Zhang Dynamics Research Articles

Finite-time solution to nonlinear equation using recurrent neural dynamics with a specially-constructed activation function

A recurrent neural dynamics (termed improved Zhang dynamics, IZD), together with a specially-constructed activation function, is proposed and investigated for finding the root of nonlinear equation in this paper. It is analyzed that the IZD model can converge to the theoretical solution within finite time. Besides, the upper bound of convergence time is estimated analytically. Compared with conventional gradient-based dynamics (GD), our proposed IZD model in the form of implicit dynamics has the following advantages: (1) has better consistency with actual situations; and (2) has a greater ability in representing dynamical systems. Besides, our model can achieve superior convergence performance (i.e., finite-time convergence) in comparison with the existing neural dynamics, specifically the original ZD (OZD) model. Both theoretical analysis and computer-simulation results substantiate the effectiveness and superiority of the IZD model for solving nonlinear equation in real-time.

Singularity-conquering ZG controllers of z2g1 type for tracking control of the IPC system

With wider investigations and applications of autonomous robotics and intelligent vehicles, the inverted pendulum on a cart (IPC) system has become more attractive for numerous researchers due to its concise and representative structure. In this article, the tracking-control problem of the IPC system is considered and investigated. Based on Zhang dynamics (ZD) and gradient dynamics (GD), a novel kind of ZG controllers are developed and investigated for achieving the tracking-control purpose, which contains controllers of z2g0 and z2g1 types according to the number of times of using the ZD and GD methods. Besides, theoretical analyses are presented to guarantee the global and exponential convergence performance of both z2g0 and z2g1 controllers. Computer simulations are further performed to substantiate the feasibility and effectiveness of ZG controllers. More importantly, comparative simulation results demonstrate that controllers of z2g1 type can conquer the singularity problem (i.e. the division-by-zero problem).

Z-type and G-type models for time-varying inverse square root (TVISR) solving

A class of neural dynamics, called Zhang dynamics (ZD), has been proposed to solve online various time-varying problems. In this paper, different Z-type (Zhang type) models based on different Zhang functions (ZFs) are proposed, investigated and simulated for solving the time-varying inverse square root (or in short, TVISR) problem. Then, for the same problem-solving task, different G-type (gradient type) models based on different energy functions (EFs) are developed and investigated as well. Moreover, the convergence analyses of Z-type and G-type models are studied in-depth for the completeness of this paper. Besides, for possible circuit and/or computer realization, Matlab Simulink modeling of Z-type and G-type models is illustrated. Through illustrative examples, the efficacy and superiority of the proposed Z-type and G-type models for TVISR problem solving are verified and substantiated.

Recurrent implicit dynamics for online matrix inversion

A novel kind of recurrent implicit dynamics together with its electronic realization is proposed and exploited for real-time matrix inversion. Compared to conventional explicit neural dynamics, our proposed model in the form of implicit dynamics has the following advantages: (a) can coincide better with systems in practice; and (b) has higher abilities in representing dynamic systems. More importantly, our model can achieve superior convergence performance in comparison with the existing dynamic systems, specifically Gradient-based dynamics (GD) and recently-proposed Zhang dynamics (ZD). Theoretical analysis and computer simulation results substantiate the effectiveness and superior efficiency of our model for online matrix inversion.

Solving time-varying nonlinear inequalities using continuous and discrete-time Zhang dynamics

A special kind of neural dynamics, termed Zhang dynamics (ZD), is proposed, generalized and investigated for the online solution of time-varying scalar-valued nonlinear inequalities by following Zhang et al.’s design method. The continuous-time ZD (CTZD) model based on an exponent-type design formula can be guaranteed to exponentially converge to the time-varying solution set of the problem in an error-free manner. For potential hardware implementation on digital circuits, the corresponding discrete-time ZD (DTZD) model is generated through the well-known Euler forward difference rule. Newton-type algorithm is also developed for comparison purposes. In addition, the simplified CTZD (S-CTZD) and DTZD (S-DTZD) models are developed for solving static scalar-valued nonlinear inequalities. Numerical simulative examples further demonstrate and verify the efficacy of the ZD models for solving time-varying and static scalar-valued nonlinear inequalities. Besides, the DTZD model possesses the lower complexity and higher accuracy, as compared with the Newton-type algorithm.

