In this paper, the stability of multi-variable fractional order nonlinear dynamic system is investigated. We propose the definition of generalized Mittag-Leffler stability with multi-variable and ...introduce the fractional Lyapunov direct method with multi-variable. Meanwhile, a novel approach is suggested to study generalized Mittag-Leffler stability in multi-variable fractional order nonlinear dynamic systems. An interesting multi-variable fractional order Lotka–Volterra predator–prey model is used to illustrate the proposed method and its effectiveness.
•Nonlinear dynamic response and buckling of sandwich porous plate are studied.•The initial imperfection of the plate is investigated.•Functionally graded porous core with graphene platelet ...reinforcement is considered.•Winkler–Pasternak foundation, thermal and damping effects are investigated.•Effects of porosity and graphene platelet weight fraction are explored.
The nonlinear vibration and the dynamic buckling of a graphene platelet reinforced sandwich functionally graded porous (GPL-SFGP) plate are thoroughly investigated in this paper. The investigated GPL-SFGP plate consists of two metal face layers and a functionally graded porous core with graphene platelet reinforcement. The effects of the Winkler–Pasternak elastic foundation, thermal environment and damping are incorporated. The open-cell metal foam model is implemented to model the mechanical properties of the porous core. Axial compressive stress is applied on the GPL-SFGP plate by exerting various compressive loading speeds at one edge of the plate. Grounded on the classical plate theory, both motion and geometric compatibility equations of the plate are deduced by introducing the Von Kármán strain-displacement relationship and stress function. Both the Galerkin and the fourth-order Runge–Kutta methods are implemented to solve the governing equation of the dynamic system. Meticulously designed numerical experiments are conducted to identify the critical influential factors of the dynamic stability of the GPL-SFGP plate. The influences of loading speed, damping ratio, temperature variation, initial imperfection, elastic foundation parameters, porosity, GPL weight fraction and the dimensions of the GPL on the overall dynamic stability of the GPL-SFGP plate are evidently demonstrated.
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1. Banhart J, García-Moreno F, Heim K, Seeliger H-W. Light-weighting in transportation and defence using aluminium foam sandwich structures. 2017
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The results of the harmonic balance method (HBM) for a nonlinear system generally show nonlinear response curves with primary, super-, and sub-harmonic resonances. In addition, the stability ...conditions can be examined by employing Hill's method. However, it is difficult to understand the practical dynamic behaviors with only their stability conditions, especially with respect to unstable regimes. Thus, the main goal of this study is to suggest mathematical and numerical approaches to determine the complex dynamic behaviors regarding the bifurcation characteristics. To analyze the bifurcation phenomena, the HBM is first implemented utilizing Hill's method where various local unstable areas are found. Second, the bifurcation points are determined by tracking the stability variational locations on the arc-length continuation scheme. Then, their points are defined for various bifurcation types. Finally, the real parts of the eigenvalues are analyzed to examine the practical dynamic behaviors, specifically in the unstable regimes, which reflect the relevance of various bifurcation types on the real part of eigenvalues. The methods employed in this study successfully explain the basic ways to examine the bifurcation phenomena when the HBM is implemented. Thus, this study suggests fundamental method to understand the bifurcation phenomena using only the HBM with Hill's method.
This paper studies the problem of sampled-data control for master-slave synchronization schemes that consist of identical chaotic Lur'e systems with time delays. It is assumed that the sampling ...periods are arbitrarily varying but bounded. In order to take full advantage of the available information about the actual sampling pattern, a novel Lyapunov functional is proposed, which is positive definite at sampling times but not necessarily positive definite inside the sampling intervals. Based on the Lyapunov functional, an exponential synchronization criterion is derived by analyzing the corresponding synchronization error systems. The desired sampled-data controller is designed by a linear matrix inequality approach. The effectiveness and reduced conservatism of the developed results are demonstrated by the numerical simulations of Chua's circuit and neural network.
This paper is concerned with the global exponential anti-synchronization of a class of chaotic memristive neural networks with time-varying delays. The dynamic analysis here employs results from the ...theory of differential equations with discontinuous right-hand side as introduced by Filippov. And by using differential inclusions theory, the Lyapunov functional method and the inequality technique, some new sufficient conditions ensuring exponential anti-synchronization of two chaotic delayed memristive neural networks are derived. The new proposed results here are very easy to verify and they also improve the earlier publications. Finally, a numerical example is given to illustrate the effectiveness of the new scheme.
