736 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 48, NO. 3, JUNE 1999 Identification of Continuous-Time Systems with Missing Data Rik Pintelon, Fellow, IEEE, and Johan Schoukens, Fellow, IEEE Abstract— This paper treats the identification of continuous- time systems with missing data in the input and output signals. A frequency-domain solution based on an extended transfer function model is given. The basic idea of the approach is to treat the missing data as unknown parameters in the identification problem. No particular pattern for the missing data is assumed. The method is illustrated on simulation and real measurement examples. Index Terms— Continuous-time systems, frequency-domain identification, missing data, transfer functions. I. INTRODUCTION D UE to temporary sensor failure and/or data transmission errors, it may happen that data samples are missing in the measured signals. The best thing to do then is to throw away the data set and to repeat the experiment. This is not always possible because, for example, the experiment is too expensive, or some of the data is collected in an irregular way using laboratory analysis. Sometimes the output is sampled at a lower rate than the input, which is a periodic missing output data problem [1], [2]. If not properly taken into account, the missing measurements can seriously deteriorate the quality (consistency, efficiency) of the estimates. The problem of identifying discrete-time systems with miss- ing output data is well understood and has been studied extensively in time series analysis [3] and system identification (see [1], [2], and [4], and the references therein). However, no solution is available when identifying continuous-time systems. This paper presents an original frequency-domain solution to the missing input and/or output data problem for continuous-time systems. The key idea is to treat the missing input–output data as unknown parameters. II. TRANSFER FUNCTION MODEL FOR ARBITRARY SIGNALS WITH MISSING DATA This section follows the lines of [5] where the theory has been developed for discrete-time systems. Consider a Manuscript received November 9, 1998. This work was supported by the Fund for Scientific Research (FWO-Vlaanderen), the Flemish Government (GOA-IMMI), and the Belgian Program on Interuniversity Poles of Attrac- tion initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming (IUAP 4/2). The authors are with the Department of Electrical Measurement, Vrije Universiteit Brussel, 1050 Brussels, Belgium. Publisher Item Identifier S 0018-9456(99)04988-8. continuous-time system with true transfer function (1) excitation and response Suppose that noiseless observations of and are available at time instants where is the sampling time. The scaled discrete Fourier transforms (DFT) of the input and output samples, and satisfy exactly the following equation: (2) with and and (see [6, Eq. (10)], divided by ). is a polynomial of order containing the initial and final conditions of the experiment, and is the spectral alias error which tends to zero as the sampling frequency tends to infinity [6]. Assume now for simplicity of notation that consecutive input samples starting at and consecutive output samples starting at are missing. Define as the signal where the missing samples are replaced by zeros elsewhere (3) and as the corresponding scaled DFT spectrum and Using these definitions the DFT spectrum can be split into the contribution of the known and the unknown samples (4) with a polynomial of order containing the missing samples (5) 0018–9456/99$10.00  1999 IEEE