Balancing and suppression of oscillations of tension and cage in dual-cable mining elevators

Dual-cable mining elevator has advantages in the transportation of heavy load to a large depth over the single cable elevator. However challenges occur when lifting a cage via two parallel compliant cables, such as tension oscillation inconformity between two cables and the cage roll, which are important physical variables relating to the fatigue fracture of mining cables. Mining elevator vibration dynamics are modeled by two pairs of 2 × 2 heterodirectional coupled hyperbolic PDEs on a time-varying domain and all four PDE bottom boundaries are coupled at one ODE. We design an output feedback boundary control law via backstepping to exponentially stabilize the dynamic system including the tension oscillation states, tension oscillation error states and the cage roll states. The control law is constructed with the estimated states from the observer formed by available boundary measurements. The exponential stability of the closed-loop system is proved via Lyapunov analysis. Effective suppression of tension oscillations, reduction of inconformity between tension oscillations in two cables, and balancing the cage roll under the proposed controller are verified via numerical simulation.