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Unformatted text preview: rom t he j w-axis die quickly compared to those arising
because o f poles located near t he jw-axis. In addition, t he coefficients of t he former
t erms a re much smaller t han unity. Hence, t hey are also very small to begin with
a nd d ecay rapidly. T he poles n ear t he j w-axis are called t he d ominant p oles. A
c riterion commonly used is t hat a ny pole which is five t imes as far from t he j w-axis
as t he d ominant poles contributes negligibly t o t he s tep response, a nd t he t ransient
behavior of a higher-order system is o ften reduced t o t hat o f a second-order system.
I n a ddition, a closely placed pole-zero p air (called d ipole), c ontributes negligibly
t o t he t ransient behavior. For this reason, m any o f t he pole-zero configurations in
practice reduce t o two or t hree poles w ith one or two zeros. Workers in t he field have
prepared c harts for t ransient b ehavior o f t hese systems for several such pole-zero
combinations, which may b e used t o design most of t he h igher-order systems. j ro 6 4 K=O -4 -6 1 ,=2 I
y (t) F ig. 6 .41 Designing a second-order system to meet a given transient specifications. S2 + 8s + K (6.87b) =0 Hence t he p oles a re
S I,2 = -4 T he poles S j a nd S 2 ( the roots of t he c haracteristic equation) move along a certain
p ath in t he s -plane as we v ary K from 0 t o 0 0. W hen K = 0, t he poles are - 8, O.
For K < 16, t he poles are real a nd b oth poles move towards a value - 4 as K
varies from 0 t o 16 (overdamping). For K = 16, b oth poles coincide a t - 4 (critical
damping). F or K > 16, t he poles become complex with values - 4 ± j JK - 16
( underdamping) Since t he r eal p art o f t he poles is - 4 for all K > 16, t he p ath of
t he poles is v ertical as illustrated in Fig. 6.41. O ne pole moves u p a nd t he o ther
( its conjugate) moves down along t he v ertical line passing thro~gh ~4.
label t he v alues of K for several points along these paths, as depicted m Fig. 6.41.
Each of these p aths r epresents a locus of a pole of T (s) or a locus of a root of the
characteristic e quation o f T (s) as K is varied from 0 t o 0 0. F or this reason this set
of p aths is c alled t he r oot l ocus. T he r oot locus gives us t he info~mati~n as. to
how t he poles o f t he closed-loop transfer function T (s) move as t he g am K IS v aned
from 0 t o 0 0. I n o ur design problem, we m ust choose a value of K s uch t hat t he
poles of T (s) l ie in t he s haded region shown in Fig. 6.41. This figure shows t hat t he
system will m eet t he given specifications [Eq. (6.86)J for 25 :s: K :s: 64. For K = 64, "!'Ie for instance we have P O = 16%, t r = 0.2 seconds, F ig. 6 .42 A feedback system with a variable gain K . (6.87c) ± J 16 - K ts 4 = - = 1 seconds
4 (6.88) 6 .7-3 Root Locus T he e xample in Sec. 6.7-2 gives a good idea of the utility of root locus in
design o f control systems. Surprisingly, root locus can be sketched quickly using
certain basic rules provided by W .R. E vans in 1948.t W ith...
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