sectorplot

R2026b

Plot sector index as function of frequency

Syntax

sectorplot(H,Q)

sectorplot(H1,H2,...,HN,Q)

sectorplot(H1,LineSpec1,...,HN,LineSpecN,Q)

sectorplot(H1,LineSpec1,...,HN,LineSpecN,Q,w)

sectorplot(___,w)

sectorplot(___,plotoptions)

sectorplot(___,Name=Value)

sectorplot(parent,___)

sp = sectorplot(___)

Description

The sectorplot function plots the sector indices of a dynamic system model. To customize the plot, you can return a SectorPlot object and modify it using dot notation. For more information, see Customize Linear Analysis Plots at Command Line.

To obtain sector indices, use the sector function.

sectorplot(H,Q) plots the relative sector indices for the dynamic system H and a given sector matrix Q. These indices measure by how much the sector bound is satisfied (index less than 1) or violated (index greater than 1) at a given frequency. (See About Sector Bounds and Sector Indices for more information about the meaning of the sector index.) sectorplot automatically chooses the frequency range and number of points based on the dynamics of H.

Let the following be an orthogonal decomposition of the symmetric matrix Q into its positive and negative parts.

$$Q=W_1{W}_1^T-W_2{W}_2^T,W_1^T{W}_2=0.$$

The sector index plot is only meaningful if $$W_2^{T}H$$ has a proper stable inverse. In that case, the sector indices are the singular values of:

$$\left(W_1^{T}H\left(j\omega \right)\right){\left(W_2^{T}H\left(j\omega \right)\right)}^{-1}.$$

If H is a model with complex coefficients, then in:

• Log frequency scale, the plot shows two branches, one for positive frequencies and one for negative frequencies. The arrows indicate the direction of increasing frequency values for each branch.

• Linear frequency scale, the plot shows a single branch with a symmetric frequency range centered at a frequency value of zero.

sectorplot(H1,H2,...,HN,Q) and plots the sector index for multiple dynamic systems H1,H2,...,HN on the same plot.

sectorplot(H1,LineSpec1,...,HN,LineSpecN,Q) and sectorplot(H1,LineSpec1,...,HN,LineSpecN,Q,w) specify a color, line style, and marker for each system in the plot.

sectorplot(___,w) plots the sector index for frequencies specified by w. You can specify a frequency range or a vector of frequencies.

You can use this syntax with any of the previous input-argument combinations.

sectorplot(___,plotoptions) plots the sector index with the options set specified in plotoptions. You can use these options to customize the plot appearance using the command line. Settings you specify in plotoptions override the preference settings in the MATLAB^® session in which you run sectorplot. Therefore, this syntax is useful when you want to write a script to generate multiple plots that look the same regardless of the local preferences.

sectorplot(___,Name=Value) specifies response properties using one or more name-value arguments. For example, sectorplot(sys,Color="r") sets the plot color to red. (since R2026b)

• When plotting responses for multiple systems, the specified name-value arguments apply to all responses.

• The following name-value arguments override values specified in other input arguments.

  • Frequency — Overrides frequencies specified using w

  • SectorGeometry — Overrides sector geometry specified using Q

  • Color — Overrides colors specified using LineSpec

  • MarkerStyle — Overrides marker styles specified using LineSpec

  • LineStyle — Overrides line styles specified using LineSpec

sectorplot(parent,___) plots the sector indices in the specified parent graphics container, such as a 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. (since R2026b)

sp = sectorplot(___) plots the sector indices and returns the corresponding chart object. To customize the appearance and behavior of the response plot, modify the chart object properties using dot notation. (since R2026b)

Examples

Plot Sector Index Versus Frequency

Plot the sector index to visualize the frequencies at which the I/O trajectories of $$G\left(s\right)=\left(s+2\right)/\left(s+1\right)$$ lie within the sector defined by:

$$S=\left\{\left(y,u\right):0.1u^2<uy<10u^2\right\}.$$

In U/Y space, this sector is the shaded region of the following diagram.

