Zhang Dynamics Research Articles

Time-varying generalized tensor eigenanalysis via Zhang neural networks

Eigenanalysis of matrices with parameters has a long history. When the parameter is time, or the matrix is time-dependent, the Zhang neural networks for the time-varying matrix problem have been developed in recent years. Motivated by tensor generalized eigenvalues and the Zhang dynamics method, we investigate the time-varying eigenpair of symmetric tensors. A continuous Zhang dynamics model is given to compute the tensor eigenpairs, such as the H- and Z-eigenpairs. In order to accelerate the convergence, a modified Zhang dynamics model is also presented. Moreover, the generalized tensor/matrix eigenpairs could also be computed by the two proposed models. Theoretical analysis of the convergence and robustness are provided. We also test some numerical examples which illustrate that the two proposed models are effective.

Four discrete-time ZD algorithms for dynamic linear matrix-vector inequality solving

Recently, a typical neural dynamics called Zhang dynamics (ZD) has been developed for online solution of dynamic linear matrix-vector inequality. This paper show a summary result by presenting the discrete-time forms of such a ZD for dynamic linear matrix-vector inequality solving. Specifically, by exploiting four different kinds of Taylor-type difference rules, the resultant discrete-time ZD (DTZD) algorithms, which are called respectively the DTZD-I, DTZD-II, DTZD-III, and DTZD-IV algorithms, are established. These algorithms can achieve excellent computational performance in solving dynamic linear matrix-vector inequality. The theoretical and numerical results are presented to further substantiate the efficacy of the presented four DTZD algorithms.

Research on a New Singularity-Free Controller for Uncertain Lorenz System

It is a challenge problem to stably control the well-known Lorenz system with uncertain parameters because of its nonlinearity and singularity. In this paper, by combining Zhang dynamics (ZD) and gradient-algorithm, a novel Zhang-Gradient (ZG) controller is designed and developed for solving the controlling problem of the uncertainty Lorenz system. In order to improve computing efficiency, the stochastic parallel gradient descent algorithm is also introduced to perform an incremental adjustment of the unknown parameters for the uncertain Lorenz system. The presented theoretical analysis in this paper shows that our such presented method could conquer the possible singularity which is a difficult problem in typical backstepping controller design. The computer simulation results exhibit that, the controlled system can be stable globally and the tracking error converges to zero asymptotically, which are further demonstrate the effectiveness and feasibility of our presented ZG controller.

The Application of Noise-Tolerant ZD Design Formula to Robots’ Kinematic Control via Time-Varying Nonlinear Equations Solving

Recently, a new formula with noise-tolerant capability has been designed by Zhang et al. , and the resultant Zhang dynamics (ZD) models have been developed to solve different types of time-varying problems. Based on previous research, this paper presents and investigates the application of such a noise-tolerant ZD design formula to kinematic control of redundant robot manipulators via time-varying nonlinear equations solving. Specifically, by exploiting this ZD design formula to solve the system of time-varying nonlinear kinematic equations involved in robot control, the redundancy-resolution scheme is established. Such a scheme contains the proportional, integral, and derivative information of the desired Cartesian path of the robot end-effector and can thus be viewed as a nonlinear proportional-integral-derivative controller for redundant robot manipulators. Then, theoretical results are given to show that the redundancy-resolution scheme has the capability of noise suppressing. Simulation results based on a four-link planar robot manipulator and a PA10 robot manipulator with zero, constant, and bounded time-varying noises further substantiate the efficacy and superiority of the redundancy-resolution scheme, and show the application prospect of the noise-tolerant ZD design formula.

A Varying-Parameter Convergent Neural Dynamic Controller of Multirotor UAVs for Tracking Time-Varying Tasks

To stably control the position and attitude angles of an unmanned aerial vehicle (UAV), a varying-parameter convergent neural dynamic (VP-CND) method is proposed and applied. First, the dynamic models of multirotor UAVs are presented. Second, to meet the requirements of high accuracy and real-time control, a VP-CND method is proposed based on an error function to derive the position and attitude angle controllers. The existing fixed-parameter CND methods (e.g., the triple-Zhang dynamics or the Zhang dynamics and gradient dynamics) and the corresponding controllers are presented, and their limitations are analyzed. The proposed VP-CND control method not only can track time-varying target values but also possesses super-exponential convergence performance. Third, simulation comparisons verify the effectiveness, stability, and fast convergence of the VP-CND controllers.

