A Homotopy Continuation Solution of the Covariance Extension Equation

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. Clyde Martin was a pioneer in applying algebraic geometry to linear systems in all three of these theaters. Perhaps the work which is closest to the results we discuss in this paper was his joint study, with Bob Hermann, of the matrix Riccati equation as a flow on a Grassmannian. In this paper we study the steady state form of a discrete-time matrix Riccati-type equation, connected to the rational covariance extension problem and to the partial stochastic realization problem. This equation, however, is nonstandard in that it lacks the usual kind of definiteness properties which underlie the solvability of the standard Riccati equation. Nonetheless, we prove the existence and uniqueness of a positive semidefinite solution. We also show that this equation has the proper geometric attributes to be solvable by homotopy continuation methods, which we illustrate in several examples.