--- title: "NyquistPlot" language: "en" type: "Symbol" summary: "NyquistPlot[lsys] generates a Nyquist plot of the transfer function for the system lsys. NyquistPlot[lsys, {\\[Omega]min, \\[Omega]max}] plots for the frequency range \\[Omega]min to \\[Omega]max. NyquistPlot[expr, {\\[Omega], \\[Omega]min, \\[Omega]max}] plots expr using the variable \\[Omega]." keywords: - Nyquist plot - Nyquist - absolute stability problem - Lure problem - Lur'e problem - passivity theorem - circle criterion - Nyquist stability criterion - exponential stability - encirclements - feedback sector - feedback path contains a memoryless nonlinearity - decompose system into a linear and nonlinear part - linear system stability - nonlinear system stability - frequency response characteristics - sinusoidal transfer function - phase margin - gain margin - bandwidth - gain crossover frequency - phase crossover frequency - classical control - control systems - control theory - nyquist canonical_url: "https://reference.wolfram.com/language/ref/NyquistPlot.html" source: "Wolfram Language Documentation" related_guides: - title: "Classical Analysis and Design" link: "https://reference.wolfram.com/language/guide/ClassicalAnalysisAndDesign.en.md" - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" - title: "Control Systems" link: "https://reference.wolfram.com/language/guide/ControlSystems.en.md" - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" - title: "Delay Control Systems" link: "https://reference.wolfram.com/language/guide/DelayControlSystems.en.md" related_functions: - title: "NyquistGridLines" link: "https://reference.wolfram.com/language/ref/NyquistGridLines.en.md" - title: "NicholsPlot" link: "https://reference.wolfram.com/language/ref/NicholsPlot.en.md" - title: "BodePlot" link: "https://reference.wolfram.com/language/ref/BodePlot.en.md" - title: "GainPhaseMargins" link: "https://reference.wolfram.com/language/ref/GainPhaseMargins.en.md" - title: "TransferFunctionModel" link: "https://reference.wolfram.com/language/ref/TransferFunctionModel.en.md" --- # 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 * ``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: [image] * For a system ``lsys`` with the corresponding transfer function $g(s)$, the following expressions are plotted: | | | | --------------------------------------------------------- | -------------------------------------------------------------------------- | | $g(i \omega )$ | continuous-time system | | $g(\exp (i \tau \omega ))$ | discrete-time system with sample time $\tau$ | * If the frequency range is not specified, the entire Nyquist contour is traversed, effectively $-\infty <\omega <\infty$ for continuous-time systems, and $-\pi /\tau <\omega <\pi /\tau$ for discrete-time systems. * ``NyquistPlot`` treats the variable ``ω`` as local, effectively using ``Block``. * The Nyquist plot can be used to infer the number of unstable poles $z$ of the closed-loop system as $z=n+p$, where $p$ is the number of unstable poles of the open-loop system, and $n$ is the number of clockwise encirclements of the point $\{-1,0\}$. » [image] * The Nyquist plot can be used to infer global exponential stability of linear systems with nonlinear feedback $u=-\phi (y)$, where $\phi$ satisfies a sector constraint $a y\leq \phi (y)\leq b y$ when $0 True`` excludes frequencies, including resonant frequencies, where the sinusoidal transfer function is discontinuous. * ``Exclusions -> {f1, f2, …}`` excludes specific frequencies ``fi``. * ``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 $u=-\phi (y)$ with $a y\leq \phi (y)\leq b y$. ### List of all options | | | | | ---------------------- | ----------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | Automatic | ratio of height to