 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor K.W. Lim under the direction of Editor F.L. Lewis. * Corresponding author Tel.: #1-819-821-8000; fax: #1-819-821- 7163. E-mail address: philippe.micheau@gme.usherb.ca (P. Micheau). Automatica 36 (2000) 1659}1664 Brief Paper Adaptive controller using "lter banks to reject multi-sinusoidal disturbance  P. Micheau*, P. Coirault GAUS, Mechanical Department, Universite & de Sherbrooke, 2500 Boul. Universite & , Sherbrooke, Que & bec, Canada J1K 2R1 E ! SIP, Universite & de Poitiers, 40 Av. du Recteur Pineau, 86022 Poitiers, France Received 10 February 1998; revised 18 June 1999; received in "nal form 10 February 2000 Abstract The purpose of this paper is to apply "lter banks to the control problem involving the rejection of multi-sinusoidal disturbance from output of slowly time-varying stable systems. The use of "lter banks allows to distribute the control e!ort in many independent adaptive controllers, each of them taking care of a sinusoidal component of the disturbance. By varying the "lter banks speci"cations, the method handles the trade-o! between a time-behavior controller, with interesting settling time and a tonal-behavior controller showing the properties of simpli"ed control, reduced computational time and on-line system identi"cation with the system output noise via the feedback loop. Numerical examples are presented to illustrate this trade-o!.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive control; Filter banks; Rejection; Sinusoidal signals; Convergence analysis 1. Introduction The motivation for this work was to design an adap- tive feedback controller for active control of pulsed #ow. In terms of control, the objective is to reject a multi- sinusoidal disturbance from output of stable system. In the "eld of active control of sound, Nelson and Elliot (1992) used an adaptive feedforward approach to control the anti-noise with high precision. However, this feedfor- ward approach needs a measured reference signal. With- out a reference signal, the problem is to design a feedback loop that is able to give the same attenuation as an adaptive feedforward approach. The feedback loop achieves desirable rejection of sinusoid despite the pres- ence of modeling uncertainty in the plant model and slowly time-varying sinusoidal disturbance. But, in the case of slowly time-varying system, the feedback control- ler must be adapted to achieve both stability and optimal reduction. In the "eld of feedback theory, a well-known Bode's theorem suggests that reduction in one frequency band must be traded o! against increase of sensitivity at other frequencies. Indeed, to obtain a perfect attenuation at one frequency, without undesirable property at other frequencies, the feedback controller must be narrow band. To deal with this control problem, di!erent con- trollers applied to real-world systems have been pro- posed. Sievers and Flotow (1992) have presented a global comparison of control methods for narrow band distur- bance rejection such as higher harmonic control, tonal control, repetitive control, learning control, LMS adap- tive feedforward "ltering, classical and modern control. The more complex design involving a LQ-based method requires an accurate model of the plant dynamics. On the other hand, the tonal controller is the simplest compen- sator because it ignores the underlying dynamics, except for the plant gain and the phase at the disturbance frequencies. Micheau, Coirault, Hardouin and Tartarin (1996) experimented with a controller working in a nar- row band to reject the preponderant harmonic of a pul- sed #ow. Good experimental results have motivated extension of the approach to the case of multiple sinusoids. For this purpose, the authors investigated "lter banks, which allow analysis and synthesis in many nar- row bands. This approach is an adaptation for a control 0005-1098/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 0 7 2 - 8