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Chapter 6: Root Locus. Basic RL Facts:. Consider standard negative gain unity feedback system T R (s) = L(s)/[1+L(s)], S(s) = 1/[1+L(s)], L=G C G, L=GH, etc Characteristic equation 1+L(s) = 0 For any point s on the root locus L(s) = -1=1e +/-j(2k+1)180 ° |L(s)|=1  magnitude criterion

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basic rl facts
Basic RL Facts:
  • Consider standard negative gain unity feedback system
  • TR(s) = L(s)/[1+L(s)], S(s) = 1/[1+L(s)], L=GCG, L=GH, etc
  • Characteristic equation 1+L(s) = 0
    • For any point s on the root locus
      • L(s) = -1=1e+/-j(2k+1)180°
        • |L(s)|=1  magnitude criterion
        • arg(L(s)) = +/- (2k+1)180°  angle criterion
        • Angle and magnitude criterion useful in constructing RL
  • RL is set of all roots (= locus of roots) of the characteristic equation (= poles of closed loop system)
  • OL poles (zeros) are poles (zeros) of L(s)
  • CL poles are poles of TR(s), or S(s), …
  • Closed loop poles start at OL poles (=poles of L(s)) when K=0
  • Closed loop poles end at OL zeros (=zeros of L(s)) when K  infinity
  • Stable CL systems have all poles in LHP (no poles in RHP)
outline
Outline
  • Graphical RL construction
  • Mathematical common knowledge
  • Motivational Examples
  • Summary of RL construction Rules
  • Matlab & RL
  • Assignments
pole zero form of l s
Pole-Zero Form of L(s)

Examples?

For any point s in the s-Plane, (s+z) or (s+p) can be expressed in polar form (magnitude and angle, Euler identity)

For use with magnitude criterion

For use with angle criterion

Graphical representation/determination.

mathematical common knowledge
Mathematical Common Knowledge

Polynomial long division

Binomial theorem

example 1
Example 1
  • RL on real axis. Apply angle criterion (AC) to various test pts on real axis.
  • RL asymptotes.
    • Angles. Apply AC to test point very far from origin, approximate L(s) = K/sm-n
    • Center. Approximate L(s) = K/(s+c)m-n, c  center
  • RL Breakaway points. Find values of s on real axis so that K = -1/L(s) is a maximum or minimum.
  • RL intersects imaginary axis. R-H criterion, auxiliary equation.
  • Complete RL plot (see Fig. 6-6, pg. 346).
  • Design. Use RL plot to set damping ratio to .5.
example 2
Example 2

New featurs: Complex roots, break-in points, departure angles.

  • Plot OL poles and zeros. Standard beginning.
  • RL on real axis. Apply angle criterion (AC) to various test pts on real axis.
  • RL asymptotes.
    • Angles. Apply AC to test point very far from origin, approximate L(s) = K/sm-n
    • Center. Approximate L(s) = K/(s+c)m-n, c  center
  • RL Break-in points. Find values of s on real axis so that K = -1/L(s) is a maximum or minimum.
  • RL intersects imaginary axis. R-H criterion, auxiliary equation.
  • Complete RL plot (see Fig. 6-6, pg. 346).
  • Design. Use RL plot to set damping ratio to .5.
root locus construction rules
Root Locus Construction Rules
  • RL on real axis. To the left of an odd number of poles & zeros
  • RL asymptotes.
    • Angles. +/- 180(2k+1)/(#poles - #zeros)
    • Center. C = -[(sum of poles)-(sum of zeros)]/(#poles - # zeros)
  • RL Break-in points. K=-1/L(s), dK/ds =0, s’s on real axis portion of RL
  • RL intersects imaginary axis. R-H criterion, auxiliary equation.
  • Other rules. We will use MatLab for details.
chapter 6 assignments
Chapter 6 Assignments

B 1, 2, 3, 4, 5, 10, 11,