HankelNormAnsatz

HankelNormAnsatz is a concept in control theory and system identification referring to a structured assumption used to construct reduced-order models that target minimization of the Hankel norm of the modeling error. The Hankel norm of a stable linear time-invariant transfer function G(s) is the largest singular value of its Hankel operator, which captures the maximum energy transfer from past inputs to future outputs. The associated Hankel singular values quantify the relative importance of the system's internal states.

In this approach, one posits a reduced-order model Gr(s) with a chosen state-space realization (A_r, B_r, C_r, D_r) of order r < n. The HankelNormAnsatz imposes constraints or a particular parameterization on these matrices so that the resulting Gr(s) is designed to minimize the Hankel-norm error ||G - Gr||_H. Practically, this involves solving optimization problems (often convex or semidefinite programs) or embedding balancing ideas to align controllability and observability properties, sometimes requiring Lyapunov equations for the Gramians.

The Hankel-norm-based reduction is relevant when preserving energy transfer characteristics is important, such as in robust control design, simulation efficiency, and model-based design for large-scale MIMO systems. It is related to, but distinct from, H2- and H-infinity-optimal reduction and is usually complemented by numerical methods that enforce stability of the reduced model.

Limitations include computational complexity, potential non-uniqueness of the solution, and the need to ensure stability and physical realizability. The term HankelNormAnsatz emphasizes the assumed structure guiding the reduction process rather than a single canonical algorithm.