Computation Of Jacobian Matrix Research Articles

Joint Physics and Data Driven Geometrical Factors of Array Laterolog in Layered Formations

Numerical geometrical factors (GFs) are developed for array laterolog measurements in one-dimensional (1D) cylindrically and planarly layered formations. The kernel of GFs assumes that laterolog responses can be expressed in terms of linear superposition of formation resistivities at different units. Two techniques, namely physics-driven modeling and data-driven approximation, have been employed to ensure both computational accuracy and efficiency of GFs. Physics-driven modeling utilizes a three-dimensional finite element algorithm to generate high-precision and fully labeled GFs data and library. Subsequently, these predetermined data are trained offline using classical neural network algorithm to establish an implicit relationship between formation parameters and GFs. The joint physics and data-driven GFs presents three main advantages. Firstly, it enables the direct representation of the spatial sensitivity of layered formations both laterally and horizontally in vertical and horizontal wells. Secondly, GFs can be used to rapidly calculate the array laterolog responses. The computation speed can be enhanced by thousands of times in comparison with traditional physic-driven modeling, allowing for real-time data processing. Another noteworthy aspect is that GFs forward modeling can handle formations with arbitrary layers, thereby resolving the fixed-layer modeling issue inherent in traditional data-driven methods. Thirdly, the GFs can be employed to approximate the derivative of logging response with respect to formation resistivity, significantly reducing the complexity of Jacobian matrix calculation for deterministic inversion. These numerical GFs not only serve as an excellent alternative simulator for array laterolog, but also will be helpful to accelerate the modeling and inversion of other logging measurement in layered structures.

Complex‐step derivative‐based extended Kalman filter for state estimation in twin rotor MIMO system

AbstractThis paper presents the design of a complex‐step extended Kalman filter (CS‐EKF) to estimate the states of the twin‐rotor MIMO system (TRMS) which is a non‐linear system. Since the model of TRMS is quite complex and contains discontinuous functions, it is very difficult to calculate the Jacobian matrix in the TRMS analytically by hand. This makes it difficult to implement control methods that require Jacobian matrix calculation for TRMS. Herein, to calculate the Jacobian matrix, the CS‐EKF uses the complex‐step derivative approach, which is a numerical technique and offers near‐analytical accuracy in a single function evaluation. The effectiveness of the CS‐EKF is demonstrated through simulation and real‐time experiments. Also, The CS‐EKF is compared to the finite‐difference extended Kalman filter (FD‐EKF) and the unscented Kalman filter (UKF) in terms of estimation accuracy, computational load, and ease of implementation.

Application of self-learning interval type-2 fuzzy neural network in PM2.5 concentration prediction

Abstract The change of PM2.5 concentration in air quality is nonlinear and difficult to predict. Therefore, a self-learning interval type-2 fuzzy neural network (SLIT2FNN) is proposed. SLIT2FNN has two parts: online structure learning and parameter learning. In structure learning, to improve the training accuracy and speed of the model, the Possibilistic Fuzzy C-Means (PFCM) algorithm is used to process the input data and obtain the number of initial rules before model training. The PFCM algorithm introduces the concept of possibility P to Fuzzy C-Means (FCM), allowing PFCM to overcome the shortcomings of FCM that cannot accurately cluster a large number of nonlinear problems. SLIT2FNN can establish an appropriate number of rules in the preparation stage, and then use the firing strength of the antecedents of the rules to judge whether to generate fuzzy rules for online self-learning, thereby optimizing its network structure. Then, the improved Levenberg–Marquardt (ILM) algorithm is used to modify the relevant parameters of SLIT2FNN. The ILM algorithm can address the challenge of numerous parameters in the Jacobian matrix and complex calculations and improve the calculation speed and adaptive ability of SLIT2FNN parameter learning. Finally, SLIT2FNN is applied to the prediction of air quality PM2.5 concentration, and the performance is compared with other models. The experiment proves that SLIT2FNN has a high prediction accuracy and fast convergence.

