Piecewise Continuous Functions Research Articles

Intermittent pinning synchronization between two hyperbolic coupled networks with time-varying delays

This paper is devoted to synchronization between two hyperbolic coupled networks (HCNs) with time-varying delays via aperiodically intermittent pinning control (AIPC). Based on the Lyapunov function with a piecewise continuous function, sufficient criteria of HCNs with internal delays only and with hybrid delays are obtained, separately. And it is found that the convergence speed of synchronization of HCNs with hybrid delays is slower than that with internal delays only. Furthermore, two simulation results are presented to verify the theoretical results.

Adaptive Non-Singular Fast Terminal Sliding Mode Trajectory Tracking Control for Robot Manipulators

In order to improve the efficiency of human–robot interaction (HRI), it is necessary to carry out research on precise control of the manipulator. In this paper, an adaptive non-singular fast terminal sliding mode control scheme is proposed for robot manipulators to solve the trajectory tracking problem with model uncertainty and external disturbances. At first, a novel non-singular fast terminal sliding mode surface is proposed, and by introducing an auxiliary function, the singularity problem caused by the inverse of the error-related matrix could be avoided in the controller design process. Then, the controller is developed by using Lyapunov synthesis. A robust adaptive strategy is used to deal with lumped uncertainty, with an adaptive update law designed to compensate for the upper bound of lumped uncertainty whose upper bound is prior unknown. Finally, a two-link robot manipulators as a simulation example is given to illustrate the effectiveness of the proposed scheme. Compared with other similar algorithms, the proposed adaptive non-singular fast terminal sliding mode control scheme has higher efficiency and smaller computational complexity for the reason that no piecewise continuous function is needed to be constructed during the controller design.

Human wealth evolution is an accelerating expansion underpinned by a decelerating optimization process

Optimization and expansion are two modes of staged evolution of complex systems where macroscopic observables change at a decreasing, respectively increasing, rate. The number of microscopic variables and their interactions are fixed in the first case but change dynamically in the second. A prime example of evolutionary expansion, Gross Domestic Product (GDP) time series gauge economic activities in changing societal structures, and the accelerating trend of their growth probably reflects a manyfold increase of the human interactions that drive change. Naively, one could think of cultural evolution as the result of an optimization process, and then expect the associated GDP growth to have a decelerating trend. We show how optimization and expansion can coexist by replacing ‘wall clock time’ t as independent variable with a measure of human interactions intensity τ. The latter was introduced in a previous work and is computed using the GDP time series itself.Our analysis of eight centuries of yearly GDP data from three regions of Western Europe, corresponding to present day UK, France and Sweden is carried out in two steps. First, a Monte Carlo algorithm is used to fit the GDP data to a piecewise continuous function comprising a sequence of exponentials with different exponents. These arguably correspond to social and technological stages of societal organization. In a second step, GDP data are plotted vs. τ and shown to display two logarithmic regimes, both decelerating, that are joined by a power-law cross-over period. We connect the end of the first regime and the beginning of the second with the dawn of the Industrial Revolution and the societal impact of new transport, communication and production technologies that became widely available after World War I. We conclude that wealth evolution in terms of τ is a decelerating process with the hallmarks of record dynamics optimization.

Asynchronously intermittent decentralized control for synchronization of stochastic delayed networks

This paper studies the asynchronously intermittent decentralized control for the synchronization of stochastic complex delayed networks, where the “asynchronously intermittent” means that the intermittent control for each subsystem is activated asynchronously, i.e., each subsystem possesses its own work or rest time. By designing an auxiliary timer and using a graph-theoretic technique as well as the Halanay’s inequality for piecewise continuous function, we obtain a novel synchronization criterion that is independent of the bound of time-varying delay. Finally, a class of coupled heterogeneous neural networks is considered to show the effectiveness and the superiority of the main result.

Adaptive neural control for nonlinear switched systems with improved MDADT and its applications

In this paper, a new framework of the robust adaptive neural control for nonlinear switched stochastic systems is established in the presence of external disturbances and system uncertainties. In the existing works, the design of robust adaptive control laws for nonlinear switched systems mainly relies on the average dwell time method, while the design and analysis based on the model-dependent average dwell time (MDADT) method remains a challenge. An improved MDADT method is developed for the first time, which greatly relaxes the requirements of Lyapunov functions of any two subsystems. Benefiting from the improved MDADT, a switched disturbance observer for discontinuous disturbances is proposed, which realizes the real-time gain adjustment. For known and unknown piecewise continuous nonlinear functions, a processing method based on the tracking differentiator and the neural network is proposed, which skillfully guarantees the continuity of the control law. The theoretical proof shows that the semiglobal uniform ultimate boundedness of all closed-loop signals can be guaranteed under switching signals with MDADT property, and simulation results of the longitudinal maneuvering control at high angle of attack are given to further illustrate the effectiveness of the proposed framework.

