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Unformatted text preview: eration d uring which certain information is lost. To invert t his o peration,
one piece of information a bout y (t) m ust b e provided t o r estore t he o riginal y (t).
Using a similar argument, we c an show t hat, given d 2 y/dt 2 , we c an determine y (t)
u niquely only if two additional pieces of information (constraints) a bout y (t) are
given. In general, t o d etermine y (t) u niquely from its n th derivative, we need n
a dditional pieces of information (constraints) a bout y (t). T hese constraints are also
called auxiliary conditions. W hen t hese conditions are given a t t = 0, t hey a re
called initial conditions. 2.2-1 S ome Insights into the Zero-Input Behavior o f a System B y definition, t he z ero-input response is t he s ystem response t o i ts internal
conditions, assuming t hat i ts i nput is zero. Understanding this phenomenon provides interesting insight into system behavior. I f a s ystem is d isturbed m omentarily
from its rest position a nd if t he d isturbance is t hen removed, t he s ystem will n ot
come back t o r est instantaneously. I n general, it will come back t o r est over a
period of time a nd only t hrough a special t ype o f motion t hat is c haracteristic of
t he s ystem. t For example, if we press on a n a utomobile fender momentarily a nd
t hen release it a t t = 0, t here is no external force on t he a utomobile for t > 0:1.
T he a uto b ody will eventually come back t o i ts rest (equilibrium) position, b ut n ot
t hrough a ny a rbitrary m otion. I t m ust do so using only a form of response which
is s ustainable by t he s ystem on its own w ithout a ny external source, because the
i nput is zero. O nly c haracteristic modes satisfy this condition. The system uses a
proper c ombination o f c haracteristic modes t o come back t o the rest p osition while
s atisfying appropriate b oundary (or i nitial) conditions.
I f t he shock absorbers of t he a utomobile are in good condition (high damping
coefficient), t he c haracteristic modes will b e m onotonically decaying exponentials,
a nd t he a uto b ody will come to rest rapidly without oscillation. I n c ontrast, for t This assumes that the system will eventually come back to its original rest (or equilibrium) position.
:I: We ignore the force of gravity, which merely causes a constant displacement of the auto body
without affecting the other motion. 114 2 Time-Domain Analysis of Continuous-Time Systems 2.3 T he U nit Impulse Response h(t) 115 t his i mportant p henomenon requires a n u nderstanding of t he z ero-state response;
for this reason we p ostpone t his topic until Sec. 2.7-7. IH 2.3 The Unit Impulse Response h (t) 20 f (t) F ig. 2.2 Modes always get a free ride.
p oor shock a bsorbers (low damping coefficients), t he c haracteristic modes will b e
e xponentially decaying sinusoids, a nd t he b ody will come t o r est through oscillatory motion. W hen a series R C c ircuit with a n initial charge on t he c apacitor is
shorted, the c apacitor will s tart t o discharge exponentially t hrough t he resistor.
T his response o f t he R C c ircuit is caused entirely by its internal conditi...
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