Order Nonlinear Dynamical Systems Research Articles

Practical fixed-time non-singular sliding mode control of second order nonlinear dynamic systems with chattering and overshooting avoidance

This paper introduces a Practical Fixed-Time Stabilization technique (PFxT) designed for a specific class of nonlinear second-order systems. The closed-loop systems, when appropriately parameterized, exhibit convergence within a predetermined time frame to a confined region near the origin. This outcome is achieved by amalgamating a nonlinear sliding mode approach with a practical fixed-time tracking virtual trajectory. The control algorithmś design incorporates considerations for overshoot reduction and chattering cancellation. Furthermore, the Lyapunov method is employed to establish the practical fixed-time stability of the proposed solution, providing insights into the limits within which the algorithm can be effectively deployed. To complete the algorithm specification, a parameter selection approach is introduced, enabling customization of the desired settling time. Comprehensive simulations are conducted to validate the effectiveness and viability of the proposed PFxT technique.

Fifth step block method and shooting constant for third order nonlinear dynamical systems

Fifth step block method and shooting constant for third order nonlinear dynamical systems

Examining reachability of fractional dynamical systems with delays in control utilizing ψ-Hilfer pseudo-fractional derivative

This research focuses on the reachability criteria of fractional dynamical systems with delays in control of order ϑ ∈ (0, 1) and type ϱ ∈ [0, 1] in the context of control theory. The study introduces the utilization of the ψ-Hilfer pseudo fractional derivative to describe the dynamics of systems with delays in control. We obtained the necessary and sufficient conditions for the reachability criteria of fractional order linear dynamical systems by employing the positive definiteness of Gramian matrices. We arrived the sufficient conditions for the reachability criteria of fractional order nonlinear dynamical systems using Banach’s fixed point theorem. A few numerical examples are made available for a better understanding of the theoretical conclusions.

Change of Measure Enhanced Near-Exact Euler–Maruyama Scheme for the Solution to Nonlinear Stochastic Dynamical Systems

The present study utilizes the Girsanov transformation based framework for solving a nonlinear stochastic dynamical system in an efficient way in comparison to other available approximate methods. In this approach, a rejection sampling is formulated to evaluate the Radon-Nikodym derivative arising from the change of measure due to Girsanov transformation. The rejection sampling is applied on the Euler Maruyama approximated sample paths which draw exact paths independent of the diffusion dynamics of the underlying dynamical system. The efficacy of the proposed framework is ensured using more accurate numerical as well as exact nonlinear methods. Finally, nonlinear applied test problems are considered to confirm the theoretical results. The test problems demonstrates that the proposed exact formulation of the Euler-Maruyama provides an almost exact approximation to both the displacement and velocity states of a second order non-linear dynamical system.

Fractional order model for yield through diagnosed/undiagnosed soil

Farming is the basic economy of any country. Soil fertility and water are key resources for yielding. The fractional order compartmental model is prepared in Caputo sense to improve the density of yield through diagnosed and undiagnosed soil. To measure the growing intensity of yield, the basic reproduction number is formulated using the next generation matrix method for integer order non-linear dynamical system. Local stability analysis is described for both the equilibrium points; undiagnosed soil free and optimum equilibrium point. Global stability is exhaustively computed by generating a Lyapunov function. With reference to the basic reproduction number, bifurcation analysis has been defined which expresses the chaotic nature of soil fertility. Moreover, optimal control theory is applied in the present fractional model to optimize yield. And optimality conditions are calculated with the help of Pontryagin maximum principle. The numerical simulation for different fractional orders is performed concerning validated data to analyze the behavior with respect to the order.

On the Bielecki–Ulam’s Type Stability Results of First Order Non-linear Impulsive Delay Dynamic Systems on Time Scales

In this manuscript, we present the existence and uniqueness of solution of first order non-linear impulsive delay dynamic systems on time scales, with the help of fixed point approach. Further, we investigate the Bielecki–Ulam–Hyers stability and Bielecki–Ulam–Hyers–Rassias stability of the proposed model. We use Gronwall’s inequality on time scales, Banach contraction principle, Picard operator and Lipschitz condition as a basic tools to develop our main results. Also, two examples are provided to illustrate the obtained results.

Dynamic characteristics analysis of time-delay fractional order dynamic system

In this paper, a kind of time-delay fractional order dynamic system is studied. The classical integral sliding mode control method is used to design a single controller to control the time-delay fractional order Liu system from chaos to fixed point, and the single controller is further designed to control the time-delay fractional order Liu system from chaotic attractor to limit loop, so the chaos control of time-delay fractional order nonlinear dynamic system is achieved. On that basis, the dynamic characteristics of the time-varying delayed fractional-order Lorenz system are analyzed by the simulation, and the chaotic phenomena of the time-varying delayed fractional-order Lorenz system are verified under the provided time delay and order.

