Descriptor systems (also known as singular systems), which are first proposed by Rosenbrock in the 1970s [1] are a class of more general dynamic systems than general ones. They have been widely used in many fields in recent years [2–4], and increasing attention has been paid to the study of descriptor systems [5–7]. Moreover, many characteristics of analysis and synthesis in general systems theory also have been extended to descriptor systems [8, 9].

Preview control (PC) is a kind of feed-forward control method. By fully utiliing the known future knowledge of references or disturbances, it can improve the tracking performance of the controlled system [10, 11]. For discrete-time systems, by using a difference operator, the preview compensation can be constructed as part of the augmented discrete-time systems. Therefore, the PC problem of discrete-time systems can be converted into a stabilisation problem. So far, the most extensive researches of PC for discrete-time systems are based on linear quadratic regulator (LQR) or linear matrix inequality (LMI) [12–14]. Moreover, the problems of H PC [15], adaptive PC [16] and sliding mode-based PC [17] are also investigated.

For continuous-time systems, the preview compensation cannot be used directly as a part of the augmented dynamic systems. An alternative method is to use the derivatives of the state equation and the tracking error. However, the obtained augmented dynamic systems are non-autonomous ones. As a result, the difficulty and complexity are greatly increased. By differentiating both the output error signal and the two sides of the state equation, the optimal PC problem of continuous-time systems is solved in [18–20]. By taking the derivative of the control input, a new optimal PC was deduced [21]. The optimal PC problem of impulse-free descriptor systems was studied [23]. More recently, for continuous-time descriptor multi-agent systems, a cooperative optimal preview tracking problem was considered in [24] by using the LQR technique.

On the other hand, repetitive control (RC) is an effective technique for improving the performance of systems that track periodic reference or reject periodic disturbance [25]. It was originated by Inoue et al. [25] and subsequently developed by many researchers, such as Hara [26] and Doh [27]. Nowadays, a great deal of research has been focused on the theory and applications of RC, and various structures and algorithms have been devised (see, for example, [28–32]). The combination of preview and RC, which is called preview RC, can significantly improve the control performance of closed-loop systems. Since the relationships between the system and the future signal as well as the output of the RC were established by the difference operator, preview RC for discrete-time systems has become very popular in many research fields.

A discrete-time preview RC method [34] was first presented based on LQ optimal control. Using ARMAX models, a new design method for discrete-time preview RC was proposed [35]. The design method of discrete-time sliding mode preview repetitive servo systems was introduced [36, 37]. Very recently, an LMI-based robust guaranteed-cost preview RC for uncertain discrete-time systems was proposed [38].

It is worth pointing out that although there have been fruitful results about discrete-time preview RC, there have been only a few studies on designing preview RC for continuous-time systems. This paper focuses on the preview RC problem of linear continuous-time descriptor systems. The main contributions are as follows:

The descriptor system is transformed into a restricted equivalent one with an algebraic system by taking non-singular linear transformation.

Based on the lift technique, an augmented system is constructed, which contains the output of the basic RC;

Different from the existing PC methods, the augmented state vector and the derivative of the tracking error, as well as the derivative of the control input, are introduced into the quadratic performance index; (3) Based on the optimal control, the regulator problem of the augmented system is solved, from which the explicit optimal preview repetitive controller (OPRC) for the original system is deduced.

The rest of this paper is organised as follows. Section 1 presents the problem formulation and assumptions. The OPRC is derived in Section 2. A numerical example is provided in Section 3. Finally, some conclusions are drawn in Section 4.

Figure 1 shows the basic configuration of a continuous-time RC system, where G(s) is the controlled plant, r(t) is a periodic reference input with period L, and C_{R}(s) is an RC. The output of the RC, v(t), is the tracking error.

This paper focuses on OPRC designing of a plant with a relative degree of zero, which is described by the following regular and impulse-free continuous-time descriptor system.
^{n} is the state, ^{q} is the control input, and ^{p} is the output of the plant. ^{p} be a periodic reference signal to be tracked by _{r}_{r}_{r}

The objective of this paper is to develop an RC with preview compensation such that in the steady-state, the output vector

The following assumptions and lemmas are needed.

