Zhang Neural Dynamics Research Articles

Novel Snap-Layer MMPC Scheme via Neural Dynamics Equivalency and Solver for Redundant Robot Arms With Five-Layer Physical Limits.

To obtain smoother kinematic control of minimum motion, a novel snap-layer minimum motion scheme, otherwise known as the minimum motion planning and control (MMPC) scheme for redundant robot arms, is proposed for the first time in this study. With the primary task of tracking planned paths and the consideration of satisfying five-layer physical limits, the snap-layer MMPC problem is transformed into a quadratic programming (QP) problem. Five-layer physical limits include angle-layer, velocity-layer, acceleration-layer, jerk-layer, and snap-layer limits, which are all considered and then transformed into a unified-layer bounded constraint through Zhang neural dynamics (ZND) equivalency. Furthermore, the snap-layer performance index and equation constraint are derived by utilizing the ZND formula. Therefore, the proposed snap-layer MMPC scheme is formulated as a standard QP that can avoid the potential physical damage of redundant robot arms. The snap-layer projection neural dynamics (PND) solver is presented and used to acquire the neural solution of the QP. Simulation results on a 6-degrees-of-freedom (DOF) planar redundant robot arm are presented to substantiate the effectiveness and superiority of the proposed snap-layer MMPC scheme by comparing it with the jerk-layer MMPC scheme and the minimum snap norm (MSN) scheme.

Robust Neural Dynamics Method for Redundant Robot Manipulator Control With Physical Constraints

Redundant robot manipulators play a significant role in modern industry. In this paper, we propose a solution scheme to the trajectory tracking problem of redundant robot manipulator with physical constraints through Zhang neural dynamics method. Such problem is integrated into a time-varying system consisting of time-varying nonlinear equation (TVNE) and time-varying linear inequality (TVLI) and solved online by the varying-parameter Zhang neural dynamics (VPZND) model. It is ensured that the redundant robot manipulator can still perform the tracking task perfectly under the coexistence of time-varying bounded noise and physical constraints. Theoretical analysis proves that this VPZND model also has an explicit fixed convergence time. Numerical experiments confirm the feasibility of our VPZND model for TVLI. The trajectory tracking problem of the redundant robot manipulator with six or three degrees of freedom under the dual influence of physical constraints and noise is perfectly solved by the VPZND model, which is enough to verify its practical value.

Time-varying Schur decomposition via Zhang neural dynamics

By applying Zhang neural dynamics method, this study proposed, analyzed, and investigated a continuous-time model for solving the time-varying Schur decomposition (SD) problem. The proposed model is an explicit dynamics model that utilizes the time derivative information. The theoretical analysis of the proposed model is presented to verify its convergence and efficiency in solving the time-varying SD problem. Furthermore, the proposed model is used to perform the SD of three time-varying matrices with different dimensions. Results confirm the effectiveness of the proposed model.

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.

Five-instant type discrete-time ZND solving discrete time-varying linear system, division and quadratic programming

Discrete time-varying linear system (LS) is a fundamental topic in science and engineering. However, conventional methods essentially designed for time-invariant LS generally assume that LS is time-invariant during a small time interval (i.e., sampling gap) for solving time-varying LS. This assumption quite limits their precision because of the existing of lagging errors. Discarding this assumption, Zhang neural dynamics (ZND) method improves the precision for LS solving, which is a great alternative for the solving of discrete time-varying problems. Note that precision solutions to discrete time-varying problems depend on discretization formulas. In this paper, we propose a new ZND model to solve the discrete time-varying LS. The discrete time-varying division is a special case of discrete time-varying LS with the solution being a scalar while it is usually studied alone. Considering the above inner connection, we further propose a special model for solving the discrete time-varying division. Moreover, as an application of discrete time-varying LS, the discrete time-varying quadratic programming (QP) subject to LS is also studied. The convergence and precision of proposed models are guaranteed by theoretical analyses and substantiated by numerous numerical experiments.

