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Measuring Degree of Controllability of a Linear Dynamical System

Emre Mengi
UCSD Department of Mathematics


A linear time-invariant dynamical system is controllable if its trajectory can be adjustedto pass through any pair of points by the proper selection of an input.Controllability canbe equivalently characterized as a rank problem and therefore cannot be verifiedreliably numerically in finite precision. To measure the degree of controllability of a systemthe distance to uncontrollability is introduced as the spectral orFrobenius norm of thesmallest perturbation yielding an uncontrollable system. For a first order system we present a polynomial time algorithm to find the nearest uncontrollable systemthat improves the computational costs of the previous techniques. The algorithm locates the globalminimum of a singular value optimization problem equivalent to the distance to uncontrollability.In the second part for higher-order systems we derive a singular-valuecharacterization and exploitthis characterization for the computation of the higher-order distance to+uncontrollability to lowprecision. Keywords: dynamical system, controllability, distance touncontrollability, Arnoldi, inverse iteration, eigenvalue optimization,polynomial eigenvalue problem

Tuesday, November 21, 2006
11:00AM AP&M 2402