A finite-difference method for linearization in nonlinear estimation algorithms
Consider a random vector \(x\) with expected value \(\overline x\) and covariance matrix \(X\). The problem is to estimate a covariance matrix \(Y\) of the vector \(y= f(x)\), where \(f\) is a given nonlinear function. A new linearization method is proposed. In the case of a unit matrix \(X\) the method is defined by \[ Y\approx FXF^T,\quad F= \{F(i,j)\}= \Biggl\{{f_i(\overline x+ e_j)- f_i(\overline x- e_j)\over 2}\Biggr\},\tag{1} \] where \(F(i,j)\) is a matrix element of \(F\), \(f_i\) is a component of \(f\), and \(e_j\) is the unit vector along coordinate axis \(j\) in the space spanned by \(x\). The procedure (1) is extended to general matrices \(X\) by a square root factorization of \(X\). This method is more accurate than the ordinary Jacobian linearization. It also has the advantage that it can be used for the models where the first derivative of \(f(x)\) is discontinuous for certain values of \(x\). A numerical example of such a model is shown in Section 4. The method is numerically efficient for large dimensions. The linearization method is applied to the extended Kalman filter algorithm, which is an optimal minimum variance estimator for nonlinear systems.
scientific article; zbMATH DE number 1192940
scientific article