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An input is said to be bounded if it is bounded above and below for all values of t. For example, cos(t) is a bounded input since it is bounded above by 1 and below by -1, while exp(t) is not a bounded input since for increasing t, the function increases without bound. A system is therefore said to be bounded if a bounded output yields a bounded input. According to Professor Boutin, mathematically this means that there exists an <math> \epsilon \!</math> such that | An input is said to be bounded if it is bounded above and below for all values of t. For example, cos(t) is a bounded input since it is bounded above by 1 and below by -1, while exp(t) is not a bounded input since for increasing t, the function increases without bound. A system is therefore said to be bounded if a bounded output yields a bounded input. According to Professor Boutin, mathematically this means that there exists an <math> \epsilon \!</math> such that | ||
| − | :<center><math>|x(t)| < \epsilon \!</math> for all values of t, | + | :<center><math>|x(t)| < \epsilon \!</math> for all values of t, |
| + | and then there exists an <math> M\!</math> for the ouput signal y(t) such that | ||
| − | <math>|y(t)| < M \!</math> | + | |
| + | <math>|y(t)| < M \!</math> | ||
Revision as of 10:26, 16 September 2008
Stable System
An input is said to be bounded if it is bounded above and below for all values of t. For example, cos(t) is a bounded input since it is bounded above by 1 and below by -1, while exp(t) is not a bounded input since for increasing t, the function increases without bound. A system is therefore said to be bounded if a bounded output yields a bounded input. According to Professor Boutin, mathematically this means that there exists an $ \epsilon \! $ such that
$ |x(t)| < \epsilon \! $ for all values of t,
and then there exists an $ M\! $ for the ouput signal y(t) such that
