Linear Time-varying Equations Research Articles

Predefined-time robust adaptive gradient neural network for solving linear time-varying equations and its applications

Adaptive gradient neural networks (AGNN) have been extensively employed as effective models for addressing time-varying problems. However, the existing AGNN are convergent in infinite time and only considered in noiseless environment. Consequently, the AGNN with a certain convergence time and noise tolerance is urgently required. Motivated by the advantages of predefined-time convergence, a novel predefined-time robust adaptive gradient neural network (PRAGNN) is developed for solving linear time-varying equations. It can be concluded that PRAGNN can efficiently obtain real-time solutions of linear time-varying equations within predefined time, independent of initial conditions, by properly selecting parameters. In the process of solving linear time-varying equations, PRAGNN can avoid matrix inversion, which reduces the computing cost compared with zeroing neural networks (ZNN). Furthermore, PRAGNN can completely resist both bounded vanishing noise and bounded non-vanishing noise. Numerical experiments demonstrate the effectiveness of PRAGNN. Finally, PRAGNN has been successfully utilized in 2-D dynamic positioning based on the angle of arrival (AOA) positioning algorithm, as well as in alternating currents (AC) computing.

Fixed-time solution of inequality constrained time-varying linear systems via zeroing neural networks

Zeroing neural network (ZNN) has been explored and applied in a variety of time-varying problems solving. However, ZNN models for inequality constrained time-varying linear equation (ICTVLE) solving are rarely reported. Generally, numerous practical problems can be mathematically modeled as ICTVLE problems. Motivated by the above mentioned issue, a nonlinear tunable activation function activated ZNN (NTAF-ZNN) model is constructed to effectively solve ICTVLE problems. To further improve the performances of NTAF-ZNN model, a fuzzy nonlinear tunable activation function (FNTAF) is also designed. Based on the FNTAF, a fuzzy nonlinear tunable activation function-based ZNN (FNTAF-ZNN) model is realized. Firstly, the fixed-time convergence property and disturbance rejection ability of the NTAF-ZNN and FNTAF-ZNN models are proved by theoretical analysis. Then, comparative simulation results of the two models with other existing ZNN models for solving the ICTVLE problem are provided to further verify their robustness and effectiveness. Additionally, the two models and other existing ZNN models are also employed to realize inequality constrained manipulator trajectory tracking under ideal and noisy environments for further comparisons. Finally, the superior performances of the FNTAF-ZNN model for manipulator trajectory tracking are further verified by physical experiment results.

Retracted: Noise-Tolerant Zeroing Neural Dynamics for Solving Hybrid Multilayered Time-Varying Linear Equation System

Retracted: Noise-Tolerant Zeroing Neural Dynamics for Solving Hybrid Multilayered Time-Varying Linear Equation System

A Predefined-Time Adaptive Zeroing Neural Network for Solving Time-Varying Linear Equations and Its Application to UR 5 Robot.

Time-varying linear equations (TVLEs) play a fundamental role in the engineering field and are of great practical value. Existing methods for the TVLE still have issues with long computation time and insufficient noise resistance. Zeroing neural network (ZNN) with parallel distribution and interference tolerance traits can mitigate these deficiencies and thus are good candidates for the TVLE. Therefore, a new predefined-time adaptive ZNN (PTAZNN) model is proposed for addressing the TVLE in this article. Unlike previous ZNN models with time-varying parameters, the PTAZNN model adopts a novel error-based adaptive parameter, which makes the convergence process more rapid and avoids unnecessary waste of computational resources caused by large parameters. Moreover, the stability, convergence, and robustness of the PTAZNN model are rigorously analyzed. Two numerical examples reflect that the PTAZNN model possesses shorter convergence time and better robustness compared with several variable-parameter ZNN models. In addition, the PTAZNN model is applied to solve the inverse kinematic solution of UR 5 robot on the simulation platform CoppeliaSim, and the results further indicate the feasibility of this model intuitively.

Motion Planning of Humanoid Upper-Body Robot Using an Integration-Enhanced Differentiator-Based Method: A Time-Varying Linear Equations Approach

Motion Planning of Humanoid Upper-Body Robot Using an Integration-Enhanced Differentiator-Based Method: A Time-Varying Linear Equations Approach

Two New Zhang Neural Networks for Solving Time-Varying Linear Equations and Inequalities Systems.

