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Unformatted text preview: t he r eady availability
of computers, root locus can b e p roduced easily. Nevertheless, understanding these
rules can b e a g reat help in developing the intuition needed for design. We shall
present here the rules, b ut o mit t he proofs of some of t hem.
We begin with a feedback system depicted in Fig. 6.42, which is i dentical to
Fig. 6.18d, except for t he explicit representation of a variable gain K . T he s ystem
in Fig. 6.36a is a special case w ith H (s) = 1. For t he s ystem in Fig. 6.42, T (s) _ K G(s) - 1 + K G(s)H(s) (6.89a) T he c haracteristic equation of t his s ystem is:j:
tThis procedure was developed as early as 1868 in Maxwell's paper "On Governors".
:j:This characteristi~ equation is also valid when the gain K is in the feedback path [I umped with
H (s) ) rather than I n the forward path. The equatlOn applies as long as the gain K is in the loop
at any point. Hence, the root locus rules discussed here apply to all such cases. 4 40 6 Continuous-Time System Analysis Using the Laplace Transform
1 + K G(s)H(s) = 0 6.7 Application t o Feedback and Control S tate Equations t he ready ~vailability o f computers and progran,s makes it much easier to draw
actual locI. . T he first four rules are still very helpful for a quick sketching j
t he root locI.
0 (6.89b) We shall consider the paths of the roots of 1 + K G(s)H(s) = 0 as K varies from
0 0. W hen t he loop is opened, the transfer function is K G(s)H(s). For this
reason, we refer t o K G(s)H(s) as the o pen-loop t ransfer f unction. T he rules
for sketching t he root locus are as follows. o to 1. R oot loci begin ( K = 0) a t t he open-loop poles and terminate on t he openloop zeros ( K = 0 0). T his fact means t hat t he number of loci is exactly n ,
t he order o f t he open-loop transfer function. Let G(s)H(s) = N (s)/D(s), where N (s) a nd D(s) a re polynomials of powers m and n, respectively. Hence,
1 + K G(s)H(s) = 0 implies D(s) + K N(s) = O. Therefore, D(s) = 0 when
K = O. I n t his case, the roots are poles of G(s)H(s); t hat is, t he open-loop
poles. Similarly, when K ~ 0 0, D(s) + K N(s) = 0 implies N (s) = O. Hence,
the roots a re t he open-loop zeros. For t he system in Fig. 6.36a, t he open-loop
transfer function is K / s(s + 8). T he open loop poles are 0 and - 8 a nd t he
zeros (where K /s(s + 8) = 0 are b oth 0 0. We can verify from Fig. 6.41 t hat
t he r oot loci do begin a t 0 a nd - 8 a nd terminate a t 0 0. Udnderstandin g these rul~s ~an be helpful in design of control systems as dem~ns(trate lat~r. T hey are an aId I II determining what modifications should be made
or w hat ~md of ~ompen~ator t o add) t o t he open loop transfer function in order
t o meet gIven deSIgn specIfications.
• E xample 6 .21 Using t he fou~ rules of t he r oot loci, sketch t he r oot locus for a system w ith
Io op transfer functIOn
open K G(s)H(s) = K
s (s + 2)(s + 4)
1. Rule 1: For this G (s)H(s) ' n - . Hence, t here are three root loci which begi t
t he poles of G (s)H(s); t hat is, a t 0, - 2 a nd - 4.
2. Rule 2: T here are odd numbers of poles t o t he right of t he real axis segment s < -...
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