Piecewise Affine Neural Networks and Nonlinear Control

Abstract

Piecewise affine neural networks are identical to feedforward neural networks except that theiractivation function is continuous piecewise affine rather than sigmoidal. These neural networks canimplement quite generic continuous piecewise affine functions on polyhedral cells. Therefore theystand between a collection of local linear controls and a nonlinear control deduced straight fromthe nonlinear system. On the other hand, since the learning phase of a neural network by gradientretropropagation in a closed loop is difficult, a neural network accurate initialization based onlinearizations of the system to be controlled is a real benefit. Besides, piecewise affine neuralnetworks can be considered as approximations of neural networks with sigmoidal activation; thisallow some results to be extended to more standard perceptrons. The aim of the paper is to exhibitthe main properties of this new kind of neural networks and to show how the training of thesenetworks can be initialized from linear functions, especially in a control environment First, theaction of a piecewise affine neural network is specified by a partition of its input space suchthat on each polyhedral cell generated by this partition, the network action is affine. Then theslopes and the constants of these affine pieces are characterized as functions of the networkweights. The number of cells are determined as a function of the number of hidden neurons. Then abasic result is that in a given hypercube any continuous piecewise affine function can be emulatedby a piecewise affine neural network. We present a methodology to initialize piecewise affineneural networks for controlling nonlinear systems and shows an illustration.