通过 Carleman 线性化对多项式系统进行矩传播以实现概率安全分析
Moment Propagation of Polynomial Systems Through Carleman Linearization for Probabilistic Safety Analysis
AI总结：
本文提出基于截断 Carleman 线性化的矩传播方法，将随机多项式系统转化为有限维线性系统以高效近似矩并给出误差界，进而通过凸优化实现概率安全分析。
摘要：
我们提出了一种近似离散时间随机多项式系统矩的方法。该方法建立在带截断的 Carleman 线性化之上。具体而言，我们将一个具有有限多个状态的随机多项式系统变换为一个具有线性确定性动力学的无穷维系统，该系统精确描述了原始多项式系统矩的演化。随后，我们截断该确定性系统以获得一个有限维线性系统，并通过沿有限维线性动力学在时间上迭代传播矩来近似矩。我们为该传播方案提供了高效的在线计算方法，并给出了若干近似误差界。我们的结果还表明，当截断系统足够大时，可以在给定时间步获得某些矩的精确值。此外，我们研究了利用约化 Kronecker 幂来降低离线计算负载的技术。基于所获得的近似矩及其误差，我们还提供了在给定概率界下安全的超椭球区域。这些界使我们能够通过凸优化在线进行概率安全分析。我们在具有随机动力学的 logistic 映射和受随机扰动影响的车辆动力学上验证了我们的结果。
英文摘要：
We develop a method to approximate the moments of a discrete-time stochastic polynomial system. Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic polynomial system with finitely many states and transform it into an infinite-dimensional system with linear deterministic dynamics, which describe the exact evolution of the moments of the original polynomial system. We then truncate this deterministic system to obtain a finite-dimensional linear system, and use it for moment approximation by iteratively propagating the moments along the finite-dimensional linear dynamics across time. We provide efficient online computation methods for this propagation scheme with several error bounds for the approximation. Our results also show that precise values of certain moments at a given time step can be obtained when the truncated system is sufficiently large. Furthermore, we investigate techniques to reduce the offline computation load using reduced Kronecker power. Based on the obtained approximate moments and their errors, we also provide hyperellipsoidal regions that are safe for some given probability bound. Those bounds allow us to conduct probabilistic safety analysis online through convex optimization. We demonstrate our results on a logistic map with stochastic dynamics and a vehicle dynamics subject to stochastic disturbance.
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