First Difference Research Articles

Fault-tolerant formation consensus control for time-varying multi-agent systems with stochastic communication protocol

Article Fault-tolerant formation consensus control for time-varying multi-agent systems with stochastic communication protocol Chunyu Li, Yifan Liu, Ming Gao, and Li Sheng * 1 College of Control Science and Engineering, China University of Petroleum (East China), Qingdao 266580, China * Correspondence: E-mail: shengli@upc.edu.cn Received: 4 September 2023 Accepted: 8 November 2023 Published: 26 March 2024 Abstract: This paper is concerned with the problem of fault-tolerant formation consensus control for linear time-varying (LTV) multi-agent systems (MASs) with stochastic communication protocol (SCP). The SCP is introduced to schedule the signal transmission, and only one neighbouring agent is allowed to transmit data at one instant. The purpose of this work is to design a fault-tolerant controller for each agent, so that, for all probabilistic scheduling behaviors, MASs can achieve the formation consensus performance. The state and fault are augmented into a new vector, meanwhile, each agent system is written as a singular one and a state observer is designed. By utilizing the estimated information of states and faults, the designed time-varying compensation term can reduce the impacts of unknown external disturbances and faults. Then, a sufficient condition is obtained to guarantee the performance constraint over the finite horizon for closed-loop systems. The parameters of observers and controllers are derived by solving coupled backward recursive Riccati difference equations. Finally, a numerical example is given to validate the effectiveness of the proposed fault-tolerant control scheme.

A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

Robust consensus tracking of one‐sided Lipschitz nonlinear multi‐agent systems under structured uncertainty and time‐varying delay

AbstractThis article delves into the intricate matter of achieving robust consensus tracking of multi‐agent systems with structured uncertainty, one‐sided Lipschitz nonlinearity and time‐varying delay. It is important to note that the time delay is both time‐varying and bounded. To solve the consensus tracking problem of multi‐agent systems, we transform it into the stability problem of an error system. Then, a Lyapunov–Krasovskii functional, incorporating the free‐weighting matrix technique and integral inequality approach, is designed to investigate the stability of the obtained error system. The Leibnitz–Newton formula and its associated relationships among terms are also taken into account. Then, two distributed control strategyies that respectively utilizes relative state and output information of neighboring agents are designed. This approach enables us to find a feasible solution to the consensus tracking problem. Finally, several numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.

Accuracy-Improved Fault Diagnosis Method for Rolling Bearing Based on Enhanced ESGMD-CC and BA-ELM Model

The current methods for early fault diagnosis of rolling bearing have some flaws, such as poor fault feature information and insufficient fault feature extraction capability, which makes it challenging to guarantee fault diagnosis accuracy. In order to increase the accuracy of fault diagnosis, it proposes a new fault diagnosis method based on enhanced Symplectic geometry mode decomposition with cosine difference factor and calculus operator (ESGMD-CC) and bat algorithm (BA) optimized extreme learning machine (ELM). The vibration signal is first decomposed into a number of Symplectic geometry components (SGCs) by SGMD. The number of iterations is reduced by the cosine difference factor, which also successfully separates the noise components from the effective components. The calculus operator is adopted to strengthen the weak fault features, making it simple to extract. The fault feature vectors are calculated by the power spectrum entropy-weighted singular values. Finally, the ELM model optimized by BA iteratively is performed as the final classifier for fault classification. The simulation and experiments demonstrate that the proposed method has a better degree of fault diagnostic accuracy and is effective at extracting the rich fault information from vibration signals.

Gaussian process-based state derivative estimator with temporal input in incremental flight control design

In this paper, a state derivative estimation technique is proposed using a derivative of Gaussian process regression method and applied to an incremental dynamics-based controller. The state derivative estimates of interest are with respect to time, and to obtain them, the temporal index array is used for Gaussian process state input. Using a Gaussian process, assumptions such as numerical differentiation of the state, parametric basis functions common in adaptive approaches, and the use of upper-bound information of the state derivative commonly used in robust methods are avoided. The proposed approach is compared with the backward difference formula (BDF) differentiator, first-order low-pass filter, second-order low-pass filter, and high-order sliding mode differentiator. Monte Carlo simulation with 1000 runs is performed under four different noise level measurement scenarios. The proposed Gaussian process-based differentiator with temporal index input resulted in better robustness and adaptability in state derivative estimation with similar or better performance.

Discrete‐time linear quadratic gaussian control with input delay and Markovian packet dropouts

AbstractThis article investigates the finite horizon linear quadratic gaussian (LQG) control problem for networked control systems with two‐way Markovian packet dropouts and input delay, where packet dropouts occur from the sensor to the estimator and from the controller to the actuator, and delay occurs from the controller to the actuator. The novelty of this work lies in providing a complete solution for the optimal control problem subject to two‐way Markovian packet dropouts and input delay. The contributions of this article are as follows: first, by applying the maximum principle to the discrete‐time linear systems and the quadratic cost function involving input delay and Markovian packet dropouts, a solution to the forward and backward stochastic difference equations (FBSDEs) is derived. Second, a sufficient and necessary condition for the optimal control problem is obtained by decoupling the coupled Riccati equations. Finally, the explicit solution of the optimal controller is presented based on the complete square method. Numerical examples are provided to demonstrate the validity of the theoretical results.

