The conditional probability density function (pdf) is the most complete statistical representation of the state
from which optimal inferences may be drawn. The transient pdf is usually infinite-dimensional and impossible
to obtain except for linear Gaussian systems. In this paper, a novel density-based filter is proposed for nonlinear
Bayesian estimation. The approach is fundamentally different from optimization- and linearization-based
methods. Unlike typical density-based methods such as probability grid filters (PGF) and sequential Monte
Carlo (SMC), the cell filter poses an off-line probabilistic modeling task and an on-line estimation task. The
probabilistic behavior is described by the Foias or Frobenius−Perron operators. Monte Carlo simulations are
used for computing these transition operators, which represent approximate aggregate Markov chains. The
approach places no restrictions on system model or noise processes. The cell filter is shown to achieve the
performance of PGF and SMC filters at a fraction of the computational cost for recursive state estimation.
Constrained state estimation in nonlinear/non-Gaussian processes has been the domain of optimization based methods such as moving horizon estimation (MHE). MHE has a Bayesian interpretation, but it is not practical to implement a recursive MHE without assumptions of Gaussianity and linearized dynamics at various stages. This paper presents the constrained cell filter (CCF) as an alternative to MHE, requiring no linearization, jacobians, or nonlinear program. The CCF computes a piecewise constant approximation of the state probability density function with support defined by constraints; thus, all point estimates are constrained. The CCF can be more accurate and orders of magnitude faster than MHE for problems of a size as investigated in this work.
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