A Unified Predefined-Time Convergent and Robust ZNN Model for Constrained Quadratic Programming

Design and Application of A Robust Zeroing Neural Network to Kinematical Resolution of Redundant Manipulators Under Various External Disturbances

The solution of a complex-valued matrix pseudoinverse is one of the key steps in various science and engineering fields. Owing to its important roles, researchers had put forward many related algorithms. With the development of research, a time-varying matrix pseudoinverse received more attention than a time-invarying one, as we know that a zeroing neural network (ZNN) is an efficient method to calculate a pseudoinverse of a complex-valued time-varying matrix. Due to the initial ZNN (IZNN) and its extensions lacking a mechanism to deal with both convergence and robustness, that is, most existing research on ZNN models only studied the convergence and robustness, respectively. In order to simultaneously improve the double features (i.e., convergence and robustness) of ZNN in solving a complex-valued time-varying pseudoinverse, this paper puts forward a double features ZNN (DFZNN) model by adopting a specially designed time-varying parameter and a novel nonlinear activation function. Moreover, two nonlinear activation types of complex number are investigated. The global convergence, predefined time convergence, and robustness are proven in theory, and the upper bound of the predefined convergence time is formulated exactly. The results of the numerical simulation verify the theoretical proof, in contrast to the existing complex-valued ZNN models, the DFZNN model has shorter predefined convergence time in a zero noise state, and enhances robustness in different noise states. Both the theoretical and the empirical results show that the DFZNN model has better ability in solving a time-varying complex-valued matrix pseudoinverse. Finally, the proposed DFZNN model is used to track the trajectory of a manipulator, which further verifies the reliability of the model.

On account of the rapid progress of zeroing neural network (ZNN) and the extensive use of fuzzy logic system (FLS), this article proposes an intelligent fuzzy robustness ZNN (IFR-ZNN) model and applies it to solving the time-variant Stein matrix equation (TVSME) problem. Be different from ZNN models before, the IFR-ZNN model uses a fuzzy parameter as the design parameter and adopts a first proposed improved nonlinear piecewise activation function. Particularly, the FLS that generates the fuzzy parameter utilizes an improved membership function of nonuniform distribution which can improve the adaptability and robustness of the IFR-ZNN model. Based on the above two optimizations, the proposed IFR-ZNN model possesses three significant advantages: (1) fixed-time convergence independent of initial states; (2) superior robustness to tolerate two kinds of noises simultaneously; and (3) better adaptiveness based on computational error. Besides, the upper bounds of fixed-time convergence of the IFR-ZNN model under noisy or non-noisy situations are calculated theoretically, and the stability as well as the excellent adaptability are analyzed in detail. Finally, simulation comparison results manifest the availability and meliority of the proposed IFR-ZNN model in solving the TVSME problem.

In industrial activities, robot arms are usually requested to perform repetitive tasks. As a special recurrent neural network, zeroing neural network (ZNN) has been successfully applied to cyclical motion planning (CMP) of redundant robots. However, most of the existing ZNN models for CMP have not taken into account physical limits of robot arms. Unfortunately, if physical limits are violated, the robot will fail to complete the task and even be damaged in practice. To overcome this problem, this article proposes a novel ZNN model for CMP of physically constrained redundant robots. The ZNN model is endowed with predefined-time convergence by using various designed activation functions. Rigorous theoretical analysis is conducted to prove the predefined-time convergence of the accelerated ZNN model. Numerical experiments are then comparatively performed, showing that the ZNN model delivers superior convergence and it is applicable to CMP of redundant robots with physical constraints considered.

Two discrete ZNN models for solving time-varying augmented complex Sylvester equation

A generalized Sylvester equation is a special formulation containing the Sylvester equation, the Lyapunov equation and the Stein equation, which is often encountered in various fields. However, the time-variant generalized Sylvester equation (TVGSE) is rarely investigated in the existing literature. In this article, we propose a noise-suppressing recurrent neural network (NSRNN) model activated by saturation-allowed functions to solve the TVGSE. For comparison, the existing zeroing neural network (ZNN) models and some improved ZNN models are introduced. Additionally, theoretical analysis on the convergence and robustness of the NSRNN model is given. Furthermore, computer simulations on illustrative examples and applications to robots and acoustic source localization are carried out. Validation results synthesized by the NSRNN model and other ZNN models are provided to illustrate the ability in solving the TVGSE and dealing with noises of the NSRNN model, and the inaction of other ZNN models to noises.

