Systems & Control Letters 102 (2017) 81–92 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Robust time-domain output error method for identifying continuous-time systems with time delay Fengwei Chen a , Marion Gilson b, c , *, Hugues Garnier b, c , Tao Liu a a School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024, China b Université de Lorraine, CRAN, UMR 7039, 2 rue Jean Lamour, F-54519 Vandoeuvre-les-Nancy, France c CNRS, CRAN, UMR 7039, France article info Article history: Received 13 June 2016 Received in revised form 12 December 2016 Accepted 27 January 2017 Keywords: Continuous-time system Identification Time delay Output error method Irregular sampling abstract In this paper, an output error method is proposed for the identification of continuous-time systems with time delay from sampled data. The challenge of time delay system identification lies in the presence of nonlinear time delays in models, then starting value-based optimization methods may be trapped easily by local minima. In order to improve the convergence performance to the choice of initial parameters, several approaches to smooth the loss function are presented. It is shown that the loss function may possess many local minima when data are regularly sampled with the inter-sample behavior of zero-order hold. Interestingly, irregular sampling can be an efficient approach to overcome these local minima. To achieve superior convergence performance, an over-parametrization approach incorporating a low-pass filtering technique is proposed to enlarge the convergence region. Theoretical and simulated results are presented to demonstrate the effectiveness of the proposed method. © 2017 Elsevier B.V. All rights reserved. 1. Introduction Control-oriented system identification, which aims to build mathematical models for characterizing system behaviors be- tween the manipulated and controlled variables from experimen- tal data, has been increasingly appealing for obtaining good control system performance. With the fast development of digital comput- ers, system identification methods have been developed based on discrete-time (DT) models in terms of shift operators to facilitate the implementation. Recently, there has been a resurgence of in- terest in the study of continuous-time (CT) model identification, which is motivated by the advantages of CT models, for example: physical insights provided by CT model parameters; flexibility in dealing with fractional time delays; invariance of model parame- ters in handling irregularly sampled data [1,2], note that in such case DT model parameters are usually time-varying even if the original systems are stationary and invariant. A common feature of industrial processes is the presence of a time delay between the input command and output response. Modeling of these sys- tems, typically by low order linear models plus time delay, has been widely recognized for obtaining good approximations of the * Corresponding author at: Université de Lorraine, CRAN, UMR 7039, 2 rue Jean Lamour, Vandoeuvre-les-Nancy, F-54519, France E-mail addresses: fengwei.chen@outlook.com (F. Chen), marion.gilson@univ-lorraine.fr (M. Gilson), hugues.garnier@univ-lorraine.fr (H. Garnier), tliu@dlut.edu.cn (T. Liu). system input–output behaviors. Identification of process models from sampled data is the problem that will be studied in this paper. The main difficulty of process model identification arises from the presence of a nonlinear parameter, i.e. the time delay, in a linear model. Here a nonlinear parameter means that the differentiation with respect to this parameter does not yield a constant. Then some traditional methods for linear system identification cannot be ap- plied directly. The study of time delay system identification has led in the literature to several methods, recent reviews can be found in [3–5]. The existing methods fall into three categories roughly: (1) identification using different experiment tests, e.g. step or relay feedback tests [5–8], persistent excitation tests [9,10]; (2) iden- tification using different strategies to estimate the whole model parameters, e.g. one-step approaches that estimate all the param- eters simultaneously [11–14], two-step approaches that estimate the rational model parameters and time delay separately [9,10]; (3) identification using different criteria for optimal model fitting, e.g. cross-correlation maximization [15,16], output or prediction error minimization [10,11]. The simplified refined instrumental variable method for CT systems (SRIVC) has been popular for direct CT modeling [17–19]. This method was first proposed in [20] and later developed in e.g. [21–23]. By extension, a SRIVC-based method for time delay systems (TDSRIVC) was proposed in [10] to identify simple process models from irregularly sampled data, in which a numerical search was incorporated to solve for the optimal time delay. Since the numerical search is quite sensitive to the choice of initial time http://dx.doi.org/10.1016/j.sysconle.2017.01.009 0167-6911/© 2017 Elsevier B.V. All rights reserved.