A heuristic Kalman filter for a class of nonlinear systems

One of the basic assumptions involved in the "optimality" of the Kalman filter theory is that the system under consideration must be linear. If the model is nonlinear, a linearization procedure is usually performed in deriving the filtering equations. This approach requires the nonlinear system dynamics to be differentiable. This note is an attempt to develop a heuristic Kalman filter for a class of nonlinear systems, with bounded first-order growth, that does not require the system dynamics to be differentiable. The proposed filter approximates the nonlinear state function by its state argument multiplied by a particular gain matrix only in the recursion of the estimation error covariance matrix. Under certain conditions, the error covariance remains bounded by bounds which can be precomputed from noise and system models, and the upper-bound tends to zero when the state noise covariance tends to zero. A numerical example, with backlash nonlinearity, is also added.