《Approximation of dynamical time-variant systems by continuous-time recurrent neural networks》.pdf

656 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 10, OCTOBER 2005 Approximation of Dynamical Time-Variant Systems by Continuous-Time Recurrent Neural Networks Xiao-Dong Li, John K. L. Ho, and Tommy W. S. Chow Abstract— This paper studies the approximation ability of con- tory to the equilibrium, there are a considerable amount of re- tinuous-time recurrent neural networks to dynamical time-variant sults on the approximation capability of the NN reported. For systems. It proves that any finite time trajectory of a given dynam- instance, it has been mathematically proved that a given con- ical time-variant system can be approximated by the internal state of a continuous-time recurrent neural network. Given several spe- tinuous mapping on a compact set could be approximately re- cial forms of dynamical time-variant systems or trajectories, this alized by using a three-layer FNN to any precision [3]–[5]. Li paper shows that they can all be approximately realized by the in- [6] showed that a discrete-time trajectory on a closed finite in- ternal state of a simple recurrent neural network. terval could be represented exactly using a discrete-time RNN. Index Terms—Approximation, dynamical time-variant systems, Jin and Nikiforuk [7] also studied the approximation problem recurrent neural networks. of approximating nonlinear discrete-time state-space trajecto- ries with input using discrete-time RNN. In the case of contin- uous-time RNN, Funahashi and Nakamura [8] studied the ap- I. INTRODUCTION