Two nonlinear zeroing neural network (ZNN) models with prescribed-time convergence for time-dependent matrix LR and QR factorization are proposed in this paper. To do so, two algorithms and two error functions are constructed to transform the time-dependent matrix LR and QR factorization problems into time-dependent linear equation systems, respectively. Simultaneously, a new activation function is introduced based on the initial ZNN models for the prescribed-time convergence of models. The excellent performance (robustness and convergence) of the two proposed ZNN models are analyzed theoretically. Furthermore, the prescribed-time convergence and anti-noise abilities of the proposed ZNN models are well demonstrated in numerical experiments. Finally, the proposed ZNN model is applied to the moving target location problem, and the results show that the location error is at the millimeter level.

Continuous and discrete zeroing neural network for a class of multilayer dynamic system

The matrix inversion problem plays a very important role in mathematics as well as practical engineering applications. In this article, unlike the traditional fixed-parameter zeroing neural network (ZNN) model, on the basis of the original varying-parameter ZNN (VPZNN) model, an improved VPZNN (IVPZNN) model is established and researched to solve time-varying matrix inversion (TVMI). Specifically, the value of the proposed novel time-varying parameter in the IVPZNN model can grow rapidly over time, which can better meet the needs of ZNN in hardware implementation. In addition, theoretical analyses of the novel time varying parameter and the proposed IVPZNN model are given to guarantee the global superexponential convergence and finite-time convergence. Numerical calculation results verify the superior property of the established IVPZNN model for addressing the TVMI problem, as compared with the existing fixed-parameter ZNN and VPZNN models.

Aircraft assembly modeling requires mass solving of quadratic programming (QP) problems. Thus, reducing the computation time becomes a very urgent task. Earlier [12], it was shown that a significant reduction in time can be achieved by reformulating the original QP problem in the so-called form of the relative QP problem. The current paper discusses the aspects of using relative QP problem related to the accuracy of the computations. Data preparation for the relative QP problem involves the matrix inversions. Since the Hessian in the considered QP problems is ill-conditioned, this operation leads to a rapid accumulation of roundoff errors. The paper proposes to use the error-free operations of addition and multiplication to increase the accuracy of the QP problem solution and to maintain high computation speed for QP problem solving. Finally, a comparison of the primal, dual, and relative QP problem formulations in terms of computation time and result accuracy for assembly problems is presented.KeywordsQuadratic programmingRelative problem formulationAssembly problemsComputational accuracy

Zeroing neural networks (ZNNs), a specialized class of recurrent neural networks, have demonstrated remarkable effectiveness in matrix computation and dynamic optimization problems due to their inherent parallel computing capabilities. In this paper, a predefined-time and noise reduction ZNN (PTNRZNN) model is proposed for solving convex quadratic programming problems with equality and inequality constraints. Additionally, a new activation function is proposed, demonstrating enhanced accelerated convergence and noise reduction performance compared to previous models. The convergence and robustness of the PTNRZNN model are effectively proven through theoretical assessment. Furthermore, the performance of the PTNRZNN model is further validated through simulation experiments. Finally, the PTNRZNN model is applied to the binary assignment problem in logistics (BAPL), yielding optimized results with an error margin as low as $$10^{-2}$$ 10 - 2 compared to theoretical values. The strong robustness of the method makes it an excellent performer in solving BAPL under noise interference.

