Optimal input design for direct data-driven tuning of model-reference controllers ⋆ Simone Formentin a , Alireza Karimi b , Sergio M. Savaresi a a Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. Da Vinci, 32, 20133 Milano (Italy) e-mail: {formentin,savaresi}@elet.polimi.it b Laboratoire d’Automatique, ´ Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), CH-1015 Lausanne (Switzerland) e-mail: alireza.karimi@epfl.ch Abstract In recent years, direct data-driven controller tuning methods have been proposed as an alternative to the standard model- based approach for model-reference control design. In this work, the problem of input design for noniterative direct data- driven techniques, namely Virtual Reference Feedback Tuning (VRFT) and noniterative Correlation-based Tuning (CbT), is investigated. For bounded input energy, the excitation signal is designed such that the expected value of the considered control cost is reduced. The above strategy is numerically tested on a benchmark example. Key words: CbT, VRFT, input design, data-driven control, identification for control 1 INTRODUCTION In system identification theory, optimal experiment de- sign is about finding the operating conditions that pro- vide the most informative data for modeling the plant. However, depending on the intended model application, the optimal experiments appear to be very different. In control applications, the model is used to design a suitable controller, and therefore the final aim for iden- tification and input design is not to accurately describe the mathematical structure of the system, but to obtain a closed-loop behavior with some desired properties. Recently, the term “identification for control” has been introduced to refer to identification from a control- oriented perspective (see [H. Hjalmarsson (2005)] for a survey). In this research area, assessing the model quality by experiment design is of primary importance, as is witnessed by a large set of contributions, see e.g. [M Gevers et al. (1986)] and [M. Gevers (1996)]. Gen- erally, to the authors’ knowledge, only the case of full- order modeling is treated, i.e. the case where the real system belongs to the model set. The only exception is [X. Bombois et al. (2006)], where upper bounds on ⋆ This work has been partially supported by MIUR project “New methods for Identification and Adaptive Control for Industrial Systems” and by the Austrian Center of Compe- tence in Mechatronics. modeling errors are considered. In general, parametric modeling errors influence the control design accuracy, and thus they might constitute a detrimental effect for the final control performance. As far as the authors are aware, input design for di- rect data-driven controller tuning, i.e. the case where a fixed-order linearly parameterized controller is di- rectly identified from data without modeling the plant, has not been considered yet. Using these methods, the typical problems related to modeling errors can be cir- cumvented. Moreover, these techniques can be very useful when a mathematical description of the plant is a costly and time-consuming undertaking. However, as in standard system-identification, a deep understanding of the asymptotic accuracy of the estimate is needed. This paper attempts to obtain some insight into statistical properties of noniterative data-driven techniques, i.e. noniterative Correlation-based Tun- ing (CbT) and Virtual Reference Feedback Tun- ing (VRFT), whereof the interesting feature is that they provide a global solution to a model-reference control issue via simple least squares techniques, when the controller is linearly parameterized. The above methodologies have been only recently intro- duced, respectively in [A. Karimi et al. (2007)] and in [M.C. Campi et al. (2002)]. Iterative data-driven methods are instead not subjects of the present work, but it should be mentioned that an analo- Preprint submitted to Automatica 18 July 2012