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:

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:

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.