width | | Axes | True | whether to draw axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | style specifications for the axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | ColorFunction | Automatic | how to apply coloring to the curve | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | ContentSelectable | Automatic | whether to allow contents to be selected | | CoordinatesToolOptions | Automatic | detailed behavior of the coordinates tool | | Epilog | {} | primitives rendered after the main plot | | 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 | | FormatType | TraditionalForm | the default format type for text | | Frame | False | whether to put a frame around the plot | | FrameLabel | None | frame labels | | FrameStyle | {} | style specifications for the frame | | FrameTicks | Automatic | frame ticks | | FrameTicksStyle | {} | style specifications for frame ticks | | GridLines | None | grid lines to draw | | GridLinesStyle | {} | style specifications for grid lines | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels etc. | | ImageSize | Automatic | the absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | 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 | | Method | Automatic | details of graphics methods to use | | NyquistGridLines | None | the Nyquist grid lines to draw | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | an overall label for the plot | | PlotLegends | None | legends for curves | | PlotPoints | Automatic | iniitial number of sample frequencies | | PlotRange | Automatic | real and imaginary range of values | | PlotRangeClipping | False | whether to clip at the plot range | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | the final display region to be filled | | PlotStyle | Automatic | graphics directives to specify the style of the plot | | PlotTheme | \$PlotTheme | overall theme for the plot | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | primitives rendered before the main plot | | RegionFunction | Automatic | how to determine if a point should be included | | RotateLabel | True | whether to rotate y labels on the frame | | 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 | | Ticks | Automatic | axes ticks | | TicksStyle | {} | style specifications for axes ticks | | WorkingPrecision | MachinePrecision | the precision used in internal computations | --- ## Examples (91) ### Basic Examples (5) A Nyquist plot of a transfer-function model: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{10*(1 + 3*s)*(1 + 4*s)}}, (1 + s)*(2 + s)*(5 + s)*(6 + s)}, s]] Out[1]= [image] ``` --- A Nyquist plot of a system with resonant frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1 + s}}, 4 + s^2}, s]] Out[1]= [image] ``` --- A Nyquist plot of a discrete-time system: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- A discrete-time system with resonant frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -1 + z}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- Another discrete-time system with resonant frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, (-1 + z)* (1 + z + z^2)}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` ### Scope (24) #### Basic Uses (11) Nyquist plot of a continuous-time system: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{10}}, (1 + 2*s)*(1 + 3*s)}, s, SamplingPeriod -> None, SystemsModelLabels -> None]] Out[1]= [image] ``` --- Nyquist plot of another continuous-time system: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{10 + s}}, (1 + s)*(100 + s)^2}, s]] Out[1]= [image] ``` --- Specify the frequency range: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{s}}, 3 + s}, s], {-5, 5}] Out[1]= [image] ``` --- Nyquist plot of a discrete-time system: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- Nyquist