Deterministic learning of nonlinear dynamical systems

In this paper, we present an approach for neural networks (NN) based identification of unknown nonlinear dynamical systems undergoing periodic or periodic-like (recurrent) motions. Among various types of NN architectures, we use a dynamical version of the localized RBF neural network, which is shown to be particularly suitable for identification in a dynamical framework. With the associated properties of localized RBF networks, especially the one concerning the persistent excitation (PE) condition for periodic trajectories, the proposed approach achieves sufficiently accurate identification of system dynamics in a local region along the experienced system trajectory. In particular, for neurons whose centers are close to the trajectories, the neural weights converge to a small neighborhood of a set of optimal values; while for other neurons with centers far away from the trajectories, the neural weights are not updated and are almost unchanged. The proposed approach implements a sort of "deterministic learning" in the sense that deterministic features of nonlinear dynamical systems are learned not by algorithms from statistical principles, but in a dynamical, deterministic manner, utilizing results from adaptive systems theory. The nature of this deterministic learning is closely related to the exponentially stability of a class of nonlinear adaptive systems. Simulation studies are included to demonstrate the effectiveness of the proposed approach.

[1]  Yuichi Nakamura,et al.  Approximation of dynamical systems by continuous time recurrent neural networks , 1993, Neural Networks.

[2]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[3]  Marios M. Polycarpou,et al.  Modelling, Identification and Stable Adaptive Control of Continuous-Time Nonlinear Dynamical Systems Using Neural Networks , 1992, 1992 American Control Conference.

[4]  Songwu Lu,et al.  Robust nonlinear system identification using neural-network models , 1998, IEEE Trans. Neural Networks.

[5]  Visakan Kadirkamanathan,et al.  Stable sequential identification of continuous nonlinear dynamical systems by growing radial basis function networks , 1996 .

[6]  C. Micchelli.  Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[7]  S. Strogatz.  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering , 1995 .

[8]  Robert M. Sanner,et al.  Gaussian Networks for Direct Adaptive Control , 1991, 1991 American Control Conference.

[9]  Manolis A. Christodoulou,et al.  Dynamical Neural Networks that Ensure Exponential Identification Error Convergence , 1997, Neural Networks.

[10]  Dimitry M. Gorinevsky,et al.  On the persistency of excitation in radial basis function network identification of nonlinear systems , 1995, IEEE Trans. Neural Networks.

[11]  Jay A. Farrell,et al.  Stability and approximator convergence in nonparametric nonlinear adaptive control , 1998, IEEE Trans. Neural Networks.

[12]  Ying Chen,et al.  Identifying chaotic systems via a Wiener-type cascade model , 1997 .

[13]  D. Broomhead,et al.  Radial Basis Functions, Multi-Variable Functional Interpolation and Adaptive Networks , 1988 .

[14]  Shuzhi Sam Ge,et al.  Direct adaptive NN control of a class of nonlinear systems , 2002, IEEE Trans. Neural Networks.

[15]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[16]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[17]  F. J. Narcowich,et al.  Persistency of Excitation in Identification Using Radial Basis Function Approximants , 1995 .

[18]  O. Rössler.  An equation for continuous chaos , 1976 .

[19]  Jun Wang,et al.  On-line learning of dynamical systems in the presence of model mismatch and disturbances , 2000, IEEE Trans. Neural Networks Learn. Syst..

[20]  Simon Haykin,et al.  Neural Networks: A Comprehensive Foundation , 1998 .

[21]  F. Girosi,et al.  Networks for approximation and learning , 1990, Proc. IEEE.

[22]  Li Yan,et al.  Nonlinear System Identification Using Lyapunov Based Fully Tuned Dynamic RBF Networks , 2000, Neural Processing Letters.

[23]  Robert M. Sanner,et al.  Stable Recursive Identification Using Radial Basis Function Networks , 1992, 1992 American Control Conference.