A comparison of homotopies for alternative formulations of the L2 optimal model order reduction problem

A fundamental problem in control systems theory is finding a reduced order model that is optimal in the L2 sense to a given (full order) system model. The numerical solution of this problem is challenging and the global convergence properties of homotopy methods are advantageous. A number of homotopy-based approaches have been developed. The primary numerical issues are the number of degrees of freedom in the homotopy parameter vector, the well-posedness of the finite-dimensional optimization problem, and the numerical robustness of the resulting homotopy algorithm. This paper develops two new homotopy algorithms for optimal model reduction and uses several examples to compare their performance with the performance of two previous algorithms. The results show that the numerical well-conditioning is inversely related to the algorithmic efficiency and that the relative performance of a given algorithm is problem dependent.