14ai Show that the constant M locus in G plane is a circle f

14.a)i) Show that the constant M locus in G- plane is a circle for all values of M except M=1 ii) The open loop transfer function of a unity feedback control system is G(s) = K / s (1+0.1s) (1+s) Draw the Bode diagram and analyze the stability of the system for K =10.

Solution

see example prob bewlow

The breakaway and break-in points are found from dK/ds = 0. Since the characteristic equation is s3 + 49\' + 5s + K = 0 we have K = -( s3 + 4s2 + 5s) . Now we set which yields s = -1, s = -1.6667 Since these points are on root loci, they are actual breakaway or break-in points. (At points = -1, the value of K is 2, and at point s = -1.6667, the value of K is 1.852.) The angle of departure from a complex pole in the upper half s plane is obtained from e = 1800 - 153.430 - go0 or 6 = -63.43\" The root-locus branch from the complex pole in the upper half s plane breaks into the real axis at s = -1.6667. Next we determine the points where root-locus branches cross the imaginary axis. By substituting s = jw into the characteristic equation, we have (j~)~ + 4(jw)\' + 5(jw) + K = 0 or (K - 4w2) + jo(5 - w2) = 0 from which we obtain w=rtfl, K=20 or w=O, K=O Chapter 6 / Root-Locus Analysis Root-locus branches cross the imaginary axis at w = fi and w = -fl. The root-locus branch on the real axis touches the jw axis at w = 0. A sketch of the root loci for the system is shown in Figure 641(b). Note that since this system is of third order, there are three closed-loop poles. The nature of the system response to a given input depends on the locations of the closed-loop poles. For 0 < K < 1.852, there are a set of complex-conjugate closed-loop poles and a real closedloop pole. For 1.852 5 K < 2, there are three real closed-loop poles. For example, the closedloop poles are located at s = -1.667, s = -1.667, s = -0.667, for K = 1.852 s = -1, s = -1, s = -2, for K = 2 For 2 < K, there are a set of complex-conjugate closed-loop poles and a real closed-loop pole Thus, small values of K (0 < K < 1.852) correspond to an underdamped system. (Since the real closed-loop pole dominates, only a small ripple may show up in the transient response.) Medium values of K (1.852 5 K < 2) correspond to an overdamped system. Large values of K (2 < K) correspond to an underdamped system. With a large v