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：本文给出了一类级联系统几乎全局渐近稳定性的充分条件，其中内环与无外力外环分别满足几乎全局渐近稳定性。我们的定性方法依赖于无外力外环非平衡点的链递归缺失、平衡点的双曲性以及前向轨道的预紧性。研究表明，所需链递归集的结构可便捷验证，并描述了两类具有该性质的重要系统。同时，我们证明预紧性要求可通过耦合子系统互连项的增长率条件进行验证。本研究结果与现有工作形成鲜明对比——后者要么要求子系统全局渐近稳定（对于在一般流形上演化的光滑系统而言，此条件无法实现），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验证中具有明确应用价值。