An extended self-organizing map for nonlinear system identification

An extended self-organizing map (ESOM) network, which consists of a self-organization phase and an optimization phase, was recently developed to construct a local model network (LMN) automatically using the plant data. However, this previous result suffers two drawbacks: (1) increased computation time in the self-organization phase as the number of local models increases, (2) lack of checking stability conditions for both local models and LMN. To overcome these problems, an improved algorithm for the ESOM network is developed in this paper by employing a competitive learning algorithm for data clustering in the self-organization phase and parametric constraints are formulated in the optimization phase to handle the stability of local models. In addition, the global stability of LMN is addressed. With LMN constructed by the ESOM network, it serves as a basis for building a nonlinear controller that combines several local controllers through the weighting functions obtained by the ESOM algorithm. Literature examples are used to illustrate the proposed ESOM-based modeling and controller design method.

Modeling of nonlinear systems using extended self-organizing map

In this paper, advanced concepts for the identification of complex nonlinear systems are discussed. Three major problems are addressed: The nonlinearity of the system, noise in the data upon which the model has to be built, and the potential to incorporate qualitative and quantitative prior knowledge about the system. As an integrated solution approach, local model networks (LMNs) with appropriate parameter estimation schemes are proposed. LMNs generally offer a versatile structure for the identification of nonlinear dynamic systems. In order to account for a realistic situation when noise is present both in input and output data, an equality constrained generalised total least squares algorithm for the local model parameter estimation of the LMN is presented; the incorporation of equality constraints allows to mathematically enforce desired system properties. As an application and benchmark problem, the vertical dynamics of a vehicle is considered. After training the LMN on a rough road, excellent predictions of the behaviour of the vehicle at crossing a single obstacle are obtained, thus proving the effectiveness of the proposed algorithm. It is illustrated how both the application of a proper parameter estimation scheme and the integration of system constraints systematically improve the performance of the model.

Local Model Networks (LMNs) provide a global representation of a nonlinear dynamical system by interpolating between a set of locally valid sub-models distributed across the operating range. Training such networks typically involves heuristic selection of the number of sub-models and their structure followed by the combined estimation of the free sub-model and interpolation function parameters. This paper describes a new genetic learning approach to the construction of LMNs comprising ARX local models and normalised Gaussian interpolation functions. In addition to allowing the simultaneous optimisation of the number of sub-models, model parameters arid interpolation function parameters, the approach provides a flexible framework for targeting transparency and generalisation. Fuzzy logic is used with special features to provide a directional and dynamic search for the genetic algorithm. Several modifications of the classical genetic algorithm are adopted to optimise each local model separately within the overall global model. A linear direct feedback control scheme is derived from the LMN representation of the nonlinear plant and local stability analysis is discussed. Simulation studies on a pH neutralisation process confirm the excellent nonlinear modelling properties of LM networks and illustrate the potential of the proposed control scheme.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

Local Model Networks (LMNs) provide a global representation of a nonlinear dynamical system by interpolating between a set of locally valid sub-models distributed across the operating range. Training such networks typically involves heuristic selection of the number of sub-model and their structure followed by the combined estimation of the free sub-model and interpolation function parameters. This paper describes a new genetic learning approach to the construction of LMNs comprising ARX local models and normalised Gaussian interpolation functions. In addition to allowing the simultaneous optimisation of the number of sub-models, model orders, model parameters and interpolation function parameters, the approach provides a flexible framework for targeting transparency and generalisation. Fuzzy logic is used with special features to provide a directional and dynamic search for the genetic algorithm. Several modifications of the classical genetic algorithm are adopted to optimise each local model separately within the overall global model. Studies on actual data from a pH neutralisation process confirm the potential of the proposed scheme.

Two-motor variable frequency speed-regulating synchronous system is a multivariable nonlinear coupling system. In order to estimate speed and tension of this system more effectively, a method of modeling based on local model networks (LMN) is given. Through using correlative testing signal to drive two-motor synchronous system firstly, series of output response about speed and tension are obtained.Then fuzzy C-means clustering algorithm is used to cluster the data of input-output according to the satisfactory performance index. Finally ,the sum of linear local model identification model for speed and tension is built . The simulation results are presented to demonstrate the validity of the proposed the method.

Adaptive nonlinear system identification of separation unit for raw materials

Inference of hidden common driver dynamics by anisotropic self-organizing neural networks.

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

Besides the feedforward neural networks, there are the recurrent networks, where the impulses can be transmitted in both directions due to some reaction connections in these networks. Recurrent Neural Networks (RNNs) are linear or nonlinear dynamic systems. The dynamic behavior presented by the recurrent neural networks can be described both in continuous time, by differential equations and at discrete times by the recurrence relations (difference equations). The distinction between recurrent (or dynamic) neural networks and static neural networks is due to recurrent connections both between the layers of neurons of these networks and within the same layer, too. The aim of this chapter is to describe a Recurrent Fuzzy Neural Network (RFNN) model, whose learning algorithm is based on the Improved Particle Swarm Optimization (IPSO) method.

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

In this paper, we consider an observer design by using a formal linearization based on Fourier expansion for nonlinear dynamic and measurement systems. A non-linear dynamic system is given by a nonlinear ordinary differential equation, and a measurement sysetm is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, a nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, a measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a new non-linear observer.

The problem of approximation of unknown dynamics of a continuous-time observable nonlinear system is considered using a feedforward neural network, operating over delayed sampled outputs of the system. Error bounds are derived that explicitly depend upon the sampling time interval and network architecture. The main result of this note broadens the class of nonlinear dynamical systems for which adaptive output feedback control and state estimation problems are solvable.

BackgroundFractional-order models have become essential tools for representing nonlinear dynamical systems with memory and hereditary effects. While constant-order fractional operators have been widely studied, many real-world systems exhibit memory characteristics that evolve over time or system state. Variable-order fractional calculus addresses this limitation but introduces significant analytical and numerical challenges, particularly in nonlinear settings where stability and convergence are difficult to guarantee. Existing numerical approaches often suffer from accuracy loss and instability when fractional orders vary rapidly. MethodsThis study develops a hybrid fractional integral framework for nonlinear dynamical systems governed by variable-order operators. The methodology combines multiple variable-order fractional integrals within a unified formulation to balance short-term and long-term memory effects. A predictor–corrector numerical scheme is employed to handle nonlinearities and evolving fractional orders efficiently. Rigorous theoretical analysis is conducted to establish existence, uniqueness, convergence, and stability under standard smoothness and Lipschitz assumptions. The framework is evaluated through numerical experiments on benchmark nonlinear systems with different variable-order profiles. ResultsThe proposed hybrid framework demonstrates consistently improved numerical performance compared with constant-order and single-operator variable-order methods. Results show lower global error, higher observed convergence order, and enhanced stability under oscillatory and discontinuous fractional-order variations. The method also exhibits greater robustness in strongly nonlinear regimes, where conventional schemes experience significant error amplification. Computational efficiency is improved through faster convergence, enabling accurate solutions with fewer time steps and reduced memory overhead. ConclusionThe hybrid fractional integral framework provides a stable, accurate, and flexible approach for solving nonlinear variable-order dynamical systems. By structurally integrating complementary memory operators, the method overcomes key limitations of existing approaches and offers a reliable computational foundation for modeling complex systems with adaptive memory dynamics.