Exponential regulation of the anti-collocatedly disturbed cage in a wave PDE-modeled ascending cable elevator

In cable elevators, large axial vibrations appear when a cage subject to disturbance is lifted up via a compliant cable. The axial vibration dynamics can be described by a wave partial differential equation (PDE) on a time-varying spatial interval with an unknown boundary disturbance. In this paper, we design an output feedback controller actuating at the boundary anti-collocated with the disturbance to regulate the state on the uncontrolled boundary of the wave PDE based on the backstepping idea and the active disturbance rejection control (ADRC) approach. The control law uses the state and disturbance information recovered from the state observer and the disturbance estimator, respectively, which are constructed via limited boundary measurements. The exponential convergence of the state on the uncontrolled boundary and uniform boundedness of all states in the closed-loop system are proved by Lyapunov analysis. Effective vibration suppression in the cable elevator with the designed controller is verified via numerical simulation.