STRUCTURED TOTAL LEAST NORM FOR NONLINEAR PROBLEMS

An extension of the recently developed structured total least norm (ST LN) problem formulation is described for solving a class of nonlinear parameter estimation problems. STLN is a problem formulation for obtai ning an approximate solution to the overdetermined linear system Ax ap proximate to b preserving the given affine structure in A or [A \ b], where errors can occur in both the vector b and the matrix A. The appr oximate solution can be obtained to minimize the error in the L-p norm , where p = 1, 2, or infinity. In the extension of STLN to nonlinear p roblems, the elements of A may be differentiable nonlinear functions o f a parameter vector, whose value needs to be approximated. We call th is extension structured nonlinear total least norm (SNTLN). The SNTLN problem is formulated and its solution by a modified STLN algorithm is described. Optimality conditions and convergence for the 2-norm case are presented. Computational tests were carried out on an overdetermin ed system with Vandermonde structure and on two nonlinear parameter es timation problems. In these problems, both the coefficients and the un known parameters were to be determined. The computational results demo nstrate that the SNTLN algorithm recovers good approximations to the c orrect values of both the coefficients and parameters, in the presence of noise in the data and poor initial estimates of the parameters. It is also shown that the SNTLN algorithm with the 1-norm minimization i s robust with respect to outliers in the data.