Recently, Xu et al. solved a class of time-varying linear equations and inequalities systems (LEIESs) by using a Zhang neural network (ZNN) model through introducing a nonnegative relaxation vector. However, the introduction of this unknown nonnegative slack vector will increase the size and complexity of the model, thereby increasing the cost of computation. In this article, we propose two new ZNN models (called traditional Zhang neural network (TZNN) and variant Zhang neural network (VZNN) models, respectively) in which no additional relaxation vector is needed. The convergence analysis of these two new models are performed, and two simulation experiments are given to illustrate their efficiency and effectiveness for solving the time-varying LEIESs, including the applicability of our proposed models to robot manipulator.

Modified Zhang neural algorithm for time-varying overdetermined linear models and applications

The challenges of the inefficiency of the existing static least-squares method and the insufficient precision of the existing neural algorithms need to be overcome. With the aid of the ten-instant Zhang time discretization formula, a novel ten-instant discrete-time neural network algorithm is proposed and generalized to the problem-solving of overdetermined system of time-varying linear equations. Theoretical analyses illustrate that the maximal steady-state residual error of the proposed algorithm has an order-6 pattern. Comparative experiment results further substantiate the superiority of the proposed algorithm. Moreover, applications of mobile localization based on the angle of arrival technique and the time difference of arrival technique are provided to display the applicability of the proposed algorithm.

Noise-Tolerant Zeroing Neural Dynamics for Solving Hybrid Multilayered Time-Varying Linear Equation System

Resource allocation problem in a wireless sensor network can be formulated as time-varying optimization, which can be further converted as time-varying linear equation system (TVLES). Hybrid multilayered time-varying linear equation system (HMTVLES) involving hybrid multilayers and time-variation characteristic is a complicated and challenging problem. Recently it has been solved by zeroing neural dynamics (ZND) method under ideal conditions, i.e., without noises. However, noises are ubiquitous, immanent and unavoidable in real-time systems. In this work, we propose a noise-tolerant zeroing neural dynamics (NTZND) model for solving HMTVLES. It can deal with different kinds of noises such as constant noise, linear-increasing noise, and random noise. Theoretical analyses guarantee the precision of NTZND model in the presence of different kinds of noises. In addition, a general NTZND model is proposed based on general activation function. Besides, classical ZND method and gradient neural dynamics (GND) method are also investigated and compared. Numerical experimental results are presented to verify the theoretical results of proposed models.

Zeroing Neural Network for Time-Varying Linear Equations With Application to Dynamic Positioning

In this article, considering the effectiveness and efficiency in solving time-varying problems, a new zeroing neural network (ZNN) is proposed to solve time-varying linear equations with column full rank coefficient matrix. In addition, two novel nonlinear activation functions are developed to enhance the comprehensive performance of the ZNN model. It is demonstrated through theoretical analysis and numerical experiments that the nonlinear activated ZNN model has better noise immunity, and faster prescribed-time convergence speed. Finally, the ZNN method is successfully applied to 2-D and 3-D dynamic positioning, with lower positioning error than the traditional pseudoinverse method.

Solving Perturbed Time-Varying Linear Equation and Inequality Problem With Adaptive Enhanced and Noise Suppressing Zeroing Neural Network

Solving time-varying linear equation and inequality (TVLEI) problem has attracted extensive attention in numerous scientific and engineered fields. In this article, it is basically considered that the commonly used dynamics neural network in the virtual environment is inevitably interfered with by the variable measurement noises while dealing with the TVLEI problem. An adaptive enhanced and noise-suppressing zeroing neural network (AENSZNN) model is proposed as an improved algorithm for solving the TVLEI problem. An adaptive scale factor based on the residual error norm is designed to make the proposed AENSZNN model converge to the theoretical solution faster. Furthermore, the momentum enhancement terms added to the model enables the AENSZNN model to effectively solve the TVLEI problem in real-time under the obstruction of different measurement noises. Besides, theoretical results and numerical experiments indicate that the AENSZNN model has advantages in convergence accuracy and robustness to noises compared with the existing algorithms. Note that, the proposed AENSZNN model is successfully exploited for the estimation of mobile object localization.