Uniqueness Solution Of Abstract Fractional Order Nonlinear Dynamical Control Problems

The aim of this paper is to investigate the Uniqueness solution of Cauchy Problem represented for fractional order nonlinear dynamical control system involving certain control input and their approach of investigated depended on commutative composite semigroup and some certain conditions in certain space.

Homotopy perturbation aided optimization procedure with applications to oscillatory fractional order nonlinear dynamical systems

This paper presents an approximate solution of nonlinear fractional differential equations (FDEs) that exhibit an oscillatory behavior by using a metaheuristic technique. The solutions of the governing equations are approximated by using homotopy perturbation method (HPM) along with the fractional derivative in the Caputo sense. The designed methodology is based on a weighted series of HPM in conjunction with a nature-inspired algorithm. The idea is instantly fascinated by the researchers on the consequent implementation of nature-inspired learning algorithms such as a Cuckoo search algorithm (CSA). The usage of CSA has accelerated the minimized search path of error to the convergent values of the solution. The validity and accuracy of the proposed technique are ascertained by calculating the approximate solution and the error norms which ensure the convergence of the approximation that can be further increased. The critical analysis is also provided by the numerical simulation of two different test models. Discussion of key points has been determined by the tabulation of numerical values and graphs. Comparative study of the results with known numerical technique is also performed.

Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

This article studies the generalized Mittag–Leffler stability of Hilfer fractional nonautonomous system by using the Lyapunov direct method. A new Hilfer type fractional comparison principle is also proved. The novelty of this article is the fractional Lyapunov direct method combined with the Hilfer type fractional comparison principle. Finally, our main results are explained by some examples.

Setting condition of surge tank based on stability of hydro-turbine governing system considering nonlinear penstock head loss

In order to provide a basis for the setting of surge tank to achieve the safe and stable operation of hydropower station, this paper studies the setting condition of surge tank (SCST) based on stability of hydro-turbine governing system (HTGS) considering nonlinear penstock head loss. Firstly, the mathematical model of the HTGS considering nonlinear penstock head loss is established. Secondly, the SCST based on stability of HTGS considering nonlinear penstock head loss is derived by using Hopf bifurcation theory. Finally, the characteristics effects of SCST are analyzed. The effect mechanism of nonlinear penstock head loss on SCST is revealed by the comparative analysis of different SCSTs based on stability. The results indicate that the HTGS considering nonlinear penstock head loss is a third order nonlinear dynamic system. There are three inequality criterions for the SCST, and the third one is the controlling inequality criterion with the highest requirement and acts as a dominant role. The flow inertia time constant of penstock Twt0 corresponding to the endpoint of inequality interval is denoted as [Twt0]. [Twt0] is the allowable value of Twt0. When Twt0 is less than [Twt0], it is not necessary to set surge tank. To obtain a more accurate SCST, the penstock head loss and its nonlinear characteristic should be considered.

Model-Free PI/PID Controller Tuning of Higher Order Nonlinear Dynamic Systems

Model-Free PI/PID Controller Tuning of Higher Order Nonlinear Dynamic Systems

Secure communication and image encryption scheme based on synchronisation of fractional order chaotic systems using backstepping

Synchronisation strategy of fractional order nonlinear dynamical systems is utilised to address the problem of secure communication by introducing additional security features. Initially, a backstepping-based controller is designed for synchronisation of two similar fractional order chaotic systems in master-slave configuration. The controller obtained is divided into two parts with one part transmitted through the communication channel whereas the other part has to be designed at the receiver end so as to obtain synchronisation between master and slave system. Further, only two signals are transmitted out of which one has an embedded message signal and the other is a complex combination of master states which will be used for synchronisation purpose at the receiver end for recovery of message signal. The initial conditions, system parameters, fractional orders, the structure of controller and the combination of master system states form a large key set which should be known is advance in order to breach the security. Here, the scheme for secure transmission of speech signal is presented first and further a scheme for image encryption is given. The simulation results and security analysis using different quantitative measures are presented at the end to demonstrate the applicability and effectiveness of the proposed scheme.

