Statistica Sinica

James C. Spall and John A. Cristion

Abstract:Neural networks (NNs) have recently attracted much attention in the mathematical modeling community. One of the most promising areas for the use of NNs is in the control of complex (multivariate) dynamic systems when the nonlinear equations governing the system are not known. In this paper, a NN is used to model the resulting unknown control lawwithoutthe need to construct a separate model (NN or other type) for the unknown process dynamics. This is a challenging statistical problem in that the weights (parameters) of the NN are estimated concurrently with controlling the system. The weight estimation uses a form of stochastic approximation that relies on an approximation to the gradient of the underlying loss function. The implementation here uses a simple smoothing operation when constructing the gradient approximation: namely, the gradient approximation at any iteration is formed as a combination of the previous approximation and a new simultaneous perturbation gradient estimate. This paper shows that this smoothing idea is often useful in improving the ability of the NN-based controller to have the system perform in the desired way. Aside from presenting the smoothing idea, this paper includes brief introductions to the fields of nonlinear adaptive control, neural networks, and stochastic approximation.

Key words and phrases:Nonlinear control systems, stochastic approximation, neural network learning, gradient estimation.