plot of a continuous-time, transfer-function model: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{2 + s^2}}, (1 + s)^2}, s]] Out[1]= [image] ``` --- Discrete-time, transfer-function model: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{3*(-0.25 + z)}}, (-2 + z)*(-0.5 + z)}, z, SamplingPeriod -> 0.5], AspectRatio -> 1] Out[1]= [image] ``` --- Nyquist plot of a state-space model: ```wl In[1]:= NyquistPlot[StateSpaceModel[{{{0, 1, 0}, {0, 0, 1}, {8, -12, 6}}, {{0}, {0}, {1}}, {{3, Rational[-5, 2], 2}}, {{Rational[1, 4]}}}, SamplingPeriod -> None, SystemsModelLabels -> None]] Out[1]= [image] ``` --- The Nyquist plots of systems with resonant frequencies have encirclements at infinity: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1 + 10*s}}, s^2*(1 + s)}, s]] Out[1]= [image] In[2]:= NyquistPlot[TransferFunctionModel[{{{1 + s}}, s*(1 + s^2)* (4 + s^2)*(16 + s^4)}, s]] Out[2]= [image] In[3]:= NyquistPlot[TransferFunctionModel[{{{1 + z}}, (-1 + z)^2}, z, SamplingPeriod -> 0.1]] Out[3]= [image] ``` --- An improper system with resonant frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{(-4 + s)*(5 + s)*(6 + s)* (-9 + s^2)}}, (1 + s)*(4 + s^2)}, s]] Out[1]= [image] ``` --- A system with a time delay: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1.}}, 0.8/E^(5.*s) + s}, s] ] Out[1]= [image] ``` --- Specify the expression of the sinusoidal transfer function of a system: ```wl In[1]:= NyquistPlot[(f + 0.1 I/40 + 4 I f - 10 f^2 - I f^3), f] Out[1]= [image] ``` #### Linear System Stability (7) There are no unstable open-loop poles ($p=0$): ```wl In[1]:= lsys = TransferFunctionModel[{{{1}}, (1 + (Complex[0, -1])*Sqrt[5] + s)* (1 + Complex[0, 1]*Sqrt[5] + s)}, s]; ``` The plot shows that $\{-1,0\}$ is not encircled ($n=0$): ```wl In[2]:= NyquistPlot[lsys] Out[2]= [image] ``` Hence the closed-loop system is stable ($z=n+p=0$); all poles are stable: ```wl In[3]:= SystemsModelFeedbackConnect[lsys] Out[3]= TransferFunctionModel[{{{1}}, {{7 + 2*s + s^2}}}, s, SamplingPeriod -> Automatic] In[4]:= TransferFunctionPoles[%]//N Out[4]= {{{-1. - 2.44949 I, -1. + 2.44949 I}}} ``` --- Here $p=0$ and $n=1$, and hence $z=n+p=1$: ```wl In[1]:= lsys = TransferFunctionModel[{{{2*(-1 + s)}}, 1 + s}, s]; In[2]:= NyquistPlot[lsys] Out[2]= [image] ``` Here $z=1$, i.e. one unstable pole in the closed-loop system: ```wl In[3]:= TransferFunctionPoles@SystemsModelFeedbackConnect[lsys]//N Out[3]= {{{0.333333}}} ``` --- Here $p=0$ and $n=2$, and hence $z=n+p=2$: ```wl In[1]:= lsys = TransferFunctionModel[{{{2}}, s^2*(1 + s)}, s]; In[2]:= NyquistPlot[lsys, PlotRange -> {{-2, 2}, Automatic}] Out[2]= [image] ``` The closed-loop system has two unstable poles: ```wl In[3]:= TransferFunctionPoles@SystemsModelFeedbackConnect[lsys]//N Out[3]= {{{-1.69562, 0.34781  - 1.02885 I, 0.34781  + 1.02885 I}}} ``` --- Now $p=1$, and $\{-1,0\}$ is encircled counterclockwise, so $n=-1$, and hence $z=n+p=0$ : ```wl In[1]:= lsys = TransferFunctionModel[{{{5 + s}}, -3 + s}, s]; In[2]:= NyquistPlot[lsys] Out[2]= [image] ``` The closed-loop system has no unstable poles: ```wl In[3]:= TransferFunctionPoles@SystemsModelFeedbackConnect[lsys]//N Out[3]= {{{-1.}}} ``` --- For positive feedback, count encirclement around $\{1,0\}$; here $p=0$, $n=2$: ```wl In[1]:= lsys = TransferFunctionModel[{{{2*(2.5 - s + s^2)}}, 1 + (1.5 + s)^2}, s]; In[2]:= NyquistPlot[lsys] Out[2]= [image] ``` There are $z=n+p=2$ unstable poles in the closed-loop system: ```wl In[3]:= TransferFunctionPoles@SystemsModelFeedbackConnect[lsys, +1]//N Out[3]= {{{0.37868, 4.62132}}} ``` --- A discrete-time system with no unstable poles ($p=0$): ```wl In[1]:= lsys = TransferFunctionModel[{{{(-0.7 + z)*z}}, (-0.9 + z)*(0.3 + z)}, z, SamplingPeriod -> 1]; ``` There are no encirclements of $\{-1,0\}$ ($n=0$): ```wl In[2]:= NyquistPlot[lsys] Out[2]= [image] ``` Hence the closed-loop system is stable ($z=n+p=0$): ```wl In[3]:= TransferFunctionPoles@SystemsModelFeedbackConnect[lsys] Out[3]= {{{-0.165535, 0.815535}}} ``` --- A stable discrete-time system ($p=0$) with positive feedback: ```wl In[1]:= lsys = TransferFunctionModel[{{{0.5 + z}}, (-0.3 + z)*(0.6 + z^2)}, z, SamplingPeriod -> 1]; ``` There is one clockwise encirclement of $\{1,0\}$ ($n=1$) : ```wl In[2]:= NyquistPlot[lsys] Out[2]= [image] ``` The closed-loop system is unstable, with one pole outside the unit circle: ```wl In[3]:= TransferFunctionPoles@SystemsModelFeedbackConnect[lsys, 1] Map[Abs, %, 3] Out[3]= {{{-0.427742 - 0.636816 I, -0.427742 + 0.636816 I, 1.15548}}} Out[3]= {{{0.767136, 0.767136, 1.15548}}} ``` #### Nonlinear System Stability (6) Starting with a stable linear system ($p=0$): ```wl In[1]:= lsys = TransferFunctionModel[{{{1 + s}}, 1 + 3*s + s^2}, s]; In[2]:= TransferFunctionPoles@lsys//N Out[2]= {{{-2.61803, -0.381966}}} ``` Using feedback $u=-\phi (y)$, where $1 y\leq \phi (y)\leq 3 y$ it is also true that $n=0$, since the disk is not encircled: ```wl In[3]:= NyquistPlot[lsys, FeedbackSector -> {1, 3}] Out[3]= [image] ``` Simulate closed-loop system constant feedback $u=-a y$ for $1\leq a\leq 3$: ```wl In[4]:= responses = Table[OutputResponse[SystemsModelFeedbackConnect[lsys, TransferFunctionModel[{{{a}}, 1}, s]], UnitStep[t], {t, 0, 10}], {a, Range[1, 3, 0.5]}]; In[5]:= Plot[responses, {t, 0, 10}, PlotRange -> All] Out[5]= [image] ``` --- The following stable system ($p=0$) has one clockwise encirclement ($n=1$) of the disk for feedback in the sector $(1.5,25)$ : ```wl In[1]:= lsys = TransferFunctionModel[{{{1 - s}}, 1 + s}, s]; In[2]:= NyquistPlot[lsys, FeedbackSector -> {1.5, 25}] Out[2]= [image] ``` Hence the closed-loop system is unstable for any feedback in that sector: ```wl In[3]:= Table[Plot[Evaluate@OutputResponse[SystemsModelFeedbackConnect[lsys, TransferFunctionModel[{{{a}}, 1}, s]], UnitStep[t], {t, 0, 1}], {t, 0, 1}], {a, {1.5, 10, 25}}]; In[4]:= GraphicsRow[%, ImageSize -> Medium] Out[4]= [image] ``` --- An unstable system ($p=1$): ```wl In[1]:= lsys = TransferFunctionModel[{{{5 + s^2}}, (-1 + s)*(3 + s)}, s]; ``` With a feedback configuration that has one counterclockwise encirclement ($n=-1$): ```wl In[2]:= NyquistPlot[lsys, FeedbackSector -> {1, 2}] Out[2]= [image] ``` The closed-loop system is stable with any feedback in the sector: ```wl In[3]:= responses = Table[OutputResponse[SystemsModelFeedbackConnect[lsys, TransferFunctionModel[{{{a}}, 1}, s]], UnitStep[t], {t, 0, 12}], {a, Range[1, 2, 0.2]}]; In[4]:= Plot[responses, {t, 0, 12}, PlotRange -> All] Out[4]= [image] ``` --- A stable system has feedback in the sector $\{0,20\}$, and its ``NyquistPlot`` is to the right of $-1/20$ : ```wl In[1]:= lsys = TransferFunctionModel[{{{1 + s}}, (2 + s)*(3 + s)}, s]; In[2]:= NyquistPlot[lsys, FeedbackSector -> {0, 20}] Out[2]= [image] ``` The closed loop is stable for any feedback in the sector $\{0,20\}$ : ```wl In[3]:= responses = Table[OutputResponse[SystemsModelFeedbackConnect[lsys, TransferFunctionModel[{{{a}}, 1}, s]], UnitStep[t], {t, 0, 4}], {a, Range[0, 20, 4]}]; In[4]:= Plot[responses, {t, 0, 4}, PlotRange -> All] Out[4]= [image] ``` --- A stable system: ```wl In[1]:= lsys = TransferFunctionModel[{{{1 + s}}, 1 + 3*s + s^2}, s]; ``` The ``NyquistPlot`` lies within the circle for feedback in the sector $\{-0.9,5\}$ : ```wl In[2]:= NyquistPlot[lsys, FeedbackSector -> {-0.9, 5}] Out[2]= [image] ``` The closed loop is stable for any feedback within the sector $\{-0.9,5\}$ : ```wl In[3]:= responses = Table[Plot[Evaluate@OutputResponse[SystemsModelFeedbackConnect[lsys, TransferFunctionModel[{{{a}}, 1}, s]], UnitStep[t], {t, 