Time-varying Matrix Inversion Research Articles

GNN Model for Time-Varying Matrix Inversion With Robust Finite-Time Convergence.

As a type of recurrent neural networks (RNNs) modeled as dynamic systems, the gradient neural network (GNN) is recognized as an effective method for static matrix inversion with exponential convergence. However, when it comes to time-varying matrix inversion, most of the traditional GNNs can only track the corresponding time-varying solution with a residual error, and the performance becomes worse when there are noises. Currently, zeroing neural networks (ZNNs) take a dominant role in time-varying matrix inversion, but ZNN models are more complex than GNN models, require knowing the explicit formula of the time-derivative of the matrix, and intrinsically cannot avoid the inversion operation in its realization in digital computers. In this article, we propose a unified GNN model for handling both static matrix inversion and time-varying matrix inversion with finite-time convergence and a simpler structure. Our theoretical analysis shows that, under mild conditions, the proposed model bears finite-time convergence for time-varying matrix inversion, regardless of the existence of bounded noises. Simulation comparisons with existing GNN models and ZNN models dedicated to time-varying matrix inversion demonstrate the advantages of the proposed GNN model in terms of convergence speed and robustness to noises.

A new recurrent neural network based on direct discretization method for solving discrete time-variant matrix inversion with application

A new recurrent neural network based on direct discretization method for solving discrete time-variant matrix inversion with application

A Fuzzy Adaptive Zeroing Neural Network Model With Event-Triggered Control for Time-Varying Matrix Inversion

Time-varying matrix inversion (TVMI) is a basic mathematical problem, which is widely involved in many scientific fields. In this paper, an event-triggered control fuzzy adaptive zeroing neural network (ETC-FAZNN) model is proposed for solving the TVMI problem, where the fuzzy adaptive convergence parameter (FACP) is got by the redesigned fuzzy logic system, which makes the ETC-FAZNN model adaptive. Meanwhile, the event-triggered control is introduced to control the update of the FACP, which improves the calculation speed of the ETC-FAZNN model. Moreover, a novel activation function called segmented predefined-time activation function (SptAF) is put forward in this paper to improve the convergence and robustness of the ETC-FAZNN model. Theoretical analysis and simulation experiments reveal that the ETC-FAZNN model can realize stability, predefined-time convergence, robustness, and adaptability performances in solving the TVMI problem.

Double integral‐enhanced Zeroing neural network with linear noise rejection for time‐varying matrix inverse

AbstractIn engineering fields, time‐varying matrix inversion (TVMI) issue is often encountered. Zeroing neural network (ZNN) has been extensively employed to resolve the TVMI problem. Nevertheless, the original ZNN (OZNN) and the integral‐enhanced ZNN (IEZNN) usually fail to deal with the TVMI problem under unbounded noises, such as linear noises. Therefore, a neural network model that can handle the TVMI under linear noise interference is urgently needed. This paper develops a double integral‐enhanced ZNN (DIEZNN) model based on a novel integral‐type design formula with inherent linear‐noise tolerance. Moreover, its convergence and robustness are verified by derivation strictly. For comparison and verification, the OZNN and the IEZNN models are adopted to resolve the TVMI under multiple identical noise environments. The experiments proved that the DIEZNN model has excellent advantages in solving TVMI problems under linear noises. In general, the DIEZNN model is an innovative work and is proposed for the first time. Satisfyingly, the errors of DIEZNN are always less than 1 × 10−3 under linear noises, whereas the error norms of OZNN and IEZNN models are not convergent to zero. In addition, these models are applied to the control of the controllable permanent magnet synchronous motor chaotic system to indicate the superiority of the DIEZNN.

