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4s -n>(s --P~)(s -p3) + K(s - zd = 0
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es - PlN - P2)(S - P3)
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As a specific example of the root-locus diagram corresponding to Eq. (15.8), let 71 = l,Q = $,rs = i, and ~-1 = i. These parameters are the same as those used in Example 14.4. The root-locus diagram is shown in Fig. 15.3. Notice that for this case there are four loci corresponding to the four roots and that they emerge (at K = 0) from the open-loop poles (0, - 1, -2, - 3). One of the loci moves toward the open-loop zero at -4 as K approaches infinity. The diagram in Fig. 15.3 should be compared with the one in Fig. 15.2 to see the effect of adding integral action to the control system. Notice that the value of K = 3.84, above which the roots move into the right half plane, is lower than the corresponding value of K = 60 for proportional control. The effect of
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FIGURE 15-3 Root-locus diagram for s(s + l)(s + 2)(s + 3)+ K(s +4) = 0;K = 6K,.
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adding integral action has been to destablize the system in terms of the amount of proportional action that can be used before instability occurs. A method for quickly sketching the root-locus diagram was developed by Evans (1954, 1948) and has been presented in many textbooks on control theory. In the next section, this method will be presented.
PLOTTING THE ROOT-LOCUS DIAGRAM
Having introduced the concept of root locus by two examples, here are some rules that were first introduced by Evans (1954, 1948) for plotting root-locus diagrams of characteristic equations of any order. Without these rules, the time and effort needed to plot root-locus diagrams would be too great to render them useful in engineering computations. The first step in applying the root-locus technique to determine the ro ts % of the characteristic equation of the closed-loop control system is to write t e open-loop transfer function (G = GiGzH) in the standard form
G=K;
where K = constant
D = ts - PI)(S - ~2). . .(s - PA
The term zi is called a zero of the open-loop transfer function. The term pi is called a pole of the open-loop transfer function. This term was defined earlier in this chapter. A zero of G(s) is any value of s for which G(s) equals zero. The factored terms (S - zi) and (S -pi) in N/D arise naturally in the open-loop transfer function. For example, in the control system considered at the beginning of this chapter, Eq. (15.2) was written in the standard form with KC K=7172 73
D = (8 - PINS -
pz)(s
N = l The second example for PI control considered earlier [Eq. (15.6)] illustrates a situation where a term (z - zi) appears in N. Using the form of G given by Eq. (15.9), the characteristic equation 1 + G = 0 may be written in the alternate form
or ,
D+KN =0
(15.10)
ROOTLOCUS
It is assumed in the remainder of this chapter that n L m, which is true for all physical systems. This being the case, the characteristic equation will be of nth order and have n roots, r 1, t2, . . . , rn . To develop the graphical method for determining the root locus, the characteristic equation is rewritten as (15.11) In terms of the poles and zeros of the open-loop transfer function, Eq. (15.11) becomes
K(s - zm - ZZ) (S - tm) = -1
(15.12)
(s - Pl)(S - P2) * * *(s - Pn)
Since the left-hand member is in general complek, we may write Eq. (15.12) in the equivalent form involving magnitude and phase angle; thus
4(s - Zl) +
4(s - z2) +
... +
40 - zd
- [4(s
- Pl) + .**
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