1. Modern Control System EKT 308 Root Locus Method (contd…)

2. Root Locus Procedure (contd…) Root Locus Procedure Step 1 (review): Locate poles and zeros in the s-plane (‘x’ for poles, ‘o’ for zeros)

3. Step 2 (review): Locate the segments of the real axis that are root loci. The root locus on the real axis lies in a segment of the real axis to the left of an odd number of poles and zeros. Magnitude and Angle Criterion

4. Magnitude and Angle Criterion (contd…)

5. Figure 1: Angle for s = s1 Note: Because complex roots appear as complex conjugate pairs, root loci must be symmetrical with respect to horizontal real axis.

6. Step 3: The loci proceed to the zeros at infinity along asymptotes centered at Where n, the order of numerator polynomial and M is the order of denominator Polynomial

7. Example for step 3. Figure 2: Root loci on real axis

8. Asymptotes are shown in Figure 3

9. Figure 3: Asymptotes

10. Step 4: Determine where the locus crosses the imaginary axis (if it does so), using Routh-Hurwitz criterion. Hint: When root locus crosses the imaginary axis from left to right, the system moves from stability to instability. Example: Complete first four steps of sketching root locus of the characteristic equation Step 1: Poles and zeros are shown in figure 4.

11. Figure 4: Poles and zeros

12. Step 2: There is a segment of root locus on the real axis between s=0 to s=-4 as shown in figure 4 above. The asymptotes are drawn in figure 5.

13. Figure 5: Asymptotes

14. Step 4.