Dynamical homotopy transient-based technique to improve the convergence of ill-posed power flow problem

This paper proposes a hybrid technique to solve the ill-posed Power Flow Problem (PFP), considering a homotopy approach. The primary proposal is to solve large-scale problems where the traditional Newton–Raphson (NR) fails to converge, as in the case of ill-posed systems. The method explores a dynamical homotopy transient-based technique to improve the convergence of the ill-conditioned problem instead of using the classical static method. Depending on the integration selected scheme and the integration step, the result furnished by the dynamical homotopy method has low accuracy. Then, the NR method is employed to refine the low-accuracy result and accurately determine the ill-posed PFP solution. The proposed approach can be implemented efficiently using only one Jacobian matrix computation and LU factorization per point of the homotopy path. In the static homotopy problem, a PFP using previous results must be solved per path point. In this case, some LU factorizations are necessary for each path point. The technique’s performance was evaluated through experiments, including a 70,000-bus large-scale system. The approximate dynamical homotopy result used as an initial estimate provided appropriate convergence quality for the NR method to determine a high-precision solution to the PFP.

Signal processing methods for crosswell electromagnetic imaging system

The principle of crosswell electromagnetic (EM) imaging is similar to that of induction logging. The difference between them is that conventional induction logging can only measure the formation conductivity within a few meters around the well. However, crosswell EM imaging can be used in two wells with a separation distance of up to 1000 m. Several novel signal processing methods for crosswell EM imaging systems are developed for calibration, casing correction, and hybrid Jacobian matrix calculation. To conveniently perform calibration on the ground, the EM field formula in a vertical layered media is derived based on Maxwell equations, and the relationships between the received signal and frequency, formation conductivity, and transceiver spacing are subsequently derived. Furthermore, a new calibration method is developed. The EM field formula for a radial layered media is derived, and the analytical expressions describing the relationship between the receiving signal and borehole parameters, metal casing parameters, formation parameters, and receiver location are obtained. The effects of mud resistivity, metal casing position, metal casing resistivity, permeability, cement ring resistivity, transmitting frequency, and receiver location on the crosswell EM receiving signal are simulated. It is determined that only the casing required a correction. The influence of the metal casing is corrected by establishing a database. The difference between the corrected data and data obtained without a metal casing is less than 1%. There is a large error in the calculation of the Jacobian matrix in some regions when Green’s function was used. A hybrid Jacobian matrix calculation method combining local perturbation and a global Green’s function is developed, which significantly reduced the error and improved the inversion accuracy.

Onboard Pointing Error Detection and Estimation of Observation Satellite Data Using Extended Kalman Filter.

The satellite communication is embellished constantly by providing information, ensuring security, and enables the communication among huge at a particular time efficiently. The satellite navigation helps in determining the people's location. Global development, natural disasters, change in climatic conditions, agriculture crop growth, etc., are monitored using satellite observation. Hence, the satellite includes detailed information data, and it must be protected confidentially. The field of the satellite is enhanced at an astonishing pace. Satellite data play an important role in this modern world; hence, the onboard-satellite data must secure through the proper selection of error detection and estimation schema. Lightweight deep learning algorithm based on Extended Kalman Filter (KFK) is proposed to detect and estimate onboard pointing error such as an error in attitude and orbit. The Extended Kalman Filter (EKF) is widely used in the satellite system. EKF is utilized in this proposed model to detect the onboard pointing error such as attitude and orbit determination. An autonomous estimation of orbit position is possible through space-borne gravity. The information obtained through the observation of satellite data is compared with the accurate gravity model in detecting the error. The utilization of EKF reduces the dependence of the ground tracking system in satellite determination. The orbital altitude and orbital position are the most important challenges faced in the satellite determination system. The satellite model using the Extended Kalman Filter is an optimum method in estimating the orbital parameters. The errors in the linearization process are detected, and this can be overcome through the proper selection of linear expansion point with the EKF algorithmic model with the Jacobian matrix calculation. The results show that the EKF implementation helps in attaining better accuracy than other methodologies. Its contribution is enormous to many space missions, autonomous rendezvous and docking for manned and unmanned missions (e.g., ISS operations and beyond, in-orbit servicing, and in-orbit refueling), routine satellite OD operations, orbital debris removal systems, Space Situational Awareness (SSA) operations, and others.

