A philosophy for the modelling of realistic nonlinear systems

First published in Proceedings of the American Mathematical Society in volume 132, number 2, by the American Mathematical Society Copyright © 2003 American Mathematical SocietyA nonlinear dynamical system is modelled as a nonlinear mapping from a set of input signals into a corresponding set of output signals. Each signal is specified by a set of real number parameters, but such sets may be uncountably infinite. For numerical simulation of the system each signal must be represented by a finite parameter set and the mapping must be defined by a finite arithmetical process. Nevertheless the numerical simulation should be a good approximation to the mathematical model. We discuss the representation of realistic dynamical systems and establish a stable approximation theorem for numerical simulation of such systems.Phil Howlett, Anatoli Torokhti, Charles Pearc

Best causal mathematical models for a nonlinear system

©2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.We provide new causal mathematical models of a nonlinear system S which are specifications of a nonlinear operator P/sub p/ of degree p=1,2,.... The operator P/sub p/ is determined from a special orthogonalization procedure and minimization of the mean squared difference between outputs of S and P/sub p/. As a result, these models have smallest possible associated errors in the class of such operators P/sub p/. The causality condition is implemented through the use of specific matrices called lower trapezoidal. The associated computational work is reduced by the use of the orthogonalization procedure. We provide a strict justification of the proposed approach including theorems on an explicit representatoin of the models' parameters, and theorems on the associated error representation. The possible extensions of the proposed approach and its potential applications are outlined.Anatoli Torokhti, Phil Howlett, and Charles Pearc

A New Technique for Multidimensional Signal Compression

The problem of efficiently compressing a large number, L, of N dimensional signal vectors is considered. The approach suggested here achieves efficiencies over current pre-processing and Karhunen-Loeve techniques when both L and N are large

Combined Reduced-Rank Transform

We propose and justify a new approach to constructing optimal nonlinear transforms of random vectors. We show that the proposed transform improves such characteristics of rank-reduced transforms as compression ratio, accuracy of decompression and reduces required computational work. The proposed transform Tp is presented in the form of a sum with p terms where each term is interpreted as a particular rank-reduced transform. Moreover, terms in Tp are represented as a combination of three operations Fk, Qk and φk with k = 1,...,p. The prime idea is to determine Fk separately, for each k = 1,...,p, from an associated rank-constrained minimization problem similar to that used in the Karhunen-Loève transform. The operations Qk andφk are auxiliary for finding Fk. The contribution of each term in Tp improves the entire transform performance. A corresponding unconstrained nonlinear optimal transform is also considered. Such a transform is important in its own right because it is treated as an optimal filter without signal compression. A rigorous analysis of errors associated with the proposed transforms is given

Optimal recursive estimation of raw data

The original publication is available at www.springerlink.comWe present a new approach to the optimal estimation of random vectors. The approach is based on a combination of a specific iterative procedure and the solution of a best approximation problem with a polynomial approximant. We show that the combination of these new techniques allow us to build a computationally effective and flexible estimator. The strict justification of the proposed technique is provided.Anatoli Torokhti, Phil Howlett and Charles Pearc

An optimal linear filter for random signals with realisations in a separable Hilbert space

© Australian Mathematical Society 2003Let u be a random signal with realisations in an infinitedimensional vector space X and v an associated observable random signal with realisations in a finitedimensional subspace Y X. We seek a pointwisebest estimate of u using a bounded linear filter on the observed data vector v. When x is a finitedimensional Euclidean space and the covariance matrix for v is nonsingular, it is known that the best estimate Ou of u is given by a standard matrix expression prescribing a linear meansquare filter. For the infinitedimensional Hilbert space problem we show that the matrix expression must be replaced by an analogous but more general expression using bounded linear operators. The extension procedure depends directly on the theory of the Bochner integral and on the construction of appropriate Hilbert Schmidt operators. An extended example is given.P. G. Howlett, C. E. M. Pearce and A. P. Torokht

On nonlinear operator approximation with preassigned accuracy

© 2003 Plenum Publishing CorporationIn this paper, we provide a state-of-the-art survey of some recent methods in nonlinear operator approximation theory and its applications. We give existence theorems for approximating operators. and discuss corresponding numerical schemes. The new results are natural but very specific extensions of known techniques to the so-called realistic operator approximation.Phil Howlett, Charles Pearce and Anatoli Torokht