The Q matrix for this sector is given by:

a = 0.1;  
b = 10; 
Q = [1 -(a+b)/2 ; -(a+b)/2 a*b];

A trajectory $$y\left(t\right)=Gu\left(t\right)$$ lies within the sector S when for all T > 0,

$$0.1{\int }_0^{T}u{\left(t\right)}^2<{\int }_0^{T}u\left(t\right)y\left(t\right)dt<10{\int }_0^{T}u{\left(t\right)}^{2}dt.$$

In the frequency domain, this same condition can be expressed as:

$${\left(\begin{array}{c}G\left(j\omega \right)\\ 1\end{array}\right)}^{H}Q\left(\begin{array}{c}G\left(j\omega \right)\\ 1\end{array}\right)<0.$$

To check whether G satisfies or violates this condition at any frequency, plot the sector index for H = [G;1].

G = tf([1 2],[1 1]); 
sectorplot([G;1],Q)

The plot shows that the sector index is less than 1 at all frequencies. Therefore, the trajectories of G(s) fit within in the specified sector Q at all frequencies.

Sector Plot with MIMO System

Examine the sector plot of a 2-output, 2-input system for a particular sector. Load the system, H1, and define the sector, Q.

load("sectorExampleSystem.mat","H1")
Q = [-5.12   2.16  -2.04   2.17
      2.16  -1.22  -0.28  -1.11
     -2.04  -0.28  -3.35   0.00
      2.17  -1.11   0.00   0.18];

Create the plot.

sectorplot(H1,Q)

Because H is 2-by-2, there are two lines on the sector plot. The largest value of the sector index exceeds 1 below about 0.5 rad/s and in a narrow band around 3 rad/s. Therefore, H does not satisfy the sector bound represented by Q.

Plot Sector Index of Model with Complex Coefficients

Plot the relative sector indices of a model with complex coefficients and a model with real coefficients on the same plot.

Create a system with complex coefficients.

A = [-3.50,-1.25-0.25i;2,0];
B = [1;0];
C = [-0.75-0.5i,0.625-0.125i];
D = 0.5;
Hc = [ss(A,B,C,D);1];

Load a system with real coefficients.

load("sectorExampleSystem.mat","Hr")

Plot the relative sector indices of both systems.

Q = [1 0.1;0.1 -1];
sectorplot(Hc,Hr,Q)
legend("Complex-coefficient model","Real-coefficient model", ...
    Location="southwest");

In log frequency scale, the plot shows two branches for models with complex coefficients, one for positive frequencies, with a right-pointing arrow, and one for negative frequencies, with a left-pointing arrow. In both branches, the arrows indicate the direction of increasing frequencies. The plots for models with real coefficients always contain a single branch with no arrows.

Set the plotting frequency scale to linear.

opt = sectorplotoptions;
opt.FreqScale = "Linear";

Plot the indices.

sectorplot(Hc,Hr,Q,opt)

legend("Complex-coefficient model","Real-coefficient model", ...
    Location="southwest");

In linear frequency scale, the plots show a single branch with a symmetric frequency range centered at a frequency value of zero. The plot also shows the negative-frequency response of a model with real coefficients when you plot the response along with a model with complex coefficients.

Input Arguments

H — Model to analyze

dynamic system model | model array

Model to analyze against sector bounds, specified as a dynamic system model such as a tf, ss, or genss model. H can be continuous or discrete. If H is a generalized model with tunable or uncertain blocks, sectorplot analyzes the current, nominal value of H.

To analyze whether all I/O trajectories (u(t),y(t) of a linear system G lie in a particular sector, use H = [G;I], where I = eye(nu), and nu is the number of inputs of G.

If H is a model array, then sectorplot plots the sector index of all models in the array on the same plot. When you use output arguments to get sector-index data, H must be a single model.

Q — Sector geometry

matrix | LTI model

Sector geometry, specified as:

• A matrix, for constant sector geometry. Q is a symmetric square matrix that is ny on a side, where ny is the number of outputs of H.

• An LTI model, for frequency-dependent sector geometry. Q satisfies Q(s)’ = Q(–s). In other words, Q(s) evaluates to a Hermitian matrix at each frequency.

The matrix Q must be indefinite to describe a well-defined conic sector. An indefinite matrix has both positive and negative eigenvalues.

For more information, see About Sector Bounds and Sector Indices.

w — Frequencies

{wmin,wmax} | vector

Frequencies at which to compute and plot indices, specified as one of the following:

• Cell array of the form {wmin,wmax} — Compute the plot indices at frequencies in the range from wmin to wmax. If wmax is greater than the Nyquist frequency of the system, the response is computed only up to the Nyquist frequency.