Power flow analysis using successive approximation and adomian decomposition methods with a new power flow formulation

The Newton-Raphson Method (NRM) is widely accepted for the power flow studies. The limitations of NRM include calculation of Jacobian matrix for each iteration, resulting in an additional calculation and an increase in computation time per iteration. Another weakness is the solution diverges when the Jacobian matrix is singular. In order to overcome these issues, this work proposes new numerical methods for the power flow studies: Successive Approximation Method (SAM) and Adomian Decomposition Method (ADM). The convergence of both methods depends on the equation formulation, contraction mapping and fixed-point theorem. Here, a new formulation of the power flow equations is introduced that satisfies the conditions for the fixed-point theorem. Both methods are coded in MATLAB platform and the NRM is also coded for as a reference method. With the standard 3-bus test system, all three methods are elaborated in detail. Later on, the methods are validated on other test systems such as 4-bus, 5-bus, 6-bus, 9-bus, 11-bus, 14-bus, 30-bus and 57-bus. Special case studies with the 14-bus system are also reported, wherein the NRM fails to provide converged solution because of singularity in Jacobian matrix. On the other hand, the SAM and ADM successfully converge in such case.

MLPNN‐RF: Software fault prediction based on robust weight based optimization and Jacobian adaptive neural network

SummarySoftware fault prediction (SFP) is a vital objective in software engineering. This might permit effective resource allocation and also enhance informed decisions about the release quality. SFP is being a critical issue for software professionals as well as the tech industry. Thus, SFP is necessary. The study intends to perform efficient error rate estimation using the proposed hybrid robust weight based optimization and Jacobian adaptive neural network (RWO‐JANN). It also aims to classify the software faults in an efficient way using multi‐layer perceptron neural network‐random forest (MLPNN‐RF). Various processes are involved to accomplish SFP. At first, the dataset is taken as input. After this, data preprocessing is performed. Subsequently, weights are initialized using the proposed RWO‐JANN. Weight initialization is performed through a series of steps. Then, the position and the weight parameter are updated to perform weight initialization. After this, the error rate is estimated and the weight is updated on the basis of the learning rate and Jacobian matrix calculation. Lastly, the decay rate is analyzed. If the error rate extends beyond the threshold value, the process repeats from weight initialization. If not, the testing process is performed and lastly the classified output for SFP is obtained by the proposed MLPNN‐RF. The proposed system is comparatively analyzed with the existing methods in terms of accuracy, precision, recall, F1 score, sensitivity, specificity, and error rate. The analytical results revealed effective outcomes of proposed system than the existing techniques with accuracy of 99.01%.

Hybrid regression model via multivariate adaptive regression spline and online sequential extreme learning machine and its application in vision servo system

To solve the problems of slow convergence speed, poor robustness, and complex calculation of image Jacobian matrix in image-based visual servo system, a hybrid regression model based on multiple adaptive regression spline and online sequential extreme learning machine is proposed to predict the product of pseudo inverse of image Jacobian matrix and image feature error and online sequential extreme learning machine is proposed to predict the product of pseudo inverse of image Jacobian matrix and image feature error. In MOS-ELM, MARS is used to evaluate the importance of input features and select specific features as the input features of online sequential extreme learning machine, so as to obtain better generalization performance and increase the stability of regression model. Finally, the method is applied to the speed predictive control of the manipulator end effector controlled by image-based visual servo and the prediction of machine learning data sets. Experimental results show that the algorithm has high prediction accuracy on machine learning data sets and good control performance in image-based visual servo.

