Identification of parameterized gray-box state-space systems: from a black-box linear time-invariant representation to a structured one: detailed derivation of the gradients involved in the cost functions

Estimating consistent parameters of a structured (gray-box) statespace representation requires a reliable initialization when the vector of parameters is computed by using a gradient-based algorithm. In the companion paper [5], the problem of supplying a reliable initial vector of parameters is tackled. More precisely, by assuming that a reliable initial fully-parameterized state-space model of the system is available, the paper [5] addresses the challenging problem of transforming this initial fully-parameterized model into the structured state-space parameterization satisfied by the system to be identified. Two solutions to solve such a parameterization problem are more precisely introduced in [5]. First, a solution based on a null-space-based reformulation of a set of equations arising from the aforementioned similarity transformation problem is considered. Second, an algorithm dedicated to non-convex optimization is presented in order to transform the initial fully-parameterized model into the structured state-space parameterization of the system to be identified. In this technical report, a specific attention is paid to the gradient computation required by the optimization algorithms used in [5] to solve the aforementioned problem. These gradient formulations are indeed necessary to apply the quasiNewton Broyden-Fletcher-Goldfarb-Shanno (BFGS) methods used for This document is available on http://www.lias-lab.fr/perso/guillaumemercere/ Guillaume Mercère (Corresponding author) is with Poitiers University, Laboratoire d’Informatique et d’Automatique pour les Systèmes, 2 rue P. Brousse, bâtiment B25, TSA 41105, 86073 Poitiers cedex 9, France. Email: [email protected] Olivier Prot is with Limoges University, Institut de Recherche XLIM, 123 avenue A. Thomas, 87060 Limoges Cedex, France. Email: [email protected] José Ramos is with Nova Southeastern University, Farquhar College of Arts and Sciences, Division of Mathematics, Science and Technology, 1301 College Avenue, Fort Lauderdale, FL 33314, Florida, USA. Email: [email protected]