Time dependent magnetic field effects on the MHD flow and heat transfer in a rectangular duct

AbstractWe consider the effect of time dependent magnetic field on the natural convection magnetohydrodynamic flow in a rectangular duct. The flow is two‐dimensional, unsteady, laminar and the fluid is electrically conducting and incompressible. The heat transfer is taken into account using the Boussinesq approximation and the buoyancy force is included in the momentum equations for the thermal coupling. The flow is considered at low magnetic Reynolds number setting and hence, the induced magnetic field is neglected. The proposed model in which the governing equations are given in terms of stream function, vorticity and temperature, is approximated by using a Chebyshev spectral collocation method for the spatial discretisation coupled with a backward difference scheme that is unconditionally stable for the temporal integration. The flow and heat transfer characteristic are analysed for several definitions of applied time varying magnetic field. Increase of the Hartmann number and the time variation of the applied magnetic field, changes the flow behaviour, especially at transient time levels and at the steady‐state.

The variable two-step BDF method for parabolic equations

The two-step backward difference formula (BDF) method on variable grids for parabolic equations with self-adjoint elliptic part is considered. Standard stability estimates for adjacent time-step ratios rj:=kj/kj-1⩽1.8685\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_j:=k_j/k_{j-1}\\leqslant 1.8685$$\\end{document} and 1.9104, respectively, have been proved by Becker (BIT 38:644–662, 1998) and Emmrich (J Appl Math Comput 19:33–55, 2005) by the energy technique with a single multiplier. Even slightly improving the ratio is cumbersome. In this paper, we present a novel technique to examine the positive definiteness of banded matrices that are neither Toeplitz nor weakly diagonally dominant; this result can be viewed as a variant of the Grenander–Szegő theorem. Then, utilizing the energy technique with two multipliers, we establish stability for adjacent time-step ratios up to 1.9398.

Fractional Lindley distribution generated by time scale theory, with application to discrete-time lifetime data

ABSTRACT The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support T = R + by T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions.

A hybrid numerical model for coupled hydro-mechanical analysis during CO2 injection into heterogeneous unconventional reservoirs

This paper presents a hybrid numerical model for simulating compressible multiphase multicomponent transport in deformable, heterogeneous unconventional reservoirs. A fully implicit theoretical framework for coupled hydromechanical behaviour is established, in which the flash calculation is avoided. To preserve the local mass conversation, an element based finite volume method is used to spatially discretize mass conversation equations. Linear momentum equilibrium equation is discretized with finite element method. Two numerical approaches are coupled through sharing the same shape functions. An implicit mid-interval backwards difference algorithm is used to handle the temporal derivative. Reliability and efficiency of the developed model have been examined by verification tests against available analytical or numerical solutions. The coupled hydromechanical processes in heterogeneous reservoirs were simulated. The results of numerical experiments show that oscillatory problem for saturation in heterogeneous media can be eliminated, demonstrating better performance of developed model for heterogeneous reservoir simulations. Reservoir performance is significantly shaped by its heterogeneity through controlling fluid flow and time for CO2–CH4 displacement. Simulation results show that the time required for completion of CO2–CH4 displacement increases by about 250 days when considering heterogeneity effect. In conclusion, this study offers a valuable tool for facilitating the development of CO2-enhanced unconventional gas recovery.

A NEW MULTI-STEP BDF ENERGY STABLE TECHNIQUE FOR THE EXTENDED FISHER-KOLMOGOROV EQUATION

The multi-step backward difference formulas of order k (BDF-k) for 3 ≤ k ≤ 5 are proposed for solving the extended Fisher–Kolmogorov equation. Based upon the careful discrete gradient structures of the BDF-k formulas, the suggested numerical schemes are proved to preserve the energy dissipation laws at the discrete levels. The maximum norm priori estimate of the numerical solution is established by means of the energy stable property. With the help of discrete orthogonal convolution kernels techniques, the L2 norm error estimates of the implicit BDF-k schemes are established. Several numerical experiments are included to illustrate our theoretical results.

Finite-time Mittag-Leffler synchronization of delayed fractional-order discrete-time complex-valued genetic regulatory networks: Decomposition and direct approaches

The complex-valued molecular model of mRNA and protein plays a crucial role in the regulatory mechanisms governing gene expression in the genetic regulatory network (GRN). In this study, we introduced a novel GRN utilizing the fractional-order discrete-time complex-valued molecular model of mRNA and protein. Both decomposition and a direct approach are employed to investigate the nonlinear regulatory function within complex-valued molecular models. New conditions are established to ensure the finite-time Mittag-Leffler synchronization (FTMLS) of the error model dynamics. Feedback controllers are applied to derive the FTMLS conditions for delayed fractional-order discrete-time complex-valued genetic regulatory networks (DFDTCVGRNs). By applying the first-order backward difference on the Lyapunov functional under the definition of fractional Caputo difference and fractional calculus theory, sufficient conditions are derived. Finally, two numerical examples are provided to illustrate the FTMLS criteria obtained.

