Automatica 46 (2010) 354–361 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper Reconstruction of the Fourier expansion of inputs of linear time-varying systems ✩ Jonathan Chauvin a , Nicolas Petit b,∗ a IFP, 1 et 4, avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France b Centre Automatique et Systèmes, Unité Mathématiques et Systèmes, MINES ParisTech, 60 bd. Saint-Michel, 75272 Paris, Cedex 06, France article info Article history: Received 8 December 2008 Received in revised form 27 May 2009 Accepted 28 October 2009 Available online 20 November 2009 Keywords: Observers Linear time-varying systems Periodic input signals Infinite-dimensional systems abstract In this paper we propose a general method to estimate periodic unknown input signals of finite- dimensional linear time-varying systems. We present an infinite-dimensional observer that reconstructs the coefficients of the Fourier decomposition of such systems. Although the overall system is infinite dimensional, convergence of the observer can be proven using a standard Lyapunov approach along with classic mathematical tools such as Cauchy series, Parseval equality, and compact embeddings of Hilbert spaces. Besides its low computational complexity and global convergence, this observer has the advantage of providing a simple asymptotic formula that is useful for tuning finite-dimensional filters. Two illustrative examples are presented. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction Linear time-varying systems driven by periodic input signals are ubiquitous in control systems. For various reasons, including disturbance rejection and diagnosis by analysis of the trajectories, estimation of their input signals is often desirable. In the present paper we propose a general method to address such problems. Consider a T 0 -periodic input signal denoted w. An easily un- derstandable idea is to aim at reconstructing it by estimating its Fourier expansion coefficients. We assume that the period T 0 is perfectly known. Previously, the case of signals w that could be written as a sum of a finite number of harmonics was consid- ered in Chauvin, Corde, Petit, and Rouchon (2007). In this context, a finite-dimensional linear time-varying observer was proposed. As a natural extension, we propose here an infinite-dimensional observer to reconstruct signals possessing an infinite Fourier ex- pansion. Besides its improved generality and global convergence, this extension provides a simple asymptotic formula that, when truncated, serves as a tuning methodology for finite-dimensional filters. This contribution is related to several research works found in the literature. Online estimation of the frequencies of a signal being the sum of a finite number of sinusoids with unknown magnitudes, ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +33 140919330; fax: +33 164694868. E-mail addresses: jonathan.chauvin@ifp.fr (J. Chauvin), nicolas.petit@mines-paristech.fr (N. Petit). frequencies, and phases has been addressed by numerous authors (one can refer to e.g. Hsu, Ortega, & Damn, 1999; Marino & Tomei, 2000; Xia, 2002). However, the problem we address is different. The signal we wish to estimate, and which is assumed to admit an infinite-dimensional Fourier decomposition, is not directly measured. It is filtered through a linear time-varying system. The filtered signal is the only available information. Secondly (and very importantly), its period is precisely known. This particularity suggests that a dedicated observation technique could be worth developing. Our approach can be considered close to the general class of methods aiming at identifying periodic disturbances in view of canceling them. Sinusoidal signals can be modeled as the output of linear exosystems (Ding, 2006). Recent progress has been made in rejecting such disturbances (Ding, 2001). Such approaches have been extended to general periodic signals (see e.g. Xi & Ding, 2007 and the references therein). Lately, in Ding (2006), estimation (and control for rejection) of general periodical disturbance has been addressed by exploiting their integral properties. Our method follows along different lines. It is focused on adapting a Fourier expansion of the signal. The main difficulty lies in determining a simple and mathematically consistent method to tune the gains of the infinite number of adaptation laws. As will appear, a simple solution is found. The paper is organized as follows. In Section 2 we state the problem, present the observer structure and describe the explicit computations of the observation gains. The observer convergence proof is provided in Section 3. This proof relies on a Lyapunov analysis and uses Cauchy series, Parseval equality and compact embeddings of Hilbert spaces. The main result of this section is Proposition 3. For illustration, we propose two examples. First a simple automotive engine application is proposed in Section 4.1. 0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.11.001