  The paper proposes a new recursive filter for non-linear systems that
inherently computes a valid bound on the mean square estimation error. The
proposed filter, bound based extended Kalman, (BEKF) is in the form of an
extended Kalman filter. The main difference of the proposed filter from the
conventional extended Kalman filter is in the use of a computed mean square
error bound matrix, to calculate the filter gain, and to serve as bound on the
actual mean square error. The paper shows that when the system is linear the
proposed filtering algorithm reduces to the conventional Kalman filter. The
theory presented in the paper is applicable to a wide class of systems, but if
the system is polynomial, then the recently developed theory of positive
polynomials considerably simplifies the filter's implementation.
