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.

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.

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

and `Q2`

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

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

and sector `Q`

, `getSectorCrossover`

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