Performance analysis of a modified Newton method for parameterized dual fuzzy nonlinear equations and its application

In this paper, we are interested in the numerical solution of nonlinear equations, particularly, when the coefficients are fuzzy numbers rather than crisp numbers. These problems often arise in numerous applications and have recently been the subject of several studies, where many researchers parameterized the fuzzy coefficients and proposed numerical method for obtaining the solutions. Most of the numerical methods applied for solving the parameterized fuzzy equations are Newton-like methods that require the computation of Jacobian matrix at every iteration or after every few iterations. However, the Newton search direction may not be a descent direction if the Jacobian J(xk) is not positive definite, and, when J(xk) is singular, the method may fail to converge. Also, there are limited literatures on numerical methods for solution of dual fuzzy nonlinear equations. This paper proposed a variant of Newton’s method for solving parameterized dual fuzzy nonlinear equations. This method introduces a new parameter μk to the Newton’s method to ensure that the search direction is a descent direction even when J(xk) is not positive definite. The proposed method was further applied to solve an application problem. Preliminary results on the considered benchmark and application problems show that the new algorithm effective and promising.

An Efficient Approach for Nodal Water Demand Estimation in Large-scale Water Distribution Systems

Real-time modeling of a water distribution system (WDS) is a critical step for the control and operation of such systems. The nodal water demand, as the most important time-varying parameter, must be estimated in real time. The computational burden of nodal water demand estimation is intensive, leading to inefficiency in the modeling of large-scale networks. The Jacobian matrix computation and Hessian matrix inversion are the main processes that dominate the computation time. To address this problem, an approach for shortening the computation time for real-time demand estimation in large-scale network is proposed. This approach allows the Jacobian matrix to be efficiently computed based on solving a system of linear equations, and a Hessian matrix inversion method based on matrix partitioning and the iterative Woodbury-Matrix-Identity Formula is proposed. The developed approach is applied to a large-scale network, in which the number of nodal water demands is 12523, and the number of measurements ranges from 10 to 2000. The results show that the time consumptions for the Jacobian computation and Hessian matrix inversion are within 465.3 ms and 1219.0 ms, respectively. The time consumption is significantly shortened compared with the existing approach, especially for nodal water demand estimation in large-scale WDSs.

Quasi-2-D Robust Inversion of Semi-Airborne Transient Electromagnetic Data With IP Effects

Semi-airborne transient electromagnetic (SATEM) method has been efficiently used recently in geophysical exploration due to its relatively portability compared to the standard airborne surveys. However, the SATEM data are often complicated by induced polarization (IP) effects manifesting as abnormal decay and sign reversal in the responses. The IP information can be extracted from SATEM data by joint inversion of the electromagnetic data into the electrical resistivity and IP parameters described in Cole-Cole model. In this paper, we introduce a quasi-two-dimensional inversion scheme to recover the resistivity and IP parameters from SATEM responses by (1) a fast semi-analytical method for Jacobian matrix calculation; and (2) a staged inversion strategy with lateral constraints. The methodology is examined on two synthetic polarized models. Our study indicates that the proposed scheme can improve inversion stability and recover the underground resistivity and IP property distributions.

ALE formulation for dynamic modeling and simulation of cable-driven mechanisms considering stick–slip frictions

This paper proposes an arbitrary Lagrange Euler (ALE) method for dynamic modeling and simulation of cable-driven mechanisms considering stick–slip frictions. The dynamic equations of the cables are presented based on the ALE formulation. The frictional forces acting at the Eulerian nodes are derived from the Coulomb dry friction model. The detections for stick–slip motions of different contact points between the cables and the holes are carried out independently through a trial-and-error method. The stiff problems yielded by the high stiffness of the cables are released by using a model smoothing method. The dynamic equations of the system are integrated by an explicit solver, avoiding the calculation of Jacobian matrix of the dynamic equations with respect to state variables, which is nonexistent in the stick–slip transition points. The effectiveness and correctness of the proposed method are verified by comparison with previous studies and with experiments. The proposed method not only can capture stick–slip motions, which are consistent well with experiments, but also has a high calculation efficiency.