Fixed-Time Filtered Adaptive Parameter Estimation and Attitude Control for Quadrotor UAVs

In this article, a fixed-time filtered adaptive parameter estimation and control scheme is proposed for attitude tracking of quadrotor unmanned aerial vehicles. A nonsingular fixed-time sliding mode surface is constructed to avoid the existence of the singularity issue resulted from the differentiation of the sliding mode variable. Through designing auxiliary filtered matrices, a fixed-time parameter update law is developed to achieve the fixed-time parameter convergence. Then, an adaptive neural controller is designed to ensure the fixed-time convergence of attitude tracking errors in the presence of the model uncertainties. Instead of using any piecewise continuous functions, the possible singularity issue in the conventional fixed-time controller design can be overcome by constructing auxiliary functions, and thus the fixed time stability analysis becomes more concise. The efficiency of the presented scheme is validated through experiments on a practical Quanser quadrotor platform.

Fractional Laplace transform for matrix valued functions with applications

We extend several fractional Laplace transform results to matrix-valued functions in this paper, which we will utilize to obtain some useful and valuable theorems, like results on the set of the piecewise continuous functions with conformable exponential order and the low and conditions to obtain the fractional Laplace transform of matrix-valued functions. As an application, we apply the obtained theorems to solve certain fractional initial value problems for vector-valued and matrix-valued functions and the solution will be victor or matrix and it will exact. Also, we establish a fractional system transfer matrix and the conformable fractional Laplace transform of the conformable exponential matrix function, which will give us directly the exact solution for a precisely type of fractional initial value problems.

A discontinuity capturing shallow neural network for elliptic interface problems

In this paper, a new Discontinuity Capturing Shallow Neural Network (DCSNN) for approximating d-dimensional piecewise continuous functions and for solving elliptic interface problems is developed. There are three novel features in the present network; namely, (i) jump discontinuities are accurately captured, (ii) it is completely shallow, comprising only one hidden layer, (iii) it is completely mesh-free for solving partial differential equations. The crucial idea here is that a d-dimensional piecewise continuous function can be extended to a continuous function defined in (d+1)-dimensional space, where the augmented coordinate variable labels the pieces of each sub-domain. We then construct a shallow neural network to express this new function. Since only one hidden layer is employed, the number of training parameters (weights and biases) scales linearly with the dimension and the neurons used in the hidden layer. For solving elliptic interface problems, the network is trained by minimizing the mean square error loss that consists of the residual of the governing equation, boundary condition, and the interface jump conditions. We perform a series of numerical tests to demonstrate the accuracy of the present network. Our DCSNN model is efficient due to only a moderate number of parameters needed to be trained (a few hundred parameters used throughout all numerical examples), and the results indicate good accuracy. Compared with the results obtained by the traditional grid-based immersed interface method (IIM), which is designed particularly for elliptic interface problems, our network model shows a better accuracy than IIM. We conclude by solving a six-dimensional problem to demonstrate the capability of the present network for high-dimensional applications.

On a Nonlinear Three-Point Subdivision Scheme Reproducing Piecewise Constant Functions

In this article, a nonlinear binary three-point non-interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the three-point subdivision scheme: A new three point approximating C2 subdivision scheme, where the convergence and the stability of this linear subdivision scheme are analyzed. It is possible to prove that this scheme does not present Gibbs oscillations in the limit functions obtained. The numerical experiments show that the linear scheme is stable even in the presence of jump discontinuities. Even though, close to jump discontinuities, the accuracy is loosed. This order reduction is equivalent to the introduction of some diffusion. Diffusion is a good property for subdivision schemes when the discontinuities are numerical, i.e., they appear when discretizing a continuous function close to high gradients. On the other hand, if the initial control points come from the discretization of a piecewise continuous function, it can be interesting that the subdivision scheme produces a piecewise continuous limit function. For instance, in the approximation of conservation laws, real discontinuities appear as shocks in the solution. The nonlinear modification introduced in this work allows to attain this objective. As far as we know, this is the first subdivision scheme that appears in the literature with these properties.

A novel kernel functions algorithm for solving impulsive boundary value problems

In this letter, to fit the character of solutions to impulsive boundary value problems (IBVPs), we firstly construct a piecewise continuous space using the reproducing kernel function (RKF) of smooth reproducing kernel Hilbert space (RKHS) W3. Then we present a fourth order convergent collocation approach for IBVPs by using the piecewise continuous basis functions yielded by RKFs in W3. Also, the convergence order of our approach is proved. The accuracy and convergence order are shown numerically.

