Design of Algebraic Observers for Brass Instruments

International audience ; Physical high-fidelity models of brass instruments are available in the literature, but controlling them to obtain a proper musical restitution is still a challenge. The inversion of the model from a unique observation, namely the sound produced by the instrument, is therefore a natural way to deal with this situation. The observer design problem consisting in an estimation of the vibro-acoustic state of the system is essential for that purpose. The observer design problem was addressed in [@AN10] for an elementary brass system using elastic player lips and straight pipe models. A neutral system representation of the system and Lyapunov methods were used ; a proof of the observer stability was obtained and simulations have demonstrated that the estimation method is robust in the presence of noisy measurements. However no adaptation to the noise power was performed, leading to a rate of convergence of the observer that was suboptimal. Moreover, as the observer dynamics was related to the uncoupled lips dynamics, the response could be slow and oscillatory. Using a representation of the same brass model as a delay-differential algebraic system [@B13], together with a sensitivity analysis and Kalman filter theory, we address these limitations through a new observer design resulting in a substantial improvement of the observer rate of convergence.