基于平方和规划的轨迹不变漏斗
Invariant Funnels around Trajectories using Sum-of-Squares Programming
摘要：
本文给出了计算多项式微分方程解周围有限时间不变域（漏斗）的数值方法。首先，我们提出了一种方法，尽管依赖于从数值积分的近似轨迹，但该方法精确地证明了不变性的充分条件。我们的第二种方法通过及时采样放松了第一种方法的约束。在应用程序中，这可以恢复几乎相同的漏斗，但计算速度要快得多。在这两种情况下，漏斗验证使用平方和编程搜索一个家庭的时变多项式李雅普诺夫函数。初始候选李雅普诺夫函数构造使用的线性化的轨迹，以及相关的时变李雅普诺夫和里卡蒂微分方程。在卫星六状态模型的稳定轨道上对这两种方法进行了比较。
英文摘要：
This paper presents numerical methods for computing regions of finite-time invariance (funnels) around solutions of polynomial differential equations. First, we present a method which exactly certifies sufficient conditions for invariance despite relying on approximate trajectories from numerical integration. Our second method relaxes the constraints of the first by sampling in time. In applications, this can recover almost identical funnels but is much faster to compute. In both cases, funnels are verified using Sum-of-Squares programming to search over a family of time-varying polynomial Lyapunov functions. Initial candidate Lyapunov functions are constructed using the linearization about the trajectory, and associated time-varying Lyapunov and Riccati differential equations. The methods are compared on stabilized trajectories of a six-state model of a satellite.
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