0, 100}], {t, 0, 100}, PlotRange -> All], {a, {-0.9, 5}}]; In[4]:= GraphicsRow[responses, ImageSize -> Medium] Out[4]= [image] ``` --- A stable system ``lsys`` with feedback in the sector $\{-7,-4\}$ : ```wl In[1]:= lsys = TransferFunctionModel[{{{0.1*(1 - s)}}, 1 + 2*s + s^2}, s]; ``` The ``NyquistPlot`` of ``-lsys`` with feedback in the sector $\{4,7\}$ shows a stable closed-loop system: ```wl In[2]:= NyquistPlot[TransferFunctionModel[{{{-0.1*(1 - s)}}, 1 + 2*s + s^2}, s], FeedbackSector -> {4, 7}] Out[2]= [image] ``` Simulate the closed loop with constant feedback: ```wl In[3]:= responses = Table[OutputResponse[SystemsModelFeedbackConnect[lsys, TransferFunctionModel[{{{a}}, 1}, s]], UnitStep[t], {t, 0, 40}], {a, Range[-7, -4, 1]}]; In[4]:= Plot[responses, {t, 0, 40}, PlotRange -> All] Out[4]= [image] ``` ### Generalizations & Extensions (1) A Nyquist plot can be obtained from a transfer-function model or directly from its expression: ```wl In[1]:= g = (0.3 + s/(2.5 + s)(0.5 - s)); In[2]:= {NyquistPlot[TransferFunctionModel[g, s]], NyquistPlot[g]} Out[2]= {[image], [image]} ``` ### Options (58) #### AspectRatio (4) By default, the ratio of the height to width for the plot is determined automatically: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- Make the height the same as the width with ``AspectRatio -> 1`` : ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> 1] Out[1]= [image] ``` --- Use numerical value to specify the height-to-width ratio: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> 1 / 2] Out[1]= [image] ``` --- ``AspectRatio -> Full`` adjusts the height and width to tightly fit inside other constructs: ```wl In[1]:= plot = NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> Full]; In[2]:= {Framed[Pane[plot, {75, 100}]], Framed[Pane[plot, {100, 100}]], Framed[Pane[plot, {100, 50}]]} Out[2]= {[image], [image], [image]} ``` #### Axes (3) By default, ``Axes`` are drawn: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- Use ``Axes -> False`` to turn off axes: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], Axes -> False] Out[1]= [image] ``` --- Turn each axis on individually: ```wl In[1]:= {NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], Axes -> {True, False}], NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], Axes -> {False, True}]} Out[1]= {[image], [image]} ``` #### AxesLabel (4) No axes labels are drawn by default: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- Place a label on the $y$ axis: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesLabel -> Im[z]] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesLabel -> {Re[z], Im[z]}] Out[1]= [image] ``` --- Use labels based on variables specified in ``NyquistPlot`` : ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesLabel -> Automatic] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1]] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesOrigin -> {1, 0}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesStyle -> {{Thick, Red}, {Thick, Blue}}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesStyle -> Green, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### CoordinatesToolOptions (1) Obtain magnitude and phase (in degrees) by selecting the graphic and typing a period (.): ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{10}}, (2 + s)*(2.5 + s)*(3 + s)}, s], CoordinatesToolOptions -> {"DisplayFunction" -> Function[pt, {Norm[pt], (Arg[First[pt] + I Last[pt]]/°)}]}] Out[1]= [image] In[2]:= NyquistPlot[TransferFunctionModel[{{{5}}, s*(1 + 5*s)}, s], CoordinatesToolOptions -> {"DisplayFunction" -> Function[pt, Switch[pt, {_ ? NumericQ, _ ? NumericQ}, {Norm@pt, Arg[First@pt + I Last@pt] / Degree}, _, {Infinity, Indeterminate}]]}] Out[2]= [image] ``` #### Exclusions (5) By