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. The result is extended inductively to upper triangular systems with an arbitrary number of subsystems. 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：本文给出了级联系统达到几乎全局渐近稳定性的充分条件，其中内环与无外力外环各自均为几乎全局渐近稳定。我们的定性分析方法基于以下三个关键要素：无外力外环非平衡点处链循环的缺失、平衡点的双曲性以及正向轨迹的预紧性。该结果可归纳递推至包含任意数量子系统的上三角系统。我们证明了链循环集所需结构的可验证性，并描述了具备该特性的两类重要系统。同时指出，通过子系统间耦合项的增长速率条件可验证预紧性要求。本研究的创新之处在于突破了以往研究的局限——既有工作通常要求子系统全局渐近稳定（对于一般流形上的光滑系统而言不可实现）、子系统间存在时间尺度分离，或外环具备强扰动鲁棒性。该方法在流形系统级联控制器的稳定性认证中具有明确应用前景。