Discretization of a continuous state-space model

I implemented the mathematical model of the inverted pendulum in Simulink following the paper http://www.uta.edu/utari/acs/ee4314/lectures/Lecture%207b.pdf (last page). My task now is to discretize such model and design a discrete controller in Simulink (discrete PID).

From the theory I know that the sampling time of the system must satisfy the Nyquist sampling theorem.

How can I estimate (theoretically or experimentally) the highest frequency of the continuous time model in order to choose a proper sampling time?

Thanks

Unless you've specifically been asked to design the controller in discrete time, then the steps you'd follow would typically be,

1. design a continuous time linear controller for the continuous time linear model, ensuring that it works well with the continuous time non-linear model (using simulation).
2. discretize the continuous time controller.

In practice the sample rate for the discretization would depend on various things including the sample rate available/used in a real-time processor for the real-time implementation, as well as the cross-over frequency of the closed loop system.

In your case, as a first cut, you should pick a sample rate so that the Bode response of the discrete time controller matches the Bode response of the continuous time controller up to a frequency that is at least an order of magnitude higher than the bandwidth of the closed-loop.