Continuous-time model identification from sampled data: implementation issues and performance evaluation H. GARNIER{*, M. MENSLER{ and A. RICHARD{ This paper deals with equation error methods that fit continuous-time transfer function models to discrete-time data recently included in the CONTSID (CONtinuous-Time System IDentification) Matlab toolbox. An overview of the methodsisfirstgivenwhereimplementationissuesarehighlighted.Theperformancesofthemethodsarethenevaluated on simulated examples by Monte Carlo simulations. The experiments have been carried out to study the sensitivity of eachapproachtothedesignparameters,samplingperiod,signal-to-noiseratio,noisepowerspectraldensityandtypeof input signal. The effectiveness of the CONTSID toolbox techniques is also briefly compared with indirect methods in which discrete-time models are first estimated and then transformed into continuous-time models. The paper does not consider iterative or recursive algorithms for continuous-time transfer function model identification. 1. Introduction Identification of continuous-time (CT) linear time- invariant (LTI) models for continuous-time dynamic processes was the initial goal in the earliest works on system identification. However, due to the success of digital computers and the availability of digital data acquisition boards, most system identification schemes usually aim at identifying the parameters of discrete- time (DT) models based on sampled input–output data. Well-established theories are available (Ljung 1987, So¨derstro¨m and Stoica 1989) and many applica- tions have been reported. Over the last few years there has been strong interest in continuous-time approaches forsystemidentificationfromsampleddata(So¨derstro¨m et al. 1997, Unbehauen and Rao 1998, Johansson et al. 1999,Garnier et al. 2000, Pintelon et al. 2000, Bastogne et al. 2001, Wang and Gawthrop 2001, Young 2002a, Young et al. 2003). Identification of CT models is in- deed a problem of considerable importance in various disciplines such as economics, control and signal processing. A simplistic way of estimating the parameters of a CTmodelbyanindirectapproachistousethesampled data to first estimate a DT model and then convert it intoanequivalentCTmodel.However,thesecondstep, i.e. obtaining an equivalent CT model from the esti- mated DT model, is not always easy. Difficulties are encountered whenever the sampling time is either too large or too small. Whereas a large sampling interval may lead to loss of information, making it very small may create numerical problems due to the fact that the polesareconstrainedtolieinaverysmallareaofthe z- plane close to the unit circle. Some conversion methods use the matrix logarithm which may produce complex arithmetic when the matrix has negative eigenvalues. Moreover, the zeros of the DT model are not as easily transformable to CT equivalents as the poles are. An alternative approach is to directly identify a continuous-time model from the discrete-time data. Since the equation error (EE) is a linear algebraic func- tion of the model parameters, EE model structure- based methods have been widely followed for direct continuous-time model identification from sampled data. The basic problem of this EE approach is with handlingofthenon-measurabletime-derivativesandthe removal of asymptotic bias on the parameter estimates whenthenoiseisatahighlevel.InanEEcontext,CT modelidentificationindeedimpliesmeasurementorgen- eration of the input–output time-derivatives. The need to generate these time-derivatives may be eliminated by applying ‘linear dynamic operations’ to the sampled input–output data according to Unbehauen and Rao (1987). These ‘linear dynamic operations’ can be inter- preted as input and output signal pre-processing. Many pre-processing approaches have been developed. Surveys (Young 1981, Unbehauen and Rao 1990, Sagara and Zhao 1991) and books (Unbehauen and Rao1987,SinhaandRao1991)offerabroadoverview of many of the available techniques. However, no com- montoolincludingthemainCTparametricmodeliden- tification approaches was available. The CONtinuous- Time System IDentification (CONTSID) Matlab tool- box has been developed to fill this gap (Garnier and Mensler2000).Itisacollectionofthemainapproaches International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/0020717031000149636 INT. J. CONTROL, 2003, VOL. 76,NO. 13, 1337–1357 Received 24 January 2002. Revised 22 January 2003. Accepted 10 May 2003. *Author for correspondence. e-mail: hugues.garnier@ cran.uhp-nancy.fr { Centre de Recherche en Automatique de Nancy (CRAN), CNRS UMR 7039, Universite´ Henri Poincare´, Nancy 1, BP 239, F-54506 Vandœuvre-les-Nancy Cedex, France. { Nissan Motor Co., Ltd. Nissan Research Center, Electronics and Information Technology Research Laboratory 1, Natsushima-cho, Yokosuka-shi, Kanagawa 237-8523, Japan.