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Unformatted text preview: 1 Outline Motivation s tems stems Uniform Stability Lyapunov Theorems for Time Varying Systems Examples ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation Many practical systems have timevarying stemsstems parameters (e.g., temperature, pressure, etc.) 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 Tracking control of nonlinear systems often results in a timevarying system Adaptive control systems are timevarying dynamic systems 2 Equilibrium point 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 if once x(t)=x* then x(t) remains equal to x* for all future time, i.e., for all t 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 Example: 0 is the equilibrium state of linear system: Without loss of generality we can take origin as the equilibrium point of interest! x A x ) ( t = & Stability Definitions 0 is said to be stable at t if for any R>0 stemsstems there exists r(R,t ) such that: x(t )<r x(t)<R for all t t . Otherwise it is called unstable . 0 is said to be asymptotically stable at t if ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys it is stable There exists r(t )>0 such that: x(t )<r x(t) 0 as t 3 Uniform Stability Definitions 0 is said to be uniformly stable if for s tems stems any R>0 there exists r(R) such that: x(t )<r x(t)<R for all t t . 0 is uniformly asymptotically stable if it is uniformly stable ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys There exists R >0 (independent of t ) such that: x(t )<R x(t) 0 uniformly in t 0 as as t Comments on Uniform Stability Stability of nonautonomous systems...
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 Spring '08
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