An Approach to Parametric Nonlinear Least Square Optimization and Application to Task-Level Learning Control

This paper considers a parametric nonlinear least square (NLS) optimization problem. Unlike a classical NLS problem statement, we assume that a 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 a numerical solution of the formulated problem. The proposed algorithm approximates the nonlinear system in a vicinity of the optimum by expanding it into a series of 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. The proposed approach is applied to task-level learning control of a two-link flexible arm. Each evaluation of the system in the optimization process means completing a controlled motion of the arm. In the simulation example, the controlled motions take only about 1.5 periods of the lowest eigenfrequency oscillations. The algorithm controls this strongly nonlinear oscillatory system very efficiently. Without any prior knowledge of the system dynamics, it achieves a satisfactory control of arbitrary arm motions after only 500 learning (optimization) iterations.