Periodic problem of first order nonlinear uncertain dynamic systems

Periodic problem of first order nonlinear uncertain dynamic systems

On the Inverse Problem of Competitive Modes and the Search for Chaotic Dynamics

Generalized competitive modes (GCM) have been used as a diagnostic tool in order to analytically identify parameter regimes which may lead to chaotic trajectories in a given first order nonlinear dynamical system. The approach involves recasting the first order system as a second order nonlinear oscillator system, and then checking to see if certain conditions on the modes of these oscillators are satisfied. In the present paper, we will consider the inverse problem of GCM: If a system of second order oscillator equations satisfy the GCM conditions, can we then construct a first order dynamical system from it which admits chaotic trajectories? Solving the direct inverse problem is equivalent to finding solutions to an inhomogeneous form of the Euler equations. As there are no general solutions to this PDE system, we instead consider the problem for restricted classes of functions for autonomous systems which, upon obtaining the nonlinear oscillatory representation, we are able to extract at least two of the modes explicitly. We find that these methods often make finding chaotic regimes a much simpler task; many classes of parameter-function regimes that lead to nonchaos are excluded by the competitive mode conditions, and classical knowledge of dynamical systems then allows us to tune the free parameters or functions appropriately in order to obtain chaos. To find new hyperchaotic systems, a similar approach is used, but more effort and additional considerations are needed. These results demonstrate one method for constructing new chaotic or hyperchaotic systems.

Asymptotic stability of distributed order nonlinear dynamical systems

In this article we present a generalization of the Lyapunov direct method for distributed order nonlinear time-varying systems. By extending recently introduced properties of the Caputo fractional derivative to the distributed order case, we provide various ways to determine the stability or asymptotic stability of certain systems. The fractional results that inspired this work are recovered if an appropriate distribution function is chosen. Some examples are given to validate the obtained results.

PID controller design for second order nonlinear uncertain systems

Although the classical PID (proportional-integral-derivative) controller is most widely and successfully used in engineering systems which are typically nonlinear with various uncertainties, almost all the existing investigations on PID controller focus on linear systems. The aim of this paper is to present a theory on PID controller for nonlinear uncertain systems, by giving a simple and analytic design method for the PID parameters together with a mathematic proof for the global stability and asymptotic regulation of the closed-loop control systems. To be specific, we will construct a 3-dimensional manifold within which the three PID parameters can be chosen arbitrarily to globally stabilize a wide class of second order nonlinear uncertain dynamical systems, as long as some knowledge on the upper bound of the derivatives of the nonlinear uncertain function is available. We will also try to make the feedback gains as small as possible by investigating the necessity of the manifold from which the PID parameters are chosen, and to establish some necessary and sufficient conditions for global stabilization of several special classes of nonlinear uncertain systems.

Force identification using correlated noise models

The problem of input identification using Bayesian estimation methods is considered. The forcing function is a time varying quantity hence it requires solution strategies that can account for the time variation, which is in contrast to methods for time invariant parameter identification. In the existing literature, in the absence of any prior information the forcing function is assumed to have a white noise model and the parameters associated with the model are taken such that the noise is uncorrelated to those associated with the response quantities. In the present work, the derivative of the force is assumed to be a white noise process and the noise parameters of this prior model and the system response quantities are taken to be correlated. The motivation to adopt this model is driven by the fact that the structure acts as a sensor in recording the load acting on it, which is available in the form of the measured structural responses. Furthermore, the estimation of the forcing function is governed by the updating of the predicted state at each time instant with respect to the measured system responses. Hence, a correlation between the force and response noise quantities may result in convergence in the estimation. The extended state vector approach, wherein the unknown force is introduced as an additional state to the system response quantities is adopted. The identification algorithm involves application of the Kalman filter and the sequential filters for the linear and nonlinear systems, respectively. The performance of the proposed algorithm will be illustrated on lower order linear and nonlinear dynamical systems.

New Results on Exponential Stability of Fractional Order Nonlinear Dynamic Systems

In this letter stability analysis of fractional order nonlinear systems is studied. An extension of Lyapunov direct method for fractional order systems is proposed by using the properties of Mittag-Leffler function and Laplace transform. Some new sufficient conditions which ensure local exponential stability of fractional order nonlinear systems are proposed firstly. And we apply these conditions to the Riemann-Liouville fractional order systems by using fractional comparison principle. Finally, three examples are provided to illustrate the validity of the proposed approach.

Design a Methodology to Model-Reference Control of First Order Delays System

Design a nonlinear controller for second order nonlinear uncertain dynamical systems is one of the most important challenging works. This research focuses on the design, and analysis of a model-reference sliding mode controller for first order delay system, in presence of uncertainties. In order to provide high performance nonlinear methodology, model-reference sliding mode controller is selected. Pure sliding mode controller can be used to control of partly known nonlinear dynamic parameters. Conversely, pure sliding mode controller is used in many applications; it has an important drawback namely; chattering phenomenon. To attenuation the chattering, new filter based high speed control technique is introduced. In this technique, two type derivative techniques are used to improve the rate of delay as well stability, robustness and chattering attenuation. This technique cased to improve the rate of delay compare with conventional PID controller and conventional sliding mode controller.