Linear fractional LPV model identification from local experiments: an H ∞ -based optimization technique Daniel Vizer 1,2 , Guillaume Merc` ere 2 , Olivier Prot 3 , Edouard Laroche 4 and Marco Lovera 5 Abstract — In this paper, a new identification tech- nique is introduced to estimate a linear fractional representation of a linear parameter-varying (LPV) system from local experiments by using a dedicated non-smooth optimization procedure. More precisely, the developed approach consists in estimating the pa- rameters of an LPV state-space model from local fully- parameterized identified state-space models through the non-smooth optimization of a specific H∞-based criterion. The method presented in this paper results directly in an LPV model whose parametric matrices can be rational functions of the scheduling variables without any interpolation step (required usually by the local approach) and without writing the local fully-parameterized LTI state-space models with re- spect to a coherent basis. A numerical example is used to illustrate the performance of the suggested technique. I. INTRODUCTION Although many identification techniques and solutions have been developed since the first attempts in 1995 [1] (see, e.g., [2], [3] and the references therein for a recent overview), identifying a linear parameter-varying (LPV) model of a system by using a local approach is still a problem which deserves attention. By local approach, it is meant that a multi-step procedure is undertaken where (1) local experiments are carried out in several operating points represented by fixed values of the scheduling signals while the inputs are excited, (2) local LTI models are estimated based on the locally gathered input-output data, (3) an LPV model is derived by interpolating of these locally-estimated models. Such an approach is considered among others in [4], [5], [6]. Notice that this approach differs from the global one which resorts to the simultaneous excitation of both the scheduling variables and the inputs of the system by 1 University of Technology and Economics of Budapest, Intelli- gent Robots Laboratory, Magyar Tud´ osok krt. 1-3 1117, Budapest, Hungary vizer@iit.bme.hu 2 University of Poitiers, Laboratoire d’Informatique et d’Automatique pour les Syst` emes, 2 rue P. Brousse, bˆ atiment B25, B.P. 633, 86022 Poitiers Cedex, France guillaume.mercere@univ-poitiers.fr 3 University of Limoges, Institut de Recherche XLIM, 123 Avenue A. Thomas, 87060 Limoges Cedex, France olivier.prot@unilim.fr 4 ICUBE Laboratory, University of Strasbourg and CNRS, Pole API, 300 Bd. Sebastien Brant, 67412 Illkirch, France laroche@unistra.fr 5 Dipartimento di Elettronica, Informazione e Bioingegneria, Po- litecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy lovera@elet.polimi.it performing one global experiment in order to capture the non-linear dynamics of the real system [7], [8], [9]. In this paper we consider the local approach because, in practice, mainly for safety and/or financial reasons, it is often difficult to perform one single experiment with a “rich” excitation of the control inputs and the schedul- ing variables simultaneously. On the contrary, applying small variations around particular operating points, as considered by the local approach, is more conceivable in many practical cases. Despite its practical simplicity, as pointed out first in [10], the interpolation step involved in the local approach can lead to a global LPV model with an inaccurate dynamic behavior even if the local LTI models are consistent. Thus, a number of issues deserve attention as far as the relation between the local models is concerned, as shown, e.g., in [2]. The main difficulty with the local approach for LPV state-space models identification is the determination of a suitable coherent basis for all of the local estimated models so that the interpolated LPV state-space representation is able to capture the dynamics of the process to identify. Many attempts to solve this problem are suggested in the liter- ature. In the black-box framework, i.e., when no prior information about the system to identify is available, interesting solutions have been developed,e.g., in [11], [4], [12], [13], [14] during the last decade. Unfortunately, as explained recently in [14], [15], whatever the black- box technique used to estimate an LPV model from local experiments, getting an LPV model able to picture the dynamic dependence on the scheduling variables is not an easy task even if the LPV system to identify satisfies a static dependence on the scheduling signal. In order to bypass this difficulty, optimization-based LPV model approximation solutions have been recently suggested in [16], [17], [18], [19], [15]. Notice however that (i) the solutions suggested in [17] are based on the H 2 -norm and focus on system matrices which are linear combination of some basis functions of p,(ii) the H ∞ -based techniques available in [18], [19] are developed for LTI, linear time- periodic and/or affine LPV models the latter identified either via a global or a local approach, (iii) the technique used in [15] still requires an interpolation step. In this paper, a solution, inspired by the develop- ments available in [16], [19], is introduced to circum- vent the aforementioned challenging problems. More pre- cisely, this solution consists in estimating the unknown parameters of LPV models directly in the so-called linear fractional representation (LFR) [20] from fully-