What is the effect of aliasing in discrete time controller?
A discrete time controller inherently depends on sample time T, therefore, also the stability margin. There is a condition which is related to Shannon's theorom. That is the so called strip condition. Your maximal possible (allowed) sampletime is calculated by $$T \leq \frac{\pi}{\omega_0} = T_{\max},$$ which is in your case $$T \leq \pi,$$ since $$\omega_0 = 1.$$ If you go with T beyond this limit you have the effect of aliasing. This is due to periodicity, then, the z-transform does not map uniquely the strip onto the complex z-plane, hence, information get lost. A possible effect that might happen to your output after violating the strip condition is that oscilations, so called hidden oscilations, arrise. The controller can not intervene if sample time is too high. 

I recommend the book from Katsuhiko Ogata "Discrete-time control systems", and there chapter 3 respective aliasing and further artefacts. Regarding strips, see chapter 4 "Mapping between the s plane and the z plane".