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
- SDP
- numerical
- unobservability
Date of Defense 2010-12-03 Availability unrestricted AbstractThe distance to uncontrollability of a controllable linear system is a measure of the degree ofperturbation 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 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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