A fixed-time convergent and noise-tolerant zeroing neural network for online solution of time-varying matrix inversion

As a common mathematical operation, the time-varying matrix inversion (TVMI) is frequently arisen in many complex problems. It has been proved in a large number of studies that the zeroing neural network (ZNN) is a reliable tool to solve TVMI problems. However, the previous reported conventional zeroing neural network (CZNN) models only achieve undesirable exponential convergence or finite-time convergence, and they are difficult to be popularized in practical applications. Therefore, by introducing a novel activation function (NAF), a fixed-time convergence and noise-tolerant zeroing neural network (FTCNTZNN) model for solving TVMI problems is proposed in this paper. Different from existing CZNN models, the proposed FTCNTZNN model possesses both fixed-time convergence and anti-noise properties for TVMI solving. To comprehensively show the superiority of the proposed FTCNTZNN model for solving TVMI, a lot of theoretical demonstrations and comparative simulation experiments have been provided in this work. Firstly, the fixed-time convergence and robustness of the proposed FTCNTZNN model are verified by detailed theoretical analysis in noiseless and noisy environment, respectively. Then, the proposed FTCNTZNN model is applied to solve two different TVMI problems in a variety of situations. In addition, a successful manipulator trajectory tracking application using the proposed FTCNTZNN model in noisy environment further validates its practicability. Both of theoretical analysis and experimental results convincingly verify that the proposed FTCNTZNN model can effectively solve TVMI problems in noiseless and noisy environment.

Prescribed-Time Convergent Adaptive ZNN for Time-Varying Matrix Inversion under Harmonic Noise

Harmonic noises widely exist in industrial fields and always affect the computational accuracy of neural network models. The existing original adaptive zeroing neural network (OAZNN) model can effectively suppress harmonic noises. Nevertheless, the OAZNN model’s convergence rate only stays at the exponential convergence, that is, its convergence speed is usually greatly affected by the initial state. Consequently, to tackle the above issue, this work combines the dynamic characteristics of harmonic signals with prescribed-time convergence activation function, and proposes a prescribed-time convergent adaptive ZNN (PTCAZNN) for solving time-varying matrix inverse problem (TVMIP) under harmonic noises. Owing to the nonlinear activation function used having the ability to reject noises itself and the adaptive term also being able to compensate the influence of noises, the PTCAZNN model can realize double noise suppression. More importantly, the theoretical analysis of PTCAZNN model with prescribed-time convergence and robustness performance is provided. Finally, by varying a series of conditions such as the frequency of single harmonic noise, the frequency of multi-harmonic noise, and the initial value and the dimension of the matrix, the comparative simulation results further confirm the effectiveness and superiority of the PTCAZNN model.

From Penrose Equations to Zhang Neural Network, Getz–Marsden Dynamic System, and DDD (Direct Derivative Dynamics) Using Substitution Technique

The time-variant matrix inversion (TVMI) problem solving is the hotspot of current research because of its frequent appearance and application in scientific research and industrial production. The generalized inverse problem of singular square matrix and nonsquare matrix can be related to Penrose equations (PEs). The PEs implicitly define the generalized inverse of a known matrix, which is of fundamental theoretical significance. Therefore, the in-depth study of PEs might enlighten problem solving of TVMI in a foreseeable way. For the first time, we construct three different matrix error-monitoring functions based on PEs and propose the corresponding models for TVMI problem solving by using the substitution technique and ZNN design formula. In order to facilitate computer simulation, the obtained continuous-time models are discretized by using ZTD (Zhang time discretization) formulas. Furthermore, the feasibility and effectiveness of the novel Zhang neural network (ZNN) multiple-multiplication model for matrix inverse (ZMMMI) and the PEs-based Getz–Marsden dynamic system (PGMDS) model in solving the problem of TVMI are investigated and shown via theoretical derivation and computer simulation. Computer experiment results also illustrate that the direct derivative dynamics model for TVMI is less effective and feasible.

A novel hybrid Zhang neural network model for time-varying matrix inversion

A new hybrid Zhang neural network (HZNN) model is formulated to solve time-varying matrix inversion problem. The network is designed such that it encompasses the goodness of both gradient neural network (GNN) and Zhang neural network (ZNN). The stability of the network and convergence results are established theoretically. The rate of convergence of HZNN model is much faster compared to the classical GNN model and ZNN model for time-varying matrix inversion problem. Nonlinear activation functions for HZNN need not have to be odd and monotonically increasing for the network to converge, however with certain restrictions. Simulations are carried out to test the efficiency of HZNN over GNN and ZNN. Also, HZNN with odd and non-odd activation functions are examined for convergence. It is observed that the network converges much faster with the proposed non-odd function 3 (nof3). Furthermore, the HZNN model with odd and non-odd functions are applied to the kinematic control of a two-link planar manipulator for tracking a circular path.

