Probability-one Homotopy Algorithms for Solving the Coupled Lyapunov Equations Arising in Reduced-Order H^2/H^(infinity) Modeling, Estimation, and Control

Optimal reduced order modeling, estimation, and control with respect to combined H^2/H^(infinity) criteria give rise to coupled Lyapunov and Riccati equations. To develop reliable numerical algorithms for these problems this paper focuses on the coupled Lyapunov equations which appear as a subset of the synthesis equations. In particular, this paper systematically examines the requirements of probability-one homotopy algorithms to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems and solve the given system by tracking the continuous curve of solutions to the family. The main emphasis is on guaranteeing transversality for several homotopy maps based upon the pseudogramian formulation of the coupled Lyapunov equations and variations based upon canonical forms. These results are essential to the probability-one homotopy approach by guaranteeing good numerical properties in the computational implementation of the homotopy algorithms.