8. Hankel-Norm Approximation
Athanasios C. Antoulas · Society for Industrial and Applied Mathematics eBooks · 2005
As mentioned in section 7.1, a balanced state-space representation of a linear system Σ can be derived by minimizing an appropriate criterion. Nevertheless, approximation by balanced truncation does not seem to minimize any norm. A refinement leads to an approximation method which is optimal with respect to the 2-induced norm of the Hankel operator, known as, optimal approximation in the Hankel-norm. The resulting theory is the generalization to dynamical systems of the optimal approximation in the 2-induced norm of finite-dimensional matrices and operators discussed in section 3.2.4. It provides explicit formulas for optimal and suboptimal approximants as well as an error bound in the related ℋ∞-norm (the 2-induced norm of the convolution operator). The developments in this chapter are due to Adamjan, Arov, and Krein [1], [2] and especially to Glover [139]. See also [237], [137] and the books [149] and [370].