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Sampling method for semidefinite programmes with non-negative Popov function constraints
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
University of Calif Los Angeles, CA 90024 USA .
2014 (English)In: International Journal of Control, ISSN 0020-7179, E-ISSN 1366-5820, Vol. 87, no 2, 330-345 p.Article in journal (Refereed) Published
Abstract [en]

An important class of optimisation problems in control and signal processing involves the constraint that a Popov function is non-negative on the unit circle or the imaginary axis. Such a constraint is convex in the coefficients of the Popov function. It can be converted to a finite-dimensional linear matrix inequality via the Kalman-Yakubovich-Popov lemma. However, the linear matrix inequality reformulation requires an auxiliary matrix variable and often results in a very large semidefinite programming problem. Several recently published methods exploit problem structure in these semidefinite programmes to alleviate the computational cost associated with the large matrix variable. These algorithms are capable of solving much larger problems than general-purpose semidefinite programming packages. In this paper, we address the same problem by presenting an alternative to the linear matrix inequality formulation of the non-negative Popov function constraint. We sample the constraint to obtain an equivalent set of inequalities of low dimension, thus avoiding the large matrix variable in the linear matrix inequality formulation. Moreover, the resulting semidefinite programme has constraints with low-rank structure, which allows the problems to be solved efficiently by existing semidefinite programming packages. The sampling formulation is obtained by first expressing the Popov function inequality as a sum-of-squares condition imposed on a polynomial matrix and then converting the constraint into an equivalent finite set of interpolation constraints. A complexity analysis and numerical examples are provided to demonstrate the performance improvement over existing techniques.

Place, publisher, year, edition, pages
Taylor and Francis: STM, Behavioural Science and Public Health Titles , 2014. Vol. 87, no 2, 330-345 p.
Keyword [en]
Kalman-Yakubovich-Popov lemma; Popov function constraint; sampling method; semidefinite programming; linear matrix inequality
National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-104286DOI: 10.1080/00207179.2013.833366ISI: 000329780500010OAI: diva2:697193
Available from: 2014-02-17 Created: 2014-02-14 Last updated: 2014-02-26

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