Analysis Of Nonlinear Systems Research Articles

The filtering-based recursive least squares identification and convergence analysis for nonlinear feedback control systems with coloured noises

The coloured noise is ubiquitous in industrial processes. This paper addresses the identification problem for the nonlinear feedback systems with coloured noise. Firstly, a direct identification scheme based on the least squares principle is developed to estimate the whole parameters of the nonlinear feedback systems and the convergence analysis is carried out through the stochastic stability theory. Secondly, for the purpose of improving the estimation accuracy, a filtering-based identification framework is proposed by constructing a linear filter for filtering the input data, output data and the coloured noise, and the coloured noise is transformed into a white noise. This identification scheme based on the filtering technique can effectively reduce the adverse effects caused by coloured noise and parameter estimation accuracy is enhanced compared with the direct least squares algorithm. Meanwhile, the convergence analysis of the filtering-based identification algorithm is given to provide a theoretical analysis. Finally, the simulation example is carried out by performance test and comparison analysis and simulation results show the effectiveness of the proposed identification methods.

Robust Consensus Analysis in Fractional-Order Nonlinear Leader-Following Systems with Delays: Incorporating Practical Controller Design and Nonlinear Dynamics

This article investigates the resilient-based consensus analysis of fractional-order nonlinear leader-following systems with distributed and input lags. To enhance the practicality of the controller design, an incorporation of a disturbance term is proposed. Our modeling framework provides a more precise and flexible approach that considers the memory and heredity aspects of agent dynamics through the utilization of fractional calculus. Furthermore, the leader and follower equations of the system incorporate nonlinear functions to explore the resulting changes. The leader-following system is expressed by a weighted graph, which can be either undirected or directed. Analyzed using algebraic graph theory and the fractional-order Razumikhin technique, the case of leader-following consensus is presented algebraically. To increase robustness in multi-agent systems, input and distributive delays are used to accommodate communication delays and replicate real-time varying environments. This study lays the groundwork for developing control methods that are more robust and flexible in complex networked systems. It does so by advancing our understanding and practical application of fractional-order multi-agent systems. Additionally, experiments were conducted to show the effectiveness of the design in achieving consensus within the system.

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

Stabilization analysis for nonlinear interconnected system with memory based coupled sampled data control via quantum-inspired genetic algorithm

In the realm of modern interconnected systems, achieving stability represents a pivotal challenge due to the complex dynamics among various components. This paper presents a new approach to stability analysis, leveraging linear matrix inequalities and Lyapunov-Krasovskii functionals. The framework of memory-based coupling sampled data control is enhanced by a quantum genetic algorithm. By incorporating linear matrix inequality approach, this study establishes a robust mathematical foundation for characterizing stability conditions and synthesizing control gains. This innovative integration provides a powerful toolset to optimize control parameters while mitigating the impact of disturbances. The simulation findings are explicitly based on experimental values of two-area interconnected power systems with doubly fed induction generator based wind farm, which guarantees the asymptotic stability of the proposed controller.

Scaled graphs for reset control system analysis

Scaled graphs allow for graphical analysis of nonlinear systems, but are generally difficult to compute. The aim of this paper is to develop a method for approximating the scaled graph of reset controllers. A key ingredient in our approach is the generalized Kalman-Yakubovich-Popov lemma to determine input specific input–output properties of a reset controller in the time domain. By combining the obtained time domain properties to cover the full input space, an over-approximation of the scaled graph is constructed. Using this approximation, we establish a feedback interconnection result and provide connections to classical input–output analysis frameworks. Several examples show the relevance of the results for the analysis and design of reset control systems.

Stochastic response and stability analysis of nonlinear vehicle energy-regenerative suspension system with time delay

With the rapid development of new energy vehicles, a novel vehicle energy-regenerative suspension for simultaneous vibration suppression and energy harvesting has become one of the key development directions of the next generation of vehicle design. In order to improve the performance of suspension system, the time-delayed feedback control is applied to the novel suspension system subjected to different external excitation. At the same time, there are few studies related to the stochastic dynamics of nonlinear vehicle energy-regenerative suspension system with time delay. This paper studies stochastic response and stability of nonlinear vehicle energy-regenerative suspension with time-delayed feedback control under the excitation of Gaussian white noise. First of all, a stochastic dynamic model of the nonlinear vehicle energy-regenerative suspension with time delay system is established. Secondly, using the stochastic averaging method based on generalized harmonic function, the steady-state probability density function (PDF) of amplitude is obtained. The accuracy of the presented approach is verified through the comparison of analytical and numerical solutions. In addition, through combining the stochastic averaging method with the Khasminskii method, the top Lyapunov exponent (TLE) of the system is obtained, and then the influence of time delay and system parameters on the asymptotic Lyapunov stability is discussed. Finally, the effects of time delay and system parameters on system performance are also studied.

