Homotopy Approaches to the H2 Reduced Order Model Problem

The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. The algorithms proposed herein utilize probability-one homotopy theory as the main tool. It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system. With a carefully chosen initial system all the systems along the homotopy path will be asymptotically stable, controllable and observable. One method, which solves the matrix equations in their original form, requires a decomposition of the projection matrix using the Drazin inverse of a matrix. An effective algorithm for computing the derivative of the projection matrix that involves solving a set of Sylvester equations is given. Several strategies for choosing the homotopy maps and the starting points (initial systems) are discussed and compared, in the context of some reduced order model problems from the literature. Numerical results are included for ten test problems, of sizes 2 through 17.