Complex dominant loci

The root locus diagram is shown in Figure 5.19. In this case the real locus occurs between. v = —5 and —3 and the complex dominant loci breakaway at rrh = —1-15. Since these loci are further to the right than the previous option, the transient response will be slower. The compensator gain that corresponds to ( = 0.7 is K = 5.3. The resulting time response is shown in Figure 5.20, where the overshoot is 5.3% and the settling time is 3.1 seconds. 

When we used root locus for controller design in Chapter 7, we chose a dominant pole (or a conjugate pair if complex). With state space representation, we have the mathematical tool to choose all the closed-loop poles. To begin, we restate the state space model in Eqs. (4-1) and (4-2) ... 

From Eq. (7.174), we see that the ideal terms obtained with infinite GB have been corrupted by additional terms caused by finite GB. To see the effects graphically, we select several values of Q and then factor the polynomial in Eq. (7.175) for many values of GB in order to draw loci. The polynomial in Eq. (7.175) is fifth order, but three of the roots are in the far left-half normalized s plane. The dominant poles are a pair of complex poles that correspond to the ideal poles in Eq. (7.172) but are shifted because of finite GB. Figure 7.117 shows the family of loci generated, one locus for each value of Q selected. Only the lod of the dominant second quadrant pole are plotted. The other dominant pole is the conjugate. 

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