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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 energylike 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 nonautonomous ) 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 ClosedLoop System Can stability analysis of â€˜openloopâ€™ 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 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 Ndim 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 2norm 2 2 2 2 1 2 n x x x + + + = L x ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ More generally pnorm (1 â‰¤ p â‰¤âˆž ) Â¡ Infinity norm 2 1 2 n i i x max = âˆž x ( )...
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 Spring '08
 Staff
 Topology, Metric space, Open set, Topological space, Nonlinear Control Systems

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