Uniform approximation of discrete-time nonlinear systems

We consider a large class of discrete-time control systems containing a dynamic linear part and a memoryless nonlinear element and show that such systems can be uniformly approximated using a TDNN, a two-stage dynamic neural structure consisting of a bank of delay elements followed by a memoryless nonlinear element. In addition, we bound the complexity of the TDNN needed to uniformly approximate the system to within a given maximum error ε. Specifically, we bound the number of delay elements a by giving constants ρ₁ and ρ₂ such that α>ρ₁ log(ρ₂ /ε) suffices, and we show that the nonlinear element satisfies a certain Lipschitz condition. Our assumptions are along the lines of the circle condition for stability, and the concept of approximately finite memory plays a central role in our results