Online Self-Learning Fuzzy Recurrent Stochastic Configuration Networks for Modeling Nonstationary Dynamics

The paper proposes a new method for identifying orders of input-output models for unknown nonlinear dynamic systems. The approach is based on the continuity property of the nonlinear functions which represent input-output models of continuous dynamic systems. The approach does not depend on any nonlinear function approximation methods and solely depends on the system's input-output data measured in experiments. By evaluating the modification of an index which is defined as Lipschitz number with the successive modification of model orders, the appropriate model orders can be determined simply and reliably. Theoretical background of the present approach is discussed. Several examples from chaotic dynamic systems, nonlinear plant models are presented to demonstrate the effectiveness of the present method.

The paper presents a review of the studies that were conducted at Energy Systems Institute (ESI) SB RAS in the field of mathematical modeling of nonlinear input-output dynamic systems with Volterra polynomials. The first part presents an original approach to identification of the Volterra kernels. The approach is based on setting special multi-parameter families of piecewise constant test input signals. It also includes a description of the respective software; presents illustrative calculations on the example of a reference dynamic system as well as results of computer modeling of real heat exchange processes. The second part of the review is devoted to the Volterra polynomial equations of the first kind. Studies of such equations were pioneered and have been carried out in the past decade by the laboratory of ill-posed problems at ESI SB RAS. A special focus in the paper is made on the importance of the Lambert function for the theory of these equations.

Multi-scale fuzzy modeling of multivariable discrete-time nonlinear dynamical systems is approached analytically in this paper. The underlying idea of this approach is to introduce the idea of multi-scale fuzzy quantization. More globally supported membership functions are used at coarser scales to capture global characteristics of the unknown system, whereas more locally supported membership functions are used at finer scales to fill in the local features. Starting from the coarsest scale down to finer ones, a projection method is used to extract a partial fuzzy-model (in terms of fuzzy membership functions and fuzzy rules) from the residual of the system at each scale until the residual drops below a prescribed modeling error bound. The multi-scale fuzzy model is then obtained by assembling the partial fuzzy models together. To reach a near optimal fuzzy model, a rule reduction technique has also been developed to delete some fuzzy rules. An illustrative example is used to demonstrate the effectiveness of the scheme.

In this paper, a model of nonlinear dynamic system with Lévy jumps based on cargo sorting DNA robot is studied. Firstly, nonlinear biochemical reaction system based on cargo sorting DNA robot model is established. Considering the influence of external disturbances on the system, nonlinear biochemical reaction system with Lévy jump is built and its dimensionality is reduced. Secondly, in order to prove that the built system conforms to the actual meaning, the existence and uniqueness of the system solution is verified. Next, the sufficient conditions for the completion of cargo pick-up of cargo sorting DNA robot and the continued sufficiency are introduced, and the progress of cargo sorting DNA robot under different noise intensities is analyzed. Then, it is proved that the positive recursion of the reaction can better describe and show the persistence of the system. Finally, numerical simulations verify the correctness of the theoretical results. The results show that the end of cargo pick-up with DNA robots for cargo sorting is closely related to the intensity of noise.KeywordsCargo sorting DNA robotNonlinear system modelLévy jumpLyapunov functionPositive recursion

This paper presents a new fuzzy inference system for modeling of nonlinear dynamic systems based on input and output data with measurement noise. The proposed fuzzy system has a number of fuzzy rules and parameter values of membership functions which are automatically generated using the extended relevance vector machine (RVM). The RVM has a probabilistic Bayesian learning framework and has good generalization capability. The RVM consists of the sum of product of weight and kernel function which projects input space into high dimensional feature space. The structure of proposed fuzzy system is same as that of the Takagi-Sugeno fuzzy model. However, in the proposed method, the number of fuzzy rules can be reduced under the process of optimizing a marginal likelihood by adjusting parameter values of kernel functions using the gradient ascent method. After a fuzzy system is determined, coefficients in consequent part are found by the least square method. Examples illustrate effectiveness of the proposed new fuzzy inference system.

In this paper we present and compare different fault diagnosis algorithms using space space models for nonlinear dynamic systems. Most fault diagnosis and isolation algorithms for dynamic systems, which can be modeled using a set of state space equations, have relied on the system being linear and the noise and disturbances being Gaussian. In such cases, optimal filtering ideas based on Kalman filtering are utilized in estimation followed by a residual analysis, for which whiteness tests are typically carried out. Linearized approximations (e.g., Extended Kalman filters) have been used in the nonlinear dynamic systems case. However, linearization techniques, being approximate, tend to suffer from poor detection or high false alarm rates. In this paper, we use the sequential Monte Carlo filtering approach where the complete posterior distribution of the estimates are represented through samples or particles as opposed to the mean and covariance of an approximated Gaussian distribution. The particle filter is combined with the innovation-based fault detection techniques to develop a fault detection and isolation scheme. The advantage of particle filters is that they are capable of handling any functional nonlinearity and system or measurement noise of any distribution. An improvement on using a single Extended Kalman filter matched to a particular model is to use the the Interacting Multiple Model estimator, which consists of a number of EKFs running in parallel. Such a multiple model estimator can handle the abrupt changes in the system dynamics, which is essential for fault diagnosis. Here, we compare the fault detection performances of these algorithms on different nonlinear systems.

