# How do I use a root locus to find a value of K such that the damping ratio of dominant closed loop poles is a specified value?

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Thomas on 9 Mar 2023
Edited: Paul on 1 May 2023
How do I use a root locus to find a value of K such that the damping ratio of dominant closed loop poles is 0.7071?
Using the following TF and associated root locus. Many thanks

Paul on 9 Mar 2023
Click on branch of the root locus and drag along the branch. The values in the datatip, which include the damping ratio, will correspond to the datamarker point on the locus.
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Paul on 1 May 2023
Edited: Paul on 1 May 2023
Here's my attempt. Can't say I've thorougly tested it, but maybe it's a start. I know for sure it will break down if the desired damping ratio is 1.
Transfer function for testing and its root locus
h = tf([2 5 1],[1 2 3]);
s = tf('s');
h = h/(s+4)/(s+5);
figure
rlocus(h);
hold on
Desired damping ratios
Loop through the zetad, compute the gains and corresponding root locations, which are added to plot as filled, black dots
plot(real(r),imag(r),'k.','MarkerSize',15);
end
Warning: The numerator or denominator of this transfer function has complex-valued coefficients.
Warning: The numerator or denominator of this transfer function has complex-valued coefficients.
Warning: The numerator or denominator of this transfer function has complex-valued coefficients. k
k = 1×3 cell array
{[214.6052]} {[15.2534 0.3113]} {[0.4787 9.2402]}
Cases with no solution
[r,k] = zetagain(tf(1,[1 1]),0.2)
Warning: The numerator or denominator of this transfer function has complex-valued coefficients.
r = 0×1 empty double column vector k = 1×0 empty double row vector
[r,k] = zetagain(tf(1,[1 2*.5*1 1]),0.7)
Warning: The numerator or denominator of this transfer function has complex-valued coefficients.
r = 0×1 empty double column vector k = 1×0 empty double row vector
Other corner cases would be where one or more branches of the root locus either lie exactly on a line of constant zetad or if the line of constant zetad is a root locus asymptote.
% compute the rotation angle
m = numel(h.num{:})-1;
n = numel(h.den{:})-1;
h1 = tf(h.num{:}.*exp(-1j*theta).^(m:-1:0),h.den{:}.*exp(-1j*theta).^(n:-1:0));
% gain margin in rotated complex plane
s = allmargin(h1);
k = s.GainMargin;
% roots at values of k
r = rlocus(h,k);
r = r(:);
% eliminate extraneous roots that aren't at the the desired zeta. use 0.02 as a tolerance,
% maybe this should be an input parameter. maybe this test could could be made more robust.
zeta = sin(atan2(-real(r),abs(imag(r))));
end

Sam Chak on 26 Apr 2023
Edited: Sam Chak on 26 Apr 2023
The steps are generally described in the following documentation, as well as in most undergrad control textbooks.
Demo:
% Plant
Gp = tf(32.5, [1 5 40])
Gp = 32.5 -------------- s^2 + 5 s + 40 Continuous-time transfer function.
% rlocus(Gp), sgrid From the Root Locus of uncompensated system Gp with the s-plane grid of constant damping factors, it is clear that no matter how we move the 'square' data marker (■) to track the gain and damping values, it cannot achieve the desired 0.7071 damping ratio. Thus, a feedback controller is needed. In this example, a PID controller is designed and its gains are tuned.
% PID Controller
kp = -0.5066;
ki = 0.8703;
kd = 0.38;
Tf = 0.7071;
Gc = pid(kp, ki, kd, Tf)
Gc = 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = -0.507, Ki = 0.87, Kd = 0.38, Tf = 0.707 Continuous-time PIDF controller in parallel form.
rlocus(Gc*Gp), sgrid From the Root Locus of the compensated system Gc*Gp, we clearly have more rooms to move the data marker (■). Now, if we zoom into the locus and move the data marker (■) until the desired 0.707 damping ratio is observed, then the corresponding gain is found to be . ### Categories

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