Example exponentially stable LTI systems Exponential stability

the impulse responses of 2 exponentially stable systems

the graph on right shows impulse response of 2 similar systems. green curve response of system impulse response



y
(
t
)
=

e

−


t
5






{\displaystyle y(t)=e^{-{\frac {t}{5}}}}

, while blue represents system



y
(
t
)
=

e

−


t
5




sin
⁡
(
t
)


{\displaystyle y(t)=e^{-{\frac {t}{5}}}\sin(t)}

. although 1 response oscillatory, both return original value of 0 on time.

real-world example

imagine putting marble in ladle. settle lowest point of ladle and, unless disturbed, stay there. imagine giving ball push, approximation dirac delta impulse. marble roll , forth resettle in bottom of ladle. drawing horizontal position of marble on time give gradually diminishing sinusoid rather blue curve in image above.

a step input in case requires supporting marble away bottom of ladle, cannot roll back. stay in same position , not, case if system marginally stable or entirely unstable, continue move away bottom of ladle under constant force equal weight.

it important note in example system not stable inputs. give marble big enough push, , fall out of ladle , fall, stopping when reaches floor. systems, therefore, proper state system exponentially stable on range of inputs.