Chapter 6: Root Locus

1 / 10

# Chapter 6: Root Locus - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Chapter 6: Root Locus' - carter

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Chapter 6: Root Locus

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

Polynomial long division

Binomial theorem

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

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
• 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

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