Documentation

# rlocus

Root locus plot of dynamic system

## Syntax

rlocus(sys)
rlocus(sys1,sys2,...)
[r,k] = rlocus(sys)
r = rlocus(sys,k)

## Description

rlocus computes the root locus of a SISO open-loop model. The root locus gives the closed-loop pole trajectories as a function of the feedback gain k (assuming negative feedback). Root loci are used to study the effects of varying feedback gains on closed-loop pole locations. In turn, these locations provide indirect information on the time and frequency responses.

rlocus(sys) calculates and plots the root locus of the open-loop SISO model sys. This function can be applied to any of the following negative feedback loops by setting sys appropriately.

If sys has transfer function

$h\left(s\right)=\frac{n\left(s\right)}{d\left(s\right)}$

the closed-loop poles are the roots of

$d\left(s\right)+kn\left(s\right)=0$

rlocus adaptively selects a set of positive gains k to produce a smooth plot. Alternatively,

rlocus(sys,k)

uses the user-specified vector k of gains to plot the root locus.

rlocus(sys1,sys2,...) draws the root loci of multiple LTI models sys1, sys2,... on a single plot. You can specify a color, line style, and marker for each model, as in

rlocus(sys1,'r',sys2,'y:',sys3,'gx').

[r,k] = rlocus(sys) and r = rlocus(sys,k) return the vector k of selected gains and the complex root locations r for these gains. The matrix r has length(k) columns and its jth column lists the closed-loop roots for the gain k(j).

## Examples

collapse all

Plot the root-locus of the following system.

$h\left(s\right)=\frac{2{s}^{2}+5s+1}{{s}^{2}+2s+3}.$

h = tf([2 5 1],[1 2 3]);
rlocus(h)

You can use the right-click menu for rlocus to add grid lines, zoom in or out, and invoke the Property Editor to customize the plot. Also, click anywhere on the curve to activate a data marker that displays the gain value, pole, damping, overshoot, and frequency at the selected point.

## Tips

You can change the properties of your plot, for example the units. For information on the ways to change properties of your plots, see Ways to Customize Plots.