A Statistical Learning Theory Approach for Uncertain Linear and Bilinear Matrix Inequalities
Abstract
In this paper, we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities, which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results in statistical learning theory, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chervonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of these problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality. The effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters.
Cite
@article{arxiv.1305.4952,
title = {A Statistical Learning Theory Approach for Uncertain Linear and Bilinear Matrix Inequalities},
author = {Mohammadreza Chamanbaz and Fabrizio Dabbene and Roberto Tempo and Venkatakrishnan Venkataramanan and Qing-Guo Wang},
journal= {arXiv preprint arXiv:1305.4952},
year = {2015}
}
Comments
19 pages, 2 figures, Accepted for Publication in Automatica