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Unformatted text preview: itions exist: (i) a t least one r oot is outside t he u nit circle; (ii) t here are
r epeated r oots o n t he u nit circle.
3. A n LTID system is marginally s table if and only if t here a re no roots outside
t he u nit circle a nd t here a re some u nrepeated r oots on t he u nit circle. tIf t he development o f discrete-time systems is parallel t o t hat o f continuous-time systems, we
w onder why t he p arallel breaks down here. Why, for instance, a ren't L HP a nd R HP t he regions
d emarcating s tability a nd i nstability? T he r eason lies in t he form o f t he c haracteristic modes. I n
c ontinuous-time systems we chose t he form o f c haracteristic mode as e A,t. I n d iscrete-time systems
we choose t he form (for c omputational convenience) t o b e 'Yf. H ad we chosen t his form t o b e e A,k
w here 'Yi = e A" t hen L HP a nd R HP ~or t he l ocation of )..;) a gain would d emarcate s tability and
instability. T he reason is t hat if'Y = e , bl = 1 implies leAl = 1, a nd t herefore)" = j w. T his shows
t hat t he u nit circle in 'Y p lane m aps i nto t he i maginary axis in t he).. p lane. F ig. 9 .7 C haracteristic r oots l ocation a nd t he c orresponding c haracteristic m odes. 6 06 9 T ime-Domain A nalysis o f D iscrete-Time S ystems 9.6 • E xample 9 .13
Determine whether t he systems specified by t he following equations are asymptoticILlly s table, marginally stable, or unstable. In each case plot t he c haracteristic roots in
t he complex plane. + IJ - 2f[kJ
= 2 f[k - IJ + 3 f[k ( c) y[k + 3J + 2y[k + 2J + ~y[k + IJ + h [kJ = J [k + IJ
( d) (E2 - E + 1)2Y[kJ = (3E + 1)J[kJ
( a) y[k + 2J + 2.5y[k + IJ + y[kJ ( b) y[kJ - y[k - IJ + 0.21y[k - 607 S ystem S tability
complex plane = f [k 2J -2 2J -\ ( a) T he characteristic polynomial is (a) (b) '"(2 + 2.5'"( + 1 = ('"( + 0.5)('"( + 2) T he c haracteristic roots are - 0.5 a nd - 2. Because 1 - 21 > 1 ( -2 lies outside t he u nit
circle), t he s ystem is u nstable (Fig. 9.8a).
( b) T he characteristic polynomial is
'"(2 _ '"( + 0.21 = ('"( - 0.3)('"( - 0.7)
-\ T he c haracteristic roots are 0.3 a nd 0.7, b oth of which lie inside t he u nit circle. T he system
is a symptotically stable (Fig. 9.8b). -\ ( c) T he c haracteristic polynomial is
'"(3 + 2'"(2 + h + ~ = ('"( + 1)('"(2 + '"( +~) = ('"( + 1)('"( + 0.5 - jO.5)('"( + 0.5 + jO.5) T he c haracteristic roots are - 1, - 0.5 ± jO.5 (Fig. 9.8c). O ne of t he c haracteristic roots
is o n t he u nit circle and t he remaining two roots a re inside t he u nit circle. T he system is
( d) T he characteristic polynomial is
('"(2 _ '"( + 1)2 = ('"( _ ~ _ j~) 2 ('"( _ ~ + j ~) 2 F ig. 9 .8 Location of characteristic roots for systems in Example 9.13.
I n c ontrast, for a m arginally s table a nd u nstable s ystem i t is p ossible t o find a
b ounded i nput t hat p roduces u nbounded o utput. T o p rove t his r esult we recall
y [k] = h[k] T he c haracteristic roots are ~ ± j~ = l e±ji r epeated twice, a nd t hey lie on t he u...
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