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

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 J₂. 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.