How piecewise affine neural networks can generate a stable nonlinear control

Abstract

Deals with the difficulty of designing an artificial neural network to control a nonlinear dynamical system. It is known that controlling a dynamical system that is not exactly modelled is difficult. The capabilities of artificial neural networks in the area of nonlinear control have been explored for instance by Jagannarthan (1998) and Sontag (1997). We have shown (1998) that piecewise affine perceptrons (PAP), a subclass of perceptrons, can be initialized to control a given nonlinear system. Besides they have the same useful properties as classical perceptrons: the universal approximation property and the generalization property. Here we give stability results for nonlinear systems controlled by PAPs. The stability results given are obtained by constructing piecewise quadratic Lyapunov functions. The paper first establishes a result about PAP that is used to adapt a result about stability of piecewise affine continuous-time systems, then a similar result is obtained for discrete-time ones, after that a methodology to tune PAP for control of nonlinear systems is given and finally this is illustrated by an example: the control of an engine combustion model by a PAP.