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Compute or plot sector index as function of frequency

`sectorplot(`

plots
the relative sector indices for the dynamic system `H`

,`Q`

)`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.

$$Q={W}_{1}{W}_{1}^{T}-{W}_{2}{W}_{2}^{T},\text{\hspace{1em}}{W}_{1}^{T}{W}_{2}=0.$$

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(`

plots
the sector index for frequencies specified by `H`

,`Q`

,`w`

)`w`

.

If

`w`

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

, then`sectorplot`

plots the sector index at frequencies ranging between`wmin`

and`wmax`

.If

`w`

is a vector of frequencies, then`sectorplot`

plots the sector index at each specified frequency. The vector`w`

can contain both negative and positive frequencies.

`sectorplot(___,`

plots the sector index with the options set specified in
`plotoptions`

)`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.

`sectorplotoptions`

| `getSectorIndex`

| `getPassiveIndex`

| `passiveplot`