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Mechanical ventilators facilitate breathing for patients who cannot breathe (sufficiently) on their own. The aim of this paper is to estimate relevant lung parameters and the spontaneous breathing effort of a ventilated patient that help keeping track of the patient's clinical condition. A key challenge is that estimation using the available sensors for typical model structures results in a non-identifiable parametrization. A sparse optimization algorithm to estimate the lung parameters and the patient effort, without interfering with the patient's treatment, using an 1 -regularization approach is presented. It is confirmed that accurate estimates of the lung parameters and the patient effort can be retrieved through a simulation case study and an experimental case study.

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