On Stability of Homogeneous Systems in Presence of Parasitic Dynamics

Usually, singularly perturbed models are used to justify the decomposition of the interconnected systems into the Main Dynamics (MD) and the Parasitic Dynamics (PD). In this paper, the effect of a homogeneous PD on the stability of a homogeneous MD, when Homogeneity Degrees (HD) are possibly different, is studied via ISS approach in the framework of singular perturbations. Thus, the possibilities to reduce the order of the interconnected system considering only Reduced-Order Dynamics (ROD) and neglecting PD are examined. Proposed analysis discovers three kinds of stability in the behavior of such an interconnection by assuming that both, ROD and unforced PD, are Globally Asymptotically Stable (GAS). In the first case, when the HD of both systems coincide and the Singular Perturbation Parameter (SPP) is small enough, GAS of the interconnection can be concluded. In the second case, when the HD of ROD is greater, only local stability of interconnection can be ensured. Moreover, proposed approach allows to estimate the domain of attraction for the trajectories of the interconnection as a function of the SPP. In the third case, when the HD of ROD is smaller than the HD of PD, only practical stability can be concluded, and a kind of chattering phenomenon can arise when the HD of ROD is negative. Furthermore, the asymptotic bound of the system's trajectories is also estimated in terms of the SPP