Open-loop and closed-loop local and remote stochastic nonzero-sum game with inconsistent information structure

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example.

Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems

The existence of boundary layers in the solutions of two-parameter singularly perturbed boundary value problems makes classical numerical methods insufficient in providing accurate approximations. Consequently, the development of layer-adapted mesh methods that achieve parameter uniform convergence and are specifically designed to accurately handle these layers becomes highly significant. The aim of this paper is to construct and examine a numerical approach for obtaining approximate solutions to a specific class of two-parameter singularly perturbed second order boundary value problems whose solutions exhibit boundary layers at both ends of the domain. The problem is discretized by employing a modified backward finite difference method on a mesh that is graded. To validate the theoretical findings, well known test problems from the existing literature are utilized. Moreover, the efficiency of the proposed method is demonstrated by comparing it to other existing methods in the literature. The stability and uniform convergence with respect to the parameters of the proposed method have been verified, revealing that it attains second-order convergence in the maximum norm. The numerical outcomes illustrate that the proposed method offers remarkably accurate approximations of the solution. Theoretical findings are in agreement with the experimental results.

An Automatic Frequency Control System for Transmission Machinery Based on Back-Propagation Neural Network Algorithm in the Internet of Things Environment

Abstract In order to reduce the error of automatic frequency control of transmission machinery under different disturbances, an automatic frequency control system of transmission machinery based on back-propagation (BP) neural network algorithm is designed. The gradient descent method and Newton method are combined to optimize the BP neural network. Aiming at the periodicity and trend of the actual mechanical vibration frequency time series, the first order backward difference is processed by the difference method, and the autoregressive sequence is derived to obtain the control model. The experimental results show that the system can realize the automatic control of transmission mechanical frequency with small error under different types of interference, and the control effect is ideal.

Koszul duality, minimal model and L∞-structure for differential algebras with weight

A differential algebra with weight is an abstraction of both the derivation (weight zero) and the forward and backward difference operators (weight ±1). In 2010 Loday established the Koszul duality for the operad of differential algebras of weight zero. He did not treat the case of nonzero weight, noting that new techniques are needed since the operad is no longer quadratic. This paper continues Loday's work and establishes the Koszul duality in the case of nonzero weight. In the process, the minimal model and the Koszul dual homotopy cooperad of the operad governing differential algebras with weight are determined. As a consequence, a notion of homotopy differential algebras with weight is obtained and the deformation complex as well as its L∞-algebra structure for differential algebras with weight are deduced.

Further analysis of stabilization of neutral type singular systems with mixed time‐varying delays and multiple random states

AbstractIn this paper, the stabilization problem of neutral type singular systems (NTSSs) with mixed time‐varying delays and multiple random states is further analyzed. Lyapunov Krasovskill functional (LKF) and generalized Itô formula, together with linear matrix inequality (LMI), are used to represent a new stability criterion for NTSSs with mixed time‐varying delays (MTVDs) in multiple random states. First, the regularity, impulse free, and stochastic stability of the system are satisfied by designing a state feedback controller (SFC). Second, on the basis of Leibniz Newton formula, the stability criterion is also feasible by integrating zero value equation technology, and then combined with singular decomposition, the gain of SFC is obtained. Finally, our paper is a further analysis of the latest literature, so two numerical examples are used to verify the feasibility and effectiveness of this method.

Exploring the Characteristics of Δh Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators

The objective of this article is to introduce the ∆h bivariate Appell polynomials ∆hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of ∆h bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the ∆h bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials ∆h are also proved by demonstrating that the ∆h bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of ∆h bivariate Appell polynomials is provided, and symmetric identities for the ∆h bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for ∆h bivariate Appell polynomials. Additionally, generating relations for the ∆h bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials.

Pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration

AbstractIn this paper, the pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration (NCSC) is considered. The artificial pressure diffusion term is introduced to reduce the constraint of the incompressible condition. The first‐order backward difference formula is adopted to approximate the temporal derivative terms, the diffusion terms in NCSC are treated implicitly, the convective terms in momentum equation are treated semi‐implicitly, the convective terms in energy equation and salt concentration equation are treated implicitly. The error estimates for the semi‐discrete pressure stabilization method are given. Some numerical experiments are presented to verify the theoretical results and show the reliability of the proposed method.

Polynomial maps and polynomial sequences in groups

Abstract This paper presents a modified version of Leibman’s group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers. We will apply it to solve Waring’s problem for general discrete Heisenberg groups in a sequel to this paper.