By Topic

A Kinship Function Approach to Robust and Probabilistic Optimization Under Polynomial Uncertainty

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

3 Author(s)
Chao Feng ; Pennsylvania State Univ., University Park, PA, USA ; Dabbene, F. ; Lagoa, C.M.

In this paper, we study a class of robust design problems with polynomial dependence on the uncertainty. One of the main motivations for considering these problems comes from robust controller design, where one often encounters systems that depend polynomially on the uncertain parameters. This paper can be seen as integrated in the emerging area of probabilistic robustness, where a probabilistic relaxation of the original robust problem is adopted, thus requiring the satisfaction of the constraints not for all possible values of the uncertainty, but for most of them. Different from the randomized approach for tackling probabilistic relaxations, which is only guaranteed to provide soft bounds on the probability of satisfaction, we present a deterministic approach based on the novel concept of kinship function introduced in this paper. This allows the development of an original framework, which leads to easily computable deterministic convex relaxations of the probabilistic problem. In particular, optimal polynomial kinship functions are introduced, which can be computed a priori and once for all and provide the “best convex bound” on the probability of constraint violation. More importantly, it is proven that the solution of the relaxed problem converges to that of the original robust optimization problem as the degree of the optimal polynomial kinship function increases. Furthermore, by relying on quadrature formulas for computation of integrals of polynomials, it is shown that the computational complexity of the proposed approach is polynomial in the number of uncertain parameters. Finally, unlike other deterministic approaches to robust polynomial optimization, the number of variables in the ensuing optimization problem is not increased by the proposed approximation. An important feature of this approach is that a significant amount of the computational burden is shifted to a one-time offline computation whose results can be stored and provided to - - the end-user.

Published in:

Automatic Control, IEEE Transactions on  (Volume:56 ,  Issue: 7 )