Type of Document Master's Thesis Author Boyce, Steven James URN etd-12142010-205618 Title The Distance to Uncontrollability via Linear Matrix Inequalities Degree Master of Science Department Mathematics Advisory Committee
Advisor Name Title Zietsman, Lizette Committee Chair Borggaard, Jeffrey T. Committee Member Day, Martin V. Committee Member Norton, Anderson H. III Committee Member Keywords
- sensor location
- LaGrange multipliers
Date of Defense 2010-12-03 Availability unrestricted 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
Numerical results are presented to compare the three algorithms for a number of bench-
mark 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.
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