Dynamic reconstruction of chaotic system based on exponential weighted online sequential extreme learning machine with kernel

Abstract

For the dynamic reconstruction of the chaotic dynamical system, a method of identifying an exponential weighted online sequential extreme learning machine with kernel(EW-KOSELM) is proposed. The kernel recursive least square (KRLS) algorithm is directly extended to an online sequential ELM framework, and weakens the effect of old data by introducing a forgetting factor. Meanwhile, the proposed algorithm can deal with the ever-increasing computational difficulties inherent in online kernel learning algorithms based on the ‘fixed-budget’ memory technique. The employed EW-KOSELM identification method is firstly applied to the numerical example of Duffing-Ueda oscillator for chaotic dynamical system based on simulated data, the qualitative and quantitative analysis for various validation tests of the dynamical properties of the original system as well as the identification model are carried out. A set of qualitative validation criteria is implemented by comparing chaotic attractors i.e. embedding trajectories, computing the corresponding Poincare mapping, plotting the bifurcation diagram, and plotting the steady-state trajectory i.e. the limit cycle between the original system and the identification model. Simultaneously, the quantitative validation criterion which includes computing the largest positive Lyapunov exponent and the correlation dimension of the chaotic attractors is also calculated to measure the closeness i.e. the approximation error between the original system and the identification model. The employed method is further applied to a practical implementation example of Chua's circuit based on the experimental data which are generated by sampling and recording the measured voltage across a capacitor, the inductor current from the double-scroll attractor, the measured voltage across a capacitor from the Chua's spiral attractor and an experimental time series from a chaotic circuit. The digital filtering technique is then used as a preprocessing approach, on the basis of wavelet denoising the measured data with lower signal-noise ratio (SNR) which can produce the double-scroll attractor or the spiral attractor, the reconstruction attractor of the identification model is compared with the reconstruction attractor from the experimental data for original system. The above experimental results confirm that the EW-FB-KOSELM identification method has a better performance of dynamic reconstruction, which can produce an accurate nonlinear model of process exhibiting chaotic dynamics. The identification model is dynamically equivalent or system approximation to the original system.