default, there are no exclusions for a system with no resonant frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{s}}, (1 + s)*(6 + s)}, s]] Out[1]= [image] ``` --- Exclude the point corresponding to 0.75: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], Exclusions -> 0.75, ExclusionsStyle -> Directive[Thick, Red]] Out[1]= [image] ``` --- Exclude multiple frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{-3 + s}}, (6 + s)* (3 + 2*s + s^2)}, s], Exclusions -> {0.3, 0.5, 0.7}, ExclusionsStyle -> Directive[Thick, Red]] Out[1]= [image] ``` --- Resonant frequencies correspond to semicircles of infinite radius: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{3 + s}}, 25 + s^2}, s]] Out[1]= [image] ``` --- Exclude only one of the resonant frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{3 + s}}, 25 + s^2}, s], Exclusions -> {5}] Out[1]= [image] ``` #### ExclusionsStyle (2) Specify the style of the exclusions: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, s^3*(1 + s)*(16 + s^2)}, s], ExclusionsStyle -> Directive[Dashed, Red]] Out[1]= [image] ``` --- A Nyquist plot without the infinite encirclements: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + z + z^2}, z, SamplingPeriod -> 1], ExclusionsStyle -> None] Out[1]= [image] ``` #### FeedbackSector (4) Stable configurations: ```wl In[1]:= Table[NyquistPlot[TransferFunctionModel[{{{2}}, (1 + s)*(2 + s)}, s], FeedbackSector -> f, PlotLabel -> "Feedback Sector: " <> ToString[f]], {f, {{0.1, 10}, {0, 1}, {-1, 1}}}]; In[2]:= TableForm[%, TableSpacing -> {6, Automatic}] Out[2]//TableForm= | | | :------ | | [image] | | [image] | | [image] | ``` --- The stable configuration of a system that is unstable: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{2*(3 + s)}}, (-1 + s)*(2 + s)}, s], FeedbackSector -> {0.4, 2}] Out[1]= [image] ``` --- An unstable configuration: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{2*(6 + s)}}, (-1 + s)*(2 + s)}, s], FeedbackSector -> {0.1, 20}] Out[1]= [image] ``` --- With feedback gain 0.1, the system is stable: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{30}}, (0.5 + s)*(1 + s)^2}, s], FeedbackSector -> {0.1, 0.1}] Out[1]= [image] ``` Rescale the gains to use the Nyquist stability criterion: ```wl In[2]:= NyquistPlot[TransferFunctionModel[{{{3}}, (0.5 + s)*(1 + s)^2}, s], FeedbackSector -> {1, 1}] Out[2]= [image] ``` #### FeedbackSectorStyle (1) Specify the style of the graphics generated by the circle criterion: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{(4 + s)*(6 + s)}}, (-4 + s)*(-3 + s)}, s], FeedbackSector -> {-2, 1}, FeedbackSectorStyle -> Directive[Thick, Green]] Out[1]= [image] ``` #### ImageSize (7) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> Tiny], NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> Small]} Out[1]= {[image], [image]} ``` --- Specify the width of the plot: ```wl In[1]:= {NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> 150], NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> {Automatic, 150}], NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= {[image], [image]} ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> UpTo[200]], NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= {[image], [image]} ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> {200, 200}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> Full, ImageSize -> {200, 200}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> {UpTo[150], UpTo[100]}], NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= {[image], [image]} ``` --- Use ``ImageSize -> Full`` to fill the available space in an object: ```wl In[1]:= Framed[Pane[NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], ImageSize -> Full, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` --- Specify