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The approximation error of Ordinary Differential Equations Based on the Moved Node Method

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Journal of Applied Mathematics and Physics 2022年第8期10卷 2450-2458页

作者： Dalabaev Umurdin Hasanova Dilfuza University of World Economy and Diplomacy Tashkent Uzbekistan

One of the central issues in solving differential equations by numerical methods is the issue of approximation. The standard way of approximating differential equations by numerical methods (particularly difference methods) is to question the degree of approximation in the form O(hp). Here h is the grid step. In this case we have an implicit approximation. Based on the difference equation approximating the differential equation, the order of approximation is obtained using the Taylor series. However, it is possible to calculate the approximation error at nodal points based on the method of moving nodes. The method of moving nodes allows obtaining an approximate analytical expression. On the basis of the approximate form, it is possible to calculate the approximation error. The analytical form of the approximation makes it possible to efficiently calculate this error. On the other hand, the property of this error allows the construction of new improved circuits. In addition, based on these types of errors, you can create a differential analog of the difference equation that gives an exact approximation.

关键词： Difference Equation Differential Equation approximation error Moving Node

Development of stability-preserving time-limited model reduction framework for 2-D and 1-D models with error bound

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Control Theory and Technology 2022年第3期20卷 371-381页

作者： Muhammad Imran Syeda Hira Ambreen Syed Nooh Hamdani Muhammad Imran Muhammad Ejaz Naveed Maria Siddiqui Department of Electrical Engineering Military College of Signals(MCS)National University of Sciences and Technology(NUST)Islamabad43000Pakistan Department of Electrical Engineering Sir Syed(Center For Advanced Studies)Institute of Technology(SSCASE)Islamabad43000Pakistan College of Science and Humanities Rumah Majmaah UniversityRumahKingdom of Saudi Arabia

Due to their complex structure,2-D models are challenging to work with;additionally,simulation,analysis,design,and control get increasingly difficult as the order of the model grows.Moreover,in particular time intervals,Gawronski and Juang’s time-limited model reduction schemes produce an unstable reduced-order model for the 2-D and 1-D models.Researchers revealed some stability preservation solutions to address this key flaw which ensure the stability of 1-D reduced-order systems;nevertheless,these strategies result in large approximation errors.However,to the best of the authors’knowledge,there is no literature available for the stability preserving time-limited-interval Gramian-based model reduction framework for the 2-D discrete-time systems.In this article,2-D models are decomposed into two separate sub-models(i.e.,two cascaded 1-D models)using the condition of minimal rank-decomposition.Model reduction procedures are conducted on these obtained two 1-D sub-models using limited-time Gramian.The suggested methodology works for both 2-D and 1-D models.Moreover,the suggested methodology gives the stability of the reduced model as well as a priori error-bound expressions for the 2-D and 1-D models.Numerical results and comparisons between existing and suggested methodologies are provided to demonstrate the effectiveness of the suggested methodology.

关键词： Model reduction Weighted Gramian Limited interval Gramian Minimal realization approximation error error bound

Hierarchical Polytope ARTMAP for Supervised Learning

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Journal of Computer Science & Technology 2010年第5期25卷 1071-1082页

作者：廖鑫鹏吴永强韩崇昭 Southwest Research Institute of Electronics and Telecommunication Technology College of Electronics and Information Engineering Xi'an Jiaotong University

The recent Polytope ARTMAP（PTAM） suggests that irregular polytopes are more flexible than the predefined category geometries to approximate the borders among the desired output predictions.However,category expansion and adjustment steps without statistical information make PTAM not robust to noise and category overlap.In order to push the learning problem towards Structural Risk Minimization（SRM）,this paper proposes Hierarchical Polytope ARTMAP （HPTAM） to use a hierarchical structure with different levels,which are determined by the complexity of regions incorporating the input pattern.Besides,overlapping of simplexes from the same desired prediction is designed to reduce category proliferation.Although HPTAM is still inevitably sensible to noisy outliers in the presence of noise,main experimental results show that HPTAM can achieve a balance between representation error and approximation error,which ameliorates the overall generalization capabilities.

关键词： structural risk minimization polytope ARTMAP hierarchical structure representation error approximation error

Carlson iterating rational approximation and performance analysis of fractional operator with arbitrary order

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Chinese Physics B 2017年第4期26卷 66-74页

作者：何秋燕余波袁晓 College of Electronics and Information Engineering Sichuan University College of Physics and Engineering Chengdu Normal University

The performance analysis of the generalized Carlson iterating process,which can realize the rational approximation of fractional operator with arbitrary order,is presented in this paper.The reasons why the generalized Carlson iterating function possesses more excellent properties such as self-similarity and exponential symmetry are also explained.K-index,P-index,O-index,and complexity index are introduced to contribute to performance analysis.Considering nine different operational orders and choosing an appropriate rational initial impedance for a certain operational order,these rational approximation impedance functions calculated by the iterating function meet computational rationality,positive reality,and operational validity.Then they are capable of having the operational performance of fractional operators and being physical realization.The approximation performance of the impedance function to the ideal fractional operator and the circuit network complexity are also exhibited.

