SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
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 complexplane 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 continuoustime systems, the Nyquist contour encloses the entire right halfplane and excludes poles on the imaginary axis. It is traversed in a clockwise direction.
 For discretetime 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:

continuoustime system discretetime system with sample time  If the frequency range is not specified, the entire Nyquist contour is traversed, effectively for continuoustime systems, and for discretetime 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 openloop 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 closedloop system is stable if , where is the number of clockwise encirclements of the disk , and is the number of openloop unstable poles. »

 NyquistPlot 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 curve ColorFunctionScaling True whether to scale arguments to ColorFunction EvaluationMonitor None expression to evaluate at every evaluation Exclusions True frequencies to exclude ExclusionsStyle Automatic what to draw at excluded frequencies FeedbackSector None the sector limits for feedback function FeedbackSectorStyle Automatic style for feedback sector disk FeedbackType "Negative" the feedback type MaxRecursion Automatic maximum recursive subdivisions allowed Mesh Automatic how many mesh divisions to draw MeshFunctions {#3&} placement of mesh divisions MeshShading Automatic how to shade regions between mesh points MeshStyle Automatic the style for mesh divisions NyquistGridLines None the Nyquist grid lines to draw PerformanceGoal $PerformanceGoal aspects of performance to try to optimize PlotPoints Automatic iniitial number of sample frequencies PlotRange Automatic real and imaginary range of values PlotStyle Automatic graphics directives to specify the style of the plot RegionFunction Automatic how to determine if a point should be included SamplingPeriod None the sampling period StabilityMargins False whether to show the stability margins StabilityMarginsStyle Automatic graphics directives to specify the style of the stability margins WorkingPrecision MachinePrecision the precision used in internal computations  The setting Exclusions>True excludes frequencies, including resonant frequencies, where the sinusoidal transfer function is discontinuous.
 Exclusions>{f_{1}, f_{2}, ...} 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 transferfunction model:
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A Nyquist plot of a system with resonant frequencies:
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A Nyquist plot of a discretetime system:
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A discretetime system with resonant frequencies:
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Another discretetime system with resonant frequencies:
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