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==Stability== | ==Stability== | ||
A system is stable if, for all bounded inputs, the output is bounded. This means that for every <math>|x(t)|\leq\epsilon</math> fed into the system, <math>|y(t)|\leq A</math>. In other words, a system is stable if <i>all</i> bounded input produces bounded output. | A system is stable if, for all bounded inputs, the output is bounded. This means that for every <math>|x(t)|\leq\epsilon</math> fed into the system, <math>|y(t)|\leq A</math>. In other words, a system is stable if <i>all</i> bounded input produces bounded output. | ||
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==Instability== | ==Instability== | ||
A system is unstable if there exists a bounded input that produces a non-bounded output. | A system is unstable if there exists a bounded input that produces a non-bounded output. | ||
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==Examples== | ==Examples== | ||
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===Stable system=== | ===Stable system=== | ||
The cruise control on your car (assuming, of course, your car is equipped with cruise control) is an example of a stable system. For a given input -- a change in vehicle speed -- the system responds with a bounded, finite, and predictable output. | The cruise control on your car (assuming, of course, your car is equipped with cruise control) is an example of a stable system. For a given input -- a change in vehicle speed -- the system responds with a bounded, finite, and predictable output. | ||
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===Unstable system=== | ===Unstable system=== | ||
The former Tacoma Narrows Bridge is an often-used example of an unstable system. For a finite input (the wind), the bridge responded in a manner that violated its physical limits; we can assume such a response to be non-bounded or infinite, because the bridge ultimately tore itself apart. | The former Tacoma Narrows Bridge is an often-used example of an unstable system. For a finite input (the wind), the bridge responded in a manner that violated its physical limits; we can assume such a response to be non-bounded or infinite, because the bridge ultimately tore itself apart. | ||
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==Note== | ==Note== | ||
Feeding a non-bounded input into a system tells us nothing useful about the boundedness of the system, or lack thereof. | Feeding a non-bounded input into a system tells us nothing useful about the boundedness of the system, or lack thereof. | ||
Latest revision as of 14:45, 16 September 2008
Stability
A system is stable if, for all bounded inputs, the output is bounded. This means that for every $ |x(t)|\leq\epsilon $ fed into the system, $ |y(t)|\leq A $. In other words, a system is stable if all bounded input produces bounded output.
Instability
A system is unstable if there exists a bounded input that produces a non-bounded output.
Examples
Stable system
The cruise control on your car (assuming, of course, your car is equipped with cruise control) is an example of a stable system. For a given input -- a change in vehicle speed -- the system responds with a bounded, finite, and predictable output.
Unstable system
The former Tacoma Narrows Bridge is an often-used example of an unstable system. For a finite input (the wind), the bridge responded in a manner that violated its physical limits; we can assume such a response to be non-bounded or infinite, because the bridge ultimately tore itself apart.
Note
Feeding a non-bounded input into a system tells us nothing useful about the boundedness of the system, or lack thereof.
