S.A. Billings

Orthogonal least squares methods and their application to non-linear system identification

Abstract Identification algorithms based on the well-known linear least squares methods of gaussian elimination, Cholesky decomposition, classical Gram-Schmidt, modified Gram-Schmidt, Householder transformation, Givens method, and singular value decomposition are reviewed. The classical Gram-Schmidt, modified Gram-Schmidt, and Householder transformation algorithms are then extended to combine structure determination, or which terms to include in the model, and parameter estimation in a very simple and efficient manner for a class of multivariate discrete-time non-linear stochastic systems which are linear in the parameters.

Input-output parametric models for non-linear systems Part II: stochastic non-linear systems

Recursive input-output models for non-linear multivariate discrete-time systems are derived, and sufficient conditions for their existence are defined. The paper is divided into two parts. The first part introduces and defines concepts such as Nerode realization, multistructural forms and results from differential geometry which are then used to derive a recursive input-output model for multivariable deterministic non-linear systems. The second part introduces several examples, compares the derived model with other representations and extends the results to create prediction error or innovation input-output models for non-linear stochastic systems. These latter models are the generalization of the multivariable ARM AX models for linear systems and are referred to as NARMAX or Non-linear AutoRegressive Moving Average models with exogenous inputs.

Correlation based model validity tests for non-linear models

Correlation based model validity tests are derived that detect omitted linear and non-linear dynamic terms in estimated models.

Orthogonal parameter estimation algorithm for non-linear stochastic systems

An orthogonal parameter estimation algorithm is derived which allows each parameter in a linear–in–the–parameters non–linear difference equation model to be estimated sequentially and quite independently of the other parameters in the model. The algorithm can be applied for any persistently exciting input and provides both unbiased estimates in the presence of correlated noise and an indication of which terms to include in the model. Several simulated examples are included to demonstrate the effectiveness of the algorithm.

Analysis and Design of Variable Structure Systems Using a Geometric Approach

Multivariable variable structure systems in the sliding mode are studies using a geometric approach. The properties of system order reduction and disturbance rejection are proved using projector theory. New design methods for the choice of switching hyperplanes are derived for the closed-loop eigenvalue/eigenvector assignment problem.

Model selection and validation methods for non-linear systems

The theory of hypothesis testing is used to select a model with the correct structure, and the relation of such a method to the AIC and FPE criteria is investigated. Parametric validation and correlation validation methods are developed for non-non-linear difference equation models. Several shortcomings of traditional methods, especially when applied to non-linear systems, are described.

A prediction-error and stepwise-regression estimation algorithm for non-linear systems

The identification of non-linear systems based on a NARMAX (Non-linear AutoRegressive Moving-Average model with exogenous inputs) model representation is considered, and a combined stepwise-regression/prediction-error estimation algorithm is derived. The stepwise-regression routine determines the model structure by detecting significant terms in the model, while the prediction-error algorithm provides optimal estimates of the final model parameters. Implementation of the algorithms is discussed in detail, and several simulated examples and industrial applications are included to illustrate that parsimonious models of non-linear systems can be identified using the algorithm.

Spectral analysis for non-linear systems, Part I: Parametric non-linear spectral analysis

Abstract A new theory of spectral analysis for non-linear systems is introduced. The method consists of estimating the parameters in a NARMAX model description of the system and then computing the generalised frequency response functions directly from the estimated model. The paper is divided into three parts. Part I introduces a methodology for estimating the models and computing the frequency response functions. Part II concentrates on the interpretation of the non-linear frequency response functions and Part III which will be published in the next issue describes a series of case study examples and illustrates in detail how to apply the algorithms to real systems.

Recursive algorithm for computing the frequency response of a class of non-linear difference equation models

Abstract A recursive algorithm is developed which is both computationally compact and enables the nth-order frequency response functions of a large class of non-linear difference equation models to be found without restriction on the order n.

Extended model set, global data and threshold model identification of severely non-linear systems

Abstract New parameter estimation algorithms, based on an extended model set, a global data model and a threshold model formulation, are derived for identifying severely non-linear systems. It is shown that in each case an integrated structure determination and parameter estimation algorithm based on an orthogonal decomposition of the regression matrix can be derived to provide procedures for identifying parsimonious models of unknown systems with complex structure. Simulation studies are included to illustrate the techniques discussed.