Maximum Sampling Period of a Second Oder Control System When controlling a second-order system, the sampling period is suggested to be at least one tenth of the rise time of the step-response of the system (See "Real-Time Systems" By H.Kopetz, 2nd Edition, Section 1.3.1. This reference does not provide a derivation.) Below I am quoting what this book say about the subject matter:

A rule of thumb says that, in a digital system which is expected to behave like a quasi-continuous system, the sampling period should be less than one-tenth of the rise time $d^\text{rise}$ of the step response function of the controlled object

I am interested on how to derive this bound. I also checked the references in the Bibliographic notes of Chapter 1 of the cited book.
Per Nyquist theorem: "a signal must be sampled at more than twice the highest frequency component of the signal". This suggests that the upper bound on the sampling period should be less than half the period of the original signal. So where does the "one tenth of $d^\text{rise}$" part come from?
One more question is how to derive a similar bound for a nonlinear control system?
 A: Unfortunately, there is no real resource that will give you a proof for that. This is more like a rule of thumb that people empirically found and passed on to the next generation. The idea is to have enough periods before the system settles, but not too much for various reasons such as computational power, noise, etc.
You cite the Nyquist theorem, but this is a signal processing result talking about the possibility of building back an analog signal from its digital counterpart. In control, you do not really care much about that. All you need is to have enough information to be able to control/stabilize the system and have enough bandwidth for the closed-loop system. Moreover, signals in control are not bandlimited, unless you filter them beforehand, of course.
For nonlinear systems, this is even worse, because there is no such thing as rise time as it will depend on the initial conditions and the input you apply to the system. So, even if you assume zero initial conditions, then this will not help. So, this is up to the control engineer to decide what a good sampling period would be. In general, this will be dictated by how fast you want your system to be and what actuators, sensors and controllers/processors you have.
