A new estimation of the lower error bound in balanced truncation method

• Published in: Automatica (Oxford) Vol. 50; no. 8; pp. 2196 - 2198
• Main Authors: Minh, Ha Binh, Batlle, Carles, Fossas, Enric
• Format: Journal Article Publication
• Language: English
• Published: Kidlington Elsevier Ltd 01-08-2014; Elsevier
• Subjects: [formula omitted]-norm; Anàlisi; Applied sciences; Balanced truncation; Computer science; control theory; systems; Control theory. Systems; Estadística matemàtica; Exact sciences and technology; H-infinity-norm; Linear systems; Linear time-invariant systems; Lower error bound; Matemàtiques i estadística; Model order reduction; model-reduction; Modelling and identification; Ober's canonical form; Sistemes lineals; Sèries temporals; Àrees temàtiques de la UPC; Linear time-invariant systems; Lower error bound; Balanced truncation; Model order reduction; Ober’s canonical form; H∞-norm; Ober's canonical form; Lower bound; Error estimation; Bounded error; H infinite control; Error bound; Canonical form; Linear time invariant system; Reduced order model
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2014.05.020

For a single-input/single-output (SISO) linear time-invariant dynamical system, the standard H∞-norm lower error bound of balanced truncation method is ‖G(s)−Gr(s)‖H∞≥σr+1, where σi,i=1,…,n, are the Hankel singular values of system in decreasing order. In this paper we provide a new estimation of the lower error, namely ‖G(s)−Gr(s)‖H∞≥max{σd,2|∑i∉Jsiσi|}, where si is the sign associated with the Hankel singular value σi in Ober’s canonical form. The subset J and the index d in the above inequality will be introduced in the paper. We show by means of an example that the new bound may be relevant in deciding which states need to be kept in the balanced truncation method, and that using the standard result does not always yield the best approximation.