Approximation of linear controlled dynamical systems with small random noise and fast periodic sampling

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ 1/\delta $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ 0 < \delta \ll 1 $\end{document}</tex-math></inline-formula>), together with small white noise perturbations of size <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M4">\begin{document}$ 0< \varepsilon \ll 1 $\end{document}</tex-math></inline-formula>) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon,\delta $\end{document}</tex-math></inline-formula>, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as <inline-formula><tex-math id="M6">\begin{document}$ \varepsilon,\delta \searrow 0 $\end{document}</tex-math></inline-formula>. The effective fluctuation process is found to vary, depending on whether <inline-formula><tex-math id="M7">\begin{document}$ \delta \searrow 0 $\end{document}</tex-math></inline-formula> faster than/at the same rate as/slower than <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon \searrow 0 $\end{document}</tex-math></inline-formula>. The most interesting case is found to be the one where <inline-formula><tex-math id="M9">\begin{document}$ \delta, \varepsilon $\end{document}</tex-math></inline-formula> 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.</p>