An Interpolation-Based Approach to Optimal H<sub>∞</sub> Model Reduction
A model reduction technique that is optimal in the H∞-norm has long been pursued due to its theoretical and practical importance. We consider the optimal H∞ model reduction problem broadly from an interpolation-based approach, and give a method for finding the approximation to a state-space symmetric dynamical system which is optimal over a family of interpolants to the full order system. This family of interpolants has a simple parameterization that simplifies a direct search for the optimal interpolant. Several numerical examples show that the interpolation points satisfying the Meier-Luenberger conditions for H₂-optimal approximations are a good starting point for minimizing the H∞-norm of the approximation error. Interpolation points satisfying the Meier-Luenberger conditions can be computed iteratively using the IRKA algorithm . We consider the special case of state-space symmetric systems and show that simple sufficient conditions can be derived for minimizing the approximation error when starting from the interpolation points found by the IRKA algorithm. We then explore the relationship between potential theory in the complex plane and the optimal H∞-norm interpolation points through several numerical experiments. The results of these experiments suggest that the optimal H∞ approximation of order r yields an error system for which significant pole-zero cancellation occurs, effectively reducing an order n+r error system to an order 2r+1 system. These observations lead to a heuristic method for choosing interpolation points that involves solving a rational Zolatarev problem over a discrete set of points in the complex plane.