Zhang Dynamics based Tracking Control of Knee Exoskeleton with Timedependent Inertial and Viscous Parameters

Knee exoskeleton plays an important role in robot-assisted rehabilitation for impaired pilots to restore their motor functionality of lower extremity through producing external movement compensation. Tracking control of knee exoskeleton often encounters time-dependent (time-varying) issues reflected in its dynamic behaviors. In many applications, inertial and viscous parameters of knee exoskeletons are measured to be time-dependent due to unexpected mechanical vibrations and contact interactions, which increases difficultly of accurate control of knee exoskeleton to follow desired joint angle trajectories. This paper proposes a novel control strategy for controlling knee exoskeleton with time-dependent (time-varying) inertial and viscous coefficients. Such controller is designed based on Zhang dynamics (ZD) method and utilizes twice Zhang function (ZF) so as to make the tracking error of joint angle exponentially converge to zero. Illustrative simulation examples and experimental validation are presented to show efficiency of this type of controller based on ZD method. Comparisons with gradient dynamic (GD) approach are also presented to demonstrate superiority of ZD-type control strategy for tracking joint angle of knee exoskeleton.

From Davidenko Method to Zhang Dynamics for Nonlinear Equation Systems Solving

The solving of nonlinear equation systems (e.g., complex transcendental dispersion equation systems in waveguide systems) is a fundamental topic in science and engineering. Davidenko method has been used by electromagnetism researchers to solve time-invariant nonlinear equation systems (e.g., the aforementioned transcendental dispersion equation systems). Meanwhile, Zhang dynamics (ZD), which is a special class of neural dynamics, has been substantiated as an effective and accurate method for solving nonlinear equation systems, particularly time-varying nonlinear equation systems. In this paper, Davidenko method is compared with ZD in terms of efficiency and accuracy in solving time-invariant and time-varying nonlinear equation systems. Results reveal that ZD is a more competent approach than Davidenko method. Moreover, discrete-time ZD models, corresponding block diagrams, and circuit schematics are presented to facilitate the convenient implementation of ZD by researchers and engineers for solving time-invariant and time-varying nonlinear equation systems online. The theoretical analysis and results on Davidenko method, ZD, and discrete-time ZD models are also discussed in relation to solving time-varying nonlinear equation systems.

Design, Analysis, and Representation of Novel Five-Step DTZD Algorithm for Time-Varying Nonlinear Optimization.

Continuous-time and discrete-time forms of Zhang dynamics (ZD) for time-varying nonlinear optimization have been developed recently. In this paper, a novel discrete-time ZD (DTZD) algorithm is proposed and investigated based on the previous research. Specifically, the DTZD algorithm for time-varying nonlinear optimization is developed by adopting a new Taylor-type difference rule. This algorithm is a five-step iteration process, and thus, is referred to as the five-step DTZD algorithm in this paper. Theoretical analysis and results of the proposed five-step DTZD algorithm are presented to highlight its excellent computational performance. The geometric representation of the proposed algorithm for time-varying nonlinear optimization is also provided. Comparative numerical results are illustrated with four examples to substantiate the efficacy and superiority of the proposed five-step DTZD algorithm for time-varying nonlinear optimization compared with the previous DTZD algorithms.

Nonconvex projection activated zeroing neurodynamic models for time-varying matrix pseudoinversion with accelerated finite-time convergence

Zeroing dynamics (ZD, or termed Zhang dynamics after its inventor), being a special type of neurodynamic methodology, has shown powerful abilities to solve a great variety of time-varying problems with monotonically increasing odd activation functions. In this paper, two limitations of existing ZD are identified, i.e., the convex restriction on projection operations of activation functions and the low convergence speed with relatively redundant formulations on activation functions. This work breaks them by proposing modified ZD models, allowing nonconvex sets for projection operations in activation functions and possessing accelerated finite-time convergence. Theoretical analyses reveal that the proposed ZD models are of global stability with timely convergence. Finally, illustrative simulation examples, including an application to the motion generation of a robot manipulator, are provided and analyzed to substantiate the efficacy and superiority of the proposed ZD models for real-time varying matrix pseudoinversion.

ZD Method Based Nonlinear and Robust Control of Agitator Tank

AbstractTo improve industrial efficiency, an agitator tank system should have not only a short response time, but also produce reagents with accurate concentration and moderate liquid level. This paper presents a new method called Zhang dynamics (ZD) to perform tracking control, that is, to make the concentration and liquid level of the agitator tank converge to the desired trajectories. Two controllers for tracking control of an agitator tank system are designed based on ZD method. In addition, the robustness of the agitator tank system equipped with ZD controllers is investigated. Theoretical analyses on the convergence performance and robustness of the agitator tank system equipped with ZD controllers are presented. To substantiate the effectiveness of the ZD method for tracking control of the agitator tank system, we perform simulations on three different groups of tracking trajectories, which reflects the corresponding application conditions in the chemical industry. To test the robustness of the agitator tank system equipped with ZD controllers, additional simulations with different disturbances added in the agitator tank system and the ZD controllers are performed. Simulation results validate the feasibility and effectiveness of the ZD method for tracking control of the agitator tank system.