Perturbed Biswas–Milovic equation with Kudryashov’s law of refractive index: analysis and solutions for nonlinear optical systems

Perturbed Biswas–Milovic equation with Kudryashov’s law of refractive index: analysis and solutions for nonlinear optical systems

A Hybrid Continuation Framework for Analyzing Nonlinear Normal Modes of Systems With Contact Nonlinearity

Abstract The prediction of the modal properties of structural systems with contact clearances and prestress presents a computational challenge, as the nonlinearity induced by piecewise-linear stiffness eliminates the use of efficient linear modal analysis techniques. The most common approach to obtaining the nonlinear normal modes (NNMs) of these structural systems is a numerical framework that integrates numerical integration, the shooting method, and the pseudo-arc-length continuation scheme. This numerical continuation framework (NCF) computes NNMs through iterative numerical calculations; thus, the computational cost of the nonlinear modal analysis of complex nonlinear systems, particularly piecewise-linear systems, becomes prohibitively expensive as the model size increases. In this study, a hybrid continuation framework (HCF) combining analytic and numerical methods is proposed to enable efficient computations of NNMs for systems with contact boundaries. This new hybrid framework utilizes a semi-analytic method to conduct the iterative shooting procedure; thus, the computational burden of the numerical continuation can be significantly reduced. The proposed method is demonstrated on spring-mass oscillators with contact elements, and the NNMs obtained using the proposed method are validated by those computed using the traditional numerical continuation framework. The modal properties of the systems can be computed using the proposed framework with significant speed-up. Furthermore, the modal properties, including internal resonance and sharp turning in NNM curves, of the piecewise-linear systems are identified and discussed.

Stability analysis for stochastic nonlinear systems with state-dependent switching and time-delay

The stability of stochastic nonlinear systems with time-delay and state-dependent switching is studied in this paper. Firstly, system maximal solution is constructed and further extended as the global solution under certain mild conditions. Secondly, several Krasovskii-type stability and Razumikhin-type theorems are proposed by uniformly asymptotically stable functions (UASFs). These stability criteria overcome some stringent constraints associated with Lyapunov functional/functions. Finally, the theoretical findings are validated through two simulation examples.

An echo state network with interacting reservoirs for modeling and analysis of nonlinear systems

An echo state network with interacting reservoirs for modeling and analysis of nonlinear systems

Proper generalized decomposition-based iterative enrichment process combined with shooting method for steady-state forced response analysis of nonlinear dynamical systems

AbstractThis paper presents a novel framework combining proper generalized decomposition (PGD) with the shooting method to determine the steady-state response of nonlinear dynamical systems upon a general periodic input. The proposed PGD approximates the response as a low-rank separated representation of the spatial and temporal dimensions. The Galerkin projection is employed to formulate the subproblem for each dimension, then the fixed-point iteration is applied. The subproblem for the spatial vector can be regarded as computing a set of reduced-order basis vectors, and the shooting problem projected onto the subspace spanned by these basis vectors is defined to obtain the temporal coefficients. From this procedure, the proposed framework replaces the complex nonlinear time integration of the full-order model with the series of solving simple iterative subproblems. The proposed framework is validated through two descriptive numerical examples considering the conventional linear normal mode method for comparison. The results show that the proposed shooting method based on PGD can accurately capture nonlinear characteristics within 10 modes, whereas linear modes cannot easily approximate these behaviors. In terms of computational efficiency, the proposed method enables CPU time savings of about one order of magnitude compared with the conventional shooting methods.

A Game Theoretical Analysis of Non-linear Blockchain System

Recent advances in blockchain research have been made in two important directions. One is refined resilience analysis utilizing game theory to study the consequences of selfish behavior of users (miners), and the other is the extension from a linear (chain) structure to a non-linear (graphical) structure for performance improvements, such as IOTA and Graphcoin. The first question that comes to mind is what improvements a blockchain system would see by leveraging these new advances. In this article, we consider three major properties for a blockchain system: α-partial verification, scalability, and finality-duration. We establish a formal framework and prove that no blockchain system can achieve α-partial verification for any fixed constant α, high scalability, and low finality-duration simultaneously. We observe that classical blockchain systems like Bitcoin achieve full verification (α =1) and low finality-duration, Ethereum 2.0 Sharding achieves low finality-duration and high scalability. We are interested in whether it is possible to partially satisfy the three properties.

Iterative learning based convergence analysis for nonlinear impulsive differential inclusion systems with randomly varying trial lengths

SummaryThis paper studies the finite‐time tracking problem for nonlinear impulsive differential inclusion systems with randomly varying trial lengths. First, we convert the set‐valued mapping in the differential inclusion systems to single‐valued mapping by a Steiner‐type selector. For the tracking problem of random discontinuous output trajectories, this paper defines a piecewise continuous variable by zero‐order holder to correct the tracking error of segmented continuity. Then, we introduce the average operator with forgetting factor to design three novel learning schemes, and establish convergence results by using the mathematical analysis tools such as impulsive Gronwall inequality and ‐norm. Finally, a numerical example verifies the validity of the theoretical results, and we compare the tracking performance of the output trajectories for different forgetting factors.

