Systems & Control Letters 127 (2019) 25–34 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Deterministic continuous-time Virtual Reference Feedback Tuning (VRFT) with application to PID design ✩ Simone Formentin a,∗ , Marco C. Campi b , Algo Carè b , Sergio M. Savaresi a a Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, P.zza L. da Vinci 32, 20133 Milano, Italy b Dipartimento di Ingegneria dell’Informazione, University of Brescia, via Branze 38, 25123 Brescia, Italy article info Article history: Received 27 June 2018 Received in revised form 12 March 2019 Accepted 23 March 2019 Available online xxxx Keywords: Autotuning PID Virtual reference feedback tuning abstract In this paper, we introduce a data-driven control design method that does not rely on a model of the plant. The method is inspired by the Virtual Reference Feedback Tuning approach for data-driven controller tuning, but it is here entirely developed in a deterministic, continuous-time setting. A PID autotuner is then developed out of the proposed approach and its effectiveness is tested on an experimental brake-by-wire facility. The final performance is shown to outperform that of a benchmark model-based design method. © 2019 Elsevier B.V. All rights reserved. 1. Introduction In the last decades, people in systems and control have inves- tigated data-driven controller tuning techniques aimed to design feedback controllers directly from data without the need of es- timating a model of the system. In the last twenty years, a large variety of methods has been created, among which Iterative Feedback Tuning (IFT [1]), data-driven loop-shaping [2], Virtual Reference Feedback Tuning (VRFT [3–6]) and Correlation-based Tuning (CbT [7,8]). In particular, for the last two methods it has been shown in [9] that their performances compare to that of standard model-based design approaches where the model has been identified from data. VRFT and CbT have been developed in the stochastic set-up described in [10], where the involved processes are stationary and evolve in discrete-time. The aim of the present work is to reformulate the VRFT approach into an autotuning method for industrial PID control. In this novel form, Input/Output (I/O) signals are not treated as stochastic processes and the theory can be fully interpreted in continuous time. This fact is important in practical problems, where control engineers are usually more familiar with continuous time models for many reasons (e.g., the fact that the settling time is directly related to the dominant poles of the system). We should also emphasize that the results ✩ The authors would like to thank F. Todeschini for his help with the experimental BBW setup. ∗ Corresponding author. E-mail addresses: simone.formentin@polimi.it (S. Formentin), marco.campi@unibs.it (M.C. Campi), algo.care@unibs.it (A. Carè), sergio.savaresi@polimi.it (S.M. Savaresi). presented here are not obtained by simple modifications of the previous contributions [3–6]. In fact, those papers are based on tools (spectral factorization and stochastic processes) that lose their validity in the present context. To derive the results of this paper, Sobolev spaces and solutions of ODEs are instead used. The resulting PID-VRFT algorithm lends itself to an easy im- plementation for the tuning of industrial PID controllers. The underlying design procedure relies on an optimization method, thoroughly developed in the paper, that establishes an equiva- lence between a model-reference control problem and the iden- tification problem the PID-VRFT algorithm is based upon. This makes the method here proposed different from the majority of the existing approaches, which are either characterized by the use of semi-empirical rules or derived from model-based methods employing low-order data-driven models, [11]. This observation is very important in relation to the use of these methods because it implies that strong guarantees on the real system cannot in general be provided: on the one hand, semi-empirical rules are not based on optimization theories and one can only hope to obtain a suitable tuning for the application at hand; on the other hand, low-order models are always approximate, and model- ing errors might jeopardize the performance of the closed-loop system, [12]. The effectiveness of the proposed approach is illustrated on a brake-by-wire (BBW) application. The BBW technology bears a promise of significant improvement over existing tools in the automotive industry, but it also poses new challenges for con- trol design. Among other requirements, BBW systems demand adaptation to aging and rapid changes of the environmental conditions, like temperature, so that data-driven PID autotun- ing represents an interesting approach for fast control system recalibration. https://doi.org/10.1016/j.sysconle.2019.03.007 0167-6911/© 2019 Elsevier B.V. All rights reserved.