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Unformatted text preview: s (s + 2)
+ 1)(8 + 10) t r S 0.5, (6.99) Let us further specify t hat t he s ystem meet t he following steady-state specifications:
e r S 0.15 e . = 0, For this case, we a lready found e .
we r equire er S 0.15. Therefore = 0, er = 8 /K a nd e p t~------..JI
' = 00 [see Eq. (6.98)]. B ut (a) c
s plane => K 2: 53.34 (6.100) T urning t o Fig. 6.41, we n ote t hat t he poles o f T (s) lie in t he a cceptable region
t o m eet t ransient specifications (25 < K < 64). Equation (6.100) shows t hat we
must use K 2: 53.34 t o meet steady-state performance. Therefore to meet b oth
t he transient a nd t he s teady-state specifications we m ust set t he gain in t he range
53.34 < K < 64. T he smallest steady-state error for a ramp i nput is obtained for
K = 64. For this case,
er 8 8 = - = - 4 = 0.125
6 T hus if t he s ystem is t o meet t he t ransient performance in Eq. (6.99), t he m inimum
er = 0.125. We can do no better. In case we a re required t o have er < 0.125 -p ( b) - C( F ig. 6 .45 A lead compensator. while mainta~ning t he s ame transient performance, we will have t o use some kind
o f compensatIOn. 6 .7-5 Compensation
. T he synthesis problem for t he position control system I·n FI·g . 6.36a I.S a very
s~mp e ex~p e where t he t ransient a nd s teady-state specifications could be met by
Simple adJu.stme~t of gain K . I n many cases, i t may be impossible t o meet b oth
sets o f specificatIOn (transi~nt a nd s teady s tate) by simple adjustment of the gain
K . "V-!e m ay ~e able t o satisfy one set o f specifications o r t he o ther b ut n ot b oth
ConSider a gam t he s ystem in Fig. 6.36a, with t he following specifications:
PO ~ - 0.15
K y (l) K G(s) K p = K /5 a nd K v = K a = O. Hence, e . = 5 /(5 + K ) a nd e r = e p = 0 0. Such
s ystems are designated as t ype 0 systems. These systems have finite e ., b ut infinite
e r a nd e p • T hese s ystems may be acceptable for s tep i nputs (position control), b ut
n ot for r amp or p arabolic i nputs (tracking velocity or acceleration).
I f G(s) has t wo poles a t t he origin, t he s ystem is designated as t ype 2 system.
In this case K p = K v = 0 0, a nd K a = finite. Hence, e . = e r = 0 a nd e p is finite.
In general, i f G (s) has q poles a t t he origin, it is a t ype q system. Clearly,
for a unity feedback system, increasing t he n umber of poles a t t he origin in G(s)
improves t he s teady-state performance. However, this procedure increases n and
reduces t he m agnitude of u , t he c entroid of t he r oot locus asymptotes. This shifts
t he r oot locus t owards t he j w-axis with consequent deterioration in t he t ransient
performance a nd t he system stability.
I t should be remembered t hat t he r esults in Eqs. (6.96) and (6.97) apply only
t o u nity feedback systems (Fig. 6.44). Steady-state error specifications in this case
are t ranslated in terms of constraints on t he open-loop transfer function K G(s).
In contrast, t he r esults in Eqs. (6.92) t hrough (6.95) apply...
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