Kronecker Product Based Second Order Approximation of Mean Value and Covariance Matrix in Nonlinear Transformations

This paper derives new formulae for the approximations of the mean value and the covariance matrix of a random vector which is derived as a nonlinear transformation of another random vector. Towards this purpose, it is first proposed to formulate the Taylor series expansion by using Kronecker product operators along with partial derivatives block matrices. This formulation facilitates the mathematical manipulation of higher order terms in multidimensional systems. The final expressions for the mean value and covariance matrix are simple in block matrix notation, convenient and easy to implement. The analytical expressions are evaluated and verified in a typical nonlinear transformation faced in navigation systems.