Estimation of boundary parameters in general heterodirectional linear hyperbolic systems

Abstract

In this paper, we extend recent results on state and boundary parameter estimation in coupled systems of linear partial differential equations (PDEs) of the hyperbolic type consisting of n rightward and one leftward convecting equations, to the general case which involves an arbitrary number of PDEs convecting in both directions. Two adaptive observers are derived based on swapping design, where one introduces a set of filters that can be used to express the system states as linear, static combinations of the filter states and the unknown parameters. Standard parameter identification laws can then be applied to estimate the unknown parameters. One observer which requires sensing at both boundaries, generates estimates of the boundary parameters and system states, while the second observer estimates the parameters from sensing limited to the boundary anti-collocated with the uncertain parameters. Proof of boundedness of the adaptive laws is offered, and sufficient conditions ensuring exponential convergence are derived. The theory is verified in simulations.