Model Reduction by Balanced Truncation for Systems with Nuclear Hankel Operators

Abstract

We prove the H-infinity error bounds for Lyapunov balanced truncation and for optimal Hankel norm approximation under the assumption that the Hankel operator is nuclear. This is an improvement of the result from Glover, Curtain, and Partington [SIAM J. Control Optim., 26 (1998), pp. 863-898], where additional assumptions were made. The proof is based on convergence of the Schmidt pairs of the Hankel operator in a Sobolev space. We also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation.

Model reduction for distributed parameter systems: A functional analytic view

“…The first question that should be asked when using this method is whether the so obtained reduced order model is close to the exact balanced truncation of the underlying partial differential equation. The following result taken from [9] shows that -under very reasonable assumptions (which however exclude our Example 1)-this is (asymptotically) true.…”

“…The general case under stronger assumptions then those mentioned here is due to Glover, Curtain and Partington [7] (the assumptions imposed by [7] for example exclude our Example 2). The improvement given here was proven in [9].…”

“…See again Glover [6] (and also [8, Section 10.5.2]) for the rational case and [7] and [9] for the general case.…”

Balancing-Related Model Reduction for Parabolic Control Systems

“…This includes a much larger class of boundary control problems. We conjecture that all results in this paper remain true in this setting, though only the BT case is considered in Guiver and Opmeer (2012).…”

“…All results in the following can also be proved using the weaker assumption that Σ(A, B, C) is a Pritchard-Salomon system (Curtain (2003)), allowing for certain unboundedness of B, C, which is required for boundary control problems. An even more general setting is considered in Guiver and Opmeer (2012), where it is only assumed that the Hankel operator related to Σ(A, B, C) is nuclear. This includes a much larger class of boundary control problems.…”