论文标题
连续时间混合符号酮系统的紧密分解功能与干扰
Tight Decomposition Functions for Continuous-Time Mixed-Monotone Systems with Disturbances
论文作者
论文摘要
混合单酮系统的矢量场可以通过分解函数分解成增加（合作）和减少（竞争性）组件，并且该分解允许例如，有效地计算可触及集合和正向不变的集合。但是，这种方法的主要挑战是确定适当的分解函数。在这项工作中，我们表明，任何具有Lipschitz连续矢量场的连续时间动力系统都是混合 - 单调的，并且在与混合酮系统的标准工具一起使用时，我们为分解函数提供了分解函数的结构。我们的构建类似于Yang和Ozay最近提出的用于计算离散时间系统的分解函数[1]的构建[1]，我们为连续时间设置进行了适当的修改，并且还通过未知的干扰输入扩展到了案例。与[1]一样，我们的分解函数构建需要解决状态空间中每个点的优化问题。但是，我们通过示例演示如何有时以封闭形式计算紧密分解功能。作为第二个贡献，我们展示了如何通过考虑向后的动力学来通过混合单调性属性有效地计算可触及集的不足。
The vector field of a mixed-monotone system is decomposable via a decomposition function into increasing (cooperative) and decreasing (competitive) components, and this decomposition allows for, e.g., efficient computation of reachable sets and forward invariant sets. A main challenge in this approach, however, is identifying an appropriate decomposition function. In this work, we show that any continuous-time dynamical system with a Lipschitz continuous vector field is mixed-monotone, and we provide a construction for the decomposition function that yields the tightest approximation of reachable sets when used with the standard tools for mixed-monotone systems. Our construction is similar to that recently proposed by Yang and Ozay for computing decomposition functions of discrete-time systems [1] where we make appropriate modifications for the continuous-time setting and also extend to the case with unknown disturbance inputs. As in [1], our decomposition function construction requires solving an optimization problem for each point in the state-space; however, we demonstrate through example how tight decomposition functions can sometimes be calculated in closed form. As a second contribution, we show how under-approximations of reachable sets can be efficiently computed via the mixed-monotonicity property by considering the backward-time dynamics.
