The Distance to Uncontrollability via Linear Matrix Inequalities

• Main Author: Boyce, Steven James
• Other Authors: Mathematics
• Format: Others
• Published: Virginia Tech 2014
• Subjects: sensor location; LaGrange multipliers; SDP; numerical; unobservability
• Online Access: http://hdl.handle.net/10919/36138; http://scholar.lib.vt.edu/theses/available/etd-12142010-205618/

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