Assume that system (3) is regular and there exists a _{0} such that det(_{0}

The matrix pair (

The matrix pair (

The matrix

In this subsection, we transform system (3) into a restricted equivalent form by non-singular linear transformation.

Taking non-singular linear transformation

Therefore, the regularity, impulse-free, stability and detectability of systems (3) and (10) are equivalent. On the other hand, it follows from assumption A2 that system (10) is impulse-free, which implies that _{22} is non-singular [3].

Since _{22} is non-singular, from the second equation of (10), we get

Substituting (11) into the first and the third equation of (10), respectively, we have

For

Eqs (22) and (23) are augmented dynamic systems. Correspondingly, we wish to find the optimal controller that minimises the following performance index [39]
^{q×q}, _{e}^{p×p} are positive-definite matrices, and _{z}_{x} Q_{xe}_{x}^{m×m}_{xe}^{p×p} are nonnegative-definite and positive-definite matrices, respectively.

Now, the design problem is transformed into the design of a control input

The following lemmas related to stabilisability (controllability) and detectability (observability) give the existence conditions of the optimal control law for the augmented dynamic system.

_{p}

_{22} is non-singular, by some calculation, we have

Clearly, the matrices

Based on Lemma 3 to Lemma, we have the following Lemma 6.

_{xe} is positive-definite and the martrix pair

_{xe}

Based on the theory of optimal control, we obtain the main result of this paper as follows.

_{e}, F_{v} and F_{x} are given by_{x} K_{e}_{c} in

^{n̅×n̅}, Δ^{n̅×1} and

Using Eq. (40) in Eq. (42), we can obtain
_{a}

Now, let _{a}_{a}_{c} is stable, with _{r}_{a}, τ

On the other hand, substituting Eq. (42) with _{x} K_{e}_{r}

_{x}x_{1} (_{v}v

_{d} and i_{q} are the d – q axis currents, u_{d} and u_{q} are the d – q axis voltages. R and L denotes the stator resistance and inductance per phase respectively, p is the number of pole pairs, φf is the permanent magnet flux, J is the rotor moment of inertia, B is the viscous friction factor and T_{L} also represents the load torque

To decouple the speed and currents, the vector control strategy of _{d}_{L}

The parameters of PMSM system are set as

Given the structure of

The numerical simulation results are shown in Figure 3 to Figure 6. Figures 3 and 4 show the output response and tracking error with different preview lengths. It can be seen that the tracking quality of the output signal can be improved by adjusting the preview lengths. But when the preview lengths reach a certain degree, there is almost no effect on the output response.

For comparison, the LMI-based RC [28, 29] simulation is also provided. The simulation results between RC and OPRC with preview lengths _{r}

This paper considered the design method of OPRC for impulse-free continuous-time descriptor systems. Based on the characteristics of impulse-free descriptor systems, the descriptor system was transformed into a restricted equivalent form. By using the lift technique and the construction of the augmented system, the OPRC designing problem was converted into a regulator problem. The optimal controller for the augmented system was designed by making use of standard optimal control theory, and the OPRC of the original system was then obtained, in which the integrator compensation was innovatively added. The effectiveness of the proposed method has been shown by a numerical example.

