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

Shivam Dhama; Chetan D. Pahlajani

doi:10.3934/mcrf.2022018

Article Contents

2023, Volume 13, Issue 3: 852-872. Doi: 10.3934/mcrf.2022018

This issue Previous Article Next Article Second-order problems involving time-dependent subdifferential operators and application to control

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

Shivam Dhama^, and
Chetan D. Pahlajani

Discipline of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382055, India

^* Corresponding author: Shivam Dhama
^* Corresponding author: Shivam Dhama

Received: September 2021

Revised: February 2022

Early access: April 20, 2022

Published online: April 20, 2022

The second author acknowledges research support from DST SERB Project No. EMR/2015/000904.

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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 \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,\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.

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Mathematics Subject Classification: Primary: 60F17.

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Figure 1. Sample paths for the components $ X_1\triangleq X_1^{ \varepsilon,\delta}(t),\; X_2 \triangleq X_2^{ \varepsilon,\delta}(t) $ of the sde defined by (10) and $ S_1 \triangleq S_1^{ \varepsilon}(t) = x_1(t)+ \varepsilon Z_1(t),\; S_2 \triangleq S_2^{ \varepsilon}(t) = x_2(t)+ \varepsilon Z_2(t) $ with $ \varepsilon = 2^{-5},\; \delta = 2^{-4}, $ and $ T = 2^3 $. Here $ Z_1(t) $ and $ Z_2(t) $ are the components of $ Z(t) $ defined in (14)

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Figure 2. For $ 1 \le i \le 7 $, let $ e_i $ be the vector $ (e_{i,1},e_{i,2}) $, where, for $ j = 1,2 $, the quantity $ e_{i,j} $ is the mean of $ |X^{ \varepsilon,\delta}_j(T)-S^ \varepsilon_j(T)| $ over 1000 sample paths generated by the Euler-Maruyama method, with $ T = 2^3 $, $ \delta = 2^{-4} $. On a $ \log_2 $-$ \log_2 $ scale, we see that as $ \varepsilon\triangleq 2^{-i} $ decreases, the corresponding error $ e_{i,j} $ also decreases

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