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Unformatted text preview: e, • E xample 1 1.10
F ind t he o utput y (t) for all t in Example 11.9.
In Example 11.9, we found t he response y[kJ only a t t he sampling instants. To
find t he o utput values between sampling instants, we use t he modified z-transform. The
procedure is t he same as before, except t hat we use modified z-transform corresponding
t o continuous-time systems a nd signals. For t he system G (s) = 1 / s + 4 w ith T = 0.5, t he
modified z-transform [Eq. (11.50b) w ith>' = - 4, a nd T = 0.5J is Z3 Y[zJ = T[zJF[zJ = 7(z---I"'-)('-z---0-=.3-94-'-)"-(z---0-.1-74""')
Moreover t o find the modified z-transform corresponding t o f (t) = u (t) [>' = 0 in Eq.
(11.50a)], we have F[z, J.iJ = z /(z - 1). S ubstitution of these expressions in those found in
Example 11.9, we o btain a nd z (z - 1)(z - 0.394)(z - 0.174)
= z - 1 - z - 0.394 + 0.083
z - 0.174 Y[ J- z, J.i - e -2" [Z 0.583z
z _ 1 - z _ 0.394 0.083z ]
0.174 +z- 7 04 11 D iscrete-Time Systems Analysis Using t he Z -Transform 1 From Eqs. (11.50), we obtain the inverse (modified) z-transform of thi~ equation as y[(k + JL)TJ = e - 2" [1 - 0.583(0.394)k t
- y k u [ -(k+ I )] + 0.083( - 0.174)k]
-7 T he complete response is also shown in Fig. U .ll. 705 11. 7 T he B ilateral Z - Transform -5 -3 -\ k- • D esign o f S ampled-Data S ystems
As with continuous-time control systems, s ampled-data systems are designed t o
m eet c ertain transient (PO, t r , ts, e tc.) and s teady-state specifications. T he design
procedure follows along the lines similar to those used for continuous-time systems.
W e begin with a general second-order system. t he r elationship between closedloop pole locations a nd t he corresponding transient p arameters PO, t r , t s ," . are
determined. Hence, for a given t ransient specifications, a n a cceptable region in the
z -plane where the dominant poles of t he closed-loop transfer function T[z] should
lie is d etermined. Next, we s ketch t he root locus for t he system. T he rules for
s ketching t he r oot locus are t he s ame as those for continuous-time systems. I f t he
r oot locus passes through t he a cceptable region, t he t ransient specifications can be
m et b y simple adjustment of t he gain K . I f not, we m ust use a compensator, which
will s teer t he r oot locus in t he a cceptable region. Fig. 1 1.12 _ ·/u[-(k + I}J and the region of convergence of its z-transform. =- = 1 1. 7 T he Bilateral Z- Transform 00 L + ...J +( 1- [1 + ~ + (~r + (~r + ...J
I~I < 1 'Y z Izl < hi Therefore f [k]z-k Izl < k =-oo T he inverse z-transform is given by
f [k] = r ~r 1
=1--1- ~ S ituations involving noncausal signals or systems c annot b e handled b y t he
(unilateral) z-transform discussed so far. Such cases c an b e analyzed by t he b ilateral (or two-sided) z - transform defined by
F[z] == [~ + ( ~ ~fF[Z]zk-ldZ
21rJ T hese e quations define t he b ilateral z-transform. T he u nilateral z-transform discussed so far is a special case, where t he i nput signals a re r estricted t o b e causal.
R estricting signals in this way results in considerable simplification in t he region of
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