Research and implementation of smart city public information mining analysis system based on mobile edge model of game theory Design of college education evaluation based on accompanying data acquisition and mathematical analysis Enterprise financial strategy and performance management analysis based on principal component analysis Study on the Dynamic Change of Regional Water Level and Climate Based on Forecast Equation Municipal Civil Engineering Construction Based on Finite Element Differential Equations Financial Risk Prevention Model of Financial Institutions Based on Linear Partial Differential Equation Application and Practice of Artificial Intelligence Technology in Interior Design Statistical Model of College Students’ Mental Health Based on the Law of Large Numbers Application of cloud accounting in enterprise financial forecasting and decision making in the era of big data The Characteristics and Monitoring System of Athletes’ Physical Strength Training Based on Tunable Neuron Mathematical Model Research on the value dimension and practical path of the construction of college soccer curriculum thinking politics under the perspective of multimedia + moral education Garden Landscape Construction Based on Landscape Urbanism and Dynamic Nonlinear System Thinking Influence of heterogeneity of local officials on the economy of resource-based cities developed with high quality Path construction of public management model based on POI social computing Analysis of influencing factors of SPOC course teaching effect using structural equation modelling Formation mechanism of financial vulnerability in rural China based on Markov model and countermeasures Secure transmission of wireless energy-carrying communication systems for the Internet of Things Dynamic Optimization of Robot Automatic Control System Based on Differential Algebraic Equations Training Model of Basketball Offensive Route Based on Nonlinear Differential Equation Research on the construction of rural interface style based on aesthetic rules Optimization Algorithm of New Media Hot Event Push Based on Nonlinear Differential Equation Mathematical Differential Equation in Calculating the Strength of Building Beam Structure Impact Resistance Stability of Building Structural Engineering Based on Fractional Differential Equations The Technical Research on the Assessment of Network Security Situation Based on D-S Evidence Theory Computer big data modeling system based on finite element mathematical equation simulation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation The Composition System of Pop Music Movement Based on Finite Element Differential Equations The Structure and Influencing Factors of Innovation and Entrepreneurship Ability of Higher Vocational Students Based on Structural Equation Model Model System Study of Accordion Score Based on Fractional Differential Equations Visual Communication Effect Modeling of Public Service Advertisements Based on Fractional Differential Equations Digital Finance in the Context of Common Wealth Helps Regional Economic Development of High Quality Data mining of Chain convenience stores location A method to predict whether middle school students will enter STEM careers in the future based on FC-Wide&Deep model Parameter Id of Metal Hi-pressure State Equation Mathematical Method to Construct Linear Programming Scheme of Football Training P-Matrix Reasoning and Information Intelligent Mining Symmetry Analysis and Exact Solutions of Extended Kadomtsev-Petviashvili Equation in Fluids Deep learning-based security situational awareness and detection technology for power networks in the context of big data Research on Detection Model of Abnormal Data in Engineering Cost List Digital Film Music Creation Model Based on Inhomogeneous First Order Constant Coefficient Linear Differential Equation Research on deformation monitoring of tunnel engineering based on 3D laser scanning Trajectory tracking control of remotely operated vehicle based on thruster dynamics Solidification treatment effect of wellsite waste mud based on physical adsorption of a composite curing agent A study of immersive technology for product usability improvement design based on comprehensive value evaluation The Calculation of Sports Injury Sensitivity of Martial Arts Athletes Based on ROC Curve On new aspects of Chebyshev polynomials for space-time fractional diffusion process Training Evaluation Model of Outdoor Cross-Country Sports Tourism Project Based on Lagrangian Differential Equation Calculation Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Application of Nonlinear Fractional Differential Equations in Computer Artificial Intelligence Algorithms Stability of equilibrium price-production of dynamic duopoly Bertrand–Cournot game in a transnational supply chain with asymmetric information Application of calculus function and value chain analysis method in the quantification of financial flexibility management indicators Analysis of the existing constraints of the “sports season teaching model” in physical education in colleges and universities Optimal Research in Piano Shape Sound and Sound Propagation Model Based on Nonlinear Differential Equations The stability of resource and environmental economics model based on hyperbolic sine function Analysis and elaboration of enterprise management innovation path based on logistic regression perspective Research on secure Internet of things gateway technology based on multi-communication methods Analysis on the application method of new media technology in film and television media in the era of big data cloud Establishment and application of traditional Wushu intelligent learning resource database under the background of big data A Review of the Design and Enhancement of Pastoral Culture Creativity in Chinese Rural Ecotourism Analysis and Research on Technical and Tactical Action Recognition in Football Based on 3D Neural Network Nonlinear Differential Equation in University Education Information Course Selection System Big Data Analytics Technology and Applications in Cloud Computing Perspective Cost management of power grid transmission and substation project based on BIM technology Analysis of IPO pricing efficiency under the registration system Case application data research of traditional