Suppose that I have a robot which is somewhere in a $100 \times 100$ arena. A Kalman filter could be used to estimate its position from noisy measurements. The estimate produced from the Kalman filter, however, applies some probability to the robot being outside the arena, which is impossible.

Is there some filter that is analogous to the Kalman filter, but that is more appropriate for continuous variables with support over bounded intervals?

  • $\begingroup$ In Simon's book 'Optimal Filtering' you have the problem of Kalman filtering with constraints discussed thoroughly. $\endgroup$
    – F. Tusell
    Oct 24, 2015 at 21:43

1 Answer 1


The Kalman filter solves an unconstrained optimization problem. The natural extension to that would be to impose constraints on that problem.

I have no experience with the algorithms, but this paper [1] suggests solving the problem while constraining the system state $x_k$ to: $$ A x_k = b\\ Cx_k < d $$

The paper takes two approaches: projecting the unconstrained estimate onto the constrained space and modifying the Kalman gain calculation to ensure that the state update satisfies the constraint.

[1] Nachi Gupta & Raphael Hauser, "Kalman Filtering with Equality and Inequality State Constraints," Report no. 07/18.


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