ZD and ZG controllers for explicit and implicit tracking of pendulum with singularity finally conquered

Zhang dynamics (ZD) and gradient dynamics (GD) are both powerful methods. Based on a pendulum system, this paper investigates both of the explicit and implicit tracking control using the ZD method. For solving the singularity-containing implicit tracking problems, this paper overcomes the singularities by using the ZD method in combination with the GD method (i.e., the ZG method). Analyses and simulations of an explicit tracking example and two implicit tracking examples show the superiority of the ZD and ZG methods.