# sigmaplot

Plot singular values for frequency response of dynamic system

## Description

The `sigmaplot`

function plots the singular values for the
frequency response of a dynamic system model and returns a `SigmaPlot`

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 singular-value data, use `sigma`

.

## Creation

### Syntax

### Description

plots the singular values (SVs) of the frequency response of the dynamic system model
`sp`

= sigmaplot(`sys`

)`sys`

and returns the corresponding chart object.

plots singular values for frequencies specified in `sp`

= sigmaplot(___,`w`

)`w`

. You can
specify a frequency range or a vector of frequencies. If you omit `w`

,
frequencies are selected based on the system dynamics. You can use
`w`

with any of the previous syntaxes.

plots the singular values using the plotting options specified in
`sp`

= sigmaplot(___,`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 singular values in the specified parent graphics container, such as a
`sp`

= sigmaplot(`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 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`

. If`wmax`

is greater than the Nyquist frequency of`sys`

, the response is computed only up to the Nyquist frequency.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.

For models with complex coefficients, if you specify a frequency range of [*w*_{min},*w*_{max}] for your plot, then in:

Log frequency scale, the plot frequency limits are set to [

*w*_{min},*w*_{max}] and the plot shows two branches, one for positive frequencies [*w*_{min},*w*_{max}] and one for negative frequencies [–*w*_{max},–*w*_{min}].Linear frequency scale, the plot frequency limits are set to [–

*w*_{max},*w*_{max}] and the plot shows a single branch with a symmetric frequency range centered at a frequency value of zero.

Specify frequencies in units of rad/`TimeUnit`

, where `TimeUnit`

is the `TimeUnit`

property of the model.

`type`

— Type of modified singular values

`1`

| `2`

| `3`

Type of modified singular values to plot, specified as one of the following values.

`1`

— Plot the singular values of the frequency response*H*^{-1}, where*H*is the frequency response of`sys`

.`2`

— Plot the singular values of the frequency response*I*+*H*.`3`

— Plot the singular values of the frequency response*I*+*H*^{-1}.

#### Dependencies

You can specify `type`

only when `sys`

has the same number of inputs and outputs.

`plotoptions`

— Singular-value plot options

`sigmaoptions`

object

Singular-value plot options, specified as a `sigmaoptions`

object. You can use these options to customize the 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 SigmaPlot Properties.

`Responses`

— Model responses

`SigmaResponse`

object | array of `SigmaResponse`

objects

Model responses, specified as a `SigmaResponse`

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 `SigmaResponse`

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.

`SingularValueType`

— Type of singular values

`0`

(default) | `1`

| `2`

| `3`

Type of singular values to plot, specified as one of the following values.

`0`

— Plot singular values of*H*, which is the frequency response of`Model`

.`1`

— Plot modified singular values of the frequency response*H*^{-1}, where*H*is the frequency response of`sys`

.`2`

— Plot modified singular values of the frequency response*I*+*H*.`3`

— Plot modified singular values of the frequency response*I*+*H*^{-1}.

When you initially create a plot:

`SingularValueType`

matches the value you specify for the`type`

argument.If you do not specify

`type`

,`SingularValueType`

is`0`

.

#### Dependencies

You can specify `type`

as `1`

, `2`

, or `3`

only when `Model`

has the same number of inputs and outputs.

`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 property.

`SigmaPeakResponse`

— 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`

.

`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.

`FrequencyScale`

— Frequency scale

`"log"`

| `"linear"`

Frequency scale, specified as either `"log"`

or `"linear"`

.

#### Dependencies

The default frequency scale depends on the 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

If

`MagnitudeScale`

is`"log"`

when you set`MagnitudeUnit`

to`"dB"`

, the software automatically changes`MagnitudeScale`

to`"linear"`

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

`MagnitudeScale`

— Magnitude scale

`"log"`

| `"linear"`

Magnitude scale, specified as either `"log"`

or `"linear"`

.

