On the Equivalence of Extended and Oblique Projected Dynamics with Applications to Hybrid Integrator-Gain Systems

The class of projected dynamical systems (PDS) provides a powerful framework for modeling dynamical systems of which the trajectories are constrained to a set by means of projection. This work is concerned with establishing equivalence results among two recent variations of PDS. These are (i) extended PDS (ePDS), which enable partial projection of dynamics, and (ii) oblique PDS, (oPDS) where projections can be done with respect to non-Euclidean norms. We present two sets of sufficient conditions for equivalence among these two system classes. These results enable the transfer of system theoretical properties and tools from one class to the other, which we illustrate in this paper. As an application, we study hybrid integrator-gain systems (HIGS), which are recently introduced hybrid control elements aiming at overcoming fundamental limitations of linear time-invariant control, and are formally described in the ePDS framework. We use our results to also describe these control elements as oPDS, thereby enabling the study of HIGS-controlled systems in this framework.