MATLAB Root Locus
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## 1 Introduction to Root Locus using the rlocus MATLAB command

Figure 1: Root Locus Diagram

MATLAB's rlocus command plots the root locus via the Evan's method. It requires the Control System Toolbox.

MATLAB's rlocus command must be passed SISO open loop systems. It will return the closed loop pole trajectories as a function of feedback gain. Where

 $LaTeX: H\left(s\right)=\frac{N\left(s\right)}{D\left(s\right)}=G\left(s\right)K\left(s\right)$

and the closed loop poles are

 $LaTeX: \Phi_{CL}=D\left(s\right)+kN\left(s\right)$

where

$LaTeX: H\left(s\right)$ is the open loop transfer function and
$LaTeX: k$ is the feedback gain.

## 2 Basic Usage of rlocus for plotting the Root Locus

The rlocus command plots the root locus of the open system sys with the following command:

 >> rlocus(sys)


where

sys is the open loop transfer function.

The root locus can also be plotted at a particular feedback gain k with the following:

 >> rlocus(sys, k)


Like plot the rlocus command can take multiple systems like this

 >> rlocus(sys1, 'b',sys2, 'r-.:',sys3, 'g--').


## 3 Required Toolboxes for rlocus

Control System Toolbox