#### Dependencies

Setting

`MagnitudeScale`

to`"log"`

is not supported when`MagnitudeUnit`

is`"dB"`

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

`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`

.

## 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 Sigma Plot

For this example, use the plot handle to change the frequency units to Hz and turn on the grid.

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

.

```
rng("default")
sys = rss(5);
sp = sigmaplot(sys);
```

Change the units to Hz and turn on the grid.

sp.FrequencyUnit = "Hz"; grid on

Alternatively, you can also use the `sigmaoptions`

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

`p = sigmaoptions('cstprefs');`

Change properties of the options set by setting the frequency units to Hz and enable the grid.

p.FreqUnits = 'Hz'; p.Grid = 'on'; sigmaplot(sys,p);

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 `Grid`

and `FreqUnits`

, override the toolbox preferences.

### Custom Sigma Plot Settings Independent of Preferences

For this example, create a sigma 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 `sigmaoptions`

.

plotoptions = sigmaoptions;

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 sigma plot using the options set `plotoptions`

.

h = sigmaplot(tf(1,[1,1]),plotoptions);

Because `plotoptions`

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

### Customized Sigma Plot of Transfer Function

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

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

Create the transfer function `sys`

.

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

Next, create the options set using `sigmaoptions`

and change the required plot properties.

plotoptions = sigmaoptions; plotoptions.Grid = 'on'; plotoptions.FreqScale = 'linear'; plotoptions.Title.String = 'Singular Value Plot of Transfer Function';

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

.

h = sigmaplot(sys,plotoptions);

`sigmaplot`

automatically selects the plot range based on the system dynamics.

### Sigma Plot with Specified Frequency Scale and Units

For this example, consider a MIMO state-space model with 3 inputs, 3 outputs and 3 states. Create a sigma plot with linear frequency scale, frequency units in Hz and turn the grid on.

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 sigma plot with with chart object `sp`

.

sp = sigmaplot(sys_mimo);

Update the plot by modifying the chart object.

sp.FrequencyScale = 'linear'; sp.FrequencyUnit = 'Hz'; grid on;

The sigma plot automatically updates when you modify the chart object

### Singular Value Plot of Identified Parametric and Nonparametric Models

For this example, compare the SV for the frequencies 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 the grid on. Then, create a sigma plot that includes both systems using this options set.

plotoptions = sigmaoptions; plotoptions.Grid = 'on'; h = sigmaplot(sys_p,'b--',sys_np,'r--',w,plotoptions); legend('Parametric Model','Non-Parametric model');

### Modified Singular Value Plot of MIMO System

Consider the following two-input, two-output dynamic system.

$$H(s)=\left[\begin{array}{cc}0& {\displaystyle \frac{3s}{{s}^{2}+s+10}}\\ {\displaystyle \frac{s+1}{s+5}}& {\displaystyle \frac{2}{s+6}}\end{array}\right].$$

Plot the singular value responses of *H*(*s*) and *I* + *H*(*s*). Set appropriate titles using the plot option set.

H = [0, tf([3 0],[1 1 10]) ; tf([1 1],[1 5]), tf(2,[1 6])]; opts1 = sigmaoptions; opts1.Grid = 'on'; opts1.Title.String = 'Singular Value Plot of H(s)'; h1 = sigmaplot(H,opts1);

Use input 2 to plot the modified SV of type, *I* + *H*(*s*).

opts2 = sigmaoptions; opts2.Grid = 'on'; opts2.Title.String = 'Singular Value Plot of I+H(s)'; h2 = sigmaplot(H,[],2,opts2);

## Version History

**Introduced before R2006a**

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

Starting in R2024b, `sigmaplot`

returns a `SigmaPlot`

chart object. Previously, the `sigmaplot`

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 singular-value plots to tiled chart layouts.

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

You can add a response you your singular-value 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 singular-value plot using the

`Children`

property of its parent figure.

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