A sum of squares optimization approach to uncertainty quantification

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

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

4 Scopus citations


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.
Number of pages7
ISBN (Electronic)9781538679265
StatePublished - Jul 2019
Event2019 American Control Conference, ACC 2019 - Philadelphia, United States
Duration: Jul 10 2019Jul 12 2019

Publication series

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


Conference2019 American Control Conference, ACC 2019
Country/TerritoryUnited States

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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