Nonlinear system

This paper is devoted to a discussion of the Discrete Fourier Transform (DFT) representation of a chaotic finite-duration sequence. This representation has the advantage that is itself a finite-duration sequence corresponding to samples equally spaced in the frequency domain. The Fast Fourier Transform (FFT) algoritm allows us an effective computation, and it can be applied to a relatively short time series. DFT representation requirements were analized and applied for determining the order-chaos transition in a nonlinear system described by the equation $x[n+1]=rx[n](1-x[n])$. Its effectiveness was demonstrated by comparing the results with those obtained by calculating the largest Lyapounov exponent for the time series set, obtained from the logistic equation.

A randomization method is developed for the calculation of covariation between multiple variables that are linked nonlinearly to a dependent variable. Covariation is a phenomenon often invoked in the study of movement coordination to capture the fact that in coordinated movement the outcome shows greater than expected consistency from the variability in the component processes. However, in most cases, the problem is that more than two variables covary in a nonlinear fashion, which makes quantification with the bivariate linear covariation and correlation coefficient inapplicable. This paper presents a generalization of the calculation of linear bivariate covariance using a variant of a randomization method that is based on the comparison between the empirically measured variability in the outcome and a covariation-free variability. The latter can be estimated by permuting data sets. A generalized correlation coefficient is derived, and it is shown how errors of estimation can be qua...

– An important part of modeling with Volterra and Wiener models is the accurate measurement of the model coefficients or kernels. The formulation of Wiener model of nonlinear systems has a major breakthrough for kernel measurement. The orthogonality of the Wiener functionals for a white Gaussian input allowed the Wiener kernels to be easily measured using cross-correlation techniques. The kernel measurement techniques are widely explored in this paper. Some properties of the third order Volterra kernel and of the input-output relation are shown. Finally original contributions have been made regarding signal reconstruction using Volterra and Wiener models.

In aceasta lucrare este prezentată problema reglarii ieşirii dacă se consideră sisteme neliniare MIMO. In acest context formele normale pentru reprezentarea sistemelor neliniare sunt de mare interes. Două forme normale pentru sisteme neliniare MIMO sunt prezentate. Prima releva vectorul de grade relative si poate fi aplicată pentru sisteme pătrate inversabile (forma normală clasică), a doua poate fi utilizată şi pentru sisteme nepatrate şi dă detalii/informaţii despre proprietatile de inversabilitate ale sistemului scris în această formă (forma normala recentă). Condiţiile necesare pentru rezolvarea problemei reglarii ieşirii pentru sisteme scrise în forma clasică şi cea recentă sunt prezentate. Din discuţia prezentată rezultă că problema pusă se poate rezolva sub presupuneri mai puţin restrictive dacă sistemul este scris în forma normală obtinută prin algoritmul structurii zerourilor la infinit (forma recenta). Această abordare permite de asemenea discutarea problemei de reglare şi...

This work proposes a methodology of identifying linear parameter varying (LPV) models for nonlinear systems. First, linear local models in some operating points, by applying standard identifications procedures for linear systems in time domain, are obtained. Next, a LPV model with linear fractional dependence (LFR) with respect to measured variables is fitted with the condition of containing all the linear models identified in previous step (differential inclusion). The fit is carried out using nonlinear least squares algorithms. Finally, this identification methodology will then be applied to a nonlinear turbocharged diesel engine.

In this work, we study the steady-states vibrations of the “Koch snowflake drum”, both numerically and by means of computer graphics. In particular, we approximate the smallest 50 eigenvalues (or “frequencies” of the drum), along with the corresponding eigenfunctions (called “snowflake harmonics”) of the Dirichlet Laplacian on the Koch snowflake domain. We describe the numerical methods used in the computations, and we display graphical representations of a selected set of the eigenfunctions (as well as of their gradients). In the case of the first harmonic, the graphical results agree with mathematically derived results (by Lapidus and Pang) concerning gradient behavior (“blow up” or “infinite stress” of the membrane) on the boundary and suggest further conjectures regarding the higher eigenfunctions. According to earlier work by the physicist Sapoval and his collaborators, this research may help better understand the formation and “stabilization” of fractal structures (e.g., coast...

Presents a globally asymptotically stabilizing (GAS) controller for regulation and dynamic positioning of ships, using only position measurements. It is assumed that these are corrupted with white noise hence a passive observer which reconstructs the rest of the states is applied. The observer produces noise-free estimates of the position, the slowly varying environmental disturbances and the velocity which are used

A universal controller is designed for cascade systems, involving dynamic uncertainty, unknown nonlinearities, exogenous disturbances and/or time-varying parameters, capable of guaranteeing prescribed performance for the output tracking error, as well as uniformly bounded signals in the closed loop. By prescribed performance we mean that the output tracking error should converge to a predefined arbitrarily small residual set, with convergence rate no less than a certain prespecified value, exhibiting maximum overshoot less than a sufficiently small preassigned constant. The proposed control scheme is of low complexity, utilizes partial state feedback and requires reduced levels of a priori system knowledge. The results can be easily extended to systems affected by bounded state measurement errors, as well as to MIMO nonlinear systems in block triangular form. Simulations clarify and verify the approach.

Abstract—In this paper a nonlinear observer which synthe-sizes sliding mode techniques and neural state space models is proposed and is applied for robust fault diagnosis in a class of nonlinear systems. The sliding mode term is utilized to eliminate the effect of system uncertainties, ...

This paper investigates the system stability of a sampled-data fuzzy-model-based control system, formed by a nonlinear plant and a sampled-data fuzzy controller connected in a closed loop. The sampled-data fuzzy controller has an advantage that it can be implemented using a microcontroller or a digital computer to lower the implementation cost and time. However, discontinuity introduced by the sampling activity complicates the system dynamics and makes the stability analysis difficult compared with the pure continuous-time fuzzy control systems. Moreover, the favourable property of the continuous-time fuzzy control systems which is able to relax the stability analysis result vanishes in the sampled-data fuzzy control systems. A Lyapunov-based approach is employed to derive the LMI-based stability conditions to guarantee the system stability. To facilitate the stability analysis, a switching fuzzy model consisting of some local fuzzy models is employed to represent the nonlinear plant to be controlled. The comparatively less strong nonlinearity of each local fuzzy model eases the satisfaction of the stability conditions. Furthermore, membership functions of both fuzzy model and sampled-data fuzzy controller are considered to alleviate the conservativeness of the stability analysis result. A simulation example is given to illustrate the merits of the proposed approach.