A Fourier/Hopfield neural network for identification of nonlinear periodic systems

This paper presents a method based on using a Hopfield neural network for the identification of nonlinear periodic systems. The system model is obtained by calculating the optimum coefficients of the expansion of the system over a set of Fourier basis functions. The identification process is accomplished using a "Fourier/Hopfield Neural Network". Fourier basis were chosen because they are best suited to analyzing periodic functions. Initially, the signals are expanded over their first harmonic Fourier base. A Hopfield neural network adapts the expansion coefficient till the relative error reaches a specified threshold value, at which point the neural network is approaching a local minimum. The global error is then computed, the number of Fourier basis is incremented by one, and the process is repeated till the global error becomes smaller than a desired minimum. The network would have then approached a global minimum. The Fourier/Hopfield Neural Network technique was applied to a nonlinear periodic function composed of sine and cosine waves of various powers. For a global and relative error threshold of 0.005, two basis functions were required with a final error of 0.004. Another simulation was run with a smaller error threshold of 10^-4. Six basis functions were then needed to obtain a final error of the order of 10^-5. In both simulations, the neural network converged to a global minimum, with a limited number of basis functions, showing thus the successful feasibility of using a Hopfield neural network in conjunction with Fourier analysis for the identification of nonlinear periodic systems.