Root locus imaginary axis intersection
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Other than using interactive data cursors, is there anyway of finding the point where the root locus intersects the imaginary axis?
Accepted Answer
Try something like this —
sys = tf([3 1],[9 7 5 6]); % Example From The Documentation
[r,k] = rlocus(sys)
fre2 = isfinite(real(r(2,:)));
fim2 = isfinite(imag(r(2,:)));
fidx2 = fre2 & fim2;
fre3 = isfinite(real(r(3,:)));
fim3 = isfinite(imag(r(3,:)));
fidx3 = fre3 & fim3;
v2 = interp1(real(r(2,fidx2)), imag(r(2,fidx2)), 0, 'linear','extrap')
k2 = interp1(imag(r(2,fidx2)), k(fidx2), v2, 'linear','extrap')
v3 = interp1(real(r(3,fidx3)), imag(r(3,fidx3)), 0, 'linear','extrap')
k3 = interp1(imag(r(3,fidx3)), k(fidx3), v3, 'linear','extrap')
figure
plot(real(r(1,:)),imag(r(1,:)), '-g')
hold on
plot(real(r(2,:)),imag(r(2,:)), '-b')
plot(real(r(3,:)),imag(r(3,:)), '-r')
plot(0, v2, 'sr')
plot(0, v3, 'sb')
hold off
grid
ylim([-6 6])
.
2 Comments
Paul
on 15 Jun 2021
Will this solution work if a branch of the root locus crosses the imaginary axis twice? For example if
sys = tf([3 1],[9 7 5 6]) * tf(20,[1 20])
Can this solution be generalized to loop over all of the rows of r?
As I understand it, this solution assumes that the rows of r are, in some sense, smooth. I think that rlocus() tries to ensure this, but I'm not sure it's guaranteed.
Star Strider
on 15 Jun 2021
This is prototype code.
It simply shows the correct approach, and would likely have to be adapted to specific situations that did not follow the same sort of loci.
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