NyquistPlot

NyquistPlot[lsys]
generates a Nyquist plot of the transfer function for the system lsys.

NyquistPlot[lsys, {min, max}]
plots for the frequency range to .

NyquistPlot[expr, {, min, max}]
plots expr using the variable .

Details and OptionsDetails and Options

  • NyquistPlot gives the complex-plane plot of the transfer function of lsys as the Nyquist contour is traversed.
  • The system lsys can be TransferFunctionModel or StateSpaceModel, including descriptor and delay systems.
  • For continuous-time systems, the Nyquist contour encloses the entire right half-plane and excludes poles on the imaginary axis. It is traversed in a clockwise direction.
  • For discrete-time systems, the Nyquist contour is the unit circle, and it encloses poles on the unit circle. It is traversed in a counterclockwise direction.
  • The arrows on the NyquistPlot show the direction the Nyquist contour is traversed.
  • The Nyquist contours:
  • For a system lsys with the corresponding transfer function , the following expressions are plotted:
  • continuous-time system
    discrete-time system with sample time
  • If the frequency range is not specified, the entire Nyquist contour is traversed, effectively for continuous-time systems, and for discrete-time systems.
  • NyquistPlot treats the variable as local, effectively using Block.
  • The Nyquist plot can be used to infer the number of unstable poles of the closed loop system as , where is the number of unstable poles of the open-loop system, and is the number of clockwise encirclements of the point . »
  • The Nyquist plot can be used to infer global exponential stability of linear systems with nonlinear feedback , where satisfies a sector constraint when . The closed-loop system is stable if , where is the number of clockwise encirclements of the disk TemplateBox[{Disk, paclet:ref/Disk}, RefLink, BaseStyle -> InlineFormula][{-(b+a)/(2 a b),0},TemplateBox[{Abs, paclet:ref/Abs}, RefLink, BaseStyle -> InlineFormula][(b-a)/(2 a b)]], and is the number of open-loop unstable poles. »
  • NyquistPlot has the same options as Graphics, with the following additions and changes:
  • AxesTruewhether to draw axes
    ColorFunctionAutomatichow to apply coloring to the curve
    ColorFunctionScalingTruewhether to scale arguments to ColorFunction
    EvaluationMonitorNoneexpression to evaluate at every evaluation
    ExclusionsTruefrequencies to exclude
    ExclusionsStyleAutomaticwhat to draw at excluded frequencies
    FeedbackSectorNonethe sector limits for feedback function
    FeedbackSectorStyleAutomaticstyle for feedback sector disk
    FeedbackType"Negative"the feedback type
    MaxRecursionAutomaticmaximum recursive subdivisions allowed
    MeshAutomatichow many mesh divisions to draw
    MeshFunctions{#3&}placement of mesh divisions
    MeshShadingAutomatichow to shade regions between mesh points
    MeshStyleAutomaticthe style for mesh divisions
    NyquistGridLinesNonethe Nyquist grid lines to draw
    PerformanceGoal$PerformanceGoalaspects of performance to try to optimize
    PlotPointsAutomaticiniitial number of sample frequencies
    PlotRangeAutomaticreal and imaginary range of values
    PlotStyleAutomaticgraphics directives to specify the style of the plot
    RegionFunctionAutomatichow to determine if a point should be included
    SamplingPeriodNonethe sampling period
    StabilityMarginsFalsewhether to show the stability margins
    StabilityMarginsStyleAutomaticgraphics directives to specify the style of the stability margins
    WorkingPrecisionMachinePrecisionthe precision used in internal computations
  • The setting Exclusions->True excludes frequencies, including resonant frequencies, where the sinusoidal transfer function is discontinuous.
  • Exclusions->{f1, f2, ...} excludes specific frequencies .
  • ExclusionsStyle->s specifies that style s should be used to render the curve joining opposite ends of each excluded point.
  • Points corresponding to exclusions at resonant frequencies are joined by semicircles at infinity.
  • FeedbackSector->{a, b} indicates feedback with .

ExamplesExamplesopen allclose all

Basic Examples (5)Basic Examples (5)

A Nyquist plot of a transfer-function model:

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A Nyquist plot of a system with resonant frequencies:

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A Nyquist plot of a discrete-time system:

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A discrete-time system with resonant frequencies:

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Another discrete-time system with resonant frequencies:

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