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

Abstract

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.

ZG Tracking Control of MI Systems

“…Those researches are meaningful and significant for the deep investigation of MI systems. Comparatively, in this chapter [7], by the presented method. Besides, a relatively different, simple and smooth form of controller is investigated, which can effectively achieve the tracking control of MI systems and can elegantly solve the DBZ problem.…”

“…As a mathematical abstraction or idealization, linear systems, which form the cornerstone of much of modern day electrical engineering, find important applications in nature, electrical engineering and economics [6,7]. Besides, the tracking-control problem is very important and has appeared in many applications [8][9][10].…”

Optimal zeroing dynamics with applications to control of serial and parallel manipulators

“…Note that t refers to the variable of time and the error vector e ( t ), which is determined by r d ( t ) and r a ( t ) in real time, is time varying. To guarantee the exponential convergence of the error vector e ( t ), e ( t ) and its time derivative are designed to meet the condition of zeroing dynamics() where Ψ(·) refers to an arbitrary monotonically increasing odd function. In general, linear function, bipolar sigmoid function, power function, and power‐sigmoid function are 4 popular activation functions widely applied in zeroing dynamics .…”

“…Moreover, several attempts to apply zeroing dynamics to control systems, eg, robot manipulators, have been conducted showing the favorable performance of the convergence as observable from the corresponding results. () For example, the time‐varying inverse kinematics problem of wheeled mobile manipulators has been solved effectively with zeroing dynamics . The combination of zeroing dynamics and gradient dynamics (GD), called zeroing GD, is