Playing with the regulation zeros in the stabilization of a double inverted pendulum

State-space methods are well adapted for solving stabilization problems. In the linear multivariable case, when the poles are placed, some degrees of freedom are left which can be used to place some zeros in the transfer functions from the perturbations to the outputs, called regulation zeros, to reject some perturbations. It is therefore interesting to use a suitable representation to compute the closed-loop transfer functions from the perturbations to the outputs. The YoulaJabr-Bongiorno parametrization of the controller turns out to be very appropriate for this task. We have applied this method to the stabilization of a double inverted pendulum, fixed on a carriage moving on an horizontal bench. We have obtained a minimal controller which stabilizes the system and rejects asymptotically on the position of the carriage some perturbations, for example: measurements noises on the angle, the slope of the bench. All the results we have obtained on a full-size realization, (which can be seen at the permanent exhibition of the Cite des Sciences et de l'lndustrie de La Villette), show that the behavior of the system closely depends on the choice of the regulation zeros.