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Estimation of an uncertain bilinear boundary condition in linear 2 × 2 hyperbolic systems with application to drilling

In this paper, unknown parameters appearing in a special bilinear boundary condition are estimated by using swapping design filters to bring a system of 2 × 2 linear hyperbolic equations to static form. Estimates of the unknown parameters can then be generated by using any standard parameter identification law, and state estimates can be generated from the static relationship and parameter estimates. Sensing is allowed at both boundaries, and the measurement collocated with the uncertain parameters is allowed to be an arbitrary linear combination of the system states. Proof of boundedness of the adaptive law and conditions for parameter convergence are given. The theory is applied to the kick and and loss detection problem in manged pressure drilling and demonstrated in a simulation.