# nicholsplot

Plot Nichols frequency response of dynamic system

## Description

The `nicholsplot`

function plots the Nichols response of a dynamic
system model and returns a `NicholsPlot`

chart object. To customize the plot,
modify the properties of the chart object using dot notation. For more information, see Customize Linear Analysis Plots at Command Line.

To obtain frequency response data, use `nichols`

.

## Creation

### Syntax

### Description

plots the frequency Nichols response of the dynamic system model
`np`

= nicholsplot(`sys`

)`sys`

and returns the corresponding chart object.

If `sys`

is a multi-input, multi-output (MIMO) model, then
`nicholsplot`

produces a grid of Nichols plots with each plot
displaying the frequency response of one input-output pair.

If `sys`

is a model with complex coefficients, then
`nicholsplot`

shows a contour comprised of both positive and negative
frequencies. For models with real coefficients, `nicholsplot`

shows only
positive frequencies.

plots Nichols responses for frequencies specified in `np`

= nicholsplot(___,`w`

)`w`

. You can
specify a frequency range or a vector of frequencies. You can use `w`

with any of the previous syntaxes.

You can use `w`

with any of the input-argument combinations in
previous syntaxes.

See `logspace`

to generate logarithmically
spaced frequency vectors.

plots the Nichols frequency response with the plotting options specified in
`np`

= nicholsplot(___,`plotoptions`

)`plotoptions`

. Settings you specify in
`plotoptions`

override the plotting preferences for the current
MATLAB^{®} session. This syntax is useful when you want to write a script to generate
multiple plots that look the same regardless of the local preferences.

plots the Nichols response in the specified parent graphics container, such as a
`np`

= nicholsplot(`parent`

,___)`Figure`

or `TiledChartLayout`

, and sets the
`Parent`

property. Use this syntax when you want to create a plot
in a specified open figure or when creating apps in App Designer.

### Input Arguments

`sys`

— Dynamic system

dynamic system model | model array

Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can use include:

Continuous-time or discrete-time numeric LTI models, such as

`tf`

,`zpk`

, or`ss`

models.Sparse state-space models, such as

`sparss`

or`mechss`

models. Frequency grid`w`

must be specified for sparse models.Generalized or uncertain LTI models such as

`genss`

or`uss`

(Robust Control Toolbox) models. Using uncertain models requires Robust Control Toolbox™ software.For tunable control design blocks, the function evaluates the model at its current value to plot the response.

For uncertain control design blocks, the function plots the nominal value and random samples of the model.

Frequency-response data models such as

`frd`

models. For such models, the function plots the response at the frequencies defined in the model.Identified LTI models, such as

`idtf`

(System Identification Toolbox),`idss`

(System Identification Toolbox), or`idproc`

(System Identification Toolbox) models. Using identified models requires System Identification Toolbox™ software.

If `sys`

is an array of models, the plot shows responses of all models in
the array on the same axes.

`LineSpec`

— Line style, marker, and color

string | character vector

Line style, marker, and color, specified as a string or character vector containing symbols. The symbols can appear in any order. You do not need to specify all three characteristics (line style, marker, and color). For example, if you omit the line style and specify the marker, then the plot shows only the marker and no line.

**Example: **`'--or'`

is a red dashed line with circle markers

Line Style | Description |
---|---|

`"-"` | Solid line |

`"--"` | Dashed line |

`":"` | Dotted line |

`"-."` | Dash-dotted line |

Marker | Description |
---|---|

`"o"` | Circle |

`"+"` | Plus sign |

`"*"` | Asterisk |

`"."` | Point |

`"x"` | Cross |

`"_"` | Horizontal line |

`"|"` | Vertical line |

`"s"` | Square |

`"d"` | Diamond |

`"^"` | Upward-pointing triangle |

`"v"` | Downward-pointing triangle |

`">"` | Right-pointing triangle |

`"<"` | Left-pointing triangle |

`"p"` | Pentagram |

`"h"` | Hexagram |

Color | Description |
---|---|

`"r"` | red |

`"g"` | green |

`"b"` | blue |

`"c"` | cyan |

`"m"` | magenta |

`"y"` | yellow |

`"k"` | black |

`"w"` | white |

`w`

— Frequencies

`{wmin,wmax}`

| vector | `[]`

Frequencies at which to compute and plot frequency response, specified as the cell array `{wmin,wmax}`

or as a vector of frequency values.

