In this work, we give sufficient conditions for the almost global asymptotic stability of a cascade in which the subsystems are only almost globally asymptotically stable. The result is extended to upper triangular systems of arbitrary size. In particular, if the unforced subsystems are almost globally asymptotically stable and their only chain recurrent points are hyperbolic equilibria, then the boundedness of forward trajectories is sufficient for the almost global asymptotic stability of the full upper triangular system. We show that unboundedness of such cascades is prohibited by growth rate conditions on the interconnection term and a Lyapunov function for the unforced outer subsystem, and the required structure for the chain recurrent set is enjoyed by classes of systems common in geometric control e.g. dissipative mechanical systems. Our results stand in contrast to prior works that require either time scale separation, prohibitively strong disturbance robustness properties, or global asymptotic stability in the subsystems.

翻译：本文给出了级联系统在子系统仅具有几乎全局渐近稳定性时实现几乎全局渐近稳定性的充分条件，并将结论推广至任意维数的上三角系统。特别地，若无外力子系统是几乎全局渐近稳定的且其唯一链回归点为双曲平衡点，则前向轨迹的有界性足以保证整个上三角系统具有几乎全局渐近稳定性。我们证明：通过互连项的增长率条件及无外力外部子系统的李雅普诺夫函数可排除此类级联系统的无界性，而几何控制中常见系统（如耗散力学系统）满足链回归集所需的结构特性。本研究区别于现有工作要求时间尺度分离、过强干扰鲁棒性特性或子系统全局渐近稳定的局限性。