Eduardo M. A. M. Mendes

Technical communique: Reducing conservativeness in recent stability conditions of TS fuzzy systems

In this correspondence a new simple strategy for reducing the conservativeness in stability analysis of continuous-time Takagi-Sugeno fuzzy systems based on fuzzy Lyapunov functions is proposed. This new strategy generalizes previous results.

Nonlinear Identification and Cluster Analysis of Chaotic Attractors from a Real Implementation of Chua's Circuit

This paper discusses the identification of nonlinear polynomial models from real data sets measured from an implementation of Chuas circuit. Data filtering issues are also discussed. The identified models are analyzed and validated and suggest that the major characteristics of the real system have been incorporated in the models. Lyapunov exponents, correlation and Lyapunov dimensions are estimated for the identified models and some practical difficulties concerning the estimation of such invariants are discussed. Cluster analysis of the identified models has been performed and it is in accord with a number of results reported in the literature. Five of the identified models are presented in the paper and because of their simple structure, it is believed that such models will be useful in the study of nonlinear dynamics of Chuas circuit.

GLOBAL NONLINEAR POLYNOMIAL MODELS: STRUCTURE, TERM CLUSTERS AND FIXED POINTS

In this paper, it is shown that the number and location of the fixed points of global nonlinear polynomials can be specified in terms of the term clusters and cluster coefficients of the respective models. It is also shown that if the structure (that is, the model basis) of a nonlinear polynomial model of degree l includes all possible terms then such a model will always have l nontrivial fixed points. In modeling problems, where the estimated polynomials are to reproduce fundamental invariants of the original system, this situation will not always be welcome and therefore suggests that the structure of polynomial models should be carefully chosen. Ways of achieving this are briefly mentioned.

Modeling Chaotic Dynamics With Discrete Nonlinear Rational Models

This paper investigates the application of discrete nonlinear rational models, a natural extension of the well-known polynomial models. Rational models are discussed in the context of two different problems: reconstruction of chaotic attractors from a time series and the estimation of static nonlinearities from dynamical data. Rational models are obtained via black box identification techniques which only need a relatively short data set. A simple modified algorithm is proposed to handle the noise thus providing a solution to one of the greatest obstacles for estimating rational models from real data. The suggested algorithm and related ideas are tested and discussed using Rosslers equations, real data collected from an implementation of Chuas circuit, logistic map, sine-map with cubic-type nonlinearities, tent map and a map of a feedback buck switching regulator model.

On Overparametrization of Nonlinear Discrete Systems

One of the subjects which has received a great deal of attention is the overparametrization problem. It is known that the dynamical performance of the model representations deteriorates if the respective model structure is too complex. This paper investigates the problem of model overparametrization. Two new types of overparametrization, fixed-point and dimension overparametrization, are introduced and based upon this a new procedure for improving structure detection of nonlinear models is developed. This procedure uses all the information from the cluster cancellation and the location of the fixed points. Numerous examples are given to illustrate the ideas.

A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions

In this letter, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. The exponent is estimated from the slope of the line derived from the lower bound error when considering two interval extensions of the original system. It is shown that the algorithm is robust, fast and easy to implement and can be considered as alternative to other algorithms available in the literature. The method has been successfully tested in five well-known systems: Logistic, Henon, Lorenz and Rossler equations and the Mackey–Glass system.

Coherence estimate between a random and a periodic signal: Bias, variance, analytical critical values, and normalizing transforms

The present work deals with recent results on the sampling distribution of the magnitude-squared coherence (also called just coherence) estimate between a random (Gaussian) and a periodic signal, in order to obtain analytical critical values, alternative expressions for the probability density function (PDF) as well as the variance and bias of the estimate. A comparison with the more general case of coherence estimation when both signals are Gaussian was also provided. The results indicate that the smaller the true coherence (TC) values the closer both distributions become. The behaviour of variance and bias as a function of the number of data segments and the TC is similar for both coherence estimates. Additionally, the effect of a normalizing function (Fishers z transform) in the coherence estimated between a random and a periodic signal was also evaluated and normality has been nearly achieved. However, the variance was less equalized in comparison with coherence estimate between two Gaussian signals.

Displacement in the parameter space versus spurious solution of discretization with large time step

In order to investigate a possible correspondence between differential and difference equations, it is important to possess discretization of ordinary differential equations. It is well known that when differential equations are discretized, the solution thus obtained depends on the time step used. In the majority of cases, such a solution is considered spurious when it does not resemble the expected solution of the differential equation. This often happens when the time step taken into consideration is too large. In this work, we show that, even for quite large time steps, some solutions which do not correspond to the expected ones are still topologically equivalent to solutions of the original continuous system if a displacement in the parameter space is considered. To reduce such a displacement, a judicious choice of the discretization scheme should be made. To this end, a recent discretization scheme, based on the Lie expansion of the original differential equations, proposed by Monaco and Normand-Cyrot will be analysed. Such a scheme will be shown to be sufficient for providing an adequate discretization for quite large time steps compared to the pseudo-period of the underlying dynamics.

Multiobjective nonlinear system identification: a case study with thyristor controlled series capacitor (TCSC)

This paper deals with multiobjective nonlinear system identification applied when modelling the relation of firing angle and equivalent reactance of a thyristor controlled series capacitor (TCSC). The mathematical representation chosen is NARMAX (Nonlinear AutoRegressive Moving Average with eXogenous inputs) due to its capability in modelling nonlinear systems and in using prior information. The methodology for incorporation of prior knowledge is presented, and particular attention is given to the case of using information about resonant static response.

On Identifying Global Nonlinear Discrete Models from Chaotic Data

This paper investigates the identification of global discrete nonlinear models from chaotic data. It is shown that when chaotic data from a nonlinear system do not contain enough information about all the fixed points, the usual model selection procedures can select different local nonlinear models.