On the transient behavior and gain design for a class of second order homogeneous systems

In this note, we consider a class of homogeneous second order systems which contains two relevant sub-classes: the first one in a controller form and the second one in an observer form. For both of them, we provide necessary and sufficient conditions on the gain parameters to guarantee oscillatory or non-oscillatory behavior of the solutions. This is in the same spirit of tuning of second order linear systems, where the gains can be designed to obtain an overdamped, critically damped or underdamped behavior. The analysis of all the cases is performed in a unified framework based on a search for invariant sets which are analogous to the invariant eigenspaces of linear systems.