Asymptotic State Observers for a Simplified Brass Instrument Model

Abstract

In this paper, a simplified model of a brass instrument is introduced. It is composed of a valve (including the mechanics of the lips), a jet (coupled with the valve dynamics), and a straight acoustic pipe excited by the jet, radiating in the air, and with frequency independent losses. This model couples an ordinary differential equation (valve) to a partial differential equation (acoustic pipe) through a static nonlinear function (Bernoulli relation on the jet). In fact, the overall system can be described by a ''so-called'' nonlinear neutral state space representation, the state of which being the position and velocity of the valve aperture and the ingoing wave of pressure at the entrance of the pipe. The measured output is the pressure at the open end of the pipe and the control is the mouth pressure. In this paper, methods of control engineering are applied to recover the state from the input and the measured output, assuming that propagation characteristics and player expression parameters are constant: a nonlinear state observer is built.} The robustness to wrong initial conditions and to noise on the measured output are analyzed.