In this work, we give sufficient conditions for the almost global asymptotic stability of a cascade in which the inner loop and the unforced outer loop are each almost globally asymptotically stable. Our qualitative approach relies on the absence of chain recurrence for non-equilibrium points of the unforced outer loop, the hyperbolicity of equilibria, and the precompactness of forward trajectories. We show that the required structure of the chain recurrent set can be readily verified, and describe two important classes of systems with this property. We also show that the precompactness requirement can be verified by growth rate conditions on the interconnection term coupling the subsystems. Our results stand in contrast to prior works that require either global asymptotic stability of the subsystems (impossible for smooth systems evolving on general manifolds), time scale separation between the subsystems, or strong disturbance robustness properties of the outer loop. The approach has clear applications in stability certification of cascaded controllers for systems evolving on manifolds.

翻译：摘要：本文给出了内环与无驱外环分别满足几乎全局渐近稳定性的级联系统实现几乎全局渐近稳定性的充分条件。我们的定性方法依赖于以下三点：无驱外环非平衡点的链回归性缺失、平衡点的双曲性，以及前向轨道的预紧性。研究表明，链回归集所需结构易于验证，并描述了两类具有该性质的重要系统。同时证明，通过子系统间互联项的增长率条件即可验证预紧性要求。本研究与现有工作的区别在于：现有工作要么要求子系统具有全局渐近稳定性（这在一般流形上演化的光滑系统中不可能实现），要么要求子系统间存在时间尺度分离，或要求外环具备强扰动鲁棒性。该方法在流形上演化系统的级联控制器稳定性认证中具有明确的应用价值。