Parallel cascade identification and kernel estimation for nonlinear systems

We consider the representation and identification of nonlinear systems through the use of parallel cascades of alternating dynamic linear and static nonlinear elements. Building on the work of Palm and others, we show that any discrete-time finite-memory nonlinear system having a finite-order Volterra series representation can be exactly represented by a finite number of parallel LN cascade paths. Each LN path consists of a dynamic linear system followed by a static nonlinearity (which can be a polynomial). In particular, we provide an upper bound for the number of parallel LN paths required to represent exactly a discrete-time finite-memory Volterra functional of a given order. Next, we show how to obtain a parallel cascade representation of a nonlinear system from a single input-output record. The input is not required to be Gaussian or white, nor to have special autocorrelation properties. Next, our parallel cascade identification is applied to measure accurately the kernels of nonlinear systems (even those with lengthy memory), and to discover the significant terms to include in a nonlinear difference equation model for a system. In addition, the kernel estimation is used as a means of studying individual signals to distinguish deterministic from random behaviour, in an alternative to the use of chaotic dynamics. Finally, an alternate kernel estimation scheme is presented.

[1]  G. Palm.  On representation and approximation of nonlinear systems , 1978, Biological Cybernetics.

[2]  Michael C. Mackey,et al.  From Clocks to Chaos , 1988 .

[3]  Larry S. Liebovitch,et al.  Introduction to the Properties and Analysis of Fractal Objects, Processes, and Data , 1989 .

[4]  Gene H. Golub,et al.  Matrix computations , 1983 .

[5]  M. Korenberg.  Identifying nonlinear difference equation and functional expansion representations: The fast orthogonal algorithm , 2006, Annals of Biomedical Engineering.

[6]  Michael J. Korenberg,et al.  Some New Approaches To Nonlinear System Identification And Time-series Analysis , 1990, [1990] Proceedings of the Twelfth Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[7]  N. Wiener,et al.  Nonlinear Problems in Random Theory , 1964 .

[8]  G. Palm,et al.  On representation and approximation of nonlinear systems , 1979, Biological Cybernetics.

[9]  M. J. Korenberg,et al.  The identification of nonlinear biological systems: Wiener and Hammerstein cascade models , 1986, Biological Cybernetics.

[10]  Hun H. Sun,et al.  Nonlineaar System Identification Via Parallel Cascaded Structure , 1990, [1990] Proceedings of the Twelfth Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[11]  J. Dieudonne.  Foundations of Modern Analysis , 1969 .

[12]  M. Fréchet.  Sur les fonctionnelles continues , 1910 .

[13]  M. J. Korenberg,et al.  A robust orthogonal algorithm for system identification and time-series analysis , 1989, Biological Cybernetics.

[14]  Y. W. Lee,et al.  Measurement of the Wiener Kernels of a Non-linear System by Cross-correlation† , 1965 .

[15]  M. Korenberg.  Statistical Identification of Parallel Cascades of Linear and Nonlinear Systems , 1982 .

[16]  Vasilis Z. Marmarelis,et al.  Analysis of Physiological Systems , 1978, Computers in Biology and Medicine.

[17]  M. Korenberg,et al.  Exact orthogonal kernel estimation from finite data records: Extending Wiener's identification of nonlinear systems , 1988, Annals of Biomedical Engineering.