Rules for Plotting root Locus G(s)H(s) K End points AtK=0,G(s0)H(S)=∞ s, are poles of the open-loop system function G(s)H(s) At|K=∞,G(0)H(S)=0 so are zeros of the open-loop system function G(s)H(s). Thus Rule #1: A root locus starts(atK=0) from a pole of G(s)H(s)and ends(at K=o)at a zero ofG(s)H(s Question: What if the number of poles t the e number oi zeros Answe Start or end at±o
Rules for Plotting Root Locus • End points — A t K = 0, G ( s o ) H( s o) = ∞ ⇒ s o are poles of the open-loop system function G ( s ) H( s). — A t |K| = ∞, G ( s o ) H( s o) = 0 ⇒ s o are zeros of the open-loop system function G ( s ) H( s). Thus: Rule #1: A root locus starts (at K = 0) from a pole of G ( s ) H( s) and ends (at |K| = ∞) at a zero of G ( s ) H( s). Question: What if the number of poles ≠ the number of zeros ? Answer: Start or end at ± ∞
Rule #2: angle criterion of the root locus K real number Thus, So is a pole for some positive value of K if: K≥0→∠G(s0)H(S0)=(27+1)r: In this case, So is a pole ifK=1/G(so H(so)l Similarly so is a pole for some negative value of K if: K≤0→∠G(0)H(s0)=2n丌 In this case, So is a pole ifK=-1/G(so) H(soI
Rule #2: Angle criterion of the root locus • Thus, s0 is a pole for some positive value of K if: In this case, s0 is a pole if K = 1/|G(s0) H(s0)|. • Similarly s0 is a pole for some negative value of K if: In this case, s0 is a pole if K = -1/|G(s0) H(s0)|
Example of root locus +2 One zero at -2 s(s+1)two poles at 0 locus g(sh(s)) K≥0 K≤0 oX 0K=0 2 Real Axis
Example of Root Locus. One zero at -2, two poles at 0, -1