Editing Exponential stability (section)

==Example exponentially stable LTI systems==

[[Image:AsymptoticStabilityImpulseScilab.png|thumb|320px|The impulse responses of two exponentially stable systems]]

The graph on the right shows the [[impulse response]] of two similar systems. The green curve is the response of the system with impulse response <math>y(t) = e^{-\frac{t}{5}}</math>, while the blue represents the system <math>y(t) = e^{-\frac{t}{5}}\sin(t)</math>. Although one response is oscillatory, both return to the original value of 0 over time. 

===Real-world example===

Imagine putting a marble in a ladle. It will settle itself into the lowest point of the ladle and, unless disturbed, will stay there. Now imagine giving the ball a push, which is an approximation to a [[Dirac delta function|Dirac delta impulse]]. The marble will roll back and forth but eventually resettle in the bottom of the ladle. Drawing the horizontal position of the marble over time would give a gradually diminishing sinusoid rather like the blue curve in the image above.

A step input in this case requires supporting the marble away from the bottom of the ladle, so that it cannot roll back. It will stay in the same position and will not, as would be the case if the system were only marginally stable or entirely unstable, continue to move away from the bottom of the ladle under this constant force equal to its weight.

In this example the system is not stable for all inputs. Give the marble a big enough push, and it will fall out of the ladle and fall, stopping only when it reaches the floor. For some systems, therefore, it is proper to state that a system is exponentially stable ''over a certain range of inputs''.