Adaptive Control of Hyperbolic PDEs Coupled with a Disturbed and Highly Uncertain ODE

We address adaptive output-feedback boundary control of coupled hyperbolic PDEs with spatially-varying coefficients and on a time-varying domain, of which the uncontrolled boundary is coupled with an uncertain and disturbed ODE. The parameters in the state matrix of this ODE and the amplitudes of the external disturbances are unknown. In our control design,the control gains can be self-turned on line to adjust the unknown state matrix of the ODE into a given target state matrix in the closed-loop system and attenuate the unmatched disturbance. The asymptotic convergence to zero of the ODE state and the boundedness of the PDE states are ensured. This study is motivated by lateral vibration suppression of a mining cable elevator, where the interaction dynamics between the cage and the flexible guide is approximate as a viscoelastic system including spring and damping with unknown stiffness and damping coefficients. The performance of the proposed controller is tested in the application of the mining cable elevator by numerical simulation.