Data-driven discovery of dynamics via machine learning is pushing the frontiers of modelling and control efforts, providing a tremendous opportunity to extend the reach of model predictive control ...(MPC). However, many leading methods in machine learning, such as neural networks (NN), require large volumes of training data, may not be interpretable, do not easily include known constraints and symmetries, and may not generalize beyond the attractor where models are trained. These factors limit their use for the online identification of a model in the low-data limit, for example following an abrupt change to the system dynamics. In this work, we extend the recent sparse identification of nonlinear dynamics (SINDY) modelling procedure to include the effects of actuation and demonstrate the ability of these models to enhance the performance of MPC, based on limited, noisy data. SINDY models are parsimonious, identifying the fewest terms in the model needed to explain the data, making them interpretable and generalizable. We show that the resulting SINDY-MPC framework has higher performance, requires significantly less data, and is more computationally efficient and robust to noise than NN models, making it viable for online training and execution in response to rapid system changes. SINDY-MPC also shows improved performance over linear data-driven models, although linear models may provide a stopgap until enough data is available for SINDY. SINDY-MPC is demonstrated on a variety of dynamical systems with different challenges, including the chaotic Lorenz system, a simple model for flight control of an F8 aircraft, and an HIV model incorporating drug treatment.
In this brief, we consider the exponential synchronization of chaotic memristive neural networks with time-varying delays using the Lyapunov functional method and inequality technique. The dynamic ...analysis here employs the theory of differential equations with discontinuous right-hand side as introduced by Filippov. The designing laws in the synchronization of neural networks are proposed via state or output coupling. In addition, the new proposed algebraic criteria are very easy to verify, and they also enrich and improve the earlier publications. Finally, an example is given to show the effectiveness of the obtained results.
A single inertial BAM neural network with time-varying delays and external inputs is concerned in this paper. First, by choosing suitable variable substitution, the original system can be transformed ...into first-order differential equations. Then, we present several sufficient conditions for the global exponential stability of the equilibrium by using matrix measure and Halanay inequality, these criteria are simple in form and easy to verify in practice. Furthermore, when employing an error-feedback control term to the response neural network, parallel criteria regarding to the exponential synchronization of the drive-response neural network are also generated. Finally, some examples are given to illustrate our theoretical results.
In this paper, a class of fractional-order neural networks is investigated. First, α-exponential stability is introduced as a new type of stability and some effective criteria are derived for such ...kind of stability of the addressed networks by handling a new fractional-order differential inequality. Based on the results, the existence and α-exponential stability of the equilibrium point are considered. Besides, the synchronization of fractional chaotic networks is also proposed. Finally, several examples with numerical simulations are given to show the effectiveness of the obtained results.
► A class of fractional-order neural networks is investigated. ► A new fractional differential inequality is handled. ► A new type of stability for fractional neural network is proposed. ► Exponential synchronization of fractional chaotic networks is considered.
This study proposes an adaptive non-singular integral terminal sliding mode control (ANITSMC) scheme for trajectory tracking of autonomous underwater vehicles (AUVs) with dynamic uncertainties and ...time-varying external disturbances. The ANITSMC is first proposed for a first-order uncertain non-linear dynamic system to eliminate the singularity problem in conventional terminal sliding mode control (TSMC) and avoid the requirement of the bound information of the lumped system uncertainty. The time taken to reach the equilibrium point from any initial error is guaranteed to be finite. The proposed ANITSMC is then applied to trajectory tracking control of AUVs. It guarantees that the velocity tracking errors locally converge to zero in finite time and after that the position tracking errors locally converge to zero exponentially. The designed ANITSMC of AUVs avoids the requirement of the prior knowledge of the lumped system uncertainty bounds as opposite to the existing globally finite-time stable tracking control (GFTSTC), provides higher tracking accuracy than the existing GFTSTC and adaptive non-singular TSMC (ANTSMC) and offers faster convergence rate and better robustness against dynamic uncertainties and time-varying external disturbances than the adaptive proportional-integral sliding mode control (APISMC). Comparative simulation results are presented to validate the superiority of the ANITSMC over the APISMC.
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