| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
== Stable System == | == 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 | + | 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 any bounded input yields a bounded output. 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> | :<center><math>|x(t)| < \epsilon \!</math> for all values of t, </center> | ||
| Line 11: | Line 11: | ||
== Unstable System == | == Unstable System == | ||
| + | Conversely, a system can be called unstable if an input signal is bounded but it yields an unbounded output signal. Assume the input signal is cos(t), which was proved above to be stable, and then assume a system behaves in this way: exp(t)*x(t). So, the output signal is exp(t)*cos(t), which increases in value as t increases, and this signal is quite obviously unbounded. Mathematically, this means that there exists an <math> \epsilon \!</math> such that | ||
| + | |||
| + | :<center><math>|x(t)| < \epsilon \!</math> for all values of t, </center> | ||
| + | |||
| + | but there is NO <math> M\!</math> for the output signal y(t) such that | ||
| + | |||
| + | <center><math>|y(t)| < M \!</math> for all values of t.</center> | ||
Latest revision as of 05:44, 19 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 any bounded input yields a bounded output. 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
Unstable System
Conversely, a system can be called unstable if an input signal is bounded but it yields an unbounded output signal. Assume the input signal is cos(t), which was proved above to be stable, and then assume a system behaves in this way: exp(t)*x(t). So, the output signal is exp(t)*cos(t), which increases in value as t increases, and this signal is quite obviously unbounded. Mathematically, this means that there exists an $ \epsilon \! $ such that
$ |x(t)| < \epsilon \! $ for all values of t,
but there is NO $ M\! $ for the output signal y(t) such that
