How to find the critical gain K from the root locus
Any suggestions would be appreciated.


You get the poles, for the closed-loop system \$ L(z) = \frac{G(z)}{1+G(z)}\$, moving on those root locus trajectories by changing the gain \$K\$. And you want to find the critical gain \$K\$ such that the poles are on the unit circle. Summing up the mathematics:


\$ G(z) = \frac{1.1353K(z+0.5232)}{(z-1)(z-0.1353)} \$,
\$ L(z) = \frac{\frac{1.1353K(z+0.5232)}{(z-1)(z-0.1353)}}{1+\frac{1.1353K(z+0.5232)}{(z-1)(z-0.1353)}} = \frac{1.1353K(z+0.5232)}{(z-1)(z-0.1353)+1.1353K(z+0.5232)}\$,
\$|z_0| = 1\$.


Now, to solve this, expand the denominator of \$L(z)\$ (since its roots are the poles of the transfer function), then find the roots of the denominator, which is a second-degree polynomial, you will get some

$$z_0 = \frac{-b ~\pm ~\sqrt{b^2-4ac}}{2a},$$

Now, from the breakpoints and a sketch of the root locus you know that \$ z_0 \$ has a real part and an imaginary one, \$b\$ is the real part and the square root will be the imaginary one. You also know that \$|z_0| = 1\$, so combine that to have

$$ |z_0|^2 = 1 = \left(\frac{b}{2a}  \right)^2 + \left(\frac{\sqrt{4ac-b^2}}{2a} \right)^2,$$

Which will only have \$ K\$ as an unknown.

obs: really long maths, if anyone ever do the numeric calculation feel free to update this.