What makes the control of discontinuous dynamical systems so complex?

The control of complex engineering systems requires combining multi-disciplinary methods. The spirit of this work is to bring together dynamical-systems analysis tools and control-engineering methods. The aim is to identify safe operating conditions and propose control methodologies to suppress non-desired phenomena in an industrial application with discontinuous elements: a conventional vertical oilwell drillstring. Due to the discontinuous bit-rock friction in the drillstring, complex phenomena are present, giving rise to different types of harmful oscillations. Torsional mechanical vibrations are studied, in particular, self-excited stick-slip oscillations at the bottom-hole assembly and other bit sticking phenomena which make the bit remain motionless. The complexity of drilling mechanisms and practices makes difficult the use of automated-type controllers. Consequently, the control system must be interpreted as an off-line safe-parameter selection method which can guide the driller in order to design the well drilling profile. From the dynamical viewpoint, the drillstring is a discontinuous or switched dynamical system and exhibits different discontinuous-induced bifurcations. The analysis of dynamical transitions and discontinuity-induced bifurcations is carried out in order to ensure good performance of the controlled system. The sticking bit phenomena are related to the existence of a sliding motion on the discontinuity surface when the bit velocity is zero. Some control solutions and parameters selection methods are presented in order to avoid non-desired transitions. Although the model used is a simplified one of two degrees of freedom, the analysis carried out can be successfully applied to multi-degree-of-freedom mechanical systems exhibiting stick-slip oscillations and dry friction.