An algorithm for on-line parametric nonlinear least square optimization

This paper considers a parametric nonlinear least square (NLS) optimization problem. In extension of a classical NLS problem statement, it is assumed that the nonlinear optimized system depends on two arguments: an input vector and a parameter vector. The input vector can be modified to optimize the system, while the parameter vector changes from one optimization iteration to another and is not controlled. The optimization process goal is to find a dependence of the optimal input vector on the parameter vector, where the optimal input vector minimizes a quadratic performance index. The paper proposes an extension of the Levenberg-Marquardt algorithm for numerical solution of the formulated problem. The proposed algorithm approximates the nonlinear system by an expansion into a series of the parameter vector functions, affine in the input vector. In particular, a radial basis function network expansion is considered. The convergence proof for the algorithm is presented