Title page for ETD etd-12142010-205618

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
  • sensor location
  • LaGrange multipliers
  • SDP
  • numerical
  • unobservability
Date of Defense 2010-12-03
Availability unrestricted
The 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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