Design of nonlinear observer by augmented linear system based on formal linearization using fourier expansion

A GENERALIZED LOCAL LINEARIZATION PRINCIPLE FOR NON-LINEAR DYNAMICAL SYSTEMS

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.

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

In this paper, the observer design problem is considered for a class of positive dynamical systems. First, we establish a set of sufficient conditions for the design of a general class of positive nonlinear observers for nonlinear positive dynamical systems. Then, the new reduced-order positive observer design techniques are provided for marginally stable and some unstable continuous positive linear systems by thoroughly analyzing the structural properties of sampled systems in terms of reducibility. In particular, a nonlinear reduced-order positive observer is proposed for general reducible positive systems.

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

Robotic and mechatronic systems, autonomous vehicles, electric power systems and smart grids, as well as manufacturing and industrial production systems can exhibit complex nonlinear dynamics or spatio-temporal dynamics which need to be controlled to ensure good functioning and performance. In this comprehensive reference, the authors present new and innovative control and estimation methods and techniques based on dynamical nonlinear and partial differential equation systems. Such results can be classified in five main domains for the control of complex nonlinear dynamical systems using respectively methods of approximate (local) linearization, methods of exact (global) linearization, Lyapunov stability approaches, control and estimation of distributed parameter systems and stochastic estimation and fault diagnosis methods. Control and Estimation of Dynamical Nonlinear and Partial Differential Equation Systems: Theory and applications will be of interest to electrical engineering, physics, computer science, robotics and mechatronics researchers and professionals working on control problems, condition monitoring, estimation and fault diagnosis and isolation problems. It will also be useful to skilled technical personnel working on applications in robotics, energy conversion, transportation and manufacturing.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

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.

Solution of nonlinear partial differential equations from base equations

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

It is well-known that the controls of neural network require that all the inputs of their input layer are available. Under this circumstance, the architecture of the neural network is constrained. Hence, this kind of neurocontrol cannot apply to a wide class of nonlinear and unknown dynamical systems, or the control system that requires more sensors. Although the unknown system state and the unknown system dynamics can be achieved from an adaptive observer and a learning neural network, respectively, the simultaneous existence of the estimated state error and the modeling error makes the control system more likely to be unstable. In this paper, a novel neurocontroller based on the concept of sliding mode with estimated state is constructed to tackle a wide class of unknown and nonlinear dynamical systems. The focal stability of the overall system can be verified by the Lyapunov stability criteria. Finally, simulations are presented to verify the usefulness of the proposed method.

This research frame work investigates the application of a clustered based Neuro-fuzzy system to nonlinear dynamic system modeling from a set of input-output training patterns. It is concentrated on the modeling via Takagi-Sugeno (T-S) modeling technique and the employment of fuzzy clustering to generate suitable initial membership functions. Hence, such created initial memberships are then employed to construct suitable T-S sub-models. Furthermore, the T-S fuzzy models have been validated and checked through the use of some standard model validation techniques (like the correlation functions). Compared to other well-known approximation techniques such as artificial neural networks, fuzzy systems provide a more transparent representation of the system under study, which is mainly due to the possible linguistic interpretation in the form of rules. Such intelligent modeling scheme is very useful once making complicated systems linguistically transparent in terms of fuzzy if-then rules. The developed T-S Fuzzy modeling system has been then applied to model a nonlinear antenna dynamic system with two coupled inputs and outputs. Validation results have resulted in a very close antenna sub-models of the original nonlinear antenna system. The suggested technique is very useful for development transparent linear control systems even for highly nonlinear dynamic systems.

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

An adaptive system for linear systems with unknown parameters is a nonlinear system. The analysis of such adaptive systems requires similar techniques to analyse nonlinear systems. Therefore it is natural to treat adaptive control as a part of nonlinear control systems. Nonlinear and Adaptive Control Systems treats nonlinear control and adaptive control in a unified framework, presenting the major results at a moderate mathematical level, suitable for MSc students and engineers with undergraduate degrees. Topics covered include introduction to nonlinear systems; state space models; describing functions for common nonlinear components; stability theory; feedback linearization; adaptive control; nonlinear observer design; backstepping design; disturbance rejection and output regulation; and control applications, including harmonic estimation and rejection in power distribution systems, observer and control design for circadian rhythms, and discrete-time implementation of continuous-time nonlinear control laws.

The problem of estimating rigid from projections may be characterized using a nonlinear dynamical system, composed of the rigid constraint and the perspective map. The time derivative of the output of such a system, which is called the motion field and approximated by the optical flow, is bilinear in the parameters, and may be used to specify a subspace constraint on either the direction of translation or the inverse depth of the observed points. Estimating may then be formulated as an optimization task constrained on such a subspace. We pose the optimization problem in a system theoretic framework as the the identification of a nonlinear implicit dynamical system with parameters on a differentiable manifold, and use techniques which pertain to nonlinear estimation and identification theory to perform the optimization task in a principled manner. The application of a general method presented in by Soatto et al. (see 33rd. IEEE conf. on Decision and Control, 1994) results in a recursive and pseudo-optimal solution of the visual estimation problem, which has robustness properties far superior to other existing techniques we have implemented. Experiments on real and synthetic image sequences show very promising results in terms of robustness, accuracy and computational efficiency. >