Ineluctability of oscillations in systems with digital implementation of derivative feedback

Differentiation in feedback is common practice in digital control. Yet, the fundamental behavior of the universally employed backward difference of quantized signals has not been studied thus far. We show that velocity always oscillates when this type of feedback is applied to a forced, linear second-order system for any system parameter. We then compute a bound for the oscillation amplitude, which can be easily computed given the parameters of the system. Experimental results are in close agreement with the theory. If the system has dry friction, our study yields a sufficient condition for the quenching of spontaneous oscillations.