If

`w`

is a cell array of the form`{wmin,wmax}`

, then the function computes the response at frequencies ranging between`wmin`

and`wmax`

.If

`w`

is a vector of frequencies, then the function computes the response at each specified frequency. For example, use`logspace`

to generate a row vector with logarithmically spaced frequency values. The vector`w`

can contain both positive and negative frequencies.`[]`

— Automatically select frequencies based on system dynamics.

For models with complex coefficients, if you specify a frequency range of [*w*_{min},*w*_{max}] for your plot, then the plot shows a contour comprised of both positive frequencies [*w*_{min},*w*_{max}] and negative frequencies [–*w*_{max},–*w*_{min}].

Specify frequencies in units of rad/`TimeUnit`

, where `TimeUnit`

is the `TimeUnit`

property of the model.

`plotoptions`

— Nichols plot options

`nicholsoptions`

object

Nichols plot options, specified as a `nicholsoptions`

object. You can use these options to customize the Nichols
plot appearance. Settings you specify in `plotoptions`

override the
preference settings for the current MATLAB session.

`parent`

— Parent container

`Figure`

object (default) | `TiledChartLayout`

object | `UIFigure`

object | `UIGridLayout`

object | `UIPanel`

object | `UITab`

object

Parent container of the chart, specified as one of the following objects:

`Figure`

`TiledChartLayout`

`UIFigure`

`UIGridLayout`

`UIPanel`

`UITab`

## Properties

**Note**

The properties listed here are only a subset. For a complete list, see NicholsPlot Properties.

`Responses`

— Model responses

`NicholsResponse`

object | array of `NicholsResponse`

objects

Model responses, specified as a `NicholsResponse`

object or an array of such objects. Use this property to modify the dynamic system model or appearance for each response in the plot. Each `NicholsResponse`

object has the following fields.

`SourceData`

— Source data

structure

Source data for the response, specified as a structure with the following fields.

`Model`

— Dynamic system

dynamic system model | model array

Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models.

When you initially create a plot, `Model`

matches the value you specify for `sys`

.

`FrequencySpec`

— Frequencies

`{wmin,wmax}`

| vector | `[]`

Frequencies at which to compute the response, specified as one of the following:

Cell array of the form

`{wmin,wmax}`

— Compute the response at frequencies in the range from`wmin`

to`wmax`

.Vector of frequencies — Compute the response at each specified frequency. For example, use

`logspace`

to generate a row vector with logarithmically spaced frequency values. The vector`w`

can contain both positive and negative frequencies.`[]`

— Automatically select frequencies based on system dynamics.

Specify frequencies in units of rad/`TimeUnit`

, where `TimeUnit`

is the `TimeUnit`

property of the model.

When you initially create a plot:

`FrequencySpec`

matches the value you specify for the`w`

argument.If you do not specify

`w`

,`FrequencySpec`

is empty and frequencies are selected based on the system dynamics.

`Name`

— Response name

string | character vector

Response name, specified as a string or character vector and stored as a string.

`Visible`

— Response visibility

`"on"`

(default) | on/off logical value

Response visibility, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Display the response in the plot.`"off"`

,`0`

, or`false`

— Do not display the response in the plot.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

`LegendDisplay`

— Option to list response in legend

`"on"`

(default) | on/off logical value

Option to list response in legend, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— List the response in the legend.`"off"`

,`0`

, or`false`

— Do not list the response in the legend.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

`MarkerStyle`

— Marker style

`"none"`

| `"o"`

| `"+"`

| `"*"`

| `"."`

| ...

Marker style, specified as one of the following values.

Marker | Description |
---|---|

`"none"` | No marker |

`"o"` | Circle |

`"+"` | Plus sign |

`"*"` | Asterisk |

`"."` | Point |

`"x"` | Cross |

`"_"` | Horizontal line |

`"|"` | Vertical line |

`"s"` | Square |

`"d"` | Diamond |

`"^"` | Upward-pointing triangle |

`"v"` | Downward-pointing triangle |

`">"` | Right-pointing triangle |

`"<"` | Left-pointing triangle |

`"p"` | Pentagram |

`"h"` | Hexagram |

`Color`

— Plot color

RGB triplet | hexadecimal color code | color name

Plot color, specified as an RGB triplet or a hexadecimal color code and stored as an RGB triplet.

Alternatively, you can specify some common colors by name. The following table lists these colors and their corresponding RGB triplets and hexadecimal color codes.

