Page 823 - The Mechatronics Handbook

066_Frame_C26 Page 8 Wednesday, January 9, 2002 1:58 PM

Root Locus Construction
There are several software packages available for generating the root locus automatically for a given F =
N/D. In particular, the related MATLAB commands are rlocus and rlocfind. In many cases, approxi-
mate root locus can be drawn by hand using the rules given below. These rules are determined from the
basic definitions (26.5), (26.7), and (26.8).
1. The root locus has n branches: r 1 (K),…,r n (K).
2. Each branch starts (K ≅ 0) at a pole p i and ends (as K → ∞ ) at a zero z j , or converges to an
ja
asymptote, Me , where M → ∞ and
2 + 1
a = --------------- × 180°, = 0,…, nm 1)
(
–
–
–
nm
3. There are (n − m) asymptotes with angles α . The center of the asymptotes (i.e., their intersection
point on the real axis) is
∑ n i=1 i – m
p
∑ z j
j=1
s a = ----------------------------------
–
nm
4. A point x ∈ is on the root locus if and only if the total number of poles p i ’s and zeros z j ’s to
the right of x (i.e., total number of p i ’s with Re(p i ) > x plus total number of z j ’s with Re(z j ) > x)
is odd. Since F(s) is a rational function with real coefficients, poles and zeros appear in complex
conjugates, so when counting the number of poles and zeros to the right of a point x ∈ we just
need to consider the poles and zeros on the real axis.
5. The values of K for which the root locus crosses the imaginary axis can be determined from the
Routh–Hurwitz stability test. Alternatively, we can set s = jω in (26.5) and solve for real ω and K
satisfying

(
(
Djw) + KN jw) = 0
Note that there are two equations here, one for the real part and one for the imaginary part.
6. The break points (intersection of two branches on the real axis) are feasible solutions (satisfying
rule 4) of

d
-----Fs() = 0 (26.11)
ds
≅
7. Angles of departure (K 0) from a complex pole, or arrival (K → +∞) to a complex zero, can be
determined from the phase rule. See example below.
Let us now follow the above rules step by step to construct the root locus for

( s + 3)
Fs() = --------------------------------------------------------------------------------------
( s 1) s + 5) s ++ j2) s + 4 j2)
(
(
(
4
–
–
First, enumerate the poles and zeros as p 1 = −4 + j2, p 2 = −4 − j2, p 3 = −5, p 4 = 1, z 1 = −3. So, n = 4
and m = 1.
1. The root locus has four branches.
2. Three branches converge to the asymptotes whose angles are 60°, 180°, and −60°, and one branch
converges to z 1 = −3.
©2002 CRC Press LLC