RootLocusPlot

RootLocusPlot[lsys,{k,k_min,k_max}]

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

Details and Options

RootLocusPlot plots the location of poles for the closed-loop system for a range of k values.
RootLocusPlot treats the parameter k as local, effectively using Block.
The systems model lsys can be a StateSpaceModel or a TransferFunctionModel.
RootLocusPlot has the same options as Graphics, with the following additions and changes:

Axes	True	whether to draw axes
ColorFunction	Automatic	how to apply coloring to the loci
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
EvaluationMonitor	None	expression to evaluate at every parameter evaluation
Exclusions	Automatic	parameter values to exclude
ExclusionsStyle	None	what to draw at excluded points
FeedbackType	"Negative"	feedback type
MaxRecursion	Automatic	the maximum number of recursive subdivisions allowed
Mesh	Automatic	how many mesh divisions to draw
MeshFunctions	Automatic	how to determine the placement of the mesh divisions
MeshShading	None	how to shade regions between mesh points
MeshStyle	None	the style for mesh divisions
Method	Automatic	the method to determine the loci
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotPoints	Automatic	initial number of sample parameter points
PlotRange	Automatic	range of values to include
PlotRangeClipping	True	whether to clip at the plot range
PlotStyle	Automatic	graphics directives to specify the style for the loci
PlotTheme	$PlotTheme	overall theme for the plot
PoleZeroMarkers	Automatic	markers for poles and zeros
RegionFunction	(True&)	how to determine whether a point should be included
PlotLegends	None	legends for root loci
WorkingPrecision	MachinePrecision	the precision used in internal computations

RootLocusPlot takes a Method option that specifies the method used for computing the root loci.
With the setting Method->"GenericSolve", the loci are determined by computing the roots at the sample points and then sorting them.
With Method->"NDSolve", RootLocusPlot uses NDSolve to solve the differential equation , where is the characteristic equation of the closed-loop system and is the complex variable.
Method->{"NDSolve",opt₁->val₁,opt₂->val₂,…} uses the specified NDSolve options.
The closed-loop poles for specific values of k can be plotted with appropriate settings for Mesh, MeshFunctions, and MeshStyle.
Markers for open-loop poles and zeros, as well as closed-loop poles, can be specified by setting the PoleZeroMarkers option.
The arguments to RegionFunction are x, y, and k.

Examples

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 k∈Interval[{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):