The Distance to Uncontrollability via Linear Matrix Inequalities

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dc.contributor.authorBoyce, Steven Jamesen
dc.contributor.committeechairZietsman, Lizetteen
dc.contributor.committeememberBorggaard, Jeffrey T.en
dc.contributor.committeememberNorton, Anderson H. IIIen
dc.contributor.committeememberDay, Martin V.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T20:49:33Zen
dc.date.adate2011-01-12en
dc.date.available2014-03-14T20:49:33Zen
dc.date.issued2010-12-03en
dc.date.rdate2011-01-12en
dc.date.sdate2010-12-14en
dc.description.abstractThe distance to uncontrollability of a controllable linear system is a measure of the degree of perturbation a system can undergo and remain controllable. The definition of the distance to uncontrollability leads to a non-convex optimization problem in two variables. In 2000 Gu proposed the first polynomial time algorithm to compute this distance. This algorithm relies heavily on efficient eigenvalue solvers. In this work we examine two alternative algorithms that result in linear matrix inequalities. For the first algorithm, proposed by Ebihara et. al., a semidefinite programming problem is derived via the Kalman-Yakubovich-Popov (KYP) lemma. The dual formulation is also considered and leads to rank conditions for exactness verification of the approximation. For the second algorithm, by Dumitrescu, Şicleru and Ştefan, a semidefinite programming problem is derived using a sum-of-squares relaxation of an associated matrix-polynomial and the associated Gram matrix parameterization. In both cases the optimization problems are solved using primal-dual-interior point methods that retain positive semidefiniteness at each iteration. Numerical results are presented to compare the three algorithms for a number of benchmark examples. In addition, we also consider a system that results from a finite element discretization of the one-dimensional advection-diffusion equation. Here our objective is to test these algorithms for larger problems that originate in PDE-control.en
dc.description.degreeMaster of Scienceen
dc.identifier.otheretd-12142010-205618en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-12142010-205618/en
dc.identifier.urihttp://hdl.handle.net/10919/36138en
dc.publisherVirginia Techen
dc.relation.haspartBoyce_SJ_T_2010.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectsensor locationen
dc.subjectLaGrange multipliersen
dc.subjectSDPen
dc.subjectnumericalen
dc.subjectunobservabilityen
dc.titleThe Distance to Uncontrollability via Linear Matrix Inequalitiesen
dc.typeThesisen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.levelmastersen
thesis.degree.nameMaster of Scienceen

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