lec5 - 1 Outline Motivation s tems stems Mathematical...

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Unformatted text preview: 1 Outline ¡ Motivation s tems stems ¡ Mathematical Background ¡ Stability Definitions ¡ Lyapunov’s Functions ¡ Lyapunov’s Direct Method ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys Motivation ¡ Stability is the minimum requirement of any control system stemsstems ¡ Lyapunov’s 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 energy-like 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 non-autonomous ) x ( f x = & ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys ¡ In this lecture we consider autonomous systems only ) , ( t x f x = & Stability of Closed-Loop System Can stability analysis of ‘open-loop’ systems (i e dx/dt=f(x)) be used for stemsstems systems (i.e., dx/dt=f(x)) be used for closed-loop 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 N-dim 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 2-norm 2 2 2 2 1 2 n x x x + + + = L x ME6402, Nonlinear Control SysME6402, Nonlinear Control Sys ¡ More generally p-norm (1 ≤ p ≤∞ ) ¡ Infinity norm 2 1 2 n i i x max = ∞ x ( )...
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lec5 - 1 Outline Motivation s tems stems Mathematical...

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