美国麻省理工大学：《航空系统的估计与控制》教学资源（讲义）Handout 4：Root-Locus Review

Handout 4: Root-Locus review Eric feron Feb17,2004 Summary of Guidelines for plotting a root-locus 1. Mark poles x and zel ros 2. Draw the locus on the real axis to the left of an odd number of real poles plus zeros. 3. Draw n-m asymptotes(n is the number of poles, m the number of zeros. The asymptotes are centered at a and leave at angles l, where ∑Pz-∑z n-n 180°+1360° l=0,1,2,,n-m-1. 4. Compute the loci departure angles from the poles and arrival angles 5. Assume so= jwo and compute the point(s) where the locus crosses he imaginary axis for positive K 6. The equation has multiple roots at points on the locus where 0. If so is on the real axis, these points are points of breakaway or break- in. Compute the angles of arrival and the angles of departure for any nts of multiple root 7. Complete the locus, using the previous steps and your erperience

� � Handout 4: Root­Locus Review Eric Feron Feb 17, 2004 Summary of Guidelines for plotting a root­locus 1. Mark Poles X and Zeros O. 2. Draw the locus on the real axis to the left of an odd number of real poles plus zeros. 3. Draw n − m asymptotes (n is the number of poles, m the number of zeros). The asymptotes are centered at α and leave at angles Φl, where zi α = pi − = −a1 + b1 , n − m n − m 180o + l360o φl = , l = 0, 1, 2, . . . n − m − 1. n − m 4. Compute the loci departure angles from the poles and arrival angles at the zeros. 5. Assume s0 = jω0 and compute the point(s) where the locus crosses the imaginary axis for positive K. 6. The equation has multiple roots at points on the locus where da db b ds − a ds = 0. If s0 is on the real axis, these points are points of breakaway or break￾in. Compute the angles of arrival and the angles of departure for any points of multiple roots. 7. Complete the locus, using the previous steps and your experience. 1

s+1 G(s) S2(8+ 2

s + 1 G(s) = s2(s + 4) 2

G(s) +1 3

s + 1 G(s) = s2(s + 12) 3

(s+0.1)2+16 (s+0.1)2+25)

(s + 0.1)2 + 16 G(s) = s((s + 0.1)2 + 25) 4