Error estimates for the numerical approximation of optimal control problems with nonsmooth pointwise-integral control constraints

Abstract The numerical approximation of an optimal control problem governed by a semilinear parabolic equation and constrained by a bound on the spatial $L^1$-norm of the control at every instant of time is studied. Spatial discretizations of the controls by piecewise constant and continuous piecewise linear functions are investigated. Under finite element approximations, the sparsity properties of the continuous solutions are preserved in a natural way using piecewise constant approximations of the control, but suitable numerical integration of the objective functional and of the constraint must be used to keep the sparsity pattern when using spatially continuous piecewise linear approximations. We also obtain error estimates and finally present some numerical examples.

Approximate-Analytic Solution of Hyperchaotic Finance System by Multistage Approach

This paper devotes to constructing an approximate analytic solution for the hyperchaotic finance model. The model describes the time variation of the interest rate, the investment demand, the price exponent, and the average profit margin. The multistage homotopy analysis method (MHAM) and multistage variational iteration method (MVIM) are utilized to generate the analytical solutions. The solutions are presented in terms of continuous piecewise functions without interpolation. These procedures prove their applicability for this kind of model due to rapidly convergent series solutions with easily computable terms, iterates, and efficiently obtained by applying it over multiple time intervals. We also provide the convergences theorem of the MHAM. Numerical comparisons are displayed with the results obtained by MHAM, MVIM, and the fourth-order Runge-Kutta method to demonstrate the validity and effectivity of this procedure.

ГЕОДИНАМІКА

The paper considers representations of the Earth external gravitational field, supplementing its traditional approximation by series in spherical functions. The necessity for additional means of describing the external potential is dictated by the need to study and use it at points in space close to the Earth's surface. It is in such areas that the need arises to investigate the convergence of series with respect to spherical functions and to adequately determine the value of the potential. The apparatus for approximating a piecewise continuous function in the middle of the ellipse is used for the representation of the Earth external gravitational field by the simple and double layer integrals. This makes it possible to expand the convergence region for the series supplying the potential to the entire space outside the integration ellipse. Therefore, as a result, the value of the gravitational potential coincides with the values of these series outside the body containing the interior masses (except for the integration ellipse). It becomes possible to evaluate the gravitational field behavior in surface areas and to carry out studies of geodynamic processes with greater reliability. Approximation of the gravitational field with the help of surface integrals also determines the geophysical aspect of the problem. Indeed, in the process of solving the problem we constructed two-dimensional integrands, which are uniquely determined by a set of Stokes constants. In this case, their expansion coefficients into series are defined by linear combinations of their function power moments. The resulting function schedules can be used to study the external gravitational field features, e.g., to study its asymmetry with respect to the equatorial plane.

Uniform exponential stability criteria with verification for nonautonomous switched nonlinear systems with uncertainty

AbstractThis article is concerned with the sufficient and necessary conditions for the global uniform exponential stability (GUES) of a class of uncertain nonautonomous switched nonlinear systems (UNSNS), that is, systems with time‐varying uncertain information. To start with, based on a piecewise continuous scalar function and a piecewise differentiable uncertainty‐dependent Lyapunov function along the trajectories, we propose alternative sufficient and necessary conditions for the GUES of UNSNS. Afterwards, for removing the dependence on uncertain information of the required Lyapunov functions, we combine an uncertainty‐independent Lyapunov function with its right upper Dini derivative and the above piecewise continuous scalar function to further obtain other alternative sufficient and necessary conditions for the GUES of UNSNS. Note that our conditions release the requirement on negative definiteness of the time‐derivatives and Dini derivatives of Lyapunov functions respectively, which is illustrated by Example 1. Further, for the rational UNSNS, we propose a linear semidefinite programming based computable approach to mechanically verify our current theoretical results. In the end, the effectiveness of our theoretical results and mechanical approach are illustrated by two examples.

Medium-Term Hourly Electricity Tariff Forecasting Using Ensemble Models

Forecasting electricity tariff rates is necessary for large suppliers, consumers, and power brokers working in the wholesale markets. Meanwhile, tariff rates of the retail market are also hourly changed for certain groups of electricity consumers. It creates more efficient electrical load regulation opportunities than the traditional load leveling approach. Power facilities that include controlled load consumers or local generation can use their capabilities by adjusting the load curve according to tariff rates. This work aims to study the potential for medium-term forecasting of retail electricity tariff rates and develop a predictive machine learning model. Hourly data on the retail market tariffs of the Novosibirsk region (Siberia) for four years were collected, several machine learning models were applied, and an analysis of the input parameters for forecasting was carried out. The most significant results are the proof of the possibility of obtaining the month ahead electricity tariff rate forecast with the mean absolute percentage error 4 %. It could be used for electricity costs reduction by regulating the load curve. It was shown that the discrete models based on ensembles of logical rules give higher accuracy than models based on continuous and piecewise continuous functions, such as neural networks. The significance of the obtained results is the proposed approach for month ahead electricity tariff rates forecasting, which was verified on a four-year dataset with an error of 4 %. The approach is based on open data and open-source machine learning models, which allow specialists with even a basic level of data science skills to put it into practice.

