Page 106 - Mechatronics with Experiments

92 MECHATRONICS
Im(s)

as parameter “a” varies j

Re(s)

FIGURE 2.37: Value of gain is
parameterized along the root locus
–j curves. The particular value of the
gain can be calculated in order to be
at a selected point on the root locus.

function and assumes that a parameter K is in the feedback loop. Therefore, in order to
use the rlocus() function to analyze the roots of a transfer function or algebraic equation as
one parameter varies, the equivalent root locus problem must be formed before calling the
rlocus() function. Below are some samples of calls to rlocus() function.

sys = tf(num,den) ; /* sys can be formed by tf, ss, zpk function calls */
sys = zpk(z,p,k) ; /* sys= (K (s-z1)(s-z2).../)(s-p1).(s-p2)....) */
sys = ss(A,B,C,D) ; /* G(s) = C (sI-A)ˆ-1B+D*/
rlocus(sys) ; /* given LTI system, plot closed loop poles as K varies
from 0 to infinity
rlocus(sys, K); /*..............for the values of the parameter given
in the vector K*/
[R]=rlocus(sys,K); /* Stores the closed loop roots in the R for numerical
reference. */
rltool(sys) ; /* Interactive graphical tools for plotting root locus */

Root Locus Sketching Rules In this section we list the rules used in approximate
hand sketching of the root locus. The derivation of the rules is given in many classic
textbooks on control systems.

1. Put the problem in the 1 + KG(s) = 0 form, where K is the varying parameter. Mark
the poles of G(s) = n(s)∕d(s)by x and the zeros by o on the s-plane. Notice that root
locus begins at x’s for K = 0 and ends at o’s as K ⟶ ∞.The x’s and o’s represent
the asymptotic location of closed loop pole locations. If K is indeed the loop transfer
function gain, the o’s are also the zeros of the open and closed loop system. Otherwise,
they only represent the asymptotic location of closed loop system poles.
2. Mark the part of the real axis to the left of odd number of poles and zeros as part
of the root locus. All the points on the real axis to the left of odd number of poles
and zeros satisfy the angle criteria, hence are part of the root locus. Notice that the