Show that for a 1st order system
$$G(s)=\frac{k}{\tau s+1}$$
a pole placement at $s = \beta \pm i\beta$ corresponds to a PI-Controller with parameters
$$K = \frac{2\beta\tau-1}{k}, \space \space T_I = \frac{Kk}{2\beta^2\tau}$$
Can someone help me get started on this? I don't know what to do to begin solving this.
On
I am not sure how $K$ and $T_I$ are exactly defined for your PI controller, but if you have some general expression for the controller $C(s)$ you can just calculate the closed loop transfer function using
$$ \frac{G(s)\,C(s)}{1 + G(s)\,C(s)} $$
and calculate the pole location for it. By applying the constraints that these poles should be located at $s=\beta\pm i\beta$ you should be able to solve for the parameters of $C(s)$.
The transfer function yields the respective ODE $$\tau\dot{y}+y=ku$$ Considering the integrator $z=\int_0^t{y(s)ds}$ as an extra state we have the following second-order dynamics $$\left[\matrix{\dot{z}\\ \dot{y}}\right]=\left[\matrix{0 & 1\\ 0 & -\frac{1}{\tau}}\right]\left[\matrix{{z}\\ {y}}\right]+\left[\matrix{0\\ \frac{k}{\tau}}\right]u$$ I assume you can now continue from here.