Singularly perturbed k-contractive linear systems
Résumé
A dynamical system is said to be k-contractive when its trajectories contract k-dimensional volumes. For k=1, this property coincides with the classical notion of contraction. However, for k>1, it allows to characterize a much richer asymptotic behavior. The property of k-contraction has been introduced only recently, thus many analysis tools that are key in relevant applications are currently lacking for k-contractive systems. Motivated by this, we study k-contraction for singularly perturbed systems (SPSs), which naturally arise in many engineering applications. In particular, we focus on singularly perturbed linear time-invariant (LTI) systems. First we show that, for a ``sufficiently large'' time-scale separation, the k-contraction properties of a SPS can be derived from those of the associated boundary-layer (fast) system, and from the dimension of the reduced order (slow) model. Then, we focus on the case in which the reduced order (slow) model is k-contractive and the boundary-layer (fast) system is 1-contractive. In this setting, we provide a stronger result by showing that the overall system is k-contractive when the time-scale separation is ``large''.
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