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A new scheme for solving the sensor failure detection, isolation and accommodation (FDIA) problem in optimally controlled linear systems has been developed. The proposed technique uses Luenberger observers with a detection logic based on the minimum value of the cost functional at any time instant. The advantages of this new technique are: (1) it can detect and isolate as many failed sensors as there are sensors, even when the failures are multiple and simultaneous; (2) FDIA can be accomplished at any time, since it does not depend on the convergence of the observer estimates to the actual system states; (3) it is not applications dependent, since it requires neither a priori information concerning likely failure modes nor the knowledge of the nominal path resulting from a specific input command. The method assumes the system is observable from each available output.