ink art elements in packaging design Research on the identification of common faults of agricultural machinery based on vibration characteristics Research on an early warning model of effectiveness evaluation in ideological and political teaching based on big data Design and implementation of computer network security protection system based on artificial intelligence technology Application of intelligent teaching resource organisation model and construction of performance evaluation model Secure transmission of simultaneous wireless information and power transfer system for Internet of things Research on the application of traditional culture in landscape design The Evaluation Model of College Students’ Mental Health Under the Fractional Differential Equation Numerical simulation of vortex vibration in main girder of cable-stayed bridge based on bidirectional fluid–structure coupling Deep learning-based foreign object detection method for aviation runways Research on location algorithm of new energy vehicle charging station based on multi-objective decision Ancient Building Surveying and Mapping Modeling Protection System Based on Lagrangian Mathematical Equation Sports Science Movement model based on fractional differential equation Training fatigue and recovery of Wushu Sanda athletes based on comprehensive environmental testing Application of grey correlation degree algorithm in the evaluation system of college students’ innovation and entrepreneurship in applied undergraduate institutions Optimisation of construction mode of residential houses based on the genetic algorithm under BIM technology The development of the integration of red gene and curriculum thinking based on BP neural network mathematical algorithm Construction of a social audit platform based on big data for “industry price, quality and credit” A study on the reform of college English education inspired by the cultural differences between China and the United States Research on innovative strategies of college students’ English teaching under the blessing of big data Structure innovation of agricultural supply chain in china: from farmers’ perspective Research on multi-dimensional optimisation design of user interface under Rhino/GH platform Quantitative Analysis of the Influence of Coal Mining on the Underflow Zone of Rivers by Using Double Integro-differential Equations Optimisation analysis of distribution network planning based on the IVIF-AHP method Exploration of “English Plus” teaching mode under the background of blockchain construction A long command subsequence algorithm for manufacturing industry recommendation systems with similarity connection technology Analysis of tea culture communication path based on the principal component analysis method Supply chain network model of cost accounting based on improved particle swarm optimisation algorithm Construction of a multimodal poetry translation corpus based on AdaBoost model Research on personalised recommendation algorithm for college students’ employment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Evaluation of spoken English self-study system considering speech knowledge recognition algorithm Design of a decentralized Internet-based control system for industrial robot robotic arms Design of a mechanical automatic transmission system for automobiles based on torsional vibration damping Application research on piano teaching in colleges and universities based on remote wireless network communication Construction of a mathematical modeling teaching quality assessment system for universities based on Eviews model Construction of Financial Risk Evaluation Index System for Biomass Graphene Fibre Industrialisation Project Innovative model of teaching in law experiment center based on LETS software A Study on the Implications of Critical Discourse Analysis Theory for College English Teaching in the Context of Big Data Computational research on flat image design and style based on perceptual feature quantification The artistic aesthetic and cultural value orientation of 3D printing A multidimensional interpretation of Marxist theory based on a decentralized Internet perspective Student management model of college student associations based on ant colony Algorithm Analysis of the evolution of China’s energy legal system under the background of discrete model theory and dual carbon Design of Morlet wavelet neural network to solve the non-linear influenza disease system An online statistical analysis of the hotel management and operation performance model Research on the relationship between the construction of school teachers’ team literacy and the educational environment Research on the post-purchase behaviour tendency of the product quality to customers in fast-selling marketing Motion about equilibrium points in the Jupiter-Europa system with oblateness Design of an embedded machine vision system for smart cameras Ultra-short-term power forecast of photovoltaic power station based on VMD–LSTM model optimised by SSA Optimal preview repetitive control for impulse-free continuous-time descriptor systems Design of information management system based on wireless communication under the background of Internet of Things The impact of global warming on the small Scottish Fishing Company Research on heat transfer properties of graphene composites Development of main functional modules for MVB and its application in rail transit Analysis of agricultural economic development and optimisation measures under the strategy of rural revitalisation Study on the impact of forest fire prevention policy on the health of forest resources Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Research on the relationship between government subsidies, R&D investment and high-quality development of manufacturing industry Research on the protection and inheritance of intangible cultural heritage under the background of rural revitalisation Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment Computer vision recognition and tracking algorithm based on convolutional neural network Innovative reform strategy of artistic practice of university folk dance education based on information fusion technology A review of the treatment techniques of VOC Study on structural parameter design and life analysis