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This paper considers estimation problems where inequality constraints are imposed on the outputs of linear systems and can be modeled by nonlinear functions. In this case, censoring functions can be designed to constrain measurements for use by filters and smoothers. It is established that the filter and smoother output estimates are unbiased, provided that the underlying probability density functions are even and the censoring functions are odd. The Bounded Real Lemma is employed to ensure that the output estimates satisfy a performance criterion. A global positioning system (GPS) and inertial navigation system (INS) integration application is discussed in which a developed solution exhibits improved performance during GPS outages when a priori information is used to constrain the altitude and velocity measurements.