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Unformatted text preview: 1 Outline Motivation s tems stems Mathematical Background Stability Definitions Lyapunovs Functions Lyapunovs Direct Method ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation Stability is the minimum requirement of any control system stemsstems Lyapunovs method (late 19 th century) is the most useful and general approach for studying stability of nonlinear systems It can also be used for designing nonlinear control systems It uses an energylike function (e g total ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys It uses an energy like function (e.g., total mechanical energy) to test stability In many cases it is physically motivated and intiutive 2 Autonomous Systems Definition : The system is said to be autonomous if f does not depend s tems stems explicitly on time, i.e., Otherwise it is called nonautonomous ) x ( f x = & ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys In this lecture we consider autonomous systems only ) , ( t x f x = & Stability of ClosedLoop System Can stability analysis of openloop systems (i e dx/dt=f(x)) be used for stemsstems systems (i.e., dx/dt=f(x)) be used for closedloop systems? ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 3 Equilibrium point The constant state x* is said to be an equilibrium state (or point) of s tems stems 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 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 for any or all s tems stems for any or all There exists Belongs to implies ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys R n Ndim Euclieadian Space Mathematical Review Vector Norms stemsstems Open and Closed Sets Continuous Functions Differentiable functions ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys 5 Vector Norm Norm (or magnitude), denoted by x measures the length of a vector s tems stems measures the length of a vector In stability analysis it is used to measure the state magnitude The most common norm is Euclidean or 2norm 2 2 2 2 1 2 n x x x + + + = L x ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys More generally pnorm (1 p ) Infinity norm 2 1 2 n i i x max = x ( )...
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 Spring '08
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