Digital Root Locus in Scilab
Using the following code we can obtain the path the poles go as you change k.


G = poly([-1],'z')/poly([0.3 1],'z');


figure;
evans(G,10)
zgrid;

y = [];
for k = logspace(-2,1,1000)
    y = [y abs(roots((k*G/(1+k*G)).den))];
end

figure;
plot2d("ll",logspace(-2,1,1000),y')
legend('a1','a2')

//pick here your optimal gain
L = 0.5*G;
L = L/(1+L);
t = 0:21;
figure;
plot(t,dsimul(tf2ss(L),ones(1,1+max(t))),'x')






module of the poles in function of k.



root locus of the system.

If you manually/algorithmically search the path in \$ y\$ you will be able to find the \$ k \$ that minimizes the time, for that you don't have to just to minimize \$ |\sigma|\$ but also get it to be near the minimum  damping \$ \zeta = 0.6\$. And get a result similar to