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.