Passivity of Linear Singularly Perturbed Systems

The passivity of singularly perturbed systems (SPSs) is generally studied without taking advantage of the time-scale separation present in this class of systems. To fill this gap, the objective of this letter is to provide easy-to-verify well-posed conditions characterizing the passivity of a perturbation variable-dependent SPS starting from the passivity of its associated reduced-order system. To achieve this goal, we rely on the connection between positive realness and passivity, as well as the notion of phase for multi-input multi-output (MIMO) systems. We use a benchmark DC motor to illustrate that classical reasoning used for stability analysis of SPSs, which is based on the stability of the reduced-order (slow) and boundary layer (fast) subsystems, cannot be applied to guarantee the passivity of an SPS. On top of that, our methodology explains how the time-scale separation can be used to analyze the passivity of general linear time-invariant (LTI) systems. The approach is illustrated on a numerical example.