I am not certain what you are starting with, so I cannot provide an exact response. However one approach may simply be to use interp1 to interpolate the existing ‘r’ and ‘k’ values that correspond to the ζ value you want.
Using an example from the rlocus documentation —
H = tf([2 5 1],[1 2 3]);
[r,kout] = rlocus(H);
zetav = cos(angle(r(1,:)));
zetaq = [0.64 0.76 0.80 0.86 0.90 0.985]; % Desired ‘zeta’ Values
for k = 1:numel(zetaq)
zidx = find(diff(sign(zetav + zetaq(k))));
idxrng = max(1,zidx-1) : min(numel(zetav),zidx+1);
rq(k,:) = interp1(zetav(idxrng), r(1,idxrng), -zetaq(k));
kq(k,:) = interp1(zetav(idxrng), kout(idxrng), -zetaq(k));
end
Result = table(zetaq(:),rq,kq, VariableNames={'zeta','r','k'})
figure
rlocus(H)
sgrid
hold on
plot(real(rq), imag(rq), 'sr') % Plot ‘zeta’ Value As Red Squares
hold off
[min_zeta,max_zeta] = bounds(zetav)
% disp(zetav)
The values corresponding to the plotted ζ l;ines plot at their intersections with the root locus.
The ‘Result’ table shows only one value for ‘r’ for convenience. (It likely has a complex conjugate that you can easily calculate.)
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