Color Name | RGB Triplet | Hexadecimal Color Code |
---|---|---|

| `[1 0 0]` | `#FF0000` |

| `[0 1 0]` | `#00FF00` |

| `[0 0 1]` | `#0000FF` |

| `[0 1 1]` | `#00FFFF` |

| `[1 0 1]` | `#FF00FF` |

| `[1 1 0]` | `#FFFF00` |

| `[0 0 0]` | `#000000` |

| `[1 1 1]` | `#FFFFFF` |

`LineStyle`

— Line style

`"-"`

| `"--"`

| `":"`

| `"-."`

Line style, specified as one of the following values.

Line Style | Description |
---|---|

`"-"` | Solid line |

`"--"` | Dashed line |

`":"` | Dotted line |

`"-."` | Dash-dotted line |

`MarkerSize`

— Marker size

positive scalar

Marker size, specified as a positive scalar.

`LineWidth`

— Line width

positive scalar

Line width, specified as a positive scalar.

`Characteristics`

— Response characteristics

`CharacteristicsManager`

object

Response characteristics to display in the plot, specified as a
`CharacteristicsManager`

object with the following properties.

`FrequencyPeakResponse`

— Visibility of peak response

`CharacteristicOption`

object

Visibility of peak response in plot, specified as a `CharacteristicOption`

object with the following property.

`Visible`

— Peak response visibility

`"off"`

(default) | on/off logical value

Peak response visibility, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Display the peak response.`"off"`

,`0`

, or`false`

— Do not display the peak response.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

`AllStabilityMargins`

— Visibility of all stability margins

`CharacteristicOption`

object

Visibility of all stability margins, specified as a `CharacteristicOption`

object with the following property.

`Visible`

— Margin visibility

`"off"`

(default) | on/off logical value

Margin visibility, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Display the margins.`"off"`

,`0`

, or`false`

— Do not display the margins.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

`MinimumStabilityMargins`

— Visibility of minimum stability margins

`CharacteristicOption`

object

Visibility of minimum stability margins, specified as a `CharacteristicOption`

object with the following property.

`Visible`

— Margin visibility

`"off"`

(default) | on/off logical value

Margin visibility, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Display the margins.`"off"`

,`0`

, or`false`

— Do not display the margins.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

`FrequencyUnit`

— Frequency units

`"rad/s"`

| `"Hz"`

| `"rpm"`

| ...

Frequency units, specified as one of the following values:

`"Hz"`

`"rad/s"`

`"rpm"`

`"kHz"`

`"MHz"`

`"GHz"`

`"rad/nanosecond"`

`"rad/microsecond"`

`"rad/millisecond"`

`"rad/minute"`

`"rad/hour"`

`"rad/day"`

`"rad/week"`

`"rad/month"`

`"rad/year"`

`"cycles/nanosecond"`

`"cycles/microsecond"`

`"cycles/millisecond"`

`"cycles/hour"`

`"cycles/day"`

`"cycles/week"`

`"cycles/month"`

`"cycles/year"`

#### Dependencies

By default, the response uses the frequency units of the plotted linear system. You can override the default units by specifying toolbox preferences. For more information, see Specify Toolbox Preferences for Linear Analysis Plots.

`MagnitudeUnit`

— Magnitude units

`"dB"`

| `"abs"`

Magnitude units, specified as one of the following:

`"dB"`

— Decibels`"abs"`

— Absolute value

#### Dependencies

The default magnitude units depend on the toolbox preferences. For more information, see Specify Toolbox Preferences for Linear Analysis Plots.

`PhaseUnit`

— Phase units

`"deg"`

| `"rad"`

Phase units, specified as one of the following:

`"deg"`

— Degrees`"rad"`

— Radians

#### Dependencies

The default phase units depend on the toolbox preferences. For more information, see Specify Toolbox Preferences for Linear Analysis Plots.

`PhaseWrappingEnabled`

— Option to enable phase wrapping

on/off logical value

Option to enable phase wrapping, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Enable phase wrapping. The phase shown in the response wraps to remain in the range defined by`PhaseWrappingBranch`

.`"off"`

,`0`

, or`false`

— Disable phase wrapping.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

#### Dependencies

The default phase-wrapping configuration depends on the toolbox preferences. For more information, see Specify Toolbox Preferences for Linear Analysis Plots.

When both phase wrapping and phase matching are enabled, the software performs the phase matching followed by the phase wrapping.

`PhaseWrappingBranch`

— Lower limit of phase-wrapping range

scalar

Lower limit of phase-wrapping range, specified as a scalar value in degrees. The phase-wrapping range is [*B,B+360*), where *B* is equal to `PhaseWrappingBranch`

.

#### Dependencies

This value is ignored when

`PhaseMatchingEnabled`

is`"off"`

.The default phase-wrapping configuration depends on the toolbox preferences. For more information, see Specify Toolbox Preferences for Linear Analysis Plots.