of large four-point contact ball bearing channel Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}Digital marketing solutions based on consumer data and homomorphic encryption The consistency method of linguistic information and other four preference information in group decision-making Research on the driving principle and guiding strategy of the public's collaborative supervision of the sharing economy in my country Influence of innovative human capital on economic development of China through the STI model New ideas of precision teaching development based on the background of big data Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development Application research of bel canto performance based on artificial intelligence technology Research on the path of integrating blockchain technology into the teaching of film and television courses in higher education institutions Combined application of employment education and big data internet technology based on the context of vocational education reform Fractal structure of magnetic island in tokamak plasma College students' innovation and entrepreneurship model based on probability theory statistics Exploring the path of network security and student privacy protection in smart campus based on Markov model Innovation and entrepreneurship model of higher vocational college students based on probability theory statistics Application of knowledge graph in smart grid fault diagnosis Research on the innovation of youth network media literacy education in the context of Internet + education Research on the optimisation of logistics parcel intelligent sorting and conveying chain combined with variable clustering mathematical method Research on power quality assessment based on ubiquitous power IoT Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Analysis of network information security issues under the background of big data Research on SOC estimation of residual power of lithium-ion batteries for electric vehicles based on extended Kalman filtering algorithm Harmonic current variation characteristics of shunt capacitors based on full-phase FFT algorithm Urban Public Epidemic Prevention and Control Model Based on Nonlinear Differential Equations Research on computer vision technology based on BP-LSTM hybrid network Research progress of computer vision tasks based on deep learning and SAE network Application of Discriminative Training Algorithm Based on Intelligent Computing in English Translation Evaluation Research on the application of GLE teaching mode in English-medium colleges The application of directional derivative in the design of animation characters and background elements Research on product process design and optimisation model based on IoT intelligent computing Simulation research on music feature recognition based on mobile big data and smart sensors Digital design and experimental testing of a compressor's suction muffler transmission loss LTE wireless network coverage optimisation based on corrected propagation model SCP model based on the new energy vehicle collaborative detection configuration method research Study on tourism development income index calculation of finite element ordinary differential mathematical equation The cardiovascular promotion of college taekwondo based on fractional differential equation Analysis and research on the economic benefits of employee labor in enterprises in the context of the Internet Application of renewable energy decorative materials in modern architectural design under low-carbon concept The boundedness of functional differential equations of hospital financial software under information Deep learning model-based brand design 3D image construction The character image setting of 3D animation works based on Poisson equation Research on the interaction between university physical culture education and college students’ physical exercise in the context of big data The optimization path of core literacy of innovative talents in universities based on the improved Markov model Analysis of the innovation path of university education management informatization in the era of big data Impact of ASEAN-China free trade area on fishery value chain based on difference-in-difference method Research on indoor environment design of comprehensive commercial shopping center based on numerical simulation Empirical analysis of the economic absolute income hypothesis based on mathematical statistics Realization of Book Collaborative Filtering Personalized Recommendation System Based on Linear Regression Equation Study on Interactive Relations between Enterprise Social Media and Decision Style Based on a vector Autoregressive Model Research on Dynamics of Flexible Multibody System with Deployable Antenna Based on Static Lagrangian Function The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations Study on spatial planning and design of learning commons in university libraries based on fuzzy matrix model The Stability Model of Piano Tone Tuning Based on Ordinary Differential Equations Construction and application of automobile user portrait based on k-mean clustering model The Homework Model of Screening English Teaching Courses Based on Fractional Differential Equations MATLAB-based simulation of industrial robots in water environment monitoring An innovative teaching model of automotive manufacturing process based on ADDIE model The train of thought and design of university talents training based on ADDIE model from the perspective of ideological and political education Research on the innovation of model based on the cultivation of talents in higher vocational art and design Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling The Mathematical Analysis Model of Educational System in Music Courses in Colleges and Universities Continuing Education Network Data Center Model Based on Fractional Differential Mathematical Equations Study on the factors influencing the evolution mechanism of cobalt trade network from the perspective of national competition Computer Art Design Model Based on Nonlinear Fractional Differential Equations The Optimization Model of Public Space Design Teaching Reform Based on Fractional Differential Equations The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation Graphical Modular