G. Calafiore – Probabilistic and Randomized Methods for Design under Uncertainty

Автор: G. Calafiore
Название книги: Probabilistic and Randomized Methods for Design under Uncertainty
Формат: PDF
Жанр: Математика
Страницы: 451
Качество: Изначально компьютерное, E-book

Probabilistic and Randomized Methods for Design under Uncertainty is a collection of contributions from the world’s leading experts in a fast-emerging branch of control engineering and operations research. The book will be bought by university researchers and lecturers along with graduate students in control engineering and operational research.

A central issue in many engineering design endeavors is the presence of uncertainty
in the problem description. Different application fields employ different
characterizations of the uncertainty (for instance, deterministic unknown-butbounded
description vs stochastic description) and correspondingly adopt different
techniques to devise ‘designs’ that are in some way insensitive, or robust,
with respect to uncertainty.
The classes of problems considered in this book refer to uncertain control
systems, as well as to generic decision or optimization problems in which the
data are not exactly known.
In the area of robust control, the approach relying on a purely deterministic
unknown-but-bounded description of the uncertainty spawned researches that
yielded significant results in the last thirty years. However, this deterministic,
or worst-case, approach also showed some inherent limitations that can be
resumed in the fundamental tradeoff between computational complexity and
‘conservatism’.
From a more philosophical perspective, a worst-case design, even if it could
come at a cheap computational cost, may not be a desirable design in practice,
since it contemplates all possible uncertainty scenarios, including those that
are extremely unlikely to happen. On the other hand, one might think that
a worst-case approach is necessary in situations where even rare events may
lead to disastrous consequences. It should nevertheless be noticed that a design
based on an unknown-but-bounded description of the uncertainty may lead
to a ‘false’ belief of safety, since it provides no guarantees for uncertainty
outcomes that, for some unforeseen reason, happen to fall outside the a-priori
assumed uncertainty set. In this respect, an alternative approach based on a
stochastic description of the uncertainty makes it clear right from the outset
that, in reality, no statement can be given with absolute certainty.
In the field of optimization, the stochastic approach is indeed a classical
one, dating back to the late fifties with the work of Dantzig [96] on linear
programming under uncertainty and Charnes and Cooper [80] on chanceconstrained
optimization. In the robust control area, instead, the stochastic approach still enjoys limited attention, since this area has been dominated
lately by the worst-case deterministic viewpoint; early exceptions include
[192, 195, 342] and later [207, 290, 347, 402]. From an historical point
of view, we notice that the two areas of optimization and control followed
two different routes: the optimization area has always been dominated by
the stochastic paradigm, and only recently the works of El Ghaoui and Lebret
[121] and Ben-Tal and Nemirovski [35] brought the worst-case approach
into this area. Conversely, since the early eighties robust control has been
mainly based on the worst-case paradigm; see the seminal works of Zames [407]
on H∞ control, of Kharitonov [183] on parametric uncertainty, and the structured
singular value theory of Doyle [114] and Safonov [316]. Lately, however,
the probabilistic paradigm gained new interest among control researchers, see,
e.g., [24, 74, 252, 359, 381] and the many references therein.
Book Scope and Structure
This book brings together leading researchers from both the optimization and
control areas, with the intent of highlighting the interactions between the two
fields, and with a focus on randomized and probabilistic techniques for solving
design problems in the presence of stochastic uncertainty.
The book is divided into three parts. The first part presents three contributions
dealing with the general theory and solution methodologies for
probability-constrained and stochastic optimization problems and a contribution
on the theory of risk measures. The second part of the book contains
five chapters devoted to explicit robust design methods based on uncertainty
randomization and sampling. The first chapter of the third part of the book
presents a novel statistical learning theory framework for system identification,
whereas the other six chapters in this part focus on applications of randomized
methods for analysis and design of robust control systems.
Acknowledgments
We sincerely thank all the outstanding researchers and friends that made this
project possible with their contributions. The financial support of the Italian
Ministry of University and Research through an FIRB grant is gratefully
acknowledged.
This book goal will be attained if it will stimulate further exchange and
dialogue among scientists from the control and the optimization areas, and if
it will foster further interest and attract new researchers to these fields.