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

Abstract

Abstract: Zhang dynamics (ZD) and gradient dynamics (GD) are bothpowerful methods. Based on a pendulum system, this paperinvestigates both of the explicit and implicit tracking controlusing the ZD method. For solving the singularity-containingimplicit tracking problems, this paper overcomes the singular-ities by using the ZD method in combination with the GDmethod (i.e., the ZG method). Analyses and simulations ofan explicit tracking example and two implicit tracking exam-ples show the superiority of the ZD and ZG methods. Keywords: Zhang dynamics (ZD); Gradient dynamics (GD); ZG (Zhang-gradient) controller; Pendulum;Explicit and implicit tracking 1. Introduction and pendulum system As two powerful methods, Zhang dynamics (ZD) and gradi-ent dynamics (GD) have recently been studied and compared[1]–[4]. While the GD method is based on a scalar-valuednonnegative energy function and has been designed original-ly for solving constant (or to say, time-invariant) problems, theZD method is based on an indefinite matrix/vector-valued er-ror function (termed Zhang function or Zhangian) and has beenproposed since 2001 for the online solution of various time-varying problems. In addition, the GD method is associatedwith explicit dynamics, while the ZD method is generally de-picted in implicit dynamics [1]–[4]. As shown later in thispaper, by using the ZD method in combination with the GDmethod [i.e., the ZG (Zhang-gradient)method], singularities inthe implicit tracking control problems can be overcome.Tracking control is widely encountered in engineering [5]-[9]. Traditionally and generally speaking, the tracking-controlproblem of a system is to design a controller in terms of theinput u(t)for the system such that the actual outputy(t)tracksthe desired output y