A Homotopy Continuation Solution of the Covariance Extension Equation

Abstract

Algebraic geometry plays an important role in the theory of linear systems for (at least) three reasons. First, the Laplace transform turns expressions about linear differential systems into expressions involving rational functions. In addition, many of the transformations studied in linear systems theory, like changes of coordinates or feedback, turn out to be the action of algebraic groups on algebraic varieties. Finally, when we study linear quadratic problems in optimization and estimation, all roads eventually lead either to the Riccati equation or to spectral factorization.