5.8.2.2 For the system of Figure 1 let G H C R + - Figure 1: Standard Closed Loop Con guration G(s)H(s)=K (s;2) s+2 C(s) R(s) = G 1+GH = G 1+ K(s;2) s+2 = G(s+2) (1 +K)s+2(1;K) Thus a(s)=G(s)(s+2)and b(s)=(1+K)s+2(1;K). Wedon'tneed to know a(s). All we need for the root locus analysis is the denominator of C(s). Clearly,the closed loop system has one pole. The position of the pole depends on K as given by s = ;2(1;K) K +1 The following table summarizes the closed loop pole location for various values of K. K 0.01 0.1 0.5 1 2 10 s -1.96 -1.64 -0.67 0 0.67 1.64 The smaller K,the closer the closed loop pole is to the pole of GH. As K increases, the closed loop pole moves away from the pole of GH at s = ;2. In this case the pole moves to the right, crosses the imaginary axis 1 X O Re(s) Im(s) Figure 2: Root locus and nishes at the zero of GH. This can be seen from the formula for the closed loop pole location. That is, lim K!1 ;2(1;K) K +1 = 2 We can get same result from the rules of root locus. 1. There is root locus on the real axis between the pole of GH at s = ;2 and the zero of GH at s =+2. 2. The root locus starts at the pole of GH (forK=0), and moves to the righttowards s =+2. GH has one pole and one zero, meaning the pole zero excess (PZE) is 0. This means no asymptote and no zeros at in nity.AsK increases towards 1, the closed loop pole moves closer and closer to the zero of GH at s =2. The root locus is shown in Figure 2 The gain to drive the system to the brink of instabilityis K = js+2j js;2j s=0 =1: 2