Dechao Chen

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

ABSTRACT In 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.

ZG Control for Ship Course Tracking with Singularity Considered and Solved

Zhang dynamics (ZD) and gradient dynamics (GD) are both effective methods for online problems solving. By combining ZD and GD methods, an innovative ZG (Zhang-gradient) control method is thus proposed and investigated in this paper, which is applied to ship course tracking for the first time. Firstly, for a constant parameter setting, we design a ZD-based controller to solve the tracking-control problem of a ship course system with no singularity appearing. Then, for a time-varying parameter setting, the ZG method is applied generally to solve the singularity-containing tracking-control problem of such a system. Simulation results further demonstrate and verify the feasibility and superiority of the unified ZG method in fulfilling the tracking-control task while conquering the singularity problem for the ship course system.

ZG controllers for output tracking of nonlinear mass-spring-damper mechanical system with division-by-zero problem solved

Recently, two classes of dynamical methods, i.e., Zhang dynamics (ZD) and gradient dynamics (GD), have been widely investigated individually and comparatively for online time-varying problems solving. In this paper, by combining ZD and GD, an innovative method called Zhang-gradient (ZG) method is proposed for tracking control of a nonlinear mass-spring-damper (MSD) mechanical system. Simulation results substantiate the feasibility and effectiveness of the combined ZG method for handling both the explicit and implicit tracking-control problems. Especially, the z1g1 controller based on the ZG method for the implicit tracking control is also capable of conquering the division-by-zero problem.

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. The design processes of the Zhang-gradient (ZG) controllers with explicit and implicit tracking control of the double-integrator (DI) system are presented.The examples of static and time-varying systems are investigated to show the effectiveness of ZG controllers for the tracking control problem solving.It is shown that different settings of simulation options in ordinary differential equation (ODE) solvers may lead to different simulation results.Successful and failed simulation practice helps establish the referential rules for users to choose appropriate settings.

GD-aided IOL (input–output linearisation) controller for handling affine-form nonlinear system with loose condition on relative degree

ABSTRACT Input–output linearisation (IOL) may encounter a singularity problem when applied to the tracking control of affine-form nonlinear system (AFNS), which may not have a well-defined relative degree. The singularity problem has occurred in the area of control for decades. In this paper, we incorporate the gradient dynamics (GD) into IOL, which leads to the GD-aided IOL method to solve the singularity problem, with the proposition of the loose condition on relative degree. Moreover, detailed theoretical analyses on tracking-error bound and convergence performance of the corresponding GD-aided IOL controller are presented. Simulations and comparisons substantiate that the proposed GD-aided IOL method is capable of completing the tracking-control task and conquering the singularity encountered in the AFNS.

Twice-Pruning Aided WASD Neuronet of Bernoulli-Polynomial Type with Extension to Robust Classification

This paper proposes a novel multi-input Bernoulli-polynomial neuronet (MIBPN) on the basis of function approximation theory. The MIBPN is trained by a weights-and-structure-determination (WASD) algorithm with twice pruning (TP). The WASD algorithm can obtain the optimal weights and structure for the MIBPN, and overcome the weaknesses of conventional BP (back-propagation) neuronets such as slow training speed and local minima. With the TP technique, the neurons of less importance in the MIBPN are pruned for less computational complexity. Furthermore, this MIBPN can be extended to a multiple input multiple output Bernoulli-polynomial neuronet (MIMOBPN), which can be applied as an important tool for classification. Numerical experiment results show that the MIBPN has outstanding performance in data approximation and generalization. Besides, experiment results based on the real-world classification data-sets substantiate the high accuracy and strong robustness of the MIMOBPN equipped with the proposed WASD algorithm for classification. Finally, the twice-pruning aided WASD neuronet of Bernoulli-polynomial type in the forms of MIBPN and MIMOBPN is established, together with the effective extension to robust classification.

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.

Solving for ZGI via ZNN and discrete-time algorithms with application to robot control

In this paper, by defining different error functions, two continuous-time models of Zhang neural net (ZNN) termed the ZNN-I and ZNN-II models are developed and investigated to solve for time-varying generalized inverse (i.e., Zhang generalized inverse, ZGI). In addition, comparing the ZNN models with the dynamic system proposed by Getz and Marsden (G-M), we show that such a G-M dynamic system can also be derived from the ZNN models. For the purpose of potential hardware (e.g., digital circuit or computer) implementation, the discrete-time algorithms (depicted by systems of difference equations) of the presented ZNN-I model are proposed and investigated in two situations, i.e., the time-derivative of the time-varying coefficient matrix being known and unknown. Simulative and numerical results further demonstrate the efficacy of the presented ZNN models for ZGI solving. Moreover, these ZNN models are applied to the kinematic control of a three-link planar robot manipulator via computing ZGI, showing their application prospects.

Optimal parameter value of Zhang equivalence for MVN redundancy resolution at velocity and acceleration levels

The equivalent relationship of two general scheme formulations at joint velocity and joint acceleration levels is established in this paper for robotic redundancy resolution, i.e., the so-called Zhang equivalence (ZE). As one representative case, ZE of the minimum velocity norm (MVN) type (i.e., MVN-type ZE) is further investigated for redundancy resolution. Moreover, computer simulations together with illustrative numerical experiments based on the widely-used robot manipulator PUMA560 show the phenomenon that an optimal equivalence value of the equivalent parameter exists within effective range from two different perspectives of equivalence; and then reveal the optimal value of the equivalent parameter in two situations from such perspectives.