Saturation in speed and speed-position

All control systems are subject to constraints on actuators, in the form of saturations, whether position or speed constraints. In the first case, these are stops (control surface stops, cylinder stroke limits, etc.), in the second case, hydraulic flow or torque limitations...

Even if these limitations only occur more or less occasionally, under disturbances or when particularly strong setpoints are applied, they are unavoidable in practice, as it is out of the question, for reasons of cost, power consumption, weight, etc., to dimension actuators in such a way as to be sure of avoiding them in all cases.

While the linear dynamics of actuators are generally taken into account when designing a control system, their non-linear behavior, linked to these saturations, is much less so, especially in the case of velocity saturations, and even more so when velocity saturation and position saturation occur simultaneously.

However, these limitations can have very significant repercussions on the performance of control systems, ranging from simple degradation, with significant overshoots, as in the phenomenon of integrator saturation (wind-up), to instability, via the generation of high-amplitude limit cycles.

We propose here a simple method (in the single-input case) for analyzing the behavior of a system subjected to velocity saturations, whether or not coupled to position saturations, and in particular for predicting the existence of limit cycles or instabilities. The method, based on the dynamic equivalent gain technique, is typically an engineering method, and is suitable for both synthesis and analysis.