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imaginary parts of z and the rationalized fraction, Now consider what happens to
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the circle x2 + y2 = 1. To show that the inside of the circle goes over to the right
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half plane, consider a convenient point inside the circle.) On the basis of this transformation, deduce an extension of the Routh criterion that will determine whether the system has roots inside the unit circle. Why might this information be of interest How can the transformation be modified to consider circles of other radii Ii-- Given the control diagram shown in Fig. P14.8, deduce by means of the Routh 4.8. criterion those values of rr for which the output C is__m.se for all inputs R and U.
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14.9. In the control system shown in Fig. P14.9, find the value of K, for which the system is on the verge of instability. The controller is replaced by a PD controller, for which the transfer function is KJTDS + 1). If K, = 10, determine the range of D for which the system is stable. 14.10. (a) Write the characteristic equation for the control system shown in Fig. P14.10. (b) Use the Routh Test to determine if the system is stable for K, = 4. (c) Determine the ultimate value of Kc, above which the system is unstable.
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14.11. For the control system in Fig. P14.11, the characteristic equation is
s4 +4s3 + 6s2 + 4s +(l+K) = 0
(a) Determine the value of K above which the system is unstable. (b) Determine the value of K for which two of the roots are on the imaginary axis, and determine the values of these imaginary roots and the remaining two roots.
FIGURE P14-11
CHAPTER
ROOT LOCUS
In the previous chapter on stability, Routh s criterion was introduced to provide an algebraic method for determining the stability of a simple feedback control system (Fig. 14.3) from the characteristic equation of the system [Eq. (14.7)]. This criterion also yields the number of roots of the characteristic equation that am located in the right half of the complex plane. In this chapter, we shall develop a graphical method for finding the actual values of the roots of the characteristic equation, from which we can obtain the transient response of the system to an arbitrary forcing function.
CONCEPT OF ROOT LOCUS
In the previous chapter, the response of the simple feedback control system, shown again in Fig. 15.1, was given by the expression C = -GG R l+G
+ -u
(15.1)
where G = GIG~H. The factor in the denominator, 1 + G, when set equal to zero, is called the characteristic equation of the closed-loop system. The roots of the characteristic equation determine the form (or character) of the response C(t) to any particular forcing function R(t) or U(t). The root-locus method is a graphical procedure for finding the roots of 1 + G = 0, as one of the parameters of G varies continuously. In our work, the parameter that will be varied is the gain (or sensitivity) K, of the controller. We can illustrate the concept of a root-locus diagram by considering the example
LINEAR CLosmmP SYSTEMS
FIGURE 15-1 Simple feedback contd system.
presented in Fig. 14.1, which is represented by the block diagram of Fig. 15.1 with G1 = K, 1 G2 = (71s + l)(QS + 1) 1 HzTjs + 1 For this case, the open-loop transfer function is G = (71s + l)(QS + l)(QS + 1) which may be written in the alternate form G(s) = where K = &
p1 = -f p2 = -& p3+
0 - PlNS - P2NS - p3)
(15.2)
The terms ~1, pz, and p3 am called the poles of the open-loop transfer function. A pole of G(s) is any value of s for which G(s) approaches infinity. For example, it is clear from Eq. (15.2) that, ifs = ~1, the denominator of Eq. (15.2) is zero and therefore G(s) approaches infinity. Hence p 1 = - l/q is a pole of G(s). The characteristic equation for the closed-loop system is 1+
(3 - PlNS - P2M - P3) = 0
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