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Chapter 6: Root Locus

Chapter 6: Root Locus. Algebra/calculus review items. Rational functions Limits of rational functions Polynomial long division Binomial theorem. Simple example 1. Unity Gain negative feedback system G p (s) =1/(s+2) G c (s)=K OL TF: L(s)=G p (s)G c (s)=K/(s+2)

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Chapter 6: Root Locus

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  1. Chapter 6: Root Locus

  2. Algebra/calculus review items • Rational functions • Limits of rational functions • Polynomial long division • Binomial theorem

  3. Simple example 1 • Unity Gain negative feedback system • Gp(s) =1/(s+2) • Gc(s)=K • OL TF: L(s)=Gp(s)Gc(s)=K/(s+2) • Plot OL poles and zeros in s-plane • CL TF: T(s)=K/(s+2+K) • Plot CL poles and zeros in s-plane as K changes • Make observations

  4. Simple example 2 • Unity Gain negative feedback system • Gp(s) =(s+1)/[s(s+2)] • Gc(s)=K • OL TF: L(s)=Gp(s)Gc(s)=K(s+1)/[s(s+2)] • Plot OL poles and zeros in s-plane • CL TF: T(s)=K(s-1)/(s2+(2+K)s+K) • Plot CL poles and zeros in s-plane as K changes • Make observations

  5. Simple example 3 • Unity Gain negative feedback system • Gp(s) =1/[s(s+5)] • Gc(s)=K • OL TF: L(s)=Gp(s)Gc(s)=K/[s(s+5)] • Plot OL poles and zeros in s-plane • CL TF: T(s)=K/(s2+5s+K) • Plot CL poles and zeros in s-plane as K changes • Make observations

  6. Learning from simple examples. • Going beyond simple examples.

  7. 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)

  8. Outline • Graphical RL construction • Mathematical common knowledge • Motivational Examples • Summary of RL construction Rules • Matlab/Sysquake & RL • Assignments

  9. 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.

  10. Mathematical Common Knowledge Polynomial long division Final remainder is lower order than divisor. Remainder = 0 => you found a factor. Binomial theorem

  11. 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.

  12. 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.

  13. 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.

  14. Matlab/Sysquake and RL

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

  16. Root locus Big Picture • Characteristic equation: 1+KL(s)=0; L(s)=N(s)/D(s) • N(s)=sm+bsm-1+…=(s+z1)…(s+zm); b=z1+…+zm • D(s)=sn+asn-1+…=(s+p1)…(s+pn); a=p1+…+pn • RL For various values of K>0 solve D(s)+KN(s)=0 • Small K RL • Real axis portion of RL • Choose a K, find the roots • Large K RL • Poly long division yields approximation L(s)=1/(s+c)n-m • binomial theorem relates c to a and b; • a determined by zeros; b determined by poles

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