• Vector of frequencies — Compute the plot indices 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.

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. For example, specify the marker and omit the line style, 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

plotoptions — Sector index plot options

SectorPlotOptions object

Sector index plot options, specified as a SectorPlotOptions object created using sectorplotoptions. 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 graphics container

Figure object | TiledChartLayout object | UIFigure object | UIGridLayout object | UIPanel object | UITab object

Parent graphics container, specified as one of these objects:

• Figure

• TiledChartLayout

• UIFigure

• UIGridLayout

• UIPanel

• UITab

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: sectorplot(sys,LegendDisplay="off") hides the response of sys from the plot legend.

Frequency — Frequencies

{wmin,wmax} | vector

Frequencies at which to compute and plot indices, specified as one of the following:

• Cell array of the form {wmin,wmax} — Compute the plot indices at frequencies in the range from wmin to wmax. If wmax is greater than the Nyquist frequency of the system, the response is computed only up to the Nyquist frequency.

• Vector of frequencies — Compute the plot indices at each specified frequency. For example, use logspace to generate a row vector with logarithmically spaced frequency values. The vector 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.

SectorGeometry — Sector geometry

matrix | LTI model

Sector geometry, Q, specified as:

• A matrix, for constant sector geometry. Q is a symmetric square matrix that is ny on a side, where ny is the number of outputs of H.

• An LTI model, for frequency-dependent sector geometry. Q satisfies Q(s)’ = Q(–s). In other words, Q(s) evaluates to a Hermitian matrix at each frequency.

The matrix Q must be indefinite to describe a well-defined conic sector. An indefinite matrix has both positive and negative eigenvalues.

For more information, see About Sector Bounds and Sector Indices.

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 these 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 the response in the legend, specified as one of these 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 these values. Specifying a marker style using a name-value argument overrides any marker style that you specify using LineSpec.

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. Specifying a color using a name-value argument overrides any color that you specify using LineSpec.

You can also specify some common colors by name. This table lists these colors and their corresponding RGB triplets and hexadecimal color codes.

Color Name	RGB Triplet	Hexadecimal Color Code
"red" or "r"	[1 0 0]	#FF0000
"green" or "g"	[0 1 0]	#00FF00
"blue" or "b"	[0 0 1]	#0000FF
"cyan" or "c"	[0 1 1]	#00FFFF
"magenta" or "m"	[1 0 1]	#FF00FF
"yellow" or "y"	[1 1 0]	#FFFF00
"black" or "k"	[0 0 0]	#000000
"white" or "w"	[1 1 1]	#FFFFFF

LineStyle — Line style

"-" | "--" | ":" | "-."

Line style, specified as one of these values. Specifying a line style using a name-value argument overrides any line style that you specify using LineSpec.

Line Style	Description
"-"	Solid line
"--"	Dashed line
":"	Dotted line
"-."	Dash-dotted line
"none"	No 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.

SeriesIndex — Series index

positive integer | "none"

Series index, specified as a positive integer or "none".

By default, the SeriesIndex property is a number that corresponds to the order in which the response was added to the chart, starting at 1. MATLAB uses the number to calculate indices for automatically assigning color, line style, or markers for responses. Any responses in the chart that have the same SeriesIndex number also have the same color, line style, and markers.

A SeriesIndex value of "none" indicates that a response does not participate in the indexing scheme.

Output Arguments

sp — Chart object

SectorPlot object

Chart object, returned as a SectorPlot object. To customize your plot appearance and behavior, modify the properties of this object using dot notation. For more information, see SectorPlot Properties.

Version History

Introduced in R2016a

R2026b: Improved customization workflows and integration with MATLAB plotting tools

Starting in R2026b, the sectorplot function returns a SectorPlot chart object.

Previously, the sectorplot function returned the computed passivity indices. To obtain computed passivity indices, use the new sector function. For more information, see Obtain passivity indices using sector function.

The new chart object allows you to customize your plot using dot notation and name-value arguments.

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

• You can now add sector plots to tiled chart layouts.

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

• You can add a response to your sector plot using the addResponse function.

R2026b: Obtain passivity indices using sector function

Starting in R2026b, you obtain computed passivity indices using the new sector function instead of the sectorplot function.

For these computation workflows, only the name of the plotting function has changed. The computation behavior remains the same.