Generalization of the IDA-PBC method for stabilization of mechanical systems

Abstract

We generalize and strengthen the method of Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) for the stabilization of mechanical systems. First, we replace the skew symmetry property of the interconnection matrix with the energy conservation property, and introduce a gyroscopic force replacing the interconnection sub-matrix that is usually denoted by J 2 . Second, we derive a new set of matching conditions where the new kinetic matching conditions are simpler than those in the literature. Third, we provide a necessary and sufficient condition for Lyapunov/exponential stabilizability by IDA-PBC for the class of all linear mechanical systems. Last, we give a necessary and sufficient condition for Lyapunov/exponential stabilizability by IDA-PBC for the class of all mechanical systems with one degree of underactuation. These conditions are easy to verify without solving any PDE's. Our results comprehend and extend most results on IDA-PBC in the literature.