Complex-valued time-dependent matrix inversion (TDMI) is extensively exploited in practical industrial and engineering fields. Many current neural models are presented to find the inverse of a matrix in an ideal noise-free environment. However, the outer interferences are normally believed to be ubiquitous and avoidable in practice. If these neural models are applied to complex-valued TDMI in a noise environment, they need to take a lot of precious time to deal with outer noise disturbances in advance. Thus, a noise-suppression model is urgent to be proposed to address this problem. In this article, a complex-valued noise-tolerant zeroing neural network (CVNTZNN) on the basis of an integral-type design formula is established and investigated for finding complex-valued TDMI under a wide variety of noises. Furthermore, both convergence and robustness of the CVNTZNN model are carefully analyzed and rigorously proved. For comparison and verification purposes, the existing zeroing neural network (ZNN) and gradient neural network (GNN) have been presented to address the same problem under the same conditions. Numerical simulation consequences demonstrate the effectiveness and excellence of the proposed CVNTZNN model for complex-valued TDMI under various kinds of noises, by comparing the existing ZNN and GNN models.

In consideration of the limitation of the communication and the possibility that redundant robots might deliver information at different power levels, cases under weight-unbalanced directed graphs from the network topology perspective are in larger accordance with those in multiple redundant robot systems. By moving forward along this direction, a distributed controller is proposed in this article to handle circumstances of collaborative control of multiple redundant robots under weight-unbalanced directed graphs. This kind of control problem is modeled into generalized quadratic programming (QP) problems with equality and inequality constraints. Then, the above QP problems are solved by a proposed neural-dynamics-based method, whose stability and convergence are theoretically proved subsequently. Besides, several experimental examples are conducted, and related comparisons are provided to demonstrate the feasibility of the proposed controller.

The zeroing neural network (ZNN) model, a powerful approach for addressing time-varying problems, has been extensively applied in the calculation and optimization fields. In this article, a new pattern activation function, the power piecewise activation function (PPAF), is proposed to establish a predefined fixed-time convergent ZNN (PFTZNN) for finding solutions to the time-varying quadratic programming problem. In comparison with the traditional activation functions, multisegmentation is a remarkable feature of the PPAF; consequently, the advantage of PPAF is that its parameters can be flexibly adjusted according to actual needs. Specifically, because of the multisegment characteristics of the PPAF, the convergence speed of the PPAF-activated PFTZNN model can be flexibly adjusted based on distinct requirements. The fixed-time convergence property of the PPAF-activated PFTZNN model is validated by detailed mathematical theoretical analysis, and its upper bound convergence time is directly calculated. Then, the comparative simulation results of the PPAF-activated PFTZNN model with other existing ZNN models for time-varying quadratic programming are provided for the further verification of its superior convergence speed and robustness. In addition, the proposed PFTZNN model is applied for dual-arm manipulator cooperative trajectory tracking, and its practical application ability is demonstrated by united simulation experiments of MATLAB and Robot studio. Finally, the PFTZNN model is also applied to control a real dual-arm manipulator to complete the trajectory tracking task, which further validates its superior performance together and widespread applicability.

Many physical process systems have performance limitation regardless the input. This limitation usually occurs in the form of input saturation of actuator as the constraint of the system. To overcome this problem, Model Predictive Control (MPC) may be used due to its capability to compute optimal control signal in the presence of input saturation. The optimal control signal is obtained by solving a quadratic programming (QP) problem with variables' constraints at each time instants. QP problem with lower and upper constraint is known to be equivalent to an algebraic loop involving diagonal upper and lower saturation. Hence, QP can be iteratively computed until its solution converges. A slow computation of QP, however, limits its applications for fast systems with simple digital processor. Therefore, in order to be able to be implemented in real-time embedded applications, a fast algorithm of QP solver is in need. In this paper, a comparison study of two QP implementation, i.e. Projected Gauss Seidel (PGS) and so called Algorithm-2, using Arduino Mega 2560 is presented. In this case, MPC with QP solver using both algorithms is implemented to control the speed of lightly loaded BLDC motor. Before implementation, system modeling, controller designing, and simulation are undertaken. The performance of both iterative algorithms are then compared in terms of average computation time in the simulation and implementation. It can be observed from the results that Algorithm-2 gives a bit faster computation than PGS under same conditions.