Root Locus Design Methods for Feedback

MAE 691

MATLAB Root Locus

• Consider aTransfer Function Model of the form G(s)=num(s)/den(s): 4(s+2)/(s2 +10s +12)

num= 4*[1 6]

den= [1 10 12]

rlocus(num,den)

-30 -25 -20 -15 -10 -5 0 5

-1.5

-1

-0.5

0

0.5

1

1.5

Root Locus

Real Axis (seconds-1)

Imagin

ary

Axis

(seconds-1

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Simple sketching rules for root locus plots

• Get characteristic equation

1+kG(s)H(s)=0

• Place open loop poles of G(s)H(s) on plot with ’x’ (n is number of poles)

• Place open loop zeros of G(s)H(s) on plot with ’o’ (m is number of zeros)

• n-m is the number if infinite closed loop zeros as k-> ∞

Simple sketching rules for root locus plots (cont.)

• On the real axis: Draw lines to the left from odd numbered real value pole/zero to the next real value pole/zero or ∞ (Note: number from the right side)

• Find the asymptotes • Number= n-m, and general Butterworth pattern

• Location of asymptotes center on real axis

mn

zpm

i

i

n

i

i

A

11

Simple sketching rules for root locus plots (cont.)

• Find break-in or break-out points on real axis by dk/ds=0 { from 1+kG(s)H(s)=0; k=-1/(G(s)H(s)) }

• Find where the root locus crosses the real axis by substituting s=jω into the characteristic equation

• Root locus is symmetric about the real axis

• Roots go from open-loop poles to open-loop zeros as k is increased from 0 to ∞