Tracking control of knee exoskeleton system with time-dependent inertial and viscous parameters

Knee exoskeleton plays an important role in robot-assisted rehabilitation for impaired pilots to restore their motor functionality of lower extremity through producing external movement compensation. Tracking control of knee exoskeleton often encounters time-dependent (time-varying) issues reflected in its dynamic behaviors. In many applications, inertial and viscous parameters of knee exoskeletons are measured to be time-dependent due to unexpected mechanical vibrations and contact interactions, which increases difficultly of accurate control of knee exoskeleton to follow desired joint angle trajectories. This paper proposes a novel control strategy for controlling knee exoskeleton with time-dependent (time-varying) inertial and viscous coefficients. Such controller is designed based on Zhang dynamics (ZD) method so as to make the tracking error of joint angle exponentially converge to zero. Illustrative examples are presented to show efficiency of this type of controller based on ZD method. Comparisons with gradient dynamic (GD) approach are also presented to demonstrate superiority of ZD-type control strategy for tracking joint angle of knee exoskeleton.

Design and analysis of two discrete-time ZD algorithms for time-varying nonlinear minimization

Nonlinear minimization, as a subcase of nonlinear optimization, is an important issue in the research of various intelligent systems. Recently, Zhang et al. developed the continuous-time and discrete-time forms of Zhang dynamics (ZD) for time-varying nonlinear minimization. Based on this previous work, another two discrete-time ZD (DTZD) algorithms are proposed and investigated in this paper. Specifically, the resultant DTZD algorithms are developed for time-varying nonlinear minimization by utilizing two different types of Taylor-type difference rules. Theoretically, each steady-state residual error in the DTZD algorithm changes in an O(τ 3) manner with τ being the sampling gap. Comparative numerical results are presented to further substantiate the efficacy and superiority of the proposed DTZD algorithms for time-varying nonlinear minimization.

ZD, ZG and IOL Controllers and Comparisons for Nonlinear System Output Tracking with DBZ Problem Conquered in Different Relative‐Degree Cases

AbstractThis paper considers the output tracking control of general‐form single‐input single‐output (SISO) nonlinear system, which may encounter the problem of division by zero (DBZ). First, via the Zhang dynamics (ZD) method, a ZD controller is proposed. Then, based on the ZD controller with the aid of gradient dynamics (GD) method, a Zhang‐gradient (ZG) controller is proposed. For comparison, the conventional input‐output linearization (IOL) controller is presented. The ZD, ZG and IOL controllers are compared in different relative‐degree cases (i.e., the standard relative‐degree case, the loose relative‐degree case and the DBZ relative‐degree case). Note that the ZG controller is valid in three relative‐degree cases, while the ZD and IOL controllers are valid only in the standard relative‐degree case and the loose relative‐degree case. In addition, performances of ZD and ZG controllers are guaranteed via theoretical analyses and computer simulations for the output tracking of general‐form nonlinear system with the DBZ problem conquered.

Theoretical analysis, numerical verification and geometrical representation of new three-step DTZD algorithm for time-varying nonlinear equations solving

To solve time-varying nonlinear equations, Zhang et al. have developed a one-step discrete-time Zhang dynamics (DTZD) algorithm with O(τ2) error pattern, where τ denotes the sampling gap. In this paper, by exploiting the Taylor-type difference rule, a new three-step DTZD algorithm with O(τ3) error pattern is proposed and investigated for time-varying nonlinear equations solving. Note that such an algorithm can achieve better computational performance than the one-step DTZD algorithm. As for the proposed three-step DTZD algorithm, theoretical results are given to show its excellent computational property. Comparative numerical results further substantiate the efficacy and superiority of the proposed three-step DTZD algorithm for solving time-varying nonlinear equations, as compared with the one-step DTZD algorithm. Besides, the geometric representation of the proposed three-step DTZD algorithm is provided for time-varying nonlinear equations solving.

Division by zero, pseudo-division by zero, Zhang dynamics method and Zhang-gradient method about control singularity conquering

ABSTRACTIn this paper, the division-by-zero (DBO) problem in the field of nonlinear control, which is traditionally termed the control singularity problem (or specifically, controller singularity problem), is investigated by the Zhang dynamics (ZD) method and the Zhang-gradient (ZG) method. According to the impact of the DBO problem on the state variables of the controlled nonlinear system, the concepts of the pseudo-DBO problem and the true-DBO problem are proposed in this paper, which provide a new perspective for the researchers on the DBO problems as well as nonlinear control systems. Besides, the two classes of DBO problems are solved under the framework of the ZG method. Specific examples are shown and investigated in this paper to illustrate the two proposed concepts and the efficacy of the ZG method in conquering pseudo-DBO and true-DBO problems. The application of the ZG method to the tracking control of a two-wheeled mobile robot further substantiates the effectiveness of the ZG method. In addition, the ZG method is successfully applied to the tracking control of a pure-feedback nonlinear system.

Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization

Online solution of time-varying nonlinear optimization problems is considered an important issue in the fields of scientific and engineering research. In this study, the continuous-time derivative (CTD) model and two gradient dynamics (GD) models are developed for real-time varying nonlinear optimization (RTVNO). A continuous-time Zhang dynamics (CTZD) model is then generalized and investigated for RTVNO to remedy the weaknesses of CTD and GD models. For possible digital hardware realization, a discrete-time Zhang dynamics (DTZD) model, which can be further reduced to Newton-Raphson iteration (NRI), is also proposed and developed. Theoretical analyses indicate that the residual error of the CTZD model has an exponential convergence, and that the maximum steady-state residual error (MSSRE) of the DTZD model has an O(ź2) pattern with ź denoting the sampling gap. Simulation and numerical results further illustrate the efficacy and advantages of the proposed CTZD and DTZD models for RTVNO.

A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients

The recently-proposed Zhang dynamics (ZD) has been proven to achieve success for solving the linear-equality constrained time-varying quadratic program ideally when time goes to infinity. The convergence performance is a significant improvement, as compared to the gradient-based dynamics (GD) that cannot make the error converge to zero even after infinitely long time. However, this ZD model with the suggested activation functions cannot reach the theoretical time-varying solution in finite time, which may limit its applications in real-time calculation. Therefore, a nonlinearly-activated neurodynamic model is proposed and studied in this paper for real-time solution of the equality-constrained quadratic optimization with nonstationary coefficients. Compared with existing neurodynamic models (specifically the GD model and the ZD model) for optimization, the proposed neurodynamic model possesses the much superior convergence performance (i.e., finite-time convergence). Furthermore, the upper bound of the finite convergence time is derived analytically according to Lyapunov theory. Both theoretical and simulative results verify the efficacy and superior of the nonlinearly-activated neurodynamic model, as compared to these of the GD and ZD models.

Zhang-Gradient Controllers for Tracking Control of Multiple-Integrator Systems

In this paper, the tracking-control problem of multiple-integrator (MI) systems is considered and investigated by combining Zhang dynamics (ZD) and gradient dynamics (GD). Several novel types of Zhang-gradient (ZG) controllers are proposed for the tracking control of MI systems (e.g., triple-integrator (TI) systems). As an example, the design processes of ZG controllers for TI systems with a linear output function (LOF) and/or a nonlinear output function (NOF) are presented. Besides, the corresponding theoretical analyses are elaborately given to guarantee the convergence performance of both z3g0 controllers (ZG controllers obtained by utilizing the ZD method thrice) and z3g1 controllers (ZG controllers obtained by utilizing the ZD method thrice and the GD method once) for TI systems. Numerical simulations concerning the tracking control of MI systems with different types of output functions are further performed to substantiate the feasibility and effectiveness of ZG controllers for tracking-control problems solving. Besides, comparative simulation results of the tracking control for MI systems with NOFs (e.g., y=cos(x1), y=x12+x22) substantiate that controllers of zmg1 type can resolve the singularity problem effectively with m being the times of using the ZD method.

Challenging simulation practice (failure and success) on implicit tracking control of double-integrator system via Zhang-gradient method

Zhang-gradient (ZG) method is a combination of Zhang dynamics (ZD) and gradient dynamics (GD) methods which are two powerful methods for online time-varying problems solving. ZG controllers are designed using the ZG method to solve the tracking control problem of a certain system. In this paper, the design process of the ZG controllers with explicit as well as implicit tracking control of the double-integrator system is presented in detail. In addition, the corresponding computer simulations are conducted with different values of the design parameter λ to illustrate the effectiveness of ZG controllers. However, even though the ZG controllers are powerful, there is still a challenge in the simulation practice. Specifically, different settings of simulation options in MATLAB ordinary differential equation (ODE) solvers may lead to different simulation results (e.g., failure and success). For better comparison, the successful and failed simulation results are both presented. The differences in simulation results remind us to pay more attention to MATLAB defaults and options when we conduct such simulations.

Singularity‐conquering tracking control of a class of chaotic systems using Zhang‐gradient dynamics

This study investigates the tracking-control problems of the Lorenz, Chen and Lu chaotic systems. Note that the input–output linearisation method cannot solve these tracking-control problems because of the existence of singularities, at which such chaotic systems fail to have a well-defined relative degree. By combining Zhang dynamics and gradient dynamics, an effective controller-design method, termed Zhang-gradient (ZG) method, is proposed for tracking control of the three chaotic systems. This ZG method, with singularities conquered, is capable of solving the tracking-control problems of the chaotic systems. Both theoretical analyses and simulative verifications substantiate that the tracking controllers based on the ZG method can achieve satisfactory tracking accuracy and successfully conquer singularities encountered during the tracking-control process.