Adaptive Vibration Control of the Moving Cage in the 4 × $\times$ 4 Hyperbolic PDE-ODE Model of the Dual-Cable Mining Elevator

This paper proposes an adaptive output-feedback boundary control scheme to stabilize the vibrations of the moving cage in the dual-cable mining elevator system assuming the damping coefficients of the cage axial and roll motions are unknown. The mathematical formulation of the system in the Riemann coordinates is described by a $4\times 4$ hyperbolic partial differential equation (PDE) on a time-varying domain coupled with an ordinary differential equation (ODE) anti-collocated with the control input. At first, the nominal non-adaptive output feedback scheme is formulated by composing a state-feedback controller with the PDE state observer, utilizing the infinite-dimensional backstepping technique. Specifically, we apply two backstepping transformations to design the nominal state-feedback controller. This significantly facilitates the adaptive solutions of the backstepping kernel equations, when unknown parameters are replaced by their time-varying estimates. Then, a Lyapunov-based approach is followed to design the update laws for the unknown damping coefficients and to prove the closed-loop stability. It is shown that all states in the closed-loop system are uniformly bounded and the cage dynamics is asymptotically stable. A numerical simulation is presented to demonstrate the performance of the proposed controller.