关键词： fractional calculus fractional operator generalized Carlson iterating process approximation error

Data-Based Optimal Tracking of Autonomous Nonlinear Switching Systems

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IEEE/CAA Journal of Automatica Sinica 2021年第1期8卷 227-238页

作者： Xiaofeng Li Lu Dong Changyin Sun the School of Automation Southeast UniversityNanjing 210096 the Key Laboratory of Measurement and Control of Complex Systems of Engineering Ministry of EducationSoutheast UniversityNanjing 210096China the College of Electronics and Information Engineering Tongji UniversityShanghai 201804China

In this paper,a data-based scheme is proposed to solve the optimal tracking problem of autonomous nonlinear switching systems.The system state is forced to track the reference signal by minimizing the performance function.First,the problem is transformed to solve the corresponding Bellman optimality equation in terms of the Q-function(also named as action value function).Then,an iterative algorithm based on adaptive dynamic programming(ADP)is developed to find the optimal solution which is totally based on sampled data.The linear-in-parameter(LIP)neural network is taken as the value function approximator.Considering the presence of approximation error at each iteration step,the generated approximated value function sequence is proved to be boundedness around the exact optimal solution under some verifiable assumptions.Moreover,the effect that the learning process will be terminated after a finite number of iterations is investigated in this paper.A sufficient condition for asymptotically stability of the tracking error is derived.Finally,the effectiveness of the algorithm is demonstrated with three simulation examples.

关键词： Adaptive dynamic programming approximation error data-based control Q-learning switching system

Estimating Subdivision Depth of Catmull-Clark Surfaces

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Journal of Computer Science & Technology 2004年第5期19卷 657-664页

作者： Hua-WeiWang Kai-HuaiQin DepartmentofComputerScienceandTechnology TsinghuaUniversityBeijing100083P.R.China

In this paper, both general and exponential bounds of the distance between a uniform Catmull-Clark surface and its control polyhedron are derived. The exponential bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Based on the exponential bound, we can predict the depth of subdivision within a user-specified error tolerance. This is quite useful and important for pre-computing the subdivision depth of subdivision surfaces in many engineering applications such as surface/surface intersection, mesh generation, numerical control machining and surface rendering.

关键词： Catmull-Clark surface subdivision depth control polyhedron approximation error

A search-free near-field source localization method with exact signal model

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Journal of Systems Engineering and Electronics 2021年第4期32卷 756-763页

作者： PAN Jingjing SINGH Parth Raj MEN Shaoyang Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space Nanjing University of Aeronautics and AstronauticsNanjing 211106China Department of Electronic Engineering Nanjing University of Aeronautics and AstronauticsNanjing 211106China International Electronics Engineering S.A. Bissen L-7795Luxembourg School of Medical Information Engineering Guangzhou University of Chinese MedicineGuangzhou 510006China

Most of the near-field source localization methods are developed with the approximated signal model,because the phases of the received near-field signal are highly non-linear.Nevertheless,the approximated signal model based methods suffer from model mismatch and performance degradation while the exact signal model based estimation methods usually involve parameter searching or multiple decomposition procedures.In this paper,a search-free near-field source localization method is proposed with the exact signal model.Firstly,the approximative estimates of the direction of arrival(DOA)and range are obtained by using the approximated signal model based method through parameter separation and polynomial rooting operations.Then,the approximative estimates are corrected with the exact signal model according to the exact expressions of phase difference in near-field observations.The proposed method avoids spectral searching and parameter pairing and has enhanced estimation performance.Numerical simulations are provided to demonstrate the effectiveness of the proposed method.

关键词： near-field source localization polynomial rooting approximation error exact signal model

Robust adaptive neural network control with supervisory controller

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Journal of Systems Engineering and Electronics 2004年第4期15卷 628-631页

作者： ZhangTianping MeiJiandong DepartmentofComputer CollegeofInformationEngineeringYangzhouUniversityYangzhou225009P.R.China

The problem of direct adaptive neural network control for a class of uncertain nonlinear systems with unknown constant control gain is studied in this paper. Based on the supervisory control strategy and the approximation capability of multilayer neural networks (MNNs), a novel design scheme of direct adaptive neural network controller is proposed. The adaptive law of the adjustable parameter vector and the matrix of weights in the neural networks and the gain of sliding mode control term to adaptively compensate for the residual and the approximation error of MNNs is determined by using a Lyapunov method. The approach does not require the optimal approximation error to be square-integrable or the supremum of the optimal approximation error to be known. By theoretical analysis, the closed-loop control system is proven to be globally stable in the sense that all signals involved are bounded, with tracking error converging to zero. Simulation results demonstrate the effectiveness of the approach.

关键词： Adaptive control Nonlinear systems Control systems approximation error Educational institutions Neural networks

Estimation of the Misclassification error for Multicategory Support Vector Machine Classification

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Acta Mathematica Sinica,English Series 2008年第3期24卷 511-528页

作者： Bing Zheng LI Department of Mathematics Zhejiang University Hangzhou 310028 P. R. China

The purpose of this paper is to provide an error analysis for the multicategory support vector machine （MSVM） classificaton problems. We establish the uniform convergency approach for MSVMs and estimate the misclassification error. The main difficulty we overcome here is to bound the offset vector. As a result, we confirm that the MSVM classification algorithm with polynomial kernels is always efficient when the degree of the kernel polynomial is large enough. Finally the rate of convergence and examples are given to demonstrate the main results.

关键词： multicategory support vector machine classifier misclassification error reproducing kernel Hilbert space approximation error