Stability analysis of oilwell drilling torsional vibrations

Compared to the known waves and transport PDEs, boundary conditions in drilling aren't trivial and depend on the system parameters. Further, nonlinearities are present due to bit-rock interaction. In this paper, torsional vibrations stabilization resulting from the bit stick-slip behavior are investigated. The second-order vibration dynamic is transformed into 2 × 2 first-order PDEs. The boundary stabilizing control law is constructed such that the transformed system tracks a designed stable target system. The backstepping techniques combined with kernel equations and the Lyapunov theory are used to prove the local exponential stability of the new system variables, consequently, the vibration suppression due to torsional dynamics. Simulations are presented to illustrate the effectiveness of the control laws.