More accurate alternative to rlocfind to analyze root locus in control systems engineering

To find the gain at the point where the root locus intersects a line of constant damping ratio, the rlocfind function can be used, but the user has to manually select a point and Matlab finds the closest point on the root locus to the selection. Is there a way to find the exact point of intersection without having to make the selection manually?

h = tf([2 5 1],[1 2 3]);
rlocus(h)         % Root locus
z = 0.707; sgrid(z,0)
k = rlocfind(h)  

[PAVI]

PROCEDURE

3.1. Given the following transfer functions, plot the root locus. For each G(s) identify number of poles, zeros, infinite zeros and poles (if exist).

3.1.1. 𝐺 (𝑠𝑠)=13𝑠𝑠2+5𝑠𝑠+1

3.1.2. 𝐺 (𝑠𝑠)=5𝑠𝑠3+3𝑠𝑠2+5𝑠𝑠+1

4

3.1.3. 𝐺 (𝑠𝑠)=𝑠𝑠2+6𝑠𝑠+1𝑠𝑠3+3𝑠𝑠2+5𝑠𝑠+1

3.2. Using command ‘[K,poles] = rlocfind(num,den)’, repeat procedure 3.1. Find the respective gain and its poles at these respective conditions.

3.2.1. At K → 0.

3.2.2. When poles break-away into the complex plane or break-in away from the complex plane.

3.2.3. When poles enter the right half plane (RHP).

3.3. Implement a unity feedback control system of G(s). Identify the proportional gain, K that will satisfy the following parameters:

3.3.1. 𝐺 (𝑠𝑠)=1𝑠𝑠2+13𝑠𝑠+15

3.3.2. %OS within 10%.

3.3.3. Peak time less than 2.0 sec.

3.3.4. Use Matlab/SIMULINK for validation.

Answer 1

[Arkadiy Turevskiy]

Hi,

Here is a way to do it (not the most efficient, but it works).

% Define the transfer function
h = tf([2 5 1], [1 2 3]);

% Define the desired damping ratio
zeta = 0.707;

% Get the poles and zeros of the transfer function
[num, den] = tfdata(h);

% Loop through different values of k to find the desired damping ratio
for k = 0:0.001:100
   
    
    % Get the poles of the transfer function at gain k
    p_k = roots(cell2mat(den) + k * cell2mat(num));
    
    % Use the damp function to calculate the damping ratio of the poles
    [wn, zeta_k] = damp(p_k);
    
    % If the damping ratio is close to the desired value, print the gain and break out of the loop
    if abs(zeta_k - zeta) < 0.001
        fprintf('The gain k for a damping ratio of %f is %f\n', zeta, k);
        break;
    end
end

The gain k for a damping ratio of 0.707000 is 0.196000

HTH

Answer 2

[Paul]

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