Chapter 6: Root Locus

1
Chapter 6 Root Locus
2
Basic RL Facts

• Consider standard negative gain unity feedback
  system
• TR(s) L(s)/1L(s), S(s) 1/1L(s), LGCG,
  LGH, etc
• Characteristic equation 1L(s) 0
• For any point s on the root locus
• L(s) -11e/-j(2k1)180
• L(s)1 ?? magnitude criterion
• arg(L(s)) /- (2k1)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 K0
• 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)

3
Outline

• Graphical RL construction
• Mathematical common knowledge
• Motivational Examples
• Summary of RL construction Rules
• Matlab RL
• Assignments

4
Pole-Zero Form of L(s)
Examples?
For any point s in the s-Plane, (sz) or (sp)
can be expressed in polar form (magnitude and
angle, Euler identity)
For use with magnitude criterion
For use with angle criterion
Graphical representation/determination.
5
Mathematical Common Knowledge
Polynomial long division
Binomial theorem
6
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/(sc)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.

7
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/(sc)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.

8
Root Locus Construction Rules

• RL on real axis. To the left of an odd number of
  poles zeros
• RL asymptotes.
• Angles. /- 180(2k1)/(poles - zeros)
• Center. C -(sum of poles)-(sum of
  zeros)/(poles - zeros)
• RL Break-in points. K-1/L(s), dK/ds 0, ss on
  real axis portion of RL
• RL intersects imaginary axis. R-H criterion,
  auxiliary equation.
• Other rules. We will use MatLab for details.

9
Matlab and RL
10
Chapter 6 Assignments
B 1, 2, 3, 4, 5, 10, 11,