generates a root locus plot of a linear time-invariant system lsys as the parameter k ranges from kmin to kmax.

Details and Options


open allclose all

Basic Examples  (2)

Root locus plot of a transfer-function model:

Root loci of a state-space model:

Scope  (3)

The root locus plot for various pole-zero configurations:

Root locus plot of a TransferFunctionModel:

Root locus plot of a StateSpaceModel:

Generalizations & Extensions  (1)

RootLocusPlot[g[s],] is taken to be RootLocusPlot[TransferFunctionModel[g[s],s],]:

Options  (21)

ColorFunction  (1)

Color stable and unstable parts green and red for a continuous-time system:

For a discrete-time system:

Epilog  (3)

Show lines corresponding to a damping ratio 0.4 on the plane:

Show the circle corresponding to the natural frequency 3 radians per time unit:

The loci of points with damping 0.4 in the plane for a system with sampling period 1:

FeedbackType  (3)

Negative feedback is assumed by default:

Positive feedback:

Root loci of an open-loop system with positive feedback:

A closed-loop system:

Method  (1)

The "NDSolve" method can be faster than "GenericSolve":

PlotLegends  (4)

Use placeholder legends for root loci:

Use a list of legend text:

Use LineLegend to add a overall legend label:

Place the legend above the plot:

PlotTheme  (2)

Use a theme with a frame and grid lines:

Change the style of the grid lines:

PoleZeroMarkers  (6)

By default, open-loop poles at zero and closed-loop poles at the mean parameter value are shown:

Show no markers:

Show the closed-loop poles only:

Use text or typeset labels:

Use graphics primitives as the pole-zero markers:

Use any 2D or 3D graphics:

RegionFunction  (1)

Show the loci only in the region where the closed-loop system is stable:

Applications  (3)

Explore and determine critical points such as break-away, break-in, and imaginary axis crossings:

Plot the roots of a polynomial as a parameter is varied:

Analyze the effect of the sensor gain on a system:

Properties & Relations  (6)

The root-locus consists of points with for negative feedback:

And points with for positive feedback:

The root locus plot does not depend on the sampling period:

For strictly proper systems, the root loci go to infinity with straight-line asymptotes:

For the strictly proper system, the number of poles is greater than the number of zeros:

The plot shows four loci going to infinity:

The slopes of the asymptotes for a negative feedback system:

Find where the asymptotes intercept the real axis:

Plot the root loci and the asymptotes:

The slopes of the asymptotes for a positive feedback system:

Plot the root loci and the asymptotes:

The break-away and break-in points on the real axis can be computed from the poles and zeros:

Select those points for which kInterval[{0,5}]:

Show the points on the root locus plot:

With the loci and closed-loop poles removed, it becomes a pole-zero plot of the open-loop system:

Compute the poles and zeros:

Use ListPlot to show the same pole-zero plot:

The complex-valued transfer function is a surface with "peaks" at the poles and "valleys" at the zeros:

The root locus plot travels from the "peaks" to the "valleys" along the lines of steepest descent:

The Bode magnitude plot is the intersection of the surface and the - plane:

Possible Issues  (1)

The root loci may not be symmetric with respect to the real axis (but the roots are):

Wolfram Research (2010), RootLocusPlot, Wolfram Language function, (updated 2014).


Wolfram Research (2010), RootLocusPlot, Wolfram Language function, (updated 2014).


Wolfram Language. 2010. "RootLocusPlot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014.


Wolfram Language. (2010). RootLocusPlot. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_rootlocusplot, author="Wolfram Research", title="{RootLocusPlot}", year="2014", howpublished="\url{}", note=[Accessed: 19-June-2024 ]}


@online{reference.wolfram_2024_rootlocusplot, organization={Wolfram Research}, title={RootLocusPlot}, year={2014}, url={}, note=[Accessed: 19-June-2024 ]}