Five-step discrete-time noise-tolerant zeroing neural network model for time-varying matrix inversion with application to manipulator motion generation

In this paper, a novel Taylor-type difference rule with O(τ4) pattern error is provided for the first-order derivative approximation. Then, a high accuracy noise-tolerant five-step discrete-time zeroing neural network (ZNN) (termed as FDNTZNN model) is proposed to solve the time-varying matrix inversion problem in real-time. In addition, to obtain the derivative value of time-varying variables in real-world applications, the backward-difference rule is exploited to develop the FD-NTZNN model when the derivative information is unknown (FD-NTZNN-U). Theoretical analysis demonstrates that the proposed FD-NTZNN models have the properties of 0−stability, consistency and convergence. For comparative analysis, the classical Euler-type discrete-time ZNN model (EDZNN), five-step Taylor-type discrete-time ZNN model (FDZNN) and Euler-type discrete-time noise-tolerant ZNN (NTZNN) model (ED-NTZNN) are reconsidered. Ultimately, two illustrative numerical simulations and an application example to motion generation of manipulator are simulated to substantiate the feasibility and effectiveness of the proposed FD-NTZNN model and FD-NTZNN-U model for online time-varying matrix inversion in the presence of different types of noise.

A Variable-Parameter Noise-Tolerant Zeroing Neural Network for Time-Variant Matrix Inversion With Guaranteed Robustness.

Matrix inversion frequently occurs in the fields of science, engineering, and related fields. Numerous matrix inversion schemes are often based on the premise that the solution procedure is ideal and noise-free. However, external interference is generally ubiquitous and unavoidable in practice. Therefore, an integrated-enhanced zeroing neural network (IEZNN) model has been proposed to handle the time-variant matrix inversion issue interfered with by noise. However, the IEZNN model can only deal with small time-variant noise interference. With slightly larger noise interference, the IEZNN model may not converge to the theoretical solution exactly. Therefore, a variable-parameter noise-tolerant zeroing neural network (VPNTZNN) model is proposed to overcome shortcomings and improve the inadequacy. Moreover, the excellent convergence and robustness of the VPNTZNN model are rigorously analyzed and proven. Finally, compared with the original zeroing neural network (OZNN) model and the IEZNN model for matrix inversion, numerical simulations and a practical application reveal that the proposed VPNTZNN model has the best robust property under the same external noise interference.

Design and Analysis of a Novel Finite-Time Convergent and Noise-Tolerant Recurrent Neural Network for Time-Variant Matrix Inversion

Matrix inversion ubiquitously arises in engineering. The so-called zeroing neural network (ZNN) is an effective recurrent neural network for solving time-variant matrix inversion. Without considering noises, the ZNN approach requires finite time or infinitely long time to converge to the exact solution. When perturbed by additive noises, the existing ZNN models exhibit limited ability to reject disturbances and are susceptible to be divergent. For instance, under time-variant bounded noises, steady-state residual errors of the existing ZNNs would be bounded. To make the steady-state residual errors arbitrarily small, infinitely long time is required and related design parameters must be set large enough or infinitely large, which is not realistic in practice. To overcome this situation, this paper for the first time systematically designs and analyses a finite-time convergent and noise-tolerant ZNN (FTNTZNN) that is capable of completely converging to the theoretical solution in finite time even under various types of noises. Theoretically, the finite-time convergence and disturbance-rejection properties of the FTNTZNN are rigorously proved. Comparative numerical results substantiate that the FTNTZNN model delivers superior convergence and robustness performance in solving time-variant matrix inversion and kinematic control of a robotic arm as compared with the existing ZNN models. The FTNTZNN model expands the current knowledge for designing neural-dynamic systems to solve matrix inversion, which can provide inspiration for other problems solving under noises.

Comprehensive Analysis of a New Varying Parameter Zeroing Neural Network for Time Varying Matrix Inversion

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.

