Krylov subspace-based model reduction for a class of bilinear descriptor systems

We consider model order reduction for bilinear descriptor systems using an interpolatory projection framework. Such nonlinear descriptor systems can be represented by a series of generalized linear descriptor systems (also called subsystems) by utilizing the Volterra-Wiener approach (Rugh, 1981). Standard projection techniques for bilinear systems utilize the generalized transfer functions of these subsystems to construct an interpolating approximation. However, the resulting reduced-order system may not match the polynomial parts of the generalized transfer functions. This may result in an unbounded error in terms of H2 or H∞ norms. In this paper, we derive an explicit expression for the polynomial part of each subsystem by assuming a special structure of the bilinear system which reduces to an index-1 linear descriptor system or differential algebraic equation (DAE) if the bilinear terms are zero. This allows us to propose an interpolatory technique for bilinear DAEs which not only achieves interpolation, but also retains the polynomial parts of the bilinear systems. The approach extends the interpolatory technique for index-1 linear DAEs (Beattie and Gugercin, 2009) to bilinear DAEs. Numerical examples are used to illustrate the theoretical results.