Power Technology of Distribution Network Based on Machine Learning Statistical Mathematical Equation Reliability analysis of reinforced concrete beams based on the stochastic model of resistance calculation Employment and Professional Education Training System of College Graduates Based on the Law of Large Numbers Economic Research on Multiple Linear Regression in Fruit Market inspection and Management Nonlinear Differential Equations in Preventing Financial Risks Lagrange’s Mathematical Equations in the Sports Training of College Students Simulation Research of Electrostatic Precipitator Power Supply Voltage Control System Based on Finite Element Differential Equation Research on the effect of generative adversarial network based on wavelet transform hidden Markov model on face creation and classification Research on Lightweight Injection Molding (CAE) and Numerical Simulation Calculate of New Energy Vehicle Power Flow Calculation in Smart Distribution Network Based on Power Machine Learning Based on Fractional Differential Equations Demonstration of application program of logistics public information management platform based on fuzzy constrained programming mathematical model Basketball Shooting Rate Based on Multiple Regression Logical-Mathematical Algorithm The Optimal Application of Lagrangian Mathematical Equations in Computer Data Analysis Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation Optimal decisions and channel coordination of a green supply chain with marketing effort and fairness concerns Game theoretic model for low carbon supply chain under carbon emissions reduction sensitive random demand The Relationship Between College Students’ Taekwondo Courses and College Health Based on Mathematical Statistics Equations Fractional Differential Equations in the Exploration of Geological and Mineral Construction AdaBoost Algorithm in Trustworthy Network for Anomaly Intrusion Detection Enterprise Financial Risk Early Warning System Based on Structural Equation Model A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment Information transmission simulation of Internet of things communication nodes under collision free probability equation Image denoising model based on improved fractional calculus mathematical equation Random Fourier Approximation of the Kernel Function in Programmable Networks The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models University Library Lending System Model Based on Fractional Differential Equations Calculation and Performance Evaluation of Text Similarity Based on Strong Classification Features Intelligent Matching System of Clauses in International Investment Arbitration Cases Based on Big Data Statistical Model Evaluation and Verification of Patent Value Based on Combination Forecasting Model Financial Institution Prevention Financial Risk Monitoring System Under the Fusion of Partial Differential Equations Prediction and Analysis of ChiNext Stock Price Based on Linear and Non-linear Composite Model Calculus Logic Function in Tax Risk Avoidance in Different Stages of Enterprises The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations Optimization Simulation System of University Science Education Based on Finite Differential Equations The Law of Large Numbers in Children's Education Optimization System of Strength and Flexibility Training in Aerobics Course Based on Lagrangian Mathematical Equation Data structure simulation for the reform of the teaching process of university computer courses A multi-factor Regression Equation-based Test of Fitness Maximal Aerobic Capacity in Athletes Garment Image Retrieval based on Grab Cut Auto Segmentation and Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multi-feature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation Construction of an enterprise financial risk management system based on F-score model Research on education management system based on machine learning and multidimensional data modeling Multi-featured multi-scale combination of high-resolution remote sensing images for building extraction Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Electronic properties of diamond semiconductor materials: based on response surface model Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching Deep learning techniques in modern women’s smart clothing design Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology A study on the aesthetic characteristics of ethnic music based on radial basis neural network model The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Analysis of the application methods of film and television media and images in the era of big data cloud Air data computer simulation and design credibility assessment considering ADC calculation model Analysis of news communication strategies in the era of full media based on big data mining On the improvement, innovation and inheritance of stage makeup styling in opera under the background of big data Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Evaluation Mechanism of Teaching Staff’s Building in Local Vocational Colleges Based on Big Data Technology Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Research on detection of parking space based on the combination of magnetoresistance and microwave radar sensor under IoT platform Research on sustainable development mechanism of regional ‘Artisan Workshop’ under big data environment Application of hybrid kernel function in economic benefit analysis and evaluation of enterprises The analysis method of the impact of missing audit information on the securities investment decisions based on Logistic regression models Research on the innovation theory of college students’ ideological and political education under the background of big data Research on classification of e-commerce customers based on BP neural network Research on the problems and reform strategies of government statistics based on big data technology The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Adaptive fusion of multi-exposure images based on perceptron model Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Exploring the Path of Integrating Excellent Traditional Culture into Civic Education in Universities in the Internet+ Era Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Design and research