Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment

TLDR

IDA‑PBC’s Hamiltonian formulation provides additional degrees of freedom for solving the required partial differential equations. The paper applies IDA‑PBC to stabilize underactuated mechanical systems and characterizes a class of systems for which it yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The approach modifies both potential and kinetic energies and defines the class through solvability of specific partial differential equations. The authors show that the original controlled‑Lagrangians method is a special case of their framework and demonstrate asymptotically stabilizing controllers for the ball‑and‑beam system and a novel inertia‑wheel pendulum.

Abstract

We consider the application of a formulation of passivity-based control (PBC), known as interconnection and damping assignment (IDA) to the problem of stabilization of underactuated mechanical systems, which requires the modification of both the potential and the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The class is given in terms of solvability of certain partial differential equations. One important feature of IDA-PBC, stemming from its Hamiltonian formulation, is that it provides new degrees of freedom for the solution of these equations. Using this additional freedom, we are able to show that the method of "controlled Lagrangians"-in its original formulation-may be viewed as a special case of our approach. As illustrations we design asymptotically stabilizing IDA-PBCs for the classical ball and beam system and a novel inertia wheel pendulum.