Stability-Constraint-Free Solutions to Solve Time-Varying Linear Equation System

Time-varying linear equation systems have been solved by the traditional zeroing neural dynamics approach in recent years. However, this method has to satisfy the stability constraint, which is a very rigorous condition. For this reason, traditional Lagrange-type finite difference formulas fail to lead to effective solutions, and we have to utilize more instants and reduce accuracy so that this condition is satisfied. In this work, we develop a new method of solving a time-varying linear equation system, which is based on theoretical solution decomposition. As a result, the proposed solutions are stability-constraint-free. We do not have to meticulously search effective time-discretization formulas because traditional Lagrange-type formulas are sufficient and especially effective. In addition, the proposed solutions have other advantages. For example, they do not need convergence procedures; they do not have storage requirements for past calculative results; and they are still effective when the sampling gap value is relatively large. Detailed comparisons are presented in this paper. Comparative numerical experiments are also shown to substantiate the effectiveness and advantages of the proposed solutions.

Impact of non-synchronous whirl on post-resonance backward whirl in vertical cracked rotors

Certain heavy-duty industrial machinery uses vertical rotating shafts. Such rotating systems typically undergo transient operation phases to reach their nominal speeds; recurrent passage through resonance generally occurs during startup and coast down operations. The Non-synchronous whirl between the shaft rotational speed and its whirling in the whirl orbit in the vicinity of the resonance zones, significantly affecting the breathing mechanism of a propagating transverse crack. Therefore, the appearance of Non-synchronous whirl during the passage through resonance rotational speed activates the crack breathing mechanism. To verify this observation, Jeffcott rotor and finite element models with induced breathing cracks are considered here for geometrically symmetrical and asymmetrical vertical rotor configurations. The mathematical models associated with these systems yield linear time-varying equations of motion incorporating the angular acceleration rate. The appearance of a Non-synchronous is found activating the crack breathing mechanism, which immediately excites the post-resonance backward whirl even with small crack depths in the rotor system. Accordingly, the excitation of post-resonance backward whirl can be considered as a strong indicator of an early crack propagation phase in vertical rotors.

Iterative Self-Tuning Minimum Variance Control of a Nonlinear Autonomous Underwater Vehicle Maneuvering Model

This paper addresses the problem of control design for a nonlinear maneuvering model of an autonomous underwater vehicle. The control algorithm is based on an iteration technique that approximates the original nonlinear model by a sequence of linear time-varying equations equivalent to the original nonlinear problem and a self-tuning control method so that the controller is designed at each time point on the interval for trajectory tracking and heading angle control. This work makes use of self-tuning minimum variance principles. The benefit of this approach is that the nonlinearities and couplings of the system are preserved, unlike in the cases of control design based on linearized systems, reducing in this manner the uncertainty in the model and increasing the robustness of the controller. The simulations here presented use a torpedo-shaped underwater vehicle model and show the good performance of the controller and accurate tracking for certain maneuvering cases.

Parameterization technique for solving variational formulation of index three linear time-varying singular control systems

This paper deals with finding an approximate solution to an index three time-varying linear differential algebraic control equations based on the theory of variational formulation. The critical points of the formulated variational functional subject to the differential- algebraic equation with the class of consistent initial conditions are shown to be the solution of the proposed index-three DAEs with control problem (singular control systems) and vise- versa. The given variational problem is also transformed from indirect method into direct method using generalized Ritz bases technique. From the numerical illustrations results, the efficiency and very good accuracy as well as the simplicity of the present approach are obtained.

A Finite-Time Convergent Neural Network for Solving Time-Varying Linear Equations with Inequality Constraints Applied to Redundant Manipulator

A Finite-Time Convergent Neural Network for Solving Time-Varying Linear Equations with Inequality Constraints Applied to Redundant Manipulator

Unified Model Solving Nine Types of Time-Varying Problems in the Frame of Zeroing Neural Network.

Many time-varying problems have been solved using the zeroing neural network proposed by Zhang et al. In this article, nine types of time-varying problems, namely time-varying nonlinear equation system, time-varying linear equation system, time-varying convex nonlinear optimization under linear equalities, unconstrained time-varying convex nonlinear optimization, time-varying convex quadratic programming under linear equalities, unconstrained time-varying convex quadratic programming, time-varying nonlinear inequality system, time-varying linear inequality system, and time-varying division, are investigated to better understand the essence of zeroing neutral network. Discrete-form time-varying problems are studied by considering the nature of unknown future and the requirement of real-time computation for time-varying problems. A unified model is proposed in the frame of zeroing neural network to uniformly solve these time-varying problems on the basis of their connections and a newly developed discretization formula. Theoretical analyses and numerical experiments, including the tracking control of PUMA560 robot manipulator, verify the effectiveness and precision of the proposed unified model.

