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Unformatted text preview: 4.3. 7 Angle Criterion The root locus for a feedback system is the path traced by the roots of the characteristic
polynomial (the poles of the transfer function) as some system parameter is varied.
Most control systems can be expressed in the form of Figure 4.4(a), where the transfer
function T(s) is ~ K C(s) ‘ 1+ KG(s)H(s) The system parameter that is to be varied is the forward path. gain K. It is normal
Closed~ and openloop practice to identify T (s) as the closedloop transfer function to distinguish it from
“3"5’5" 'ilndmm 3’9 dEﬁned anothertransferfunction, which willnowbeintroduced. In Figure 4.40)), the feedback
loop is broken at the variable called Y1 (s). We can deﬁne a second transfer function. the open» loop transfer function, relating Y I (s) to R(s). That second transfer function is
Y1 (S )
R (s ) As always, each transfer function is deﬁned with initial conditions equal to zero.
Notice that the closedloop transfer function depends heavily on the OpelHoop trans—
fer function KG(s)H(s). It is standard practice to deﬁne the zeros and poles of
There are openloop poles G(s)H(s) as being openloop zeros and open—loop poles, while the zeros and poles
and zeros and {WSW’00}? of T(s) are called the closedloop zeros and closedloop poles. For convenience,
pales and zeros‘ the openloop zeros and openloop poles may be simply called GH zeros and GH poles. The term mots generally applies to the poles of T(s) which are the roots of the
characteristic polynomial. T(s) Tim = Kcrsmo) = Since the closedloop poles are the roots of the characteristic polynomial, then
those closedloop poles satisfy 1+ KG(s)H(s) = 0 [41] Ru) (1’) Figure 4.4 Feedback system with a variable gain K. (a) Closedloop system. (b) Openloop system with loop broken at
Y} “mm—Mm Table 4.2 Some Root Locus Plots
M 1m _ 1m
6(5)}!(5) = G(s)H(3) =
.ww ' 31
{S'PIJR‘PN {S‘PﬂU'pz}
WWW
p2 2. p: “6 P2 17. in R”
i (a) (b) G(:)H{s) =
1
(s ' p; )(s ‘ P2)(S ’ P3) (d) In:
G(s')H(.9) =
u.._wi:&__,
(s + on +j,8)(s + (l‘jB) 1m
G(5)H(s) =
S "‘ Z] (s+a+jB)(s+aj) “0mm i
f
(a! (f) 1m
G(s)H(s} =
1 Wm (5”p.)(s+ a+jﬁ)(s+ a—jﬁ) (g) (h) WWW
Table 4.2 Some Root Locus Plats {ConlinuedJ Im 1m
G(s)H(s) = (7(5)H(5) : ‘
S M 2']
H mm ‘92)“ “93! }
(5“p1}(s + a +jﬂ)(s + (x‘jﬁ) (i) (j) 1 [m G(s)H(s) = ~ 5 z! (5 “ P1)“ ' P33“ " [93) m
G{5}H(S) =
s '” z;
(5 " P1X! " P2)” ‘ P3) (k) (I) m G(5)H(s) =
5 * z, (5 “mm “pzﬁs "123) (T"pot‘s "1’2st m) (m) (:2) lm
“(W/B Gummy: maI‘jB 1 \j (S""i’1)($""‘l’2}{5 ‘9' was)“ + a "113) (Hymn): l
(5 "m )(~"'I’2K~“ "” 6* MB)"
‘(,s‘+a'jB) Re Re ~u~j§5 "HUB (0) 4p} ...
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 Spring '09

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