# getSectorCrossover

Crossover frequencies for sector bound

## Description

returns the frequencies at which the following matrix
`wc`

= getSectorCrossover(`H`

,`Q`

)*M*(*ω*) is singular:

$$M\left(\omega \right)=H{\left(j\omega \right)}^{H}Q\text{\hspace{0.17em}}H\left(j\omega \right).$$

When a frequency-domain sector plot exists, these frequencies are the frequencies
at which the relative sector index (R-index) for `H`

and
`Q`

equals 1. See About Sector Bounds and Sector Indices for details.

## Examples

### Find Sector Crossover Frequency

Find the crossover frequencies for the dynamic system $$G\left(s\right)=\left(s+2\right)/\left(s+1\right)$$ and the sector defined by:

$$S=\left\{\left(y,u\right):a{u}^{2}<uy<b{u}^{2}\right\},$$

for various values of *a* and *b*.

In U/Y space, this sector is the shaded region of the following diagram (for *a*, *b* > 0).

The `Q`

matrix for this sector is given by:

$$Q=\left[\begin{array}{cc}1& -\left(a+b\right)/2\\ -\left(a+b\right)/2& ab\end{array}\right];\phantom{\rule{1em}{0ex}}a=0.1,\phantom{\rule{0.2777777777777778em}{0ex}}b=10.$$

`getSectorCrossover`

finds the frequencies at which $$H(s{)}^{H}QH(s)$$ is singular, for $$H\left(s\right)=\left[G\left(s\right);I\right]$$. For instance, find these frequencies for the sector defined by `Q`

with *a* = 0.1 and *b* = 10.

G = tf([1 2],[1 1]); H = [G;1]; a = 0.1; b = 10; Q = [1 -(a+b)/2 ; -(a+b)/2 a*b]; w = getSectorCrossover(H,Q)

w = 0x1 empty double column vector

The empty result means that there are no such frequencies.

Now find the frequencies at which $${H}^{H}QH$$ is singular for a narrower sector, with *a* = 0.5 and *b* = 1.5.

a2 = 0.5; b2 = 1.5; Q2 = [1 -(a2+b2)/2 ; -(a2+b2)/2 a2*b2]; w2 = getSectorCrossover(H,Q2)

w2 = 1.7321

Here the resulting frequency is where the R-index for `H`

and `Q2`

is equal to 1, as shown in the sector plot.

sectorplot(H,Q2)

Thus, when a sector plot exists for a system `H`

and sector `Q`

, `getSectorCrossover`

finds the frequencies at which the R-index is 1.

## Input Arguments

`H`

— Model to analyze

dynamic system model

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, `getSectorCrossover`

analyzes the current,
nominal value of `H`

.

To get the frequencies at which the I/O trajectories (*u*,*y*) of a linear system *G* lie in a
particular sector, use `H = [G;I]`

, where ```
I =
eyes(nu)
```

, and `nu`

is the number of inputs of
`G`

.

`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.

## Output Arguments

`wc`

— Sector crossover frequencies

vector | `[]`

Sector crossover frequencies, returned as a vector. The frequencies are
expressed in rad/`TimeUnit`

, relative to the
`TimeUnit`

property of `H`

. If the
trajectories of `H`

never cross the boundary, ```
wc
= []
```

.

## Version History

**Introduced in R2016a**

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