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

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

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

Magnitude and Angle Criterion (contd…)

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.

Step 3: The loci proceed to the zeros at infinity along asymptotes centered at

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

Asymptotes are shown in Figure 3

Figure 3: Asymptotes

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.

Figure 4: Poles and zeros

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.

Figure 5: Asymptotes

Step 4.

Root Locus (contd…) Step 5: Determine the break away point on the real axis (if any). The locus breakaway from the real axis occurs where there is a multiplicity of roots. At the breakaway point the angle of the tangent to the locus does not change for small change in s (fig 1). Fig 1: Breakaway point

Procedure for finding breakaway point Graphical method.

Breakaway point is expected between -4 and -2 Poles: -2, -4 Zeros: none. Breakaway point is expected between -4 and -2 Fig 2: Poles (no zeros) Let us plot p(s) between s=-4 to s=-2. Fig 3: p(s) versus s

Find the maximum point in the ps-s curve. It occurs at s = -3 Find the maximum point in the ps-s curve. It occurs at s = -3. So the locus breaks away at s=-3 as shown in figure 3. Fig 3: Breakaway at s=-3. b) Analytical method

Example. The angle of locus departure from a pole = Step 6: Determine the angle of departure from pole and of arrival at zero The angle of locus departure from a pole = Difference between net angle due to all other poles and zeros, and the criterion angle Similar formula for calculating angle of arrival at zero. Example. Poles are zeros are shown in figure 4.

In order to find angle of departure at, say complex pole -p1, place a test s1 at infinitesimal distance from -p1. From angle criterion, we get Step 7: Complete the sketch Complete all the sections of the locus not covered in the previous six steps.

Figure 4: Angle of departure