of environmentally friendly field sewage purification mechanical device 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System An analysis of time commitment for college students to online English writing and peer assessment learning Research on network security management monitoring platform based on Internet of things technology Study of frictional wear properties of materials for mechanical seals Fractional Differential Equations in Electronic Information Models Exploring the construction of an all-round cost management system for investment control in university engineering construction BIM Engineering Management Oriented to Curve Equation Model Relationship Between Physical Activity Sedentary Behavior and Body Mass Index in Children and Adolescents Based on FIT Modelo Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Grid data asset relationship and intelligent classification integrating knowledge graph and Internet of Things Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Analysis of digital humanistic knowledge production based on natural language processing Design of language assisted learning model and online learning system under the background of artificial intelligence Study on the influence of adolescent smoking on physical training vital capacity in eastern coastal areas Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering The Construction of Judgment Model for Cultivating Intelligent Accounting Professionals in Universities Based on Logarithmic Periodic Power Law Model Action of Aut( G ) on the set of maximal subgroups ofp -groupsDigital Regeneration and Database Construction of Hunan Embroidery Needlework under the Perspective of Artificial Intelligence Research on the practice of innovation and entrepreneurship education in universities under the environment of big data Application of digital technology in environmental art design Research on loyalty prediction of e-commerce customer based on data mining Research on the evaluation system of tourism management classroom teaching based on multimedia big data analysis The application of ARCS motivation model in college oral English teaching Research on Innovation of Translation Teaching and Translation Strategies for College Students in Multimedia Background A multi-channel convolutional neural network based on attention mechanism fusion for facial expression recognition Exploring the development of cultural and creative tourism under the perspective of ecological theory of tourism destination management Modeling Analysis of User Consumption Data of Internet Culture Industry in the Context of Emerging Technologies Reference: An algorithm for recognizing the main melody of orchestral music based on artificial intelligence of music melody contour Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Evaluation and prediction of regional human capital based on optimised BP neural network Study on inefficient land use determination method for cities and towns from a city examination perspective A study of local smoothness-informed convolutional neural network models for image inpainting The construction of a practical platform for party building and ideological and political education based on information fusion A study on the influence of the spread of Yangming Studies in Japan on the psychology of the Japanese people based on big data analysis Research on personalized recommendation algorithm for micromechanical sensors based on cloud model Research on the value of modern vernacular literature in contemporary rural construction based on ADDIE model Robust speech recognition based on deep learning for sports game review Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Model Construction of Influencing Factors of Physical Education Core Quality Cultivation Based on Multiple Linear Regression Application of Nonlinear Differential Equation in Electric Automation Control System Research on the operation and management mechanism of cross-border e-commerce shared practice training base Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Strategies for improving Chinese language proficiency based on artificial intelligence technology The success rate model of college students’ new entrepreneurship based on nonlinear differential equations Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System Research on the application of marketing strategy of national fitness exercise and dance events in the construction of sports culture based on big data technology Decisions of competing supply chain with altruistic retailer under risk aversion Strategies for constructing mathematical models of nonlinear systems based on multiple linear regression models Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations Research on human resource management performance evaluation based on BP algorithm The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training Nonlinear Channel Estimation for Internet of Vehicles Some Necessary Conditions for Feedback Functions of de Bruijn Sequences The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model An empirical research on economic growth from industrial structure optimisation in the Three Gorges Reservoir area The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Study of agricultural finance policy information extraction based on ELECTRA-BiLSTM-CRF Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Innovative research of vertical video creation under the background of mobile communication Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model Genetic algorithm-based congestion control optimisation for mobile data network Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Badminton players’ trajectory under numerical calculation method Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model Statistical Model of College Ideological and Political Learning Based on Fractional Differential Equations Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials Mechanics of Building Structural Materials Based on Lagrangian Mathematical Model Analysis The Mental Health Education Management of Higher Vocational Students Based on Fractional Differential Equations Application of regression function model based on panel data in financial risk management of bank resource allocation University Ideological and