Discrete-time ZD, GD and NI for solving nonlinear time-varying equations

A special class of neural dynamics called Zhang dynamics (ZD), which is different from gradient dynamics (GD), has recently been proposed, generalized, and investigated for solving time-varying problems by following Zhang et al.'s design method. In view of potential digital hardware implemetation, discrete-time ZD (DTZD) models are proposed and investigated in this paper for solving nonlinear time-varying equations in the form of $f(x,t)=0$ . For comparative purposes, the discrete-time GD (DTGD) model and Newton iteration (NI) are also presented for solving such nonlinear time-varying equations. Numerical examples and results demonstrate the efficacy and superiority of the proposed DTZD models for solving nonlinear time-varying equations, as compared with the DTGD model and NI.

ZG Controllers of z2g0 and z2g1 Types for Tracking Control of IPC Mathematical Model

Recently, the Zhang dynamics (ZD) and the gradient dynamics (GD) have been used for solving online problems, but they are usually investigated separately. In this paper, we firstly illustrate the ZD method and the GD method by employing them separately to solve the time-varying matrix inversion problem. Then, to solve the tracking-control problem of the mathematical model of the inverted pendulum on a cart (IPC) system, the ZD-based (i.e., z2g0) controller and the ZD-GD combined (i.e., z2g1) controller are designed. These two types of controllers with greatly simplified design procedure can achieve good performance in terms of tiny tracking error and quick response. The simulation results further substantiate the feasibility and superiority of the z2g0 controller and the smoother z2g1 controller for the output tracking of the mathematical model of the IPC system.

Solving for time-varying and static cube roots in real and complex domains via discrete-time ZD models

Different from conventional gradient-based neural dynamics, a special type of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and/or static (or termed, time-invariant) problems. The design of Zhang dynamics (ZD) is based on the elimination of an indefinite error function, instead of the elimination of a square-based positive (or at least lower-bounded) energy function usually associated with gradient dynamics (GD). In this paper, we generalize, propose and investigate the continuous-time ZD model and its discrete-time models in two situations (i.e., the time-derivative of the coefficient being known or unknown) for time-varying cube root finding, including the complex-valued continuous-time ZD model for finding cube roots in complex domain. In addition, to find the static scalar-valued cube root, a simplified continuous-time ZD model and its discrete-time model are generated. By focusing on such a static problem solving, Newton-Raphson iteration is found to be a special case of the discrete-time ZD model by utilizing the linear activation function and fixing the step-size value to be 1. Computer-simulation and testing results demonstrate the efficacy of the proposed ZD models (including real-valued ZD models and complex-valued ZD models) for time-varying and static cube root finding, as well as the link and new explanation to Newton-Raphson iteration.

Different Zhang functions leading to different Zhang-dynamics models illustrated via time-varying reciprocal solving

Along with neural dynamics (based on analog solvers) widely arising in scientific computation and optimization fields in recent decades which attracts extensive interest and investigation of researchers, a novel type of neural dynamics, called Zhang dynamics (ZD), has been formally proposed by Zhang et al. for the online solution of time-varying problems. By following Zhang et al.’s neural-dynamics design method, the ZD model, which is based on an indefinite Zhang function (ZF), can guarantee the exponential convergence performance for the online time-varying problems solving. In this paper, different indefinite Zhang functions, which can lead to different ZD models, are proposed and developed as the error-monitoring functions for the time-varying reciprocal problem solving. Additionally, for the goal of developing the floating-point processors or coprocessors for the future generation of computers, the MATLAB Simulink modeling and simulative verifications of such different ZD models are further presented for online time-varying reciprocal solving. The modeling results substantiate the efficacy of such different ZD models for time-varying reciprocal solving.

