Balanced Averaging of Bilinear Systems with Applications to Stochastic Control

Abstract

Abstract. We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics can be split into a fast and a slow subspace and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.

Optimal control of multiscale systems using reduced-order models

“…We therefore conclude that the optimal controlû for the Langevin equation (27) converges to the optimal control of the overdamped equation (32) as → 0. Moreover, Theorem 3 guarantees that the control value is asymptotically exact if we replaceû with the controlû = − √ 2∇ x V 0 in the multiscale dynamics (30).…”

“…Equation (32) is called the overdamped Langevin equation that is obtained from (27) by letting the inertial second-order term tend to zero [45]. We now derive an explicit asymptotic expression for the optimal feedback lawû t :=û …”

“…MOR aims at approximating the response of a controlled system to any given control input from a certain class, e.g., piecewise constant or square integrable functions; see, e.g., [25,4] and the references given there. A very popular MOR method is balanced truncation that gives easily computable error bounds in terms of the Hankel norm of the corresponding transfer functions [44,23], and which has recently been extended to deterministic and stochastic slow-fast systems, using averaging and homogenization techniques [29,26,27]. In applications MOR is often used to drastically reduce the system dimension, before a possibly computational expensive optimal control problem is solved.…”

Model Reduction of Bilinear System using Genetic Algorithm

“…Designing control for the bilinear system have been proposed by some authors [6,11,12], and model reduction method to obtain the reduced bilinear system have been published in literatures, such as, the balanced truncation [1,2,5,7,13], moment matching through Krylov subspaces [3,4,8,9,15,16]. The other methods can be found in [13].…”

“…The other methods can be found in [13]. The generalization of the balanced singular perturbation approximation for bilinear systems as the extension work of Liu [16] for the linear system has been published in [18].…”