the image size as a fraction of the available space: ```wl In[1]:= Framed[Pane[NyquistPlot[TransferFunctionModel[{{{z}}, -0.5 + z}, z, SamplingPeriod -> 1], AspectRatio -> Full, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 100}]] Out[1]= [image] ``` #### NyquistGridLines (2) Automatically chosen values of closed-loop magnitude and phase: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, (1 + s)*(3 + s)*(6 + s)}, s], NyquistGridLines -> Automatic] Out[1]= [image] ``` --- Draw specific contours: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1 + 10*s}}, 1 + 2*s + s^2}, s], NyquistGridLines -> {{0.6, 0.7, 0.8}, Range[-20, 20, 4] Degree}] Out[1]= [image] ``` #### PlotLegends (4) Use placeholder legends for multiple systems: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1, s}}, {{1 - 2*s + s^2, 1 + 2*s + s^2}}}, s], PlotLegends -> Automatic] Out[1]= [image] ``` --- Use the systems as the legend text: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1 + s, -1 + s}}, {{1 - 2*s + s^2, 1 + 2*s + s^2}}}, s], PlotLegends -> "Expressions"] Out[1]= [image] ``` --- Use ``LineLegend`` to add an overall legend label: ```wl In[1]:= NyquistPlot[Table[TransferFunctionModel[{{{1}}, 1 + s^2 + 2*s*ζ}, s], {ζ, 0.2, 0.8, 0.2}], PlotLegends -> LineLegend[Range[0.2, 0.8, 0.2], LegendLabel -> "ζ"]] Out[1]= [image] ``` --- Place the legend above the plot: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1 + s, -1 + s}}, {{1 - 2*s + s^2, 1 + 2*s + s^2}}}, s], PlotLegends -> Placed[{"one", "two"}, Above]] Out[1]= [image] ``` #### PlotRange (2) Specify the range of coordinates to include in a plot: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{2*(1 + s)}}, s*(4 + s^2)* (16 + s^2)}, s], PlotRange -> 0.5, PlotRangePadding -> 0] Out[1]= [image] ``` Points at infinity are shown in the region specified by ``PlotRangePadding`` : ```wl In[2]:= NyquistPlot[TransferFunctionModel[{{{2*(1 + s)}}, s*(4 + s^2)* (16 + s^2)}, s], PlotRange -> 0.5, PlotRangePadding -> Automatic] Out[2]= [image] In[3]:= AbsoluteOptions[%, PlotRange] Out[3]= {PlotRange -> {{-0.5, 0.5}, {-0.5, 0.5}}} In[4]:= AbsoluteOptions[%%, PlotRangePadding] Out[4]= {PlotRangePadding -> {{0.604365, 0.64353}, {0.56182, 0.64353}}} ``` --- Specify the plot range explicitly: ```wl In[1]:= Table[NyquistPlot[TransferFunctionModel[{{{10}}, s^3}, s], PlotRange -> pr], {pr, {Automatic, 40}}] Out[1]= {[image], [image]} In[2]:= Table[NyquistPlot[TransferFunctionModel[{{{2}}, (-1 + z)*(Rational[-1, 2] + z)}, z, SamplingPeriod -> 1], PlotRange -> pr], {pr, {Automatic, {{-15, 5}, {-30, 30}}}}] Out[2]= {[image], [image]} ``` #### PlotTheme (2) Use a theme with a frame and grid lines: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1 + s, -1 + s}}, {{1 - 2*s + s^2, 1 + 2*s + s^2}}}, s], PlotTheme -> "Detailed"] Out[1]= [image] ``` --- Change the style of the grid lines: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1 + s, -1 + s}}, {{1 - 2*s + s^2, 1 + 2*s + s^2}}}, s], PlotTheme -> "Detailed", GridLinesStyle -> LightGray] Out[1]= [image] ``` #### StabilityMargins (2) Show the stability margins: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{6}}, (2 + s)*(2 + 2*s + s^2)}, s], StabilityMargins -> True, StabilityMarginsStyle -> ( Directive[Thick, #]& /@ {Green, Red})] Out[1]= [image] ``` --- Stability margins for a system with resonant frequencies: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, s*(1 + s)*(5 + s)}, s], StabilityMargins -> True, StabilityMarginsStyle -> Directive[Green, Thick], PlotRange -> {{-0.08, 0}, {-0.01, 0.01}}, PlotRangePadding -> 0] Out[1]= [image] ``` #### StabilityMarginsStyle (1) Specify the style of stability margins: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{6}}, (2 + s)*(2 + 2*s + s^2)}, s], StabilityMargins -> True, StabilityMarginsStyle -> {Directive[Green, Thick], Red}] Out[1]= [image] ``` #### Ticks (4) Ticks are placed automatically in each plot: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s]] Out[1]= [image] ``` --- Use ``Ticks -> None`` to not draw any tick marks: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], Ticks -> None] Out[1]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], Ticks -> {{-.25, .4, .7}, {-.5, -1, .5}}] Out[1]= [image] ``` --- Draw tick marks at the specified positions with the specified labels: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], Ticks -> {{{-.25, -a}, {.4, b}, {.7, c}}, {{-.5, -d}, {-1, -e}, {.5, d}}}] Out[1]= [image] ``` #### TicksStyle (4) Specify overall ticks style, including the tick labels: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], TicksStyle -> Directive[Bold, Red]] Out[1]= [image] ``` --- Specify tick style for each of the axes: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], TicksStyle -> {Directive[Black, Bold], Directive[Bold, Red]}] Out[1]= [image] ``` --- Specify tick marks with scaled lengths: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], Ticks -> {{{-.25, -a, .14}, {.4, b, .75}, {.7, c, .65}}, {{-.5, -d, .7}, {-1, -e, .42}, {.5, d, .7}}}] Out[1]= [image] ``` --- Customize each tick with position, length, labeling and styling: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{1}}, 1 + s + s^2}, s], Ticks -> {{{-.25, -a, .14, Directive[Red, Thick]}, {.4, b, .75, Directive[Red, Dashed]}, {.7, c, .65, Directive[Red, Thick, Dashed]}}, {{-.5, -d, .7, Directive[Blue, Thick]}, {-1, -e, .42, Directive[Blue, Dashed]}, {.5, d, .7, Directive[Blue, Thick, Dashed]}}}] Out[1]= [image] ``` ### Applications (2) Compute gain and phase margins: ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{100}}, s*(100 + 3*s + s^2)}, s], StabilityMargins -> True, StabilityMarginsStyle -> {Green, Red}]; Show[%, PlotRange -> {{-0.4, 0}, {-1, 0}}, PlotRangePadding -> 0] Out[1]= [image] ``` --- With no encirclements of $-1$ and no poles in the right half-plane, the closed loop with unity negative feedback is stable (Nyquist stability criterion): ```wl In[1]:= NyquistPlot[TransferFunctionModel[{{{100}}, s*(100 + 3*s + s^2)}, s]] Out[1]= [image] In[2]:= s /. NSolve[s(s^2 + 3 s + 100) + 100 == 0, s] Out[2]= {-1.02062, -0.989691 - 9.84887 I, -0.989691 + 9.84887 I} ``` ### Properties & Relations (1) The sinusoidal transfer function of a discrete-time system is periodic: ```wl In[1]:= Table[NyquistPlot[TransferFunctionModel[{{{1}}, 1 + z + z^2}, z, SamplingPeriod -> 1], Evaluate@freqRange], {freqRange, {{-π, π}, {-2 π, 2 π}}}] Out[1]= {[image], [image]} ``` ## See Also * [`NyquistGridLines`](https://reference.wolfram.com/language/ref/NyquistGridLines.en.md) * [`NicholsPlot`](https://reference.wolfram.com/language/ref/NicholsPlot.en.md) * [`BodePlot`](https://reference.wolfram.com/language/ref/BodePlot.en.md) * [`GainPhaseMargins`](https://reference.wolfram.com/language/ref/GainPhaseMargins.en.md) * [`TransferFunctionModel`](https://reference.wolfram.com/language/ref/TransferFunctionModel.en.md) ## Related Guides * [Classical Analysis and Design](https://reference.wolfram.com/language/guide/ClassicalAnalysisAndDesign.en.md) * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) * [Control Systems](https://reference.wolfram.com/language/guide/ControlSystems.en.md) * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) * [Delay Control Systems](https://reference.wolfram.com/language/guide/DelayControlSystems.en.md) ## History * [Introduced in 2010 (8.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn80.en.md) \| [Updated in 2012 (9.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn90.en.md) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md)