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@ -P& - P~)(s - ~3) + K = 0
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(15.3)
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Using the same numerical values for the poles that were used at the beginning of Chap. 14 (-1, -2, -3) gives (15.4) (s + l)(s + 2)(s + 3) + K = 0 ~
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Expanding the product of this equation gives s3 + 6s2 + 11s + (K + 6) = 0 (15.5)
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which is third-order. For any particular value of controller gain K,, we can obtain the roots of the characteristic equation [Eq. (15.5)]. For example, if K, = 4.41(K = 26.5), Eq. (15.5) becomes s3 + 6s + 11s + 32.5 = 0 Solving* this equation for the three roots gives rl = -5.10
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r* = -0.45 - j2.5
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r3 = -0.45 +
j2.5
By selecting other values of K, other sets of roots are obtained as shown in Table 15.1. For convenience, we may plot the roots r 1, r2, and r3 on the complex plane as K changes continuously. Such a plot is called a root-locus diagram and is shown in Fig. 15.2. Notice that there are three loci or brunches corresponding to the three roots and that they emerge or begin (for K = 0) at the poles of the open-loop transfer function (- 1, - 2, -3). The direction of increasing K is indicated on the diagram by an arrow. Also the values of K are marked on each locus. The root-locus diagram for this system and others to follow is symmetrical with respect to the real axis, and only the portion of the diagram in the upper half plane need be drawn. This follows from the fact that the characteristic equation for a physical system contains coefficients that are real, and therefore complex roots of such an equation must appear in conjugate pairs. The root-locus diagram has the distinct advantage of giving at a glance the character of the response as the gain of the controller is continuously changed. The diagram of Fig. 15.2 reveals two critical values of K, one is at K2 where two of the roots become equal, and the other is at K3 where two of the roots are pure imaginary. It should be clear from the discussion in Chap. 14 that the nature of the response C(t) will depend only on the roots r 1, r 2, r3. Thus, if the roots are all real, which occurs for K < K2 in Fig. 15.2, the response will be nonoscillatory.
*The procedure for obtaining the roots of a higher-order equation, such as Eq. (15.5), is covered in any text on advanced algebra. In a later section of this chapter, we shall find the roots by a graphical technique called the root-locus method. There am also numerical methods for finding the roots. In Appendix 15A of this chapter, a BASIC computer program for computing the roots of a polynomial equation is given.
180 LINEAR
TABLE 15.1
CLOSED-LOOP
SYSTEMS
Roots of the characteristic equation (s + l)(s + 2)(s + 3) + K = 0
K = 6K,
0.23 0.39 1.58 6.6 26.5 60.0 100.0
-3 -3.10 -3.16 -3.45 -4.11 -5.10 -6.00 -6.72
-2 -1.75 -1.42 - 1.28 - j0.75 -0.95 - j 1.5 -0.45 - j2.5 0.0 - j3.32 0.35 - j4
-1 -1.15 - 1.42 -1.28 + j0.75 -0.95 + jl.5 -0.45 + j2.5 0.0 + j3.32 0.35 + j4
If two of the roots are complex and have negative real parts (K2 < K < Ks), the response will include damped sinusoidal terms, which will produce an oscillatory response. If K > K3, two of the roots are complex and have positive real parts, \ and the response is a growing sinusoid. Some of these types of response were shown in Fig. 14.2. As another example of a root-locus diagram, let the proportional controller be replaced with a PI controller, for which case G t in Fig. 15.1 is G1 = K,(l + $)
For this case, the open-loop transfer function is
G(s) = Kc(vs + 1)
qs(71s + l)(QS + l)(qs + 1)
KS=60 a
FIGURE 15-2
Root-locus diagram for (s + l)(s + 2)(s + 3) + K = 0.
ROOTL~XUS
which may be written in an alternate form G(s) = where K =
717273 ' z1= -+ p2 = -I 72 p3 = -1 T3
K(s - Zl> es - Plm - P2)O - P3)
(15.6)
p1 = - I , 71
The term zr is called a zero of the open-loop transfer function. A zero of G(s) is any value of s for which G(s) approaches zero. By comparing Eq. (15.6) with Eq.(15.2), we see that the addition of integral action contributes to the open-loop transfer function one zero at z1 and one additional pole at the origin. The characteristic equation corresponding to Eq. (15.6) is 1+ This expression may be written
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