To estimate a reasonable sampling time for discretizing a DT controller designed for a system to achieve a closed-loop bandwidth of 10 Hz, we can use the phase margin rule and the rise time rule.
The phase margin rule states that the sampling time should be less than or equal to one-tenth of the dominant closed-loop pole's time constant. Assuming that the dominant closed-loop pole is at 10 Hz (based on the closed-loop bandwidth), the time constant is 0.1 seconds. Therefore, a reasonable sampling time would be less than or equal to 0.01 seconds.
The rise time rule states that the sampling time should be less than or equal to one-tenth of the system's rise time. Assuming that the system's rise time is less than or equal to 0.1 seconds (based on the closed-loop bandwidth), a reasonable sampling time would be less than or equal to 0.01 seconds.
Therefore, a reasonable sampling time to discretize such a controller would be less than or equal to 0.01 seconds, based on both the phase margin rule and the rise time rule.