Time scale observability and constructibility of linear dynamic equations

This paper investigates the observability and constructibility problems of time-varying linear dynamic equations using time scale theory. First, we define observability, reachability and constructibility operators on time scales. Some necessary and sufficient conditions are proposed to ensure the observability on non-uniform time domains based on some linear algebra tools. Then, constructibility is also examined using the same approach. Moreover, the link between observability and constructibility concepts on arbitrary time sets is discussed. Further, the observability and reachability duality relationship for time-varying linear systems on time scales is established. The current work unifies and extends some existing results given for standard cases (i.e. the continuous line and the discrete time domain) to non-uniform time domains. Finally, the obtained results are described with an illustrative example.

Application of Physical Function Model to State Estimations of Nonlinear Mechanical Systems

The physical function model has been effectively used for model-based development (MBD) of automobile systems. This research demonstrates a novel application of this modeling method to the state estimation of nonlinear mechanical systems based on the Kalman filtering theory. The physical function model is a block diagram that describes each engineering field by a common rule, which focuses on the energy flow. Compared to traditional modeling approaches, this model has the flexibility to incorporate a wide range of nonlinear characteristics and the know-how accumulated by the manufacturers. Hence, it has a quite high affinity with the industrial world. The purpose of this research is to pioneer a new application of the physical function model beyond simulation analysis. In particular, physical function modeling offers a model of a system with multiple nonlinearities in the form of a time-varying linear state equation. By focusing on this feature, this study applies it to the Kalman filtering theory. The proposed approach is applicable to a wide range of nonlinearities, reduces the calculation load, and considers the background of the current MBD. Finally, verifications using an experimental apparatus, which simplifies an automotive drivetrain with backlash, demonstrate the effectiveness of the proposed approach.

Co-Design of Finite-Time Convergence and Noise Suppression: A Unified Neural Model for Time Varying Linear Equations With Robotic Applications

Computing time-varying linear systems is widely encountered in engineering practice and scientific computation. Dynamic neural networks, as a class of modeling approaches, have been intensively explored in recent decades for solving linear equations. The time-varying nature of this problem and the noisy workspace for many engineering practice require two features of practical design: 1) fast convergence in time and 2) robustness against noises and disturbance. Existing solutions usually decouple the problem into two steps by designing a fast-convergent neural controller and then topped with an additional low-pass filter to reach noise robustness. However, due to the interplay of the mentioned two dynamical parts, the overall system may lose stability if the parameters are not well tuned. In this paper, we establish the first dynamical neural model for simultaneously achieving fast-convergence, particularly finite-time convergence, and noise-robustness, with the capability to reject the unknown noise when it is constant or varies slowly. To do so, a superior design formula activated by noise-tolerant nonlinear functions is proposed to enhance the capability of zeroing neural networks (ZNNs), achieving denoising and finite-time convergence in a unified design. According to this design formula, a novel recurrent neural network (RNN) with finite-time convergence and inherently noise-suppression performance [thus termed the finite-time robust RNN (FTRRNN)] is developed and applied to robotic motion tracking illustrated via time-varying linear equation system solving. Furthermore, theoretical analyses on the global stability, the finite-time convergence and the denoising ability of the proposed design formula and the corresponding FTRRNN model are presented in details. The upper bound on the convergence time is also analytically derived. A numerical example is supplied to verify the superior property of the FTRRNN model to the ZNN model according to the results of computing time-varying linear equation system in the presence of additive noises. Finally, an application to robotic motion tracking is presented to show that the presented FTRRNN model can successfully realize the ellipse-path tracking control of a planar two-link manipulator in front of the external disturbances, while the conventional ZNN model fails under the same conditions.

General and Improved Five-Step Discrete-Time Zeroing Neural Dynamics Solving Linear Time-Varying Matrix Equation with Unknown Transpose

In this paper, a general five-step discrete-time zeroing neural dynamics (DTZND) model is proposed to solve linear time-varying matrix equation with unknown transpose. Specifically, the explicit continuous-time zeroing neural dynamics (CTZND) model is derived from the time-varying matrix equation with unknown transpose via Kronecker product and vectorization technique. Furthermore, a general five-step discretization formula is designed to approximate the first-order derivative of the target point, and the convergence condition is given. Thus, the general five-step DTZND model is obtained by using the general five-step discretization formula to discretize the CTZND model. Theoretical analyses present the stability and convergence of the proposed general five-step DTZND model. Numerical experiment results substantiate that the proposed DTZND model for solving linear time-varying matrix equation is stable and convergent with the theoretically analyzed errors. In addition, the improved DTZND models are provided in terms of accuracy and computational complexity, and verified by numerical experiments.