A Kernel Method for Non-Linear Systems Identification-Infinite Degree Volterra Series Estimation

Volterra series expansions are widely used in analysing and solving the problems of nonlinear dynamical systems. However, the problem that the number of terms to be determined increases exponentially with the order of the expansion restricts its practical application. In practice, Volterra series expansions are truncated severely so that they may not give accurate representations of the original system. To address this problem, kernel methods are shown to be deserving of exploration. In this report, we make use of an existing result from the theory of approximation in reproducing kernel Hilbert space (RKHS) that has not yet been exploited in the systems identification field. An exponential kernel method, based on an RKHS called a generalised Fock space, is introduced, to model non linear dynamical systems and to specify the corresponding Volterra series expansion. In this way a nonlinear dynamical system can be modelled using a finite memory length, infinite degree Volterra series expansion, thus reducing the source of approximation error soley to truncation in time. We can also, in principle, recover any coefficient in the Volterra series.