View Full DocumentRight Arrow Icon
4 Beyond Lyapunov Stability We can use Lyapunov methodologies to show that states are bounded even without having an equilibrium point at the origin. Given (, ) :[0, ) is piecewise continuous in , and locally Lipschitz in . n xf t x fD t x = ∞× → i \
Background image of page 4
5 Uniform Boundedness Definition: 0 0 0 The solutions of ( , ) are 1) Uniformly bounded if there exists a positive constant independent of 0, and for every (0, ), ( ) 0, independent of , such that: || ( ) | xf t x c ta c a t xt ββ = ≥∈ = > i 0 | || ( ) || , ( ) 2) Globally uniformly bounded if ( ) holds for arbitrarly large . ax t t t a β ⇒≤
Background image of page 5

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

View Full DocumentRight Arrow Icon
6 Uniform Boundedness () xt 0 a a β
Background image of page 6
7 Uniform Ultimate Boundedness 0 The solutions of ( , ) are 1) Uniformly ultimately bounded with ultimate bound if there exists positive constants , independent of 0, and for every (0, ), ( , ) 0,independent xf t x b bc t ac T T a b = ∈∃ = i 0 00 of , such that: || ( ) || || ( ) || , + ( ) 2) Globally uniformly ultimately bounded if ( ) holds for arbitrarly large . t xt a b t t T a ≤⇒ ∗∗ Definition:
Background image of page 7

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

View Full DocumentRight Arrow Icon
8 Uniform Ultimate Boundedness () xt 0 a b 0 tT +
Background image of page 8
9 Lyapunov Results Theorem (4.18 in Khalil): 12 3 Let be a domain that contains the origin and let :[0, ) be a continuously differentiable function: (|| ||) ( , ) (|| ||) ( , ) ( ), || || 0 n D VD R xV t x x VV ftx W x x tx t αα µ ∞× → ≤≤ ∂∂ +≤ > ∀≥ \ 3 1 21 0,
Background image of page 9

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

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/06/2010 for the course ECE 514 taught by Professor Chaoukit.abdallah during the Spring '09 term at University of New Brunswick.

Page1 / 24

Module 4 p4 - ECE 514 Nonlinear & Adaptive Control Module...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online