Simple approximations of semialgebraic sets and their applications to control

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F17%3A00311735" target="_blank" >RIV/68407700:21230/17:00311735 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.1016/j.automatica.2016.11.021" target="_blank" >http://dx.doi.org/10.1016/j.automatica.2016.11.021</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.automatica.2016.11.021" target="_blank" >10.1016/j.automatica.2016.11.021</a>

Result language

angličtina

Original language name

Simple approximations of semialgebraic sets and their applications to control

Original language description

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the Schur and Hurwitz stability domains. These sets often have very complicated shapes (nonconvex, and even non-connected), which render difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrectangles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly nonconvex yet still simple approximations, based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encountered in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of convex linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Finally, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

20205 - Automation and control systems

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2017

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Automatica

ISSN

0005-1098

e-ISSN

1873-2836

Volume of the periodical

78

Issue of the periodical within the volume

April

Country of publishing house

GB - UNITED KINGDOM

Number of pages

9

Pages from-to

110-118

UT code for WoS article

000398010500013

EID of the result in the Scopus database

2-s2.0-85010460383