A LAPACK Implementation of the Dynamic Mode Decomposition

The Dynamic Mode Decomposition (DMD) is a method for computational analysis of nonlinear dynamical systems in data driven scenarios. Based on high fidelity numerical simulations or experimental data, the DMD can be used to reveal the latent structures in the dynamics or as a forecasting or a model order reduction tool. The theoretical underpinning of the DMD is the Koopman operator on a Hilbert space of observables of the dynamics under study. This paper describes a numerically robust and versatile variant of the DMD and its implementation using the state-of-the-art dense numerical linear algebra software package LAPACK . The features of the proposed software solution include residual bounds for the computed eigenpairs of the DMD matrix, eigenvectors refinements and computation of the eigenvectors of the Exact DMD, compressed DMD for efficient analysis of high dimensional problems that can be easily adapted for fast updates in a streaming DMD. Numerical analysis is the bedrock of numerical robustness and reliability of the software, that is tested following the highest standards and practices of LAPACK . Important numerical topics are discussed in detail and illustrated using numerous numerical examples.

Stability analysis of nonlinear systems with delayed impulses: a generalised average dwell-time scheme

This paper studies the input-to-state stability (ISS) and uniform stability for impulsive nonlinear systems with delayed impulses, where delays are flexible between two consecutive impulsive instants. Illustrating effects caused by delays, concepts of generalised average impulsive delay and generalised reverse average impulsive delay are introduced, and more applicable in practice. Moreover, sufficient conditions for stability properties are formulated with destabilising and stabilising impulses through a general candidate ISS-Lyapunov function and the generalised conditions. It is shown that the double effects of delays in impulses (i.e. stabilising an initial unstable system, or contrarily, destabilising an initial stable system) coincide with the existing results using the exponential candidate ISS-Lyapunov function. Finally, two examples are provided to demonstrate our developed techniques.

Stability criteria of random nonlinear systems with non‐instantaneous multiple impulse

AbstractThis paper comprehensively explores the stability analysis of random nonlinear systems subjected to random non‐instantaneous multiple impulses. The random non‐instantaneous multiple impulses refer to various impulsive intensities, with the type of impulse at each moment determined by a Markov chain. Drawing on the properties of stochastic processes, a series of stability criteria are developed for the random non‐instantaneous multiple impulsive systems within the constraint of fixed impulsive intervals. To validate the practicality of the proposed conceptual methods, the tracking control problem of network surface vessels under deception attacks is investigated, demonstrating feasibility through simulation results.

Nonlinear Optimal Control for Stochastic Dynamical Systems

This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear system’s asymptotic stability in probability is ensured through a Lyapunov function, identified as the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation. This dual assurance guarantees both stochastic stability and optimality. Additionally, optimal feedback controllers for affine nonlinear systems are developed using an inverse optimality framework tailored to the stochastic stabilization problem. Furthermore, the paper derives stability margins for optimal and inverse optimal stochastic feedback regulators. Gain, sector, and disk margin guarantees are established for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton–Jacobi–Bellman controllers.

On the qualitative analysis of nonlinear q-fractional delay descriptor systems

On the qualitative analysis of nonlinear q-fractional delay descriptor systems

Mathematical Analysis of P-Stability Maps for Parametric Conic Vector Optimization

Stability analysis for nonlinear programming systems deals with the possible changes of the system parameters and/or equations that maintain the stability of the solutions. It is a crucial requirement to study the nonlinear system and its practical values, specifically the economic impact in most real-world applications. This paper presents some outcomes in connection with stability analysis corresponding to parametric conic vector optimization problems. For these last optimization problems, two novel types of P-Stability maps, which are the P-Stability notion map and the P-Stability perturbation map, are considered based on six kinds of sets: P-feasible set, P-solvability set, the first, second, third, and fourth kinds of P-Stability notion sets with respect to a specific domination cone P. Furthermore, qualitative characteristics of the P-Stability maps under some continuity and convexity assumptions on the objective function are provided and proved. Specifically, the connections between the P-Stability maps and the P-Stability notion set are investigated. Accordingly, these characteristics were extended to the P-perturbation maps. In addition, the idea of Pstability has heavily used in different applications like network privacy, engineering fields, and some business financial models.

Solvability and Ulam-Hyers stability analysis for nonlinear piecewise fractional cancer dynamic systems

We examine a nonlinear dynamical model that depicts the interaction between cancerous cells and an oncolytic virus. For best modelling the disease, we use the Caputo fractional derivative in piecewise approaches. By employing piecemeal techniques, we treat a compartment in the body that contains infectious and non-infectious cells. More precisely, the solvability and Ulam-Hyers (U-H) stability results are considered using standard concepts. Further, to support our investigation with numerical results, we apply the Euler method to develop an approximation solution. It connected with numerous graphical representations of the system using various arbitrary ordering and varying values of the isolation parameters. Here we remark that the multi-step behavior that certain problems exhibit, is one of important issues naturally. This paper introduces the idea of piecewise derivative with the goal of modeling real-world issues that follow multiples processes. With the help of the used approach, we investigate the cancer disease model and its transmission dynamical behavior with crossover effect.