Structural Optimization of Neural Networks for Systems Modelling

This book presents a systematic framework targeting at fuzzy modeling and fuzzy control of nonlinear systems with uncertainties. The book is organized into three major parts incorporating 13 chapters. The first part contains four chapters focusing on the modeling of nonlinear dynamical systems by using fuzzy logic. The second part includes five chapters in which fuzzy inference and control techniques are addressed. The final part consists of four chapters, which covers several advanced topics in fuzzy control ranging from controller and filter design to chaotification of fuzzy systems and feedforward fuzzy control of nonlinear systems via Fourier integrals. One of the major contributions of this book is that it presents the latest original research developments in the field. It provides rich examples and references, too. It is a valuable resource for those researchers and practitioners interested in expanding their knowledge from fuzzy logic and applications to nonlinear dynamical systems.

The problem of creating mathematical models of nonlinear dynamical systems does not have an unambiguous solution and requires the creation of a separate synthesis method for each such object. To develop a method for synthesizing mathematical models of an extensive class of nonlinear dynamical systems with polynomial nonlinearities. The work uses a method based on the solution of the Volterra integral equation in the ideology set forth in Van Trees H.L., according to which the structure of a nonlinear dynamical object present47s a series connection of the linear part, characterizing the inertial properties of the system, and the nonlinear element, given by static characteristic. The difference of the suggested version of the method from the classical one, proposed in the works of Van Trees H.L., is an expansion of their input and output signals into Fourier series and a representation of the inertial part of these systems by their Bode plots, connected into one structure with input and output signals and non-linearity by Volterra integral equation. The algorithm of the proposed method is disclosed by the example of solving the problem of identifying a nonlinear dynamical system which impulse response of the inertial part satisfies the separability requirement, the order of the polynomial nonlinearity is three, and the model of the input signal has the form of a sinusoid "raised" over the time axis on a priori given constant level. A computational experiment was carried out on the example of nonlinear dynamical systems with the third order of the nonlinear characteristic and the first and second orders of the model of the inertial part of these systems with the specified algorithms of their parametric identification. The suggested method allows to synthesis the mathematical model of a nonlinear dynamical system with the polynomial static characteristic to the case when the input signal has an arbitrary number of harmonics, and the model of the inertial part and the nonlinear polynomial function have an arbitrary order.

Reduced models of nonlinear dynamical systems require closure, or the modelling of the unresolved modes. The Mori-Zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the time history of the resolved variables. While this procedure does not reduce the complexity of the original system, these equations can serve as a mathematically consistent basis to develop closures based on memory approximations. In this scenario, knowledge of the memory kernel is paramount in assessing the validity of a memory approximation. Unravelling the memory kernel requires solving the orthogonal dynamics, which is a high-dimensional partial differential equation that is intractable, in general. A method to estimate the memory kernel a priori, using full-order solution snapshots, is proposed. The key idea is to solve a pseudo orthogonal dynamics equation, which has a convenient Liouville form, instead. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics is a composition operator for one observable. The method is exact for linear systems. Numerical results on the Burgers and Kuramoto-Sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel.

A nonlinear dynamical system model that approximates a microscopic Gibbs field model for the yielding of a viscoplastic material subjected to varying external stresses recently reported in R. Sainudiin, M. Moyers-Gonzalez and T. Burghelea, Soft Matter, 2015, 11(27), 5531-5545 is presented. The predictions of the model are in fair agreement with microscopic simulations and are in very good agreement with the micro-structural semi-empirical model reported in A. M. V. Putz and T. I. Burghelea, Rheol. Acta, 2009, 48, 673-689. With only two internal parameters, the nonlinear dynamical system model captures several key features of the solid-fluid transition observed in experiments: the effect of the interactions between microscopic constituents on the yield point, the abruptness of solid-fluid transition and the emergence of a hysteresis of the micro-structural states upon increasing/decreasing external forces. The scaling behaviour of the magnitude of the hysteresis with the degree of the steadiness of the flow is consistent with previous experimental observations. Finally, the practical usefulness of the approach is demonstrated by fitting a rheological data set measured with an elasto-viscoplastic material.

Learning soft computing control strategies in a modular neural network architecture

Chapter 3 - Neural Networks: A Methodology for Modeling and Control Design of Dynamical Systems

This paper proposes a general framework for modeling nonlinear dynamical systems based on support vector machine (SVM), firstly provides a short introduction to regression SVM, then uses standard support vector machine to model nonlinear dynamical system, and gives a theoretic analysis about its robustness under noise. The simulation results indicate that the SVM method can reduce the effect of sample's number and noise for modeling, and its performance is better than that of neural network modeling method.

The authors study the stability and convergence properties of recurrent high-order neural networks (RHONNs) as models of nonlinear dynamical systems. The overall structure of the RHONN consists of dynamical elements distributed throughout the network in the form of dynamical neurons, which are interconnected by high-order connections. It is shown that if a sufficiently large number of high-order connections between neurons is allowed, the RHONN model is capable of approximating the input-output behavior of general dynamical systems to any degree of accuracy. Based on the linear-in-the-weights property of the RHONN model, the authors have developed identification schemes and derived weight adaptive laws for adjustment of the weights. The convergence and stability properties of these weight adaptive laws have been analyzed. In the case of no modeling error, the state error between the system and the RHONN model converges to zero asymptotically. If modeling errors are present, the sigma -modification is proposed as a method of guaranteeing the stability of the overall scheme. The feasibility of applying these techniques has been demonstrated by considering the identification of a simple rigid robotic system. >