Model Reduction of Descriptor Systems by Interpolatory Projection Methods

In this paper, we investigate an interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, we first illustrate that directly applying the subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded ${{\mathcal H}_2}$ or ${{\mathcal H}_{\infty}}$ errors due to the mismatch of the polynomial parts of the full and reduced-order transfer functions. We then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error. For the special cases of index-$1$ and index-$2$ descriptor systems, we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation. The question of how to choose interpolation points optimally naturally arises as in the standard state space setting. We answer this question in the framework of the ${{\mathcal H}_2}$-norm by extending the iterative rational Krylov algorithm to descriptor systems. Several numerical examples are used to illustrate the theoretical discussion.