论文标题
线性控制的动力系统的近似，随机噪声较小，快速元素采样
Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling
论文作者
论文摘要
在本文中，我们研究了一个线性控制系统的动力学，具有给定的状态反馈控制法，在时间频率下以$ 1/δ$（$ 0 <δ\ ll 1 $）进行快速周期性采样，以及大小$ \ varepsilon $（$ 0 <\ varepsilon \ ll 1 $）的小白噪声扰动在状态动力学中。对于随之而来的连续时间随机过程，由两个小参数$ \ varepsilon（δ$）索引，我们获得了有效的普通和随机微分方程，描述了平均行为和关于平均值的典型波动，以限制为$ \ varepsilon，δ\ searrow 0 $。发现有效的波动过程会有所不同，具体取决于$δ\ searrow 0 $ 0 $比/以/较慢的速度快于$ \ varepsilon \ searrow 0 $。发现最有趣的案例是$δ，\ varepsilon $的大小相当的情况。在这里，波动的限制随机微分方程既具有扩散项，这是由于噪声较小和有效的漂移项，从而捕获了快速采样的累积效应。在此制度中，我们的结果产生了一个固定的马尔可夫过程，该过程提供了原始非马克维亚过程的强（路线）近似，以及对随之而来的误差的估计。一个简单的示例涉及无限的时间范围线性二次调节问题说明了结果。
In this paper, we study the dynamics of a linear control system with given state feedback control law in the presence of fast periodic sampling at temporal frequency $1/δ$ ($0 < δ\ll 1$), together with small white noise perturbations of size $\varepsilon$ ($0<\varepsilon \ll 1$) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters $\varepsilon,δ$, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as $\varepsilon,δ\searrow 0$. The effective fluctuation process is found to vary, depending on whether $δ\searrow 0$ faster than/at the same rate as/slower than $\varepsilon \searrow 0$. The most interesting case is found to be the one where $δ,\varepsilon$ are comparable in size; here, the limiting stochastic differential equation for the fluctuations has both a diffusive term due to the small noise and an effective drift term which captures the cumulative effect of the fast sampling. In this regime, our results yield a time-inhomogeneous Markov process which provides a strong (pathwise) approximation of the original non-Markovian process, together with estimates on the ensuing error. A simple example involving an infinite time horizon linear quadratic regulation problem illustrates the results.