GMDS-ZNN Model 3 and its Ten-Instant Discrete Algorithm for Time-Variant Matrix Inversion Compared With Other Multiple-Instant Ones

The online time-variant matrix inversion problem has attracted extensive attention and study, because of its considerable appearance and application in scientific research and industrial production. For various control optimization problems, the demand for the high-precision and rapid-convergence of matrix inversion algorithm is increasing. It remains an ongoing challenge due to the rigorous requirements of precision and convergence of the algorithm. In this paper, on the basis of our previous works, by using Zhang neural network (ZNN) method, a continuous time-variant matrix inversion model, which is also a Getz-Marsden dynamic system (i.e., GMDS-ZNN model 3), is proposed. Besides, a general ten-instant Zhang et al discretization (ZeaD) formula is presented and investigated, with corresponding theoretical results being provided. Next, by applying this general formula to discretize the continuous time-variant matrix inversion model, a general ten-instant discrete time-variant matrix inversion (DTVMI) algorithm with the sixth-order precision is proposed. For comparison, four other DTVMI algorithms, with the second-, third-, fourth-, and fifth-order precisions, are also proposed and presented, respectively, by using other ZeaD formulas to discretize the continuous time-variant matrix inversion model. Besides, for the situation of coefficient matrix derivative being unknown, we provided the formula of estimating it with the fifth-order precision. With the help of the proposed matrix derivative estimation formula, the actual application field of GMDS-ZNN model 3 is expanded evidently. Finally, theoretical analyses and simulation experiment results highlight the effectiveness and accuracy of the proposed GMDS-ZNN model 3 and DTVMI algorithms.

Analysis and Application of Modified ZNN Design With Robustness Against Harmonic Noise

The Zhang neural network (ZNN) has recently realized remarkable success in solving time-varying problems. Harmonic noise widely exists in industrial applications and can severely affect the solution computed by ZNN models. This article attempts to solve the aforementioned limitations by providing the first ZNN design with an inherent capability to prohibit harmonic noise. Moreover, it opens new opportunities to shift the research on ZNNs in ideal situations to that with theoretical consideration on nonideal working environments. We establish a modified ZNN design formula in a noisy environment by incorporating the dynamics of harmonic signals. Theoretical analysis shows the convergence of the proposed ZNN design. An application case study for the new ZNN model verifies its effectiveness for time-varying matrix inversion in the presence of harmonic noise. The simulation further substantiates the effectiveness, superiority, and application prospects of the proposed ZNN design.

A Novel Recurrent Neural Network-Based Ultra-Fast, Robust, and Scalable Solver for Inverting a "Time-Varying Matrix".

The concept presented in this paper is based on previous dynamical methods to realize a time-varying matrix inversion. It is essentially a set of coupled ordinary differential equations (ODEs) which does indeed constitute a recurrent neural network (RNN) model. The coupled ODEs constitute a universal modeling framework for realizing a matrix inversion provided the matrix is invertible. The proposed model does converge to the inverted matrix if the matrix is invertible, otherwise it converges to an approximated inverse. Although various methods exist to solve a matrix inversion in various areas of science and engineering, most of them do assume that either the time-varying matrix inversion is free of noise or they involve a denoising module before starting the matrix inversion computation. However, in the practice, the noise presence issue is a very serious problem. Also, the denoising process is computationally expensive and can lead to a violation of the real-time property of the system. Hence, the search for a new ‘matrix inversion’ solving method inherently integrating noise-cancelling is highly demanded. In this paper, a new combined/extended method for time-varying matrix inversion is proposed and investigated. The proposed method is extending both the gradient neural network (GNN) and the Zhang neural network (ZNN) concepts. Our new model has proven that it has exponential stability according to Lyapunov theory. Furthermore, when compared to the other previous related methods (namely GNN, ZNN, Chen neural network, and integration-enhanced Zhang neural network or IEZNN) it has a much better theoretical convergence speed. To finish, all named models (the new one versus the old ones) are compared through practical examples and both their respective convergence and error rates are measured. It is shown/observed that the novel/proposed method has a better practical convergence rate when compared to the other models. Regarding the amount of noise, it is proven that there is a very good approximation of the matrix inverse even in the presence of noise.

Higher-Order ZNN Dynamics

Several improvements of the Zhang neural network (ZNN) dynamics for solving the time-varying matrix inversion problem are presented. Introduced ZNN dynamical design is termed as ZNN models of the order p, $$p\ge 2$$, and it is based on the analogy between the proposed continuous-time dynamical systems and underlying discrete-time pth order hyperpower iterative methods for computing the constant matrix inverse. Such ZNN design is denoted by $$\hbox {ZNN}_H^p$$. Particularly, the $$\hbox {ZNN}_H^2$$ design coincides with the standard ZNN design. Moreover, $$\hbox {ZNN}_H^3$$ design represents a time-varying generalization of the previously defined ZNNCM model. In addition, an integration-enhanced noise-handling $$\hbox {ZNN}_H^p$$ model, termed as $$\hbox {IENHZNN}_H^p$$, is introduced. In the time-invariant case, we present a hybrid enhancement of the $$\hbox {ZNN}_H^p$$ model, shortly termed as $$\hbox {HZNN}_H^p$$, and investigate it theoretically and numerically. Theoretical and numerical comparisons between the improved and standard ZNN dynamics are considered.

An exponential-enhanced-type varying-parameter RNN for solving time-varying matrix inversion

In order to compute time-varying matrix inversion faster, a novel exponential-enhanced-type varying-parameter recurrent neural network (EVP-RNN) is proposed and investigated in this paper. First, the detailed design process of the proposed EVP-RNN is stated and presented. Then, mathematical analysis proves that the proposed EVP-RNN has superior exponential convergence property than the conventional fixed-parameter recurrent neural network (FP-RNN) with four kinds of specific activation functions. Meanwhile, the guideline of choosing an activation function is provided to achieve a better convergence property. Third, theoretical analysis shows that the upper-bounds of calculation error of EVP-RNN are always smaller than those of FP-RNN and actual calculation error of EVP-RNN always converges faster than that of the FP-RNN. Simultaneously, an idea of designing a time-varying parameter is given. Finally, the results of comparative simulations verify the effectiveness, high accuracy, and superiority of the EVP-RNN compared with the traditional FP-RNN for solving time-varying matrix inversion.

Discrete-time neural network with two classes of bias noises for solving time-variant matrix inversion and application to robot tracking

It is well known that noise is inevitable in real world, especially in the case of solving time-variant matrix inversion. Therefore, it is more necessary to study the algorithm with bias noises to solve time-variant matrix inversion. This paper investigates discrete-time neural network with two classes of bias noises for solving time-variant matrix inversion, and its application to robot tracking based on the property of second-order differential equation. Firstly, the model is presented and some indispensable propaedeutics are given. Then, continuous-time and discrete-time neural network with two classes of bias noises is designed, respectively. Their convergence and finite-time stability are also theoretically analyzed. Finally, the proposed models are applied to a five-link robot tracking. Numerical simulations demonstrate the superiority and effectiveness of our method.

A novel recurrent neural network and its finite-time solution to time-varying complex matrix inversion

A complex-valued nonlinear recurrent neural network is designed and researched for time-varying matrix inversion solving in complex field. Unlike the design methods of the conventional gradient neural network (CGNN) and the previous Zhang neural network (ZNN), the proposed complex-valued nonlinear recurrent neural network (CVNRNN) model is established on basis of a nonlinear evolution formula and possesses a better finite-time convergence Besides, the detailed theoretical analysis provides a guarantee for the finite-time convergence achievement of the CVNRNN model. In addition, the theoretical analysis is also verified by numerical simulations, which comparatively show that the proposed CVNRNN model is faster and more accurate than the ZNN model and the CGNN model in solving time-varying complex matrix inversion.

Zhang Neural Dynamics Approximated by Backward Difference Rules in Form of Time-Delay Differential Equation

Time-varying matrix inversion (TVMI) and tracking control of vehicular inverted pendulum (VIP) system are usually considered as two important benchmarking examples in mathematical science and control fields. In this paper, Zhang neural dynamics (ZND) method is used for solving these two problems. Besides, in order to investigate the effect of time delay (especially the time delay caused by the derivative approximation) on the ZND method when solving the above-mentioned two problems, backward difference rules in the form of time-delay differential equation are elegantly applied to approximating the ZND method, which is simply termed time-delay ZND method. Moreover, two theorems about time-delay ZND method are presented together with theoretical proofs. Finally, simulative tests of TVMI and tracking control of VIP system are conducted to show the effectiveness and accuracy of non-time-delay and time-delay ZND methods.