`PhaseMatchingEnabled`

— Option to enable phase matching

`"off"`

(default) | on/off logical value

Option to enable phase matching, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Enable phase matching such that the phase response matches the value specified in`PhaseMatchingValue`

at the frequency specified in`PhaseMatchingFrequency`

. The remaining phase response shifts to maintain the same phase profile.`"off"`

,`0`

, or`false`

— Disable phase matching.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

#### Dependencies

When both phase wrapping and phase matching are enabled, the software performs the phase matching followed by the phase wrapping.

`PhaseMatchingFrequency`

— Phase matching frequency

`0`

(default) | scalar

Phase matching frequency, specified as a scalar.

#### Dependencies

This value is ignored when `PhaseMatchingEnabled`

is `"off"`

.

`PhaseMatchingValue`

— Phase matching response value

`0`

(default) | scalar

Phase matching response value, specified as a scalar.

#### Dependencies

This value is ignored when `PhaseMatchingEnabled`

is `"off"`

.

`MinimumGainEnabled`

— Option to enable minimum gain

`"off"`

(default) | on/off logical value

Option to enable minimum gain for plotting, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Set the minimum gain for plotting to the`MinimumGainValue`

property value.`"off"`

,`0`

, or`false`

— Set the minimum gain for plotting automatically based on the system dynamics.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

#### Dependencies

The default minimum-gain configuration depends on the toolbox preferences. For more information, see Specify Toolbox Preferences for Linear Analysis Plots.

`MinimumGainValue`

— Minimum gain value

`0`

(default) | scalar

Minimum gain value for plotting, specified as a scalar.

#### Dependencies

To apply the minimum gain value, set the

`MinimumGainEnabled`

property to`"on"`

.The default minimum-gain configuration depends on the toolbox preferences. For more information, see Specify Toolbox Preferences for Linear Analysis Plots.

#### Dependencies

`Visible`

— Chart visibility

`"on"`

(default) | on/off logical value

Chart visibility, specified as one of the following logical on/off values:

`"on"`

,`1`

, or`true`

— Display the chart.`"off"`

,`0`

, or`false`

— Hide the chart without deleting it. You still can access the properties of chart when it is not visible.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

.

`IOGrouping`

— Grouping of inputs and outputs pairs

`"none"`

(default) | `"inputs"`

| `"outputs"`

| `"all"`

Grouping of inputs and outputs pairs, specified as one of the following:

`"none"`

— Do not group inputs or outputs.`"inputs"`

— Group only inputs.`"outputs"`

— Group only outputs.`"all"`

— Group all input-output pairs.

`InputVisible`

— Option to display inputs

on/off logical value | array of on/off logical values

Option to display inputs, specified as one of the following logical on/off values or an array of such values:

`"on"`

,`1`

, or`true`

— Display the corresponding input.`"off"`

,`0`

, or`false`

— Hide the corresponding input.

`InputVisible`

is an array when the plotted system has multiple inputs. By
default, all inputs are visible in the plot.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

or an array of such values.

`OutputVisible`

— Option to display outputs

on/off logical value | array of on/off logical values

Option to display outputs, specified as one of the following logical on/off values or an array of such values:

`"on"`

,`1`

, or`true`

— Display the corresponding output.`"off"`

,`0`

, or`false`

— Hide the corresponding output.

`OutputVisible`

is an array when the plotted system has multiple outputs.
By default, all outputs are visible in the plot.

The value is stored as an on/off logical value of type `matlab.lang.OnOffSwitchState`

or an array of such values.

## Object Functions

`addResponse` | Add dynamic system response to existing response plot |

`showConfidence` (System Identification Toolbox) | Display confidence regions on response plots for identified models |

## Examples

### Customize Nichols Plot

For this example, use the plot handle to change the title, turn on the grid, and set axis limits.

Generate a random state-space model with 5 states and create the Nichols plot with chart object `np`

.

```
rng("default")
sys = rss(5);
np = nicholsplot(sys);
```

Change the title, enable the grid, and set axis limits.

np.Title.String = "Nichols Frequency Response"; xlim([-2 4]) ylim([3.3 4.3]) grid on

Alternatively, you can also use the `nicholsoptions`

command to specify the required plot options. First, create an options set based on the toolbox preferences.

`plotoptions = nicholsoptions('cstprefs');`

Change the desired properties of the options set.

plotoptions.Title.String = 'Nichols Frequency Response'; plotoptions.Grid = 'on'; plotoptions.XLim = {[-2,4]}; plotoptions.YLim = {[3.3,4.3]}; nicholsplot(sys,plotoptions);

Depending on your own toolbox preferences, the plot you obtain might look different from this plot. Only the properties that you set explicitly, in this example `Title`

, `Grid`

, `XLim`

and `YLim`

, override the toolbox preferences.

