lec7 - 1 Outline Â Motivation s tems stems Â Uniform...

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Unformatted text preview: 1 Outline Â¡ Motivation s tems stems Â¡ Uniform Stability Â¡ Lyapunov Theorems for Time Varying Systems Â¡ Examples ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation Â¡ Many practical systems have time-varying stemsstems parameters (e.g., temperature, pressure, etc.) Â¡ Analyzing a system about a nominal trajectory (rather than an equilibrium point) results in a time varying system ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ Tracking control of nonlinear systems often results in a time-varying system Â¡ Adaptive control systems are time-varying dynamic systems 2 Equilibrium point Â¡ The constant state x* is said to be an equilibrium state (or point) of a non- s tems stems equilibrium state (or point) of a non autonomous system Â¡ if once x(t)=x* then x(t) remains equal to x* for all future time, i.e., for all t â‰¥ t (t : initial time) Example: 0 is the equilibrium state of ) , ( t x f x = & ) , ( * = t x f ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ Example: 0 is the equilibrium state of linear system: Â¡ Without loss of generality we can take origin as the equilibrium point of interest! x A x ) ( t = & Stability Definitions Â¡ 0 is said to be stable at t if for any R>0 stemsstems there exists r(R,t ) such that: Â¡ ||x(t )||<r â‡’ ||x(t)||<R for all t â‰¥ t . Â¡ Otherwise it is called unstable . Â¡ 0 is said to be asymptotically stable at t if ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ it is stable Â¡ There exists r(t )>0 such that: ||x(t )||<r â‡’ ||x(t)|| â†’ 0 as t â†’ âˆž 3 Uniform Stability Definitions Â¡ 0 is said to be uniformly stable if for s tems stems any R>0 there exists r(R) such that: Â¡ ||x(t )||<r â‡’ ||x(t)||<R for all t â‰¥ t . Â¡ 0 is uniformly asymptotically stable if Â¡ it is uniformly stable ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Â¡ There exists R >0 (independent of t ) such that: Â¡ ||x(t )||<R â‡’ ||x(t)|| â†’ 0 uniformly in t 0 as as t â†’ âˆž Comments on Uniform Stability Â¡ Stability of non-autonomous systems...
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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.

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lec7 - 1 Outline Â Motivation s tems stems Â Uniform...

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