This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)


generates the root locus plot of a rational function g of k ranging from to .
plots the root loci of a TransferFunctionModel or StateSpaceModel object sys.
AxesTruewhether to draw axes
ColorFunctionAutomatichow to apply coloring to the loci
ColorFunctionScalingTruewhether to scale arguments to ColorFunction
EvaluationMonitorNoneexpression to evaluate at every parameter evaluation
ExclusionsAutomaticparameter values to exclude
ExclusionsStyleNonewhat to draw at excluded points
FeedbackType"Negative"feedback type
MaxRecursionAutomaticthe maximum number of recursive subdivisions allowed
MeshAutomatichow many mesh divisions to draw
MeshFunctionsAutomatichow to determine the placement of the mesh divisions
MeshShadingNonehow to shade regions between mesh points
MeshStyleNonethe style for mesh divisions
MethodAutomaticthe method to determine the loci
PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
PlotPointsAutomaticinitial number of sample parameter points
PlotRangeAutomaticrange of values to include
PlotRangeClippingTruewhether to clip at the plot range
PlotStyleAutomaticgraphics directives to specify the style for the loci
PoleZeroMarkersAutomaticmarkers for poles and zeros
RegionFunction(True&)how to determine whether a point should be included
WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • RootLocusPlot takes a Method option that specifies the method used for computing the root loci.
  • With the setting Method, the loci are determined by computing the roots at the sample points and then sorting them.
  • With Method, RootLocusPlot uses NDSolve to solve the differential equation , where is the characteristic equation of the closed-loop system and is the complex variable.
  • Markers for open-loop poles and zeros, as well as closed-loop poles, can be specified by setting the PoleZeroMarkers option.
The root locus plot of a rational polynomial:
Root locus plot of a transfer-function model:
Root loci of a state-space model:
The root locus plot of a rational polynomial:
Click for copyable input
Root locus plot of a transfer-function model:
Click for copyable input
Root loci of a state-space model:
Click for copyable input
The root locus plot for various pole-zero configurations:
Root locus plot of a transfer-function object:
Root locus plot of a state-space model object:
Label the axes:
Specify the origin as the axes origin:
Color stable and unstable parts green and red, respectively:
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:
Negative feedback is assumed by default:
Positive feedback:
Root loci of an open-loop system with positive feedback:
A closed-loop system:
The method can be faster than :
By default, closed-loop poles at the midpoint of the parameter range as well as the open-loop poles and zeros 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:
Use more points to get a smoother curve:
Show the loci only in the region where the closed-loop system is stable:
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:
For negative feedback systems, the root loci are points on the complex plane where the phase of the transfer function is , and for positive feedback systems it is 0:
The root locus plot does not depend on the sampling period:
If the number of poles is greater than the number of zeros, the root loci go to infinity with straight-line asymptotes as the parameter is increased:
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 of the transfer function:
Select those points for which kInterval[{0, 5}]:
Show the points on the root locus plot:
The complex-valued transfer function is a surface with "peaks" at the poles and "valleys" at the zeros:
The root locus plot projected on the surface 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:
The system must be proper:
The root loci may not be symmetric with respect to the real axis (but the roots are):
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