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Chapter 6: Root Locus

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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Chapter 6: Root Locus

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  1. Chapter 6: Root Locus

  2. 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)

  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, (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.

  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/(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.

  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/(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.

  8. 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.

  9. Matlab and RL

  10. Chapter 6 Assignments B 1, 2, 3, 4, 5, 10, 11,

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