Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)

A pragmatic and systematic statistical analysis for identification of industrial robots

Identification of industrial robots is a prolific topic that has been deeply investigated over the last three decades. The standard method is based on the use of the inverse dynamic model and the least-squares estimation (IDIM-LS method) while robots are operating in closed loop by tracking exciting trajectories. Recently, in order to secure the consistency of the parameters estimates, an instrumental variable (IV) approach, called IDIM-IV method, has been designed and experimentally validated. However, the statistical analysis of estimates was not treated. Surprisingly, this topic is rarely addressed in mechatronics whereas it has been deeply investigated in automatic control. This paper aims at bridging the gap between these two communities by presenting a pragmatic statistical analysis of the IDIM-IV estimates. This analysis consists of a two-step procedure: first, the consistency of the IDIM-IV estimates is validated by the Revised Durbin-Wu-Hausman test, and then the statistical analysis of the IDIM-IV residuals is treated. This two-step approach is experimentally validated on the TX40 robot

Some aspects of estimation for vector time series models

This thesis is primarily concerned with some aspects of estimation for vector autoregressive moving average models which are in their appropriate echelon canonical forms. We restrict attention to the most straightforward part of the modelling procedure, namely, the estimation for fixed values of the Kronecker indices of the structural parameters using ordinary least squares, Gaussian (maximum) likelihood and generalized least squares methods, respectively. Our primary objective here is to give a systematic account of these procedures for handling data and also to provide a thorough exposition of the mathematical details that underlie the techniques. In addition to these abstract mathematical derivations, emphasis will be placed on the practical aspects of the procedures. The discussion of these various issues is organized into six chapters as follows: In Chapter 1 we introduce the class of models and assumptions upon which the results obtained in the thesis are based, and the justification for adopting an echelon structure for such models is also provided. This introductory chapter concludes with a description of the identification procedures for echelon canonical forms. Chapter 2 considers the estimation of the structural parameters using maximum (Gaussian) likelihood procedure and the asymptotic properties of the corresponding estimators are presented. In the evaluation of the parameter estimates, however, explicit expressions are derived for the gradient vector and (approximate) Hessian matrix of the log likelihood function in relatively simple terms. Chapter 3 commences with a procedure for evaluating the least squares estimators. Consistency and asymptotic normality results are established. Chapter 4 assesses the asymptotic relative efficiency of the Gaussian and least squares estim ators via the variance-covariance matrices of the limiting normal distributions obtained in Chapters 2 and 3, respectively. Situations under which substantial loss or gain in efficiency could be expected are discussed and illustrated with some numerical examples. Chapter 5 is devoted to a detailed discussion of the generalized least squares (GLS) procedure for param eter estimation. In particular, the theoretical aspect of the relationship between the GLS and Gaussian estimation methods is investigated and the asymptotic convergence of the GLS estim ator to the Gaussian estim ator is established. Also, an alternative numerical method for implementing the GLS procedure is proposed and some simulation results are presented to illustrate the theory. Finally, in Chapter 6, a method for generating a stable spectral factor from an unstable v x v full rank polynomial operator using closed form algebraic manipulations is proposed. An application of the technique is illustrated and the implementation of the method in the statistical context of system estimation is discussed

A new kernel-based approach for overparameterized Hammerstein system identification

In this paper we propose a new identification scheme for Hammerstein systems, which are dynamic systems consisting of a static nonlinearity and a linear time-invariant dynamic system in cascade. We assume that the nonlinear function can be described as a linear combination of $p$ basis functions. We reconstruct the $p$ coefficients of the nonlinearity together with the first $n$ samples of the impulse response of the linear system by estimating an $np$-dimensional overparameterized vector, which contains all the combinations of the unknown variables. To avoid high variance in these estimates, we adopt a regularized kernel-based approach and, in particular, we introduce a new kernel tailored for Hammerstein system identification. We show that the resulting scheme provides an estimate of the overparameterized vector that can be uniquely decomposed as the combination of an impulse response and $p$ coefficients of the static nonlinearity. We also show, through several numerical experiments, that the proposed method compares very favorably with two standard methods for Hammerstein system identification.Comment: 17 pages, submitted to IEEE Conference on Decision and Control 201

Instrumental variables quantile regression for panel data with measurement errors

This paper develops an instrumental variables estimator for quantile regression in panel data with fixed effects. Asymptotic properties of the instrumental variables estimator are studied for large N and T when Na/T ! 0, for some a > 0. Wald and Kolmogorov-Smirnov type tests for general linear restrictions are developed. The estimator is applied to the problem of measurement errors in variables, which induces endogeneity and as a result bias in the model. We derive an approximation to the bias in the quantile regression fixed effects estimator in the presence of measurement error and show its connection to similar effects in standard least squares models. Monte Carlo simulations are conducted to evaluate the finite sample properties of the estimator in terms of bias and root mean squared error. Finally, the methods are applied to a model of firm investment. The results show interesting heterogeneity in the Tobin’s q and cash flow sensitivities of investment. In both cases, the sensitivities are monotonically increasing along the quantiles

Sparse modeling of categorial explanatory variables

Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $L_1$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS355 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Regularization and Model Selection with Categorial Effect Modifiers

The case of continuous effect modifiers in varying-coefficient models has been well investigated. Categorial effect modifiers, however, have been largely neglected. In this paper a regularization technique is proposed that allows for selection of covariates and fusion of categories of categorial effect modifiers in a linear model. It is distinguished between nominal and ordinal variables, since for the latter more economic parametrizations are warranted. The proposed methods are illustrated and investigated in simulation studies and real world data evaluations. Moreover, some asymptotic properties are derived

Using the partial least squares (PLS) method to establish critical success factor interdependence in ERP implementation projects

This technical research report proposes the usage of a statistical approach named Partial Least squares (PLS) to define the relationships between critical success factors for ERP implementation projects. In previous research work, we developed a unified model of critical success factors for ERP implementation projects. Some researchers have evidenced the relationships between these critical success factors, however no one has defined in a formal way these relationships. PLS is one of the techniques of structural equation modeling approach. Therefore, in this report is presented an overview of this approach. We provide an example of PLS method modelling application; in this case we use two critical success factors. However, our project will be extended to all the critical success factors of our unified model. To compute the data, we are going to use PLS-graph developed by Wynne Chin.Postprint (published version