This paper studies the asymptotic behavior of switched linear systems, beyond classical stability. We focus on systems having a low-dimensional asymptotic behavior, that is, systems whose trajectories converge to a common time-varying low-dimensional subspace. We introduce the concept of path-complete $p$-dominance for switched linear systems, which generalizes the approach of quadratic Lyapunov theory by replacing the contracting ellipsoids by families of quadratic cones whose contraction properties are dictated by an automaton. We show that path-complete $p$-dominant switched linear systems are exactly the ones that have a $p$-dimensional asymptotic behavior. Then, we describe an algorithm for the computation of the cones involved in the property of $p$-dominance. This allows us to provide an algorithmic framework for the analysis of switched linear systems with a low-dimensional asymptotic behavior.

本文研究了超越经典稳定性的切换线性系统的渐近行为。我们关注具有低维渐近行为的系统，即其轨迹收敛到公共时变低维子空间的系统。我们引入路径完备的概念$磷$- 切换线性系统的优势，它通过用二次锥族替换收缩椭球来概括二次李雅普诺夫理论的方法，二次锥的收缩特性由自动机决定。我们证明路径完整$磷$-占主导地位的切换线性系统正是那些具有 $磷$维渐近行为。然后，我们描述了一种计算涉及性质的锥体的算法$磷$-支配地位。这使我们能够为分析具有低维渐近行为的切换线性系统提供算法框架。