### Custom Nichols Plot Settings Independent of Preferences

For this example, create a Nichols plot that uses 15-point red text for the title. This plot should look the same, regardless of the preferences of the MATLAB session in which it is generated.

First, create a default options set using `nicholsoptions`

.

plotoptions = nicholsoptions;

Next, change the required properties of the options set `plotoptions`

.

plotoptions.Title.FontSize = 15; plotoptions.Title.Color = [1 0 0]; plotoptions.FreqUnits = 'Hz'; plotoptions.Grid = 'on';

Now, create a Nichols plot using the options set `plotoptions`

.

nicholsplot(tf(1,[1,1]),{0,15},plotoptions);

Because `plotoptions`

begins with a fixed set of options, the plot result is independent of the toolbox preferences of the MATLAB session.

### Customized Nichols Plot of Transfer Function

For this example, create a Nichols plot of the following continuous-time SISO dynamic system. Then, turn the grid on and rename the plot.

$$sys(s)=\frac{{s}^{2}+0.1s+7.5}{{s}^{4}+0.12{s}^{3}+9{s}^{2}}.$$

Create the transfer function `sys`

.

sys = tf([1 0.1 7.5],[1 0.12 9 0 0]);

Next, create the options set using `nicholsoptions`

and change the required plot properties.

plotoptions = nicholsoptions; plotoptions.Grid = 'on'; plotoptions.Title.String = 'Nichols Plot of Transfer Function';

Now, create the Nichols plot with the custom option set `plotoptions`

.

nicholsplot(sys,plotoptions)

`nicholsplot`

automatically selects the plot range based on the system dynamics.

### Customized Nichols Plot of MIMO System

For this example, consider a MIMO state-space model with 3 inputs, 3 outputs and 3 states. Create a Nichols plot with phase units in radians.

Create the MIMO state-space model `sys_mimo`

.

J = [8 -3 -3; -3 8 -3; -3 -3 8]; F = 0.2*eye(3); A = -J\F; B = inv(J); C = eye(3); D = 0; sys_mimo = ss(A,B,C,D); size(sys_mimo)

State-space model with 3 outputs, 3 inputs, and 3 states.

Create a Nichols plot with chart object `np`

.

np = nicholsplot(sys_mimo);

Set the phase unit ad radians.

`np.PhaseUnit = "rad";`

The Nichols plot automatically updates when you modify the chart object. For MIMO models, `nicholsplot`

produces an array of Nichols plots, each plot displaying the frequency response of one I/O pair.

### Nichols Response of Identified Parametric and Nonparametric Models

For this example, compare the Nichols response of a parametric model, identified from input/output data, to a non-parametric model identified using the same data. Identify parametric and non-parametric models based on the data.

Load the data and create the parametric and non-parametric models using `tfest`

and `spa`

, respectively.

load iddata2 z2; w = linspace(0,10*pi,128); sys_np = spa(z2,[],w); sys_p = tfest(z2,2);

`spa`

and `tfest`

require System Identification Toolbox™ software. The model `sys_np`

is a non-parametric identified model while, `sys_p`

is a parametric identified model.

Create an options set to turn phase matching and the grid on. Then, create a Nichols plot that includes both systems using this options set.

plotoptions = nicholsoptions; plotoptions.PhaseMatching = 'on'; plotoptions.Grid = 'on'; plotoptions.XLim = {[-240,0]}; h = nicholsplot(sys_p,'r.-.',sys_np,'b.-.',w,plotoptions); legend('Parametric Model','Non-Parametric model');

## Version History

**Introduced before R2006a**

### R2024b: Improved customization workflows and integration with MATLAB plotting tools

Starting in R2024b, `nicholsplot`

returns a `NicholsPlot`

chart object. Previously, the `nicholsplot`

function returned a handle to the
resulting plot.

The new chart object allows you to customize your plot using dot notation.

The new chart object also improves integration with MATLAB plotting tools. For example:

You can now add Nichols plots to tiled chart layouts.

Saving and loading a parent figure now maintains full plot interactivity.

You can add a response you your Nichols plot using the new

`addResponse`

function.

The following functionality changes might require updates to your code.

The

`gca`

function now returns the chart object rather than an axes within the plot.You can no longer access the graphics objects within a Nichols plot using the

`Children`

property of its parent figure.

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