Randomized Methods for Control of Uncertain Systems

Abstract

In this article, we study the tools and methodologies for the analysis and design of control systems in the presence of random uncertainty. For analysis, the methods are largely based on the Monte Carlo simulation approach, while for design new randomized algorithms have been developed. These methods have been successfully employed in various application areas, which include systems biology; aerospace control; control of hard disk drives; high-speed networks; quantized, embedded, and electric circuits; structural design; and automotive and driver assistance. PreliminariesRandomized methods for control deal with the design of uncertain and complex systems. They have been originally developed for linear systems affected by structured uncertainty, usually expressed in the so-called M configuration. A similar approach may be followed when dealing with uncertainty in other contexts, such as uncertainty in the environment (random disturbances) or even when there is no uncertainty in the problem formulation, but the complexity of the problem is such that randomized methods may be the best approach, since these methods are known to break the curse of dimensionality, see Tempo et al. (2013) for details.For the sake of simplicity, we consider here an uncertain plant transfer function P .s; q/ affected by parametric uncertainty q D OEq 1 : : : q` T bounded in a set Q R`. The objective is to design the parameters  2 R n of a controller transfer function C.s; Â/ so to guarantee robustly some desired performance. This is reformulated as the problem of finding a design satisfying some uncertain constraints of the form f .Â; q/ Ä for all q 2 Q:In other words, the goal is to design a robust controller which satisfies the uncertain constraints. Specific examples of these constraints include an H 1 or H 2 norm bound on the closed-loop sensitivity function or time-domain specifications.

An adaptive Monte Carlo test for average performance of uncertain systems

“…The following algorithm describes the classical Monte Carlo Estimation Algorithm [9,10,11,23] for risk evaluation.…”