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

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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 time-varying 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 time-varying system Adaptive control systems are time-varying 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 non-autonomous systems...
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lec7 - 1 Outline Motivation s tems stems Uniform Stability...

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