# lec5 - 1 Outline Â Motivation s tems stems Â Mathematical...

This preview shows pages 1–6. 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 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: 1 Outline Â¡ Motivation s tems stems Â¡ Mathematical Background Â¡ Stability Definitions Â¡ Lyapunovâ€™s Functions Â¡ Lyapunovâ€™s Direct Method ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation Â¡ Stability is the minimum requirement of any control system stemsstems Â¡ Lyapunovâ€™s method (late 19 th century) is the most useful and general approach for studying stability of nonlinear systems Â¡ It can also be used for designing nonlinear control systems Â¡ It uses an energy-like function (e g total ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ It uses an energy like function (e.g., total mechanical energy) to test stability Â¡ In many cases it is physically motivated and intiutive 2 Autonomous Systems Â¡ Definition : The system is said to be autonomous if f does not depend s tems stems explicitly on time, i.e., Â¡ Otherwise it is called non-autonomous ) x ( f x = & ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ In this lecture we consider autonomous systems only ) , ( t x f x = & Stability of Closed-Loop System Can stability analysis of â€˜open-loopâ€™ systems (i e dx/dt=f(x)) be used for stemsstems systems (i.e., dx/dt=f(x)) be used for closed-loop systems? ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 3 Equilibrium point Â¡ The constant state x* is said to be an equilibrium state (or point) of s tems stems Â¡ If once x(t)=x* then x(t) remains equal to x* for all future time, i.e., ) ( x f x = & * ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ Without loss of generality we can take origin as the equilibrium point of interest! ) ( = x f Physical Example Stable Unstable stemsstems Unstable ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 4 Standard Mathematical Notations Notation Explanation âˆ€ for any or all s tems stems âˆ€ for any or all âˆƒ There exists âˆˆ Belongs to â‡’ implies ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys R n N-dim Euclieadian Space Mathematical Review Â¡ Vector Norms stemsstems Â¡ Open and Closed Sets Â¡ Continuous Functions Â¡ Differentiable functions ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 5 Vector Norm Â¡ Norm (or magnitude), denoted by ||x|| measures the length of a vector s tems stems measures the length of a vector Â¡ In stability analysis it is used to measure the state magnitude Â¡ The most common norm is Euclidean or 2-norm 2 2 2 2 1 2 n x x x + + + = L x ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ More generally p-norm (1 â‰¤ p â‰¤âˆž ) Â¡ Infinity norm 2 1 2 n i i x max = âˆž x ( )...
View Full Document

## This note was uploaded on 09/22/2011 for the course ME 6402 taught by Professor Staff during the Spring '08 term at Georgia Tech.

### Page1 / 20

lec5 - 1 Outline Â Motivation s tems stems Â Mathematical...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online