A sum of squares optimization approach to uncertainty quantification

Brendon K. Colbert, Luis G. Crespo, Matthew Peet

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

This paper proposes a Sum of Squares (SOS) optimization technique for using multivariate data to estimate the probability density function of a non-Gaussian generating process. The class of distributions over which we optimize, result from using a polynomial map to lift the data into a higher-dimensional space, solving for an optimal Gaussian fit in this space, and then projecting a polynomial slice of the resulting joint density into physical space. The resulting distribution, to be called Sliced Normal, is able to characterize multimodal responses and strong parameter dependencies. We investigate several formulations of the problem, first maximizing a log-likelihood function, then a worst-case log-likelihood function, and finally using a heuristic to increase sparsity within the maximum log-likelihood formulation - thereby identifying independent subsets of the random variables. Using the optimal density functions in each scenario, we then propose semi-algebraic sets representing confidence regions, or 'safe sets,' for future data. Finally, we show numerically that these 'safe sets' are reliable and, hence, can be used for system identification, fault detection, robustness analysis, and robust control design.

Original languageEnglish (US)
Title of host publication2019 American Control Conference, ACC 2019
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages5378-5384
Number of pages7
ISBN (Electronic)9781538679265
StatePublished - Jul 1 2019
Event2019 American Control Conference, ACC 2019 - Philadelphia, United States
Duration: Jul 10 2019Jul 12 2019

Publication series

NameProceedings of the American Control Conference
Volume2019-July
ISSN (Print)0743-1619

Conference

Conference2019 American Control Conference, ACC 2019
CountryUnited States
CityPhiladelphia
Period7/10/197/12/19

Fingerprint

Probability density function
Polynomials
Robust control
Set theory
Robustness (control systems)
Fault detection
Random variables
Identification (control systems)
Uncertainty

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

Cite this

Colbert, B. K., Crespo, L. G., & Peet, M. (2019). A sum of squares optimization approach to uncertainty quantification. In 2019 American Control Conference, ACC 2019 (pp. 5378-5384). [8814931] (Proceedings of the American Control Conference; Vol. 2019-July). Institute of Electrical and Electronics Engineers Inc..

A sum of squares optimization approach to uncertainty quantification. / Colbert, Brendon K.; Crespo, Luis G.; Peet, Matthew.

2019 American Control Conference, ACC 2019. Institute of Electrical and Electronics Engineers Inc., 2019. p. 5378-5384 8814931 (Proceedings of the American Control Conference; Vol. 2019-July).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Colbert, BK, Crespo, LG & Peet, M 2019, A sum of squares optimization approach to uncertainty quantification. in 2019 American Control Conference, ACC 2019., 8814931, Proceedings of the American Control Conference, vol. 2019-July, Institute of Electrical and Electronics Engineers Inc., pp. 5378-5384, 2019 American Control Conference, ACC 2019, Philadelphia, United States, 7/10/19.
Colbert BK, Crespo LG, Peet M. A sum of squares optimization approach to uncertainty quantification. In 2019 American Control Conference, ACC 2019. Institute of Electrical and Electronics Engineers Inc. 2019. p. 5378-5384. 8814931. (Proceedings of the American Control Conference).
Colbert, Brendon K. ; Crespo, Luis G. ; Peet, Matthew. / A sum of squares optimization approach to uncertainty quantification. 2019 American Control Conference, ACC 2019. Institute of Electrical and Electronics Engineers Inc., 2019. pp. 5378-5384 (Proceedings of the American Control Conference).
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