计算混合单调系统的鲁棒前向不变集
Computing Robustly Forward Invariant Sets for Mixed-Monotone Systems
摘要：
本文提出了一种新的工具来研究连续时间混合单调动力系统在扰动输入下的可达性和集不变性。混合单调系统的向量场可以通过分解函数分解为递增和递减分量，并且这种分解能够将原始动力学嵌入到更高维的嵌入系统中。当原系统受到未知干扰输入时，嵌入系统没有干扰，其轨迹为原动态的有限时间可达集提供了边界。我们的主要贡献是展示如何可以有效地确定鲁棒的前向不变和吸引集的混合单调系统，通过研究这个嵌入系统的某些平衡。我们还展示了这种方法，当应用到向后的时间动态，建立不同的鲁棒性向前不变集的原始动态。最后，我们提出了一个独立的结果计算分解函数的系统多项式动力学。这些工具和结果证明通过几个例子和案例研究。
英文摘要：
This work presents new tools for studying reachability and set invariance for continuous-time mixed-monotone dynamical systems subject to a disturbance input. The vector field of a mixed-monotone system is decomposable via a decomposition function into increasing and decreasing components, and this decomposition enables embedding the original dynamics in a higher-dimensional embedding system. While the original system is subject to an unknown disturbance input, the embedding system has no disturbances and its trajectories provide bounds for finite-time reachable sets of the original dynamics. Our main contribution is to show how one can efficiently identify robustly forward invariant and attractive sets for mixed-monotone systems by studying certain equilibria of this embedding system. We show also how this approach, when applied to the backward-time dynamics, establishes different robustly forward invariant sets for the original dynamics. Lastly, we present an independent result for computing decomposition functions for systems with polynomial dynamics. These tools and results are demonstrated through several examples and a case study.
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