k-contraction analysis for discrete-time systems
Résumé
The definition of k-contraction promises a useful generalization of the classical notion of
contraction for dynamical systems. However, most of the k-contraction literature focuses on
continuous-time systems. In this work, we derive conditions for k-contractivity of discrete-
time dynamics. Our first result follows traditional lines for k-contraction analysis, and provides
Lyapunov-like sufficient conditions based on matrix compounds and state-dependent metrics.
However, our subsequent results avoid the complexities related to matrix compounds. Inspired by
recent findings in the context of k-contraction for continuous-time systems, we provide conditions
on the system’s dynamics that rely on generalized Lyapunov inequalities and quadratic cone
fields. The proposed conditions are also shown to be necessary for linear time-invariant systems.
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