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

Abstract

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/\delta $ ($ 0 < \delta \II 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,\delta $, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as $ \varepsilon,\delta \searrow 0 $. The effective fluctuation process is found to vary, depending on whether $ \delta \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 $ \delta, \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.