Hankelnorm

The Hankel norm, or Hankel norm of a linear time-invariant system, is a measure of how much energy can flow from past inputs to future outputs. It is defined via the Hankel operator associated with the system and is the largest gain achieved by mapping past input signals into future output signals. The norm is particularly used in control theory and model reduction to quantify how strongly past behavior influences future behavior.

For a stable, minimal state-space realization with continuous-time dynamics described by

ẋ = Ax + Bu, y = Cx + Du

the Hankel norm can be computed from the controllability and observability Gramians. Solve the Lyapunov equations

A P + P A^T + B B^T = 0 and

A^T Q + Q A + C^T C = 0

to obtain P (controllability) and Q (observability). The Hankel singular values are the square roots of the eigenvalues of the product P Q. The Hankel norm is the largest Hankel singular value. In a balanced realization, P and Q are equal and diagonal, with these values on the diagonal, making the norm easy to read off.

The Hankel norm is closely tied to model reduction. In balanced truncation, states associated with the smallest Hankel singular values are discarded to obtain a reduced model. The standard error bound states that the H-infinity distance between the original and reduced systems is bounded by twice the sum of the neglected Hankel singular values, giving a practical control over approximation quality.

Historically, the concept derives from the Hankel operator, named after Hermann Hankel, and is widely used in signal processing and systems engineering to assess and bound how past behavior influences future responses.