On infinitesimal contraction analysis for hybrid systems
On infinitesimal contraction analysis for hybrid systems
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DOI：
10.1109/cdc51059.2022.9992825
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发表时间：
2018-11
期刊：
影响因子：
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通讯作者：
Samuel A. Burden;S. Coogan
中科院分区：
文献类型：
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作者：
Samuel A. Burden;S. Coogan
无穷小压缩分析是一种强大的分析工具，它的全局收敛结果是由局部动力学性质得到的。在这篇文章中，我们将无穷小收缩分析推广到混杂系统，其中状态依赖的守卫触发由模式之间的重置映射定义的跃迁，这些模式可能具有不同的范数并且不需要具有相同的维度。与现有文献不同的是，我们不限制模式序列或驻留时间，我们工作的环境中，混杂系统流几乎处处可微，其导数是跳跃-线性-时变微分方程解，其跳跃由由保护、重置映射和矢量场确定的跃迁矩阵定义。我们的主要结果表明，如果矢量场是无限收缩的，如果跃迁矩阵是非扩张的，那么任意两条轨迹之间的内禀距离随时间呈指数减小。当驻留时间的界限可用时，我们的方法产生关于轨迹之间的内在距离的界限，而不管动力学是扩张的还是收缩的。我们用两个例子来说明我们的结果：一个受约束的机械系统和一个带有理想二极管的电路。
Infinitesimal contraction analysis, wherein global convergence results are obtained from properties of local dynamics, is a powerful analysis tool. In this paper, we generalize infinitesimal contraction analysis to hybrid systems in which state-dependent guards trigger transitions defined by reset maps between modes that may have different norms and need not be of the same dimension. In contrast to existing literature, we do not restrict mode sequence or dwell time.We work in settings where the hybrid system flow is differentiable almost everywhere and its derivative is the solution to a jump-linear-time-varying differential equation whose jumps are defined by a saltation matrix determined from the guard, reset map, and vector field. Our main result shows that if the vector field is infinitesimally contracting, and if the saltation matrix is non-expansive, then the intrinsic distance between any two trajectories decreases exponentially in time. When bounds on dwell time are available, our approach yields a bound on the intrinsic distance between trajectories regardless of whether the dynamics are expansive or contractive. We illustrate our results using two examples: a constrained mechanical system and an electrical circuit with an ideal diode.