How can I show the stability graph?

Hey guys,

I got a transfer function that is my school project :

How can I calculate 'K' constant and how can I show plot window ?

[Ameer Hamza]

Since this is a school project, what have you already tried?

[Can Yilmaz]

I tried routh hurwitz criterion but I can't find 'K'

[Ameer Hamza]

What is the issue with the Routh Hurwitz criterion? It should help in finding a value of K, which makes the system stable.

Answer 1

[Sam Chak]

Hi @Can Yilmaz,

The transfer function

can be rewritten as

where where the characteristic equation

defines the behavior of the system. A 2nd-order linear time-invariant system is considered stable if all of its poles have negative real parts.

In algebra, we learned that for both terms and s to be positive, the sign of K must also be positive in order to produce the negative roots. In other words, the exponential stability condition is . Thus, we can plot a graph that shows the stability region where yields negative real parts of the poles.

K = -0.2:0.025:0.6;

for j = 1:numel(K)
    
    % characteristic polynomial
    p = [1 156.1e-6/25.1e-9 K(j)/25.1e-9];
    
    % find the roots of the polynomial
    s = real(roots(p));
    
    % plot the real part of the poles
    plot(K(j), s, '.'), hold on 
    
end

v = [0 -8e3; 0.6 -8e3; 0.6 2e3; 0 2e3];
f = [1 2 3 4];
patch('Faces', f, 'Vertices', v, 'FaceColor', '#ffb7c5', 'FaceAlpha', 0.35)

hold off, grid on, 

xlabel('K'), 
ylabel('Re(s)')
title('Stability region (sakura patch)')
xline(0, '--', 'K = 0', 'color', '#7F7F7F')
yline(0, '--', 'Re(s) = 0', 'color', '#7F7F7F')