290 Chapter 6 / Control Systems Analysis and Design by the Root-Locus MethodNote that once we have some experience with the method, we can easily evaluate the
changes in the root loci due to the changes in the number and location of the open-loop
poles and zeros by visualizing the root-locus plots resulting from various pole–zero
configurations.
Summary. From the preceding discussions, it should be clear that it is possible to
sketch a reasonably accurate root-locus diagram for a given system by following simple
rules. (The reader should study the various root-locus diagrams shown in the solved
problems at the end of the chapter.) At preliminary design stages, we may not need the
precise locations of the closed-loop poles. Often their approximate locations are all that
is needed to make an estimate of system performance. Thus, it is important that the
designer have the capability of quickly sketching the root loci for a given system.
6–3 Plotting Root Loci with MATLAB
In this section we present the MATLAB approach to the generation of root-locus plots
and finding relevant information from the root-locus plots.
Plotting Root Loci with MATLAB. In plotting root loci with MATLAB we
deal with the system equation given in the form of Equation (6–11), which may be
written as
where num is the numerator polynomial and den is the denominator polynomial.
That is,
Note that both vectors num and den must be written in descending powers of s.
A MATLAB command commonly used for plotting root loci is
rlocus(num,den)
Using this command, the root-locus plot is drawn on the screen. The gain vector Kis au-
tomatically determined. (The vector Kcontains all the gain values for which the closed-
loop poles are to be computed.)
For the systems defined in state space,rlocus(A,B,C,D)plots the root locus of the
system with the gain vector automatically determined.
Note that commands
rlocus(num,den,K) and rlocus(A,B,C,D,K)
use the user-supplied gain vector K.
=sn+Ap 1 +p 2 +p+pnBsn-^1 +p+p 1 p 2 p pn
den=As+p 1 BAs+p 2 BpAs+pnB
=sm+Az 1 +z 2 +p+zmBsm-^1 +p+z 1 z 2 p zm
num=As+z 1 BAs+z 2 BpAs+zmB
1 +K
num
den
= 0
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