Hybrid Observer for multi-frequency signals ⋆ Daniele Carnevale ∗ Sergio Galeani ∗ Alessandro Astolfi ∗∗ ∗ Dipartimento di Informatica, Sistemi e Produzione (DISP), Universit`a di Roma “Tor Vergata”, 00133 Roma, Italy. (E-mail: carnevale@disp.uniroma2.it). ∗∗ Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, United Kingdom, and with the Dipartimento di Informatica, Sistemi e Produzione, Universit`a di Roma “Tor Vergata”, 00133 Roma, Italy (E-mail: a.astolfi@imperial.ac.uk). Abstract: We proposed a hybrid observer to estimate the frequencies of a signal even in the presence of saturation. Semi-global exponential convergence of the estimation error is provided, and the problem of retrieve dynamically the angular frequencies is addressed. Keywords: Hybrid Observer, frequency estimation, identification, saturation. 1. INTRODUCTION The problem of estimating the n unknown angular fre- quencies ω i of the signal y(t)= n  i=1 E i sin (ω i t + φ i ), (1) amplitudes E i and phases φ i , for i =1, .., n, has been widely studied in the past given its importance within dif- ferent scientific fields as identification and control, acous- tic, signal analysis, telecommunication. Classic off-line solutions make use of Fourier transform to process sets of batch data ( see S.M.Kay and Marple (1981)). Afterwards on-line methods, suitable for many engineering applications, have been firstly proposed in the case of a single frequency signal employing infinite impulse response filter in Regalia (1991) yielding local results, and then combined with adaptation mechanism in Hsu et al. (1999) to yield global results. In Bittanti and Savaresi (2000) a modified extended Kalman filter allows to estimate the frequency of signal in the presence of additive broad-band noise. In the sequel, just to name a few, global multi-frequency estimator have been proposed in Obregon-Pulido et al. (2002) and Xia (2002) exploiting adaptive identifiers, and in Marino and Tomei (2002), mainly relaying on a filtered transformation of co-ordinates, with improved performances. In Marino et al. (2003), the asymptotic estimates of the angular frequencies have been used to cancel out the noise affecting the feedback signals. The amplitudes E i have been reconstructed in Hou (2007) via adaptive identifiers. Within the general framework of Immersion and Invari- ance observers proposed in Karagiannis et al. (2008), a reduced order observer of dimension (3n-1) has been pro- ⋆ Supported in part by ASI, ENEA-Euratom. posed to solve the same problem in Carnevale and Astolfi (2008), also in the case of a single frequency saturated signal Carnevale and Astolfi (2009). In this work we propose an hybrid observer, having discrete-time and continuous-time dynamics, which allows to reduce the complexity of the continuous time dynamics of the observers usually proposed to solve this problem, and to solve the case of multifrequency saturated signal, i.e. when the measured signal is of the form y(t) = sat σ  n  i=1 E i sin (ω i t + φ i )  , (2) where σ> 0 is the saturation level and sat(·) is the satu- ration function defined as sat σ (x) = max (−σ, min(σ, x)), extending the result in Carnevale and Astolfi (2009). Since the structure we propose exploits sampling of the signals (1) and (2), with a specific sampling time, the results we derive are semi-global given that only signal frequencies lower than half of the sampling frequency can be reconstructed (aliasing ). However, in practice, sampling is mandatory and the same limitations applies for the implementation of global observers too. From a numerical point of view, the algorithm in Sec- tion 4.2 supplies estimates of ˆ ω i ’s with improved transient with respect to the one in Theorem 1 which, as the greater part of the observers devoted to this problem, provides indirect estimate of ω i ’s estimating the characteristic poly- nomial of the LTI system whose output is (1). 2. PRELIMINARIES To estimate the unknown frequencies ω i of the signal (1), we propose an hybrid observer of the form given in Goebel et al. (2009). Some of the main definitions for this class of hybrid system are recalled next. The reader should refer to Goebel et al. (2009) for further details. Adaptation and Learning in Control and Signal Processing — ALCOSP 2010 Antalya, Turkey, August 26-28, 2010 ISBN 978-3-902661-85-2/11/$20.00 © 2010 IFAC 1