Political Learning Model Based on Statistical Memory Curve Mathematical Equation Research on Resonance Properties of Semantic Wave Fractal Fractals Based on Quantitative Analysis of English Corpus Research on urban landscape big data information processing system based on ordinary differential equations Modeling of fractional differential equation in cloud computing image fusion algorithm Conventional Algorithms in Sports Training Based on Fractional Differential Equations Dynamic Nonlinear System Based on Complex System Theory in the Development of Vocational Education The Algorithm Accuracy of Mathematical Model to Improve the Transmission Speed of E-commerce Platform The Security of Database Network Model Based on Fractional Differential Equations Electric Vehicle Mechanical Transmission System Based on Fractional Differential Equations The Practice System of Physics and Electronics Courses in Higher Vocational Colleges Based on Fractional Differential Equations The Economic Model of Rural Supply and Demand Under the Data Analysis Function Based on Ordered Probit 3D Modeling System of Indoor Environment Art Landscape Design under Statistical Equation Electronic Information Security Model of Nonlinear Differential Equations The Optimization Model of College Students' Physical Exercise Motivation and Self-control Ability Based on the Mathematical Model of Probability Theory Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas Criminal law imputation path for biometric information Research on composite dynamic disaster prevention and control system of mine earthquake and shock in thick and hard rock mines Research on urban energy Internet planning based on fuzzy comprehensive method Research on innovative strategies of college students’ English teaching under the background of artificial intelligence Electromagnetic interference prediction technology of new energy motor drive system Research on the intelligent path of college students’ network ideological and political education based on big data mining technology Research on the application of PLC technology in electrical automation engineering Application of matrix multiplication in signal sensor image perception Analysing the variation of metadiscourse verb patterns in English academic papers from within and between disciplines Impact of COVID-19 policy on stock prices of listed property companies Research on the experimental principle of deep integration of LETS software and criminal procedure under the background of artificial intelligence VR-based computer maintenance practical training platform development design and application research Design and implementation of an integrated management system for discipline construction and postgraduate education in universities Research on innovative human capital for China’s economic development based on STI model In-depth analysis of the artistic expression of paper-cut elements in the design of boat space Knowledge graph construction and Internet of Things optimisation for power grid data knowledge extraction Modeling the pathway of breast cancer in the Middle East Construction and intelligent analysis of power grid physical data knowledge graph based on Internet of Things for power system Research on industrial Internet of Things and power grid technology application based on knowledge graph and data asset relationship model Research on the effects of financial literacy on rural household credit constraint Research on educational resource recommendation system based on MRLG Rec Visual analysis of the trend of mental health changes among Chinese college students in the context of the normalisation of the new crown epidemic A study on the phenomenon of anaphoric correction in college students’ English conversation Limit cycles of a generalised Mathieu differential system Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Application of data mining in basketball statistics The nonlinear effects of ageing on national savings rate – An Empirical Study based on threshold model Design of fitness walker for the elderly based on ergonomic SAPAD model Research on the innovation of collaborative education model of scientific research and teaching in colleges and universities based on the theory of process reengineering AtanK-A New SVM Kernel for Classification The innovative path of diversified vocal music education reform in colleges and universities under the advanced educational concepts of China and the West Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Rhino application in ceramic packaging design teaching Study of a linear-physical-programming-based approach for web service selection under uncertain service quality Analysis and countermeasures of cultivating independent learning ability in colleges teaching English based on OBE theory A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Projection of Early Warning Identification of Hazardous Sources of Gas Explosion Accidents in Coal Mines Based on NTM Deep Learning Network Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Can policy coordination facilitate unimpeded trade? An empirical study on factors influencing smooth trade along the Belt and Road Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Research on design of customer portrait system for E-commerce Research on rule extraction method based on concept lattice of intuitionistic fuzzy language Fed-UserPro: A user profile construction method based on federated learning Design and evaluation of intelligent teaching system on basic movements in PE Research on identifying psychological health problems of college students by logistic regression model based on data mining Research on evaluation system of cross-border E-commerce platform based on the combined model A mathematical model of PCNN for image fusion with non-sampled contourlet transform Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Retraction: Wei, Wei and Lu, Liu. “The influence of X fuzzy mathematics method in basketball tactics scoring” Applied Mathematics and Nonlinear Sciences, vol.7, no.1, 2022, pp.677-684 Retraction Retraction: Retraction of: Mundewadi, R. A. and S, Kumbinarasaiah. “Numerical Solution of Abel's Integral Equations using Hermite Wavelet” Applied Mathematics and Nonlinear Sciences, vol.4, no.2, 2019, pp.395-406