The Distance to Uncontrollability via Linear Matrix Inequalities
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 deï¬ nition of the distance to uncontrollability leads to a non-convex optimization problem in two variables. In 2000 Gu proposed the ï¬ r...
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Virginia Tech
2014
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| Online Access: | http://hdl.handle.net/10919/36138 http://scholar.lib.vt.edu/theses/available/etd-12142010-205618/ |
| Summary: | 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 deï¬ nition of the distance
to uncontrollability leads to a non-convex optimization problem in two variables. In 2000
Gu proposed the ï¬ rst 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 ï¬ rst algorithm, proposed by Ebihara et. al., a semideï¬ nite programming problem
is derived via the Kalman-Yakubovich-Popov (KYP) lemma. The dual formulation is also
considered and leads to rank conditions for exactness veriï¬ cation of the approximation.
For the second algorithm, by Dumitrescu, Å icleru and Å tefan, a semideï¬ nite 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 semideï¬ niteness 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 ï¬ nite 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. === Master of Science |
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