R_Locus_add - 4.3. 7 Angle Criterion The root locus for a...

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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 open-loop practice to identify T (s) as the closed-loop 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 closed-loop transfer function depends heavily on the Opel-Hoop trans— fer function KG-(s)H(s). It is standard practice to define the zeros and poles of There are open-loop poles G(s)H(s) as being open-loop zeros and open—loop poles, while the zeros and poles and zeros and {WSW-’00}? of T(s) are called the closed-loop zeros and closed-loop poles. For convenience, pales and zeros‘ the open-loop zeros and open-loop 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 closed-loop poles are the roots of the characteristic polynomial, then those closed-loop poles satisfy 1+ KG(s)H(s) = 0 [4-1] Ru) (1’) Figure 4.4 Feedback system with a variable gain K. (a) Closed-loop system. (b) Open-loop 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+a-j) “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 "12-3) (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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R_Locus_add - 4.3. 7 Angle Criterion The root locus for a...

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