Discrete time-variant nonlinear optimization and system solving via integral-type error function and twice ZND formula with noises suppressed

In this paper, by using integral-type error function and twice zeroing neural-dynamics (or termed, Zhang neural-dynamics, ZND) formula, continuous-time advanced zeroing neural-dynamics (CT-AZND) model is proposed for solving the continuous time-variant nonlinear optimization problem. Furthermore, a discrete-time advanced zeroing neural-dynamics (DT-AZND) model is first proposed, analyzed, and investigated for solving the discrete time-variant nonlinear optimization (DTVNO) problem. Theoretical analyses show that the proposed DT-AZND model is convergent, and its steady-state residual error has an \(O(g^3)\) pattern with g denoting the sampling gap. In addition, in the presence of various kinds of noises, the proposed DT-AZND model possesses advantaged performance. In detail, the proposed DT-AZND model converges toward the time-variant theoretical solution of the DTVNO problem with \(O(g^3)\) residual error in the presence of an arbitrary constant noise and has excellent ability to suppress linear-form time-variant noise and bounded random noise. Illustrative numerical experiments further substantiate the efficacy and advantage of the proposed DT-AZND model for solving the DTVNO problem.

Tracking control of modified Lorenz nonlinear system using ZG neural dynamics with additive input or mixed inputs

The tracking-control problem of a special nonlinear system (i.e., the extension of a modified Lorenz chaotic system) with additive input or the mixture of additive and multiplicative inputs is considered in this paper. It is worth pointing out that, with the parameters fixed at some particular values, the modified Lorenz nonlinear system degrades to the modified Lorenz chaotic system. Note that, due to the existence of singularities at which the nonlinear system fails to have a well-defined relative degree, the input–output linearization method and the backstepping design technique cannot solve the tracking-control problem. By combining Zhang neural dynamics and gradient neural dynamics, a new effective controller-design method, termed Zhang-gradient (ZG) neural dynamics, is proposed for the tracking control of the modified Lorenz nonlinear system. With singularities conquered, this ZG neural dynamics is able to solve the tracking-control problem of the modified Lorenz nonlinear system via additive input or mixed inputs (i.e., the mixture of additive and multiplicative inputs). Both theoretical analyses and simulative verifications substantiate that the tracking controllers based on the ZG neural dynamics with additive input or mixed inputs not only achieve satisfactory tracking accuracy but also successfully conquer the singularities encountered during the tracking-control process. Moreover, the applications to the synchronization, stabilization and tracking control of other nonlinear systems further illustrate the effectiveness and advantages of the ZG neural dynamics.

A nonlinearly activated neural dynamics and its finite-time solution to time-varying nonlinear equation

This paper proposes a nonlinearly activated neural dynamics to solve time-varying nonlinear equations in real time. Different from most existing neural dynamics for solving time-varying nonlinear equations, the proposed neural dynamics can converge in finite time. In addition, the upper bound of convergence time is estimated analytically in theory. Simulations are performed to evaluate the performance of the proposed neural dynamics, which substantiate the effectiveness and superiority of the finite-time convergent neural dynamics for solving time-varying nonlinear equations in real time, as compared with the conventional gradient-based neural dynamics and the recently proposed Zhang neural dynamics.

Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation

Different from gradient-based neural dynamics, a special kind of recurrent neural dynamics has recently been proposed by Zhang et al. for solving online time-varying problems. Such a recurrent neural dynamics is designed based on an indefinite error-monitoring function instead of a usually norm- or square-based energy function. In addition, Zhang neural dynamics (ZND) are depicted generally in implicit dynamics, whereas gradient-based neural dynamics (GND) are associated with explicit dynamics. In this paper, we generalize the ZND design method to solving online nonlinear time-varying equations in the form of f (x, t) = 0. For comparative purposes, the GND model is also employed for such time-varying equations’ solving. Computer-simulation results via power-sigmoid activation functions substantiate the theoretical analysis and efficacy of the ZND model for solving online nonlinear time-varying equations.