A modified forward and backward reaching inverse kinematics based incremental control for space manipulators

Forward and backward reaching inverse kinematics (FABRIK) is an efficient two-stage iterative solver for inverse kinematics of spherical-joint manipulator without the calculation of Jacobian matrix. Based on FABRIK, this paper presents an incremental control scheme for a free-floating space manipulator consists of revolute joints and rigid links with the consideration of joint constraints and dynamic coupling effect. Due to the characteristics of FABRIK, it can induce large angular movements on specific joints. Apart from that, FABRIK maps three dimensional (3D) problem into two dimensional (2D) problem by a simple geometric projection. This operation can cause infinite loops in some cases. In order to overcome these issues and apply FABRIK on space manipulators, an increments allocation method is developed to constrain the angular movements as well as to re-orient the end-effector. The manipulator is re-positioned based on the momentum conservation law. Instead of pure target position tracking, the orientation control of the end-effector is also considered. Numerical simulation is performed to testify and demonstrate the effectiveness and reliability of the proposed incremental control approach.

Non-conventional 1D and 2D finite elements based on CUF for the analysis of non-orthogonal geometries

When dealing with innovative materials – such as composites and metamaterials with complex microstructure – or structural components with non-orthogonal beam/plate geometry, the Finite Element Method can become very costly in calculations and time because of the use of very fine 3D meshes. By exploiting the Node-Dependent Kinematic approach of the Carrera Unified Formulation and using Lagrange expanding functions, this work presents the implementation of non-conventional 1D and 2D elements mainly based on the 3D integration of the approximating functions and computation of 3D Jacobian matrix inside the element for the derivation of stiffness and mass matrices; substantially, the resulting elements are 3D elements in which the order of expansion can be different in the three spatial directions. The free vibration analysis of some typical components is performed and the results are provided in terms of natural frequencies. The present elements allow us to accurately study beam-like and plate-like structures with non-orthogonal geometries by employing much less degrees of freedom with respect to the use of classical 3D finite elements.

A new parameter-free method for Toeplitz systems of weakly nonlinear equations

Based on the fact that the linear term is strongly dominant over the nonlinear term, by using the approximate inverse-free preconditioned conjugate gradient (AIPCG) iteration technique, we establish the Picard-AIPCG iteration method to solve Toeplitz systems of weakly nonlinear equations. Since the storage and the accurate computation of Jacobian matrix are not necessary in this method and the only need is to solve the constant Toeplitz sub-systems, there may be a great saving in computer storage and calculation workload when facing the practical problems. The global convergence has been proved under some suitable conditions. Some numerical examples demonstrate the feasibility and effectiveness of this new iteration.

An efficient DY-type spectral conjugate gradient method for system of nonlinear monotone equations with application in signal recovery

<abstract> Many problems in engineering and social sciences can be transformed into system of nonlinear equations. As a result, a lot of methods have been proposed for solving the system. Some of the classical methods include Newton and Quasi Newton methods which have rapid convergence from good initial points but unable to deal with large scale problems due to the computation of Jacobian matrix or its approximation. Spectral and conjugate gradient methods proposed for unconstrained optimization, and later on extended to solve nonlinear equations do not require any computation of Jacobian matrix or its approximation, thus, are suitable to handle large scale problems. In this paper, we proposed a spectral conjugate gradient algorithm for solving system of nonlinear equations where the operator under consideration is monotone. The search direction of the proposed algorithm is constructed by taking the convex combination of the Dai-Yuan (DY) parameter and a modified conjugate descent (CD) parameter. The proposed search direction is sufficiently descent and under some suitable assumptions, the global convergence of the proposed algorithm is proved. Numerical experiments on some test problems are presented to show the efficiency of the proposed algorithm in comparison with an existing one. Finally, the algorithm is successfully applied in signal recovery problem arising from compressive sensing. </abstract>

