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A system is called stable if for any bounded input <math>\,x(t)\,</math> | A system is called stable if for any bounded input <math>\,x(t)\,</math> | ||
| − | <math>\,\exists \epsilon \in \mathbb{R}\,</math> such that <math>\,|x(t)|<\epsilon , \forall t\in\mathbb{R}\,</math> | + | ( <math>\,\exists \epsilon \in \mathbb{R}\,</math> such that <math>\,|x(t)|<\epsilon , \forall t\in\mathbb{R}\,</math> ) |
| + | |||
yields a bounded output <math>\,y(t)\,</math>. | yields a bounded output <math>\,y(t)\,</math>. | ||
| − | <math>\,\exists \delta \in \mathbb{R}\,</math> such that <math>\,|y(t)|<\delta , \forall t\in\mathbb{R}\,</math> | + | ( <math>\,\exists \delta \in \mathbb{R}\,</math> such that <math>\,|y(t)|<\delta , \forall t\in\mathbb{R}\,</math> ) |
== Formal Definition of an Unstable System == | == Formal Definition of an Unstable System == | ||
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| + | A system is called unstable if there exists a bounded input <math>\,x(t)\,</math> | ||
| + | |||
| + | ( <math>\,\exists \epsilon \in \mathbb{R}\,</math> such that <math>\,|x(t)|<\epsilon , \forall t\in\mathbb{R}\,</math> ) | ||
| + | |||
| + | |||
| + | that yields a unbounded output <math>\,y(t)\,</math>. | ||
| + | |||
| + | ( there is no <math>\,\delta \in \mathbb{R}\,</math> such that <math>\,|y(t)|<\delta , \forall t\in\mathbb{R}\,</math> ) | ||
Revision as of 14:53, 17 September 2008
Formal Definition of a Stable System
A system is called stable if for any bounded input $ \,x(t)\, $
( $ \,\exists \epsilon \in \mathbb{R}\, $ such that $ \,|x(t)|<\epsilon , \forall t\in\mathbb{R}\, $ )
yields a bounded output $ \,y(t)\, $.
( $ \,\exists \delta \in \mathbb{R}\, $ such that $ \,|y(t)|<\delta , \forall t\in\mathbb{R}\, $ )
Formal Definition of an Unstable System
A system is called unstable if there exists a bounded input $ \,x(t)\, $
( $ \,\exists \epsilon \in \mathbb{R}\, $ such that $ \,|x(t)|<\epsilon , \forall t\in\mathbb{R}\, $ )
that yields a unbounded output $ \,y(t)\, $.
( there is no $ \,\delta \in \mathbb{R}\, $ such that $ \,|y(t)|<\delta , \forall t\in\mathbb{R}\, $ )