Collocation Approach Based on an Extended Cubic B -Spline for a Second-Order Volterra Partial Integrodifferential Equation

This paper focuses on an efficient spline-based numerical technique for numerically addressing a second-order Volterra partial integrodifferential equation. The time derivative is discretized using a finite difference scheme, while the space derivative is approximated using the extended cubic B -spline basis. The scheme is also tested for stability study to ensure that the errors do not accumulate. The convergence of the proposed scheme is also investigated. The scheme’s key benefit is that the approximate solution is produced as a smooth piecewise continuous function allowing us to approximate the solution at any location in the domain. Numerical study is performed, and the comparison of results is made to previously reported results in the literature to show the efficiency of the suggested scheme.

Topological structure of solution sets for hybrid impulsive fractional differential inclusion in banach algebra

The aim of this paper is to study the topological structure of solution sets of hybrid impulsive fractional differential inclusions on the weighted space of piecewise continuous functions, we apply dhage fixed point theorem for multivalued operator with analysis tools to prove existence result. The paper concludes with an example to illustrate the feasibility of our main result.

An efficient collocation method based on Hermite formula and cubic B-splines for numerical solution of the Burgers’ equation

In this paper, an efficient computational technique for the numerical solution of the Burgers’ equation (BE) is presented. The derivative in space is approximated using cubic B-splines and the Hermite formula whereas time discretization is performed by finite differences. The stability of the proposed scheme is derived using the standard von Neumann method to establish the fact that the errors do not amplify. Convergence analysis for the proposed scheme is also discussed. A sufficiently smooth piecewise continuous function is obtained as an approximation to the exact solution which enables us to approximate the solution at any wanted position in the domain. Numerical tests are carried out to further confirm the accuracy and stability of the method. The outcomes of this study are compared with those previously presented in literature.

First failure load of sandwich beams under transient loading using a space–time coupled finite element method

We numerically analyze transient plane stress deformations of linearly elastic laminates and sandwich structures by using a coupled space–time finite element method with the weak form of governing equations derived by the least-squares method. The governing equations are written as seven first-order partial differential equations as in the state space formulation, and each variable is approximated by using piece-wise continuous basis functions. The sum of the integral over the problem (space-time) domain of squares of residuals of the governing equations, and initial and boundary conditions is minimized with respect to the nodal values of the seven variables to deduce coupled linear algebraic equations. The developed software is verified by comparing computed results for sample problems with their either analytical or numerical solutions obtained with a commercial software, ABAQUS. Using statistical method and the Tsai–Wu failure criterion, sensitivities of the maximum deflection and the first failure load to beam’s aspect ratio (AR), the facesheet-core thickness ratio (FCTR), and the facesheet-core in-plane and transverse stiffness ratios (FCISR, FCTSR) are elucidated. For sandwich beams with equal FCISR and FCTSR, the results are sensitive to the AR and the FCTR. In general, they strongly depend upon the FCTSR and its interaction with the AR and the FCTR.

New criteria of almost sure exponential stability and instability of nonlinear stochastic systems with a generalization to stochastic coupled systems

This paper focuses on the almost sure exponential stability and instability problems for nonlinear stochastic systems. In contrast with previous works, combining with Lyapunov’s second method and inequalities techniques, two classes of less conservative criteria are obtained, depending on piecewise continuous scalar functions (PCSFs). It is highly desirable that the PCSF can not only replace the constant coefficient of the upper bound estimation which belongs to the diffusion operator of a Lyapunov function in available literature but also even be unbounded. Especially, when PCSFs are assumed to be periodic, our theoretical results can be transformed into the case of almost sure exponential stability and instability of nonlinear systems with intermittent noises, and corresponding criteria are derived. Furthermore, the sufficient criteria we present are popularized to stochastic coupled systems taking advantage of graph theory and Lyapunov method. As an application of theoretical results, we put forward an exploration to the synchronization analysis of a class of single-link robot arms system via intermittent noise control. Finally, several numerical examples consisting of a time-varying stochastic system and a stochastic coupled oscillators system as well as single-link robot arms systems are provided to illustrate the feasibility of the stated theoretical results.