3-D Time-Domain Airborne EM Inversion for a Topographic Earth

The topography has serious effects on time-domain airborne electromagnetic (AEM) signal, and the EM responses resulted from the topography frequently overwhelm those from the underground abnormal bodies. This brings big challenges to the traditional AEM interpretations based on a flat ground model. In this article, we develop a 3-D AEM inversion algorithm for a topographic earth model. The time-domain finite-element algorithm based on unstructured mesh is used to model the AEM responses. The tetrahedral grids provide the flexibility to fit the rugged topography. Furthermore, we adopt the Gauss–Newton method for our inversion of time-domain AEM data. In the forward modeling and the calculation of Jacobian matrix, we introduce an unstructured local mesh and decouple the meshes for forward modeling and inversion to improve the computational efficiency. For that purpose, we first set up an unstructured inversion mesh and then extract those cells corresponding to the sensitive area of AEM system for each survey station from the inversion mesh and construct a local forward mesh with the extracted cells as the core. After that, we take advantage of the spatial relationship between the local forward meshes and the inversion ones to set up the global sensitivity matrix for Gauss–Newton inversion. We test the effectiveness of our algorithm by applying our 3-D inversion code to both synthetic and survey data. The numerical experiments show that the Earth topography can have big influence on AEM inversions, and ignoring the topography can create serious distortion to AEM inversion results.

Teleoperation Control Design with Virtual Force Feedback for the Cable-Driven Hyper-Redundant Continuum Manipulator

Hyper-redundant continuum manipulators present dexterous kinematic skills in complicated tasks and demonstrate promising potential in underground exploration, intra-cavity inspection, surgery, etc. However, the hyper-redundancy, which endows much dexterity and flexibility, brings a huge challenge to the kinematics solution and control of the continuum manipulators. Due to the pseudoinverse calculation of high-order Jacobian matrix or iteration, many inverse kinematic solution approaches of continuum manipulators are very time-consuming, which extremely limit their applicability in real-time control. Additionally, it is often difficult for the manipulators to perform the tasks well in complex scenarios due to lack of human intervention. Therefore, in this paper, a simplified kinematics model of a typical hyper-redundant manipulator is proposed based on its unique geometry relationships, where the mapping relationships between the actuators’ rotation and the end-effector’s position are derived through the analysis of its driving subsystem and motion subsystem, in particular the joint modules. To perform the tasks of manipulators with the help of operators, a teleoperation control scheme with modified wave transmission structure is designed to achieve the guaranteed stability and improved transparency, and the leader’s trajectory and generated force feedback are the transmitted signals in the communication channel. Specifically, a virtual force feedback generation algorithm is developed in the teleoperation control scheme via the processing tracking errors, which can improve the operators’ assistance and perception during the teleoperation process. The practical experiments with comparative wave variable structures in two different sets are implemented to verify the effectiveness of proposed kinematics model and control scheme.

A multilevel index optimization method for fast kinematic calibration configuration of serial manipulators based on Compute Unified Device Architecture parallel computing

The kinematic calibration accuracy of serial manipulators is affected by the error expression ability of the selected measurement configurations and non-geometric errors such as joint disturbance, measurement noise, etc. Based on the observability of configurations, deviation of identifiable parameters, and calibration robustness, this paper proposes a multilevel evaluation criterion for measurement configuration optimization. In addition, based on the Compute Unified Device Architecture (CUDA) parallel computing technique, the most time-consuming Jacobian matrix calculation program in the algorithm is modified, and an efficient optimization algorithm for measurement configurations is established, to guarantee the feasibility of the evaluation criterion. Combined with CUDA algorithm, fast calibration is implemented with fewer measurement points and relatively higher accuracy, by means of multilevel optimization. The results illustrate the effectiveness and the universality of the proposed multilevel evaluation criterion. The criterion can be applied in calibration experiments of multi-degree of freedom (DOF) serial manipulators with complex structures.