Design of a Port-Hamiltonian Control for an Alt-Azimuth Liquid–Mirror Telescope

Abstract

In this work, we design a control strategy to be applied in a port-Hamilton representation of a liquid-mirror telescope for an alt-azimuth configuration. Starting from a dynamical model for an alt-azimuth liquid-mirror telescope based on Lagrange mechanics, a transformation to the port-Hamilton form is made. Such a dynamical model is obtained by computing the kinetic and potential energy of the telescope and substituting them in the Euler–Lagrange equation of motion. Then, for the transformation to the port-Hamiltonian form, we obtain the relation between the Hamiltonian and the Lagrangian. The resulting open-loop model based on the Hamiltonian function is controlled using an extension of the interconnection and damping-assignment passivity-based control aiming for a robust and accurate steady behavior in the closed loop while tracking a star’s position. For comparison purposes, two different control strategies are applied to the Lagrangian model, inverse-dynamics control and sliding mode super-twisting control. Since the light is collected by the principal mirror of the telescope while tracking a star, we make a description of the liquid mirror’s behavior. The tracking star’s position is described as a function of the observer’s position and the star’s coordinates as well as the date of observation. The simulations’ results show that the port-Hamilton control has a good transitory and steady response as well as great accuracy competing with that of inverse-dynamics control but with greater robustness and no chattering drawback.