On the output regulation problem: The generalized second-order underactuated linear system case

In this work, we aimed to solve the control regulation problem for a generalized second-order underactuated linear system in order to induce a periodic or chaotic behavior or to cancel the external perturbations, generated by an exogenous system, in the nonactuated coordinate. Further, we showed that, in some cases, it is possible to bring to zero the regulation output errors of the underactuated linear plant, depending on the structure of the plant itself and the exogenous system. In the first stage, the solution was developed for the ideal scenario, in which the whole states of the plant and of the exogenous system were available. Secondly, we showed that in some cases it was possible to solve the regulation output problem when only the observable plant output was measurable. That is, the whole plant state and the exogenous signal could be recovered, if some assumptions were fulfilled. The Lyapunov method was used to perform the stability analysis.The proposed solution was assessed through numerical simulations.