Time-varying square roots finding via Zhang dynamics versus gradient dynamics and the former's link and new explanation to Newton–Raphson iteration

Different from conventional gradient-based neural dynamics, a special class of neural dynamics have been proposed by Zhang et al. since 12 March 2001 for online solution of time-varying and static (or termed, time-invariant) problems (e.g., nonlinear equations). The design of Zhang dynamics (ZD) is based on the elimination of an indefinite error-function, instead of the elimination of a square-based positive or at least lower-bounded energy-function usually associated with gradient dynamics (GD) and/or Hopfield-type neural networks. In this paper, we generalize, develop, investigate and compare the continuous-time ZD (CTZD) and GD models for online solution of time-varying and static square roots. In addition, a simplified continuous-time ZD (S-CTZD) and discrete-time ZD (DTZD) models are generated for static scalar-valued square roots finding. In terms of such scalar square roots finding problem, the Newton iteration (also termed, Newton–Raphson iteration) is found to be a special case of the DTZD models (by focusing on the static-problem solving, utilizing the linear activation function and fixing the step-size to be 1). Computer-simulation results via a power-sigmoid activation function further demonstrate the efficacy of the ZD solvers for online scalar (time-varying and static) square roots finding, in addition to the DTZD's link and new explanation to Newton–Raphson iteration.

Continuous and discrete time Zhang dynamics for time-varying 4th root finding

Recently, a special class of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and/or static nonlinear equations. Different from eliminating a square-based positive error-function associated with gradient-based dynamics (GD), the design method of Zhang dynamics (ZD) is based on the elimination of an indefinite (unbounded) error-function. In this paper, for the purpose of online solution of time-varying 4th root, both continuous-time ZD (CTZD) and discrete-time ZD (DTZD) models are developed and investigated. In addition, power-sigmoid activation function is exploited in Zhang dynamics, which makes ZD models possess the property of superior convergence and better accuracy. To summarize generalization for possible widespread application, such approach is further extended to general time-varying nonlinear equations solving. Computer-simulation results demonstrate the efficacy of the ZD models for finding online time-varying 4th root and solving general time-varying equations.

Phase Separation Driven by a Fluctuating Two-Dimensional Self-Affine Potential Field

We study phase separation in a system of hard-core particles driven by a fluctuating two-dimensional self-affine potential landscape which evolves through Kardar–Parisi–Zhang (KPZ) dynamics. We find that particles tend to cluster together on a length scale which grows in time. The final phase-separated steady state is characterized by an unusual cusp singularity in the scaled correlation function and a broad distribution for the order parameter. Unlike the one-dimensional case studied earlier, the cluster-size distribution is asymmetric between particles and holes, reflecting the broken reflection symmetry of the KPZ dynamics, and has a contribution from an infinite cluster in addition to a power law part. A study of the surface in terms of coarse-grained depth variables helps understand many of these features.

Coating and active surface of Ni–Zn alloys studied by atomic force microscopy

The topology analysis of a real material surface undertaken here runs into a new approach of the active surface parameter, S a, based on the scaling behaviour of the interface width, w. This approach is experimentally illustrated from the coating of a copper substrate with 2 μm-thick Ni–Zn alloy electodeposits investigated varying their growth rate, r, located below 0.50 μm/s. The imaging of their surface by atomic force microscopy (AFM) enables the following of their Hurst exponent, α, and their correlation length, L c, versus r. It is shown that their interface growth may proceed from the Kardar–Parisi–Zhang (KPZ) dynamics, the restricted surface diffusion effect increasing with the increase of r. Both, α and L c stabilise from a critical rate r c′=0.16 μm/s. The study evidences a peculiar surface microstructure of the Ni–Zn alloy favourable to catalytic activity at another critical point r c″=0.08 μm/s which corresponds to a Ni (80 at.%)-rich specimen leading to the maximal active surface obtained S a=132 nm.