Application of on-line algebraic identication in active vibration control F. BeltrÆn-Carbajal  , G. Silva-Navarro, H. Sira-Ramrez Centro de Investigacin y de Estudios Avanzados del I.P.N. Departamento de Ingeniera ElØctrica - Seccin de Mecatrnica Apdo. Postal 14-740 C.P. 07300 MØxico, D.F. MØxico E-mail: fbeltran@mail.cinvestav.mx Abstract In this paper is described the application of an on-line algebraic identication methodology for parameter and signal estimation in vibrating systems. An important property of this algebraic iden- tication is that the parameter identication is not asymptotic but algebraic, that is, the parameters are computed as fast as the system dynamics is being excited by some external input or changes in its initial conditions, in contrast to traditional identication methods which need persistent excita- tion. The algebraic identication is employed to estimate the mass, sti/ness and damping in simple mechanical systems using only position measurements. This approach is also used in the identi- cation of the frequency and amplitude of exogenous vibrations a/ecting the mechanical system. The algebraic identication is combined with a certainty equivalence controller to asymptotically stabilize the sytem response and, simultaneously, cancel the harmonic vibrations. The adaptive-like control scheme results quite fast and robust against parameter uncertainty and frequency variations. Numerical and experimental results illustrate the dynamic and robust performance of the algebraic identication and the active vibration control scheme. 1 Introduction The identication of dynamical systems, involving explicitly the parameter identication, is a well- known process to develop or improve the mathematical description of a physical system by means of a proper use of experimental data. System identication methods provide the analytical tools, algorithms, computational programs and real-time implementation to get good approximations to an adequate model for analysis and control purposes. There exists a vast literature on the area, although most of the identication and estimation methods are essentially asymptotic, recursive or complex that lead to unrealistic implementations. See, e.g., Ljung [9] and Soderstrom [12]. This paper deals with the application of an on-line algebraic identication approach to estimate the physical parameters on mass-spring-damper systems as well as the amplitude and excitation frequency of harmonic perturbations a/ecting directly the mechanical system. The proposed results are strongly based on a theoretical framework on algebraic identication methods reported recently by Fliess and Sira-Ramrez [7], which employ di/erential algebra, module theory and operational calculus. The main virtue of the proposed identication and adaptive-like control scheme for vibrating systems is that only measurements of the transient input/output behavior are used during the identication process, in contrast to the well-known persisting excitation condition and complex algorithms required by most of the traditional identication methods (Ljung [9] and Soderstrom [12]). It is important to emphasize that the proposed results are now possible thanks to the existence of high speed DSP boards with high computational performance operating at high sampling rates. 157