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We show that the Lagrangian dual of a constrained linear estimation problem is a particular nonlinear optimal control problem. The result has an elegant symmetry, which is revealed when the constrained estimation problem is expressed as an equivalent nonlinear optimisation problem. The results extend and enhance known connections between the linear quadratic regulator and linear quadratic state estimation problems.

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This paper investigates the structure present in constrained linear state estimation problems formulated as a quadratic optimization program subject to linear inequality constraints. Polyhedral constraints on the system disturbance, the measurement noise, and the initial state are considered. The result interprets the measurement data and prior estimate as parameters and the parameter space is partitioned into multiple regions. Within each region the state estimate can be calculated as a piece-wise affine function of the measurement data and prior estimate. The parameterized regions and coefficients of the piece-wise affine function can be precomputed offline allowing a simplified approach in implementing a moving horizon estimation scheme. Copyright

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