Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

摘要

This paper deals with the tracking approach of a self-generated stable limit cycle for an underactuated mechanical system: The inertia wheel inverted pendulum (IWIP). Such system is subject to unilateral constraints limiting its swing motion. It is known that an interconnection and damping assignment passivity-based control (IDA-PBC) can be employed to control such pendulum to its upright position. In this work, we briefly show first that the IWIP can generate a stable period-1 limit cycle through a Hopf bifurcation by varying some gain parameter of the IDA-PBC. Thus, such self-generated limit cycle is used as a reference trajectory, which is chosen to be tracked by the IWIP. To achieve the tracking problem, a supplementary control input is added. Such tracking problem is reformulated as an asymptotic stabilization of the tracking error. Our fundamental approach hinges mainly on the use of the S-procedure to introduce the unilateral constraints, and the Schur complement and the matrix inversion lemma to transform bilinear matrix inequalities into linear matrix inequalities. Several simulations have been presented to corroborate the mathematical results and to show the efficiency of the proposed tracking scheme of the self-generated stable limit cycle of the controlled IWIP, even if it is subject to external disturbances, or in the presence of uncertainties in the friction parameters.