This paper presents three discrete observer designs for cycle-by-cycle estimation of the in-cylinder oxygen mass fraction at the intake valve closing (IVC) on diesel engines equipped with exhaust gas recirculation systems. These observers can provide critical in-cylinder condition oxygen fraction information that is useful for control of combustion, in particular advanced combustion modes on a cycle-by-cycle basis. The observers were designed based on Lyapunov analysis and linear matrix inequality techniques. By input-to-state stability analysis, robust properties of the three observers were revealed. Simulations using a high fidelity, computational, GT-Power engine model exhibited that the observers can effectively estimate the in-cylinder oxygen mass fraction at IVC on a cycle-by-cycle basis at both steady-state and transient operations. In addition, comparisons were made to evaluate the observers' robustness against measurement and parametric uncertainties/inaccuracies, such as volumetric efficiency uncertainty, exhaust pressure sensor uncertainty, and temperature sensor dynamics. Experimental results from a medium-duty diesel engine were provided to show the effectiveness of the observers and to validate the in-cylinder oxygen fraction estimation in an indirect and innovative way by utilizing the measured cylinder pressure signals.

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