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Unformatted text preview: e very cell o f t he plant.
I n o rder t o u nderstand t his interesting phenomenon, recall t hat t he c haracteristic modes of a system are very special to t hat s ystem because it can sustain these
signals w ithout t he a pplication of a n e xternal input. In o ther words, t he s ystem
offers a free ride and ready access t o t hese signals. Now imagine w hat would h appen
if we a ctually drove t he s ystem w ith a n i nput h aving t he form of a characteristic
mode! We would expect t he s ystem to respond strongly (this is, in fact, t he resonance phenomenon discussed l ater in this section). I f t he i nput is n ot e xactly a
characteristic mode b ut is close t o s uch a mode, we would still expect t he s ystem
response to b e s trong. However, if t he i nput is very different from any of t he characteristic modes, we would expect t he s ystem to respond poorly. We shall now show
t hat t hese intuitive deductions are indeed true. 154 2 T ime-Domain Analysis of Continuous-Time Systems A lthough we have devised a measure of similarity of signals (correlation) l ater
in C hapter 3, we s hall t ake a s impler approach here. Let us restrict t he s ystem's
i nputs only t o e xponentials of t he form e (t, where ( is generally a complex number.
T he s imilarity o f two exponential signals e(t a nd e At will t hen b e m easured by t he
closeness of ( a nd ),. I f t he difference ( - ), is small, t he signals are similar; if ( - ),
is large, t he s ignals a re dissimilar.
Now c onsider a first-order system w ith a single characteristic m ode e At a nd
t he i nput e (t. T he i mpulse response o f t his system is t hen given by Ae At , w here
t he e xact v alue o f A is n ot i mportant for this qualitative discussion. T he s ystem
response y (t) is given by * f (t)
AeAtu(t) * e (tu(t) y(t) = h (t)
= 155 2.7 Intuitive Insights into S ystem B ehavior c onstant. We arrive a t t he s ame conclusion via a nother a rgument. T he o utput is
a convolution of t he i nput w ith h(t). I f a n i nput is a pulse of w idth T f, t hen t he
o utput p ulse w idth is T f + Th a ccording t o t he w idth p roperty o f convolution. This
conclusion shows t hat t he s ystem r equires Th seconds t o r espond fully t o a ny i nput.
T he s ystem t ime c onstant i ndicates h ow fast the system is. A s ystem with a smaller
t ime c onstant is a faster s ystem t hat responds quickly t o a n input. A s ystem with
a relatively large t ime c onstant is a sluggish s ystem t hat c annot respond well to
r apidly v arying signals.
S trictly s peaking, t he d uration o f t he i mpulse response h (t) is 0 0 b ecause t he
c haracteristic modes approach zero asymptotically as t - + 0 0. However, beyond
some value of t , h (t) b ecomes negligible. I t is t herefore necessary t o use some
suitable measure of t he i mpulse response's effective width. h ( t) F rom t he c onvolution table (Table 2.1), we o btain
Clearly, if t he i nput e (t is similar t o e At, ( - ), is small, a nd t he s ystem response is
large. T he closer the i nput f (t) is t o t he c haracteristic m ode, the stronger is t he
s ystem response. I...
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