NyquistPlot

NyquistPlot[lsys]

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

NyquistPlot[lsys,{ωmin,ωmax}]

plots for the frequency range ωmin to ωmax.

NyquistPlot[expr,{ω,ωmin,ωmax}]

plots expr using the variable ω.

Details and Options

Examples

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Basic Examples  (5)

A Nyquist plot of a transfer-function model:

A Nyquist plot of a system with resonant frequencies:

A Nyquist plot of a discrete-time system:

A discrete-time system with resonant frequencies:

Another discrete-time system with resonant frequencies:

Scope  (24)

Basic Uses  (11)

Nyquist plot of a continuous-time system:

Nyquist plot of another continuous-time system:

Specify the frequency range:

Nyquist plot of a discrete-time system:

Nyquist plot of a continuous-time, transfer-function model:

Discrete-time, transfer-function model:

Nyquist plot of a state-space model:

The Nyquist plots of systems with resonant frequencies have encirclements at infinity:

An improper system with resonant frequencies:

A system with a time delay:

Specify the expression of the sinusoidal transfer function of a system:

Linear System Stability  (7)

There are no unstable open-loop poles ():

The plot shows that is not encircled ():

Hence the closed-loop system is stable (); all poles are stable:

Here and , and hence :

Here , i.e. one unstable pole in the closed-loop system:

Here and , and hence :

The closed-loop system has two unstable poles:

Now , and is encircled counterclockwise, so , and hence :

The closed-loop system has no unstable poles:

For positive feedback, count encirclement around ; here , :

There are unstable poles in the closed-loop system:

A discrete-time system with no unstable poles ():

There are no encirclements of ():

Hence the closed-loop system is stable ():

A stable discrete-time system () with positive feedback:

There is one clockwise encirclement of () :

The closed-loop system is unstable, with one pole outside the unit circle:

Nonlinear System Stability  (6)

Starting with a stable linear system ():

Using feedback , where it is also true that , since the disk is not encircled:

Simulate closed-loop system constant feedback for :

The following stable system () has one clockwise encirclement () of the disk for feedback in the sector :

Hence the closed-loop system is unstable for any feedback in that sector:

An unstable system ():

With a feedback configuration that has one counterclockwise encirclement ():

The closed-loop system is stable with any feedback in the sector:

A stable system has feedback in the sector , and its NyquistPlot is to the right of :

The closed loop is stable for any feedback in the sector :

A stable system:

The NyquistPlot lies within the circle for feedback in the sector :

The closed loop is stable for any feedback within the sector :

A stable system lsys with feedback in the sector :

The NyquistPlot of -lsys with feedback in the sector shows a stable closed-loop system:

Simulate the closed loop with constant feedback:

Options  (26)

CoordinatesToolOptions  (1)

Obtain magnitude and phase (in degrees) by selecting the graphic and typing a period (.):

Exclusions  (5)

By default, there are no exclusions for a system with no resonant frequencies:

Exclude the point corresponding to 0.75:

Exclude multiple frequencies:

Resonant frequencies correspond to semicircles of infinite radius:

Exclude only one of the resonant frequencies:

ExclusionsStyle  (2)

Specify the style of the exclusions:

A Nyquist plot without the infinite encirclements:

FeedbackSector  (4)

Stable configurations:

The stable configuration of a system that is unstable:

An unstable configuration:

With feedback gain 0.1, the system is stable:

Rescale the gains to use the Nyquist stability criterion:

FeedbackSectorStyle  (1)

Specify the style of the graphics generated by the circle criterion:

NyquistGridLines  (2)

Automatically chosen values of closed-loop magnitude and phase:

Draw specific contours:

PlotLegends  (4)

Use placeholder legends for multiple systems:

Use the systems as the legend text:

Use LineLegend to add an overall legend label:

Place the legend above the plot:

PlotRange  (2)

Specify the range of coordinates to include in a plot:

Points at infinity are shown in the region specified by PlotRangePadding:

Specify the plot range explicitly:

PlotTheme  (2)

Use a theme with a frame and grid lines:

Change the style of the grid lines:

StabilityMargins  (2)

Show the stability margins:

Stability margins for a system with resonant frequencies:

StabilityMarginsStyle  (1)

Specify the style of stability margins:

Applications  (2)

Compute gain and phase margins:

With no encirclements of and no poles in the right half-plane, the closed loop with unity negative feedback is stable (Nyquist stability criterion):

Properties & Relations  (1)

The sinusoidal transfer function of a discrete-time system is periodic:

Wolfram Research (2010), NyquistPlot, Wolfram Language function, https://reference.wolfram.com/language/ref/NyquistPlot.html (updated 2014).

Text

Wolfram Research (2010), NyquistPlot, Wolfram Language function, https://reference.wolfram.com/language/ref/NyquistPlot.html (updated 2014).

CMS

Wolfram Language. 2010. "NyquistPlot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/NyquistPlot.html.

APA

Wolfram Language. (2010). NyquistPlot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NyquistPlot.html

BibTeX

@misc{reference.wolfram_2023_nyquistplot, author="Wolfram Research", title="{NyquistPlot}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/NyquistPlot.html}", note=[Accessed: 25-February-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_nyquistplot, organization={Wolfram Research}, title={NyquistPlot}, year={2014}, url={https://reference.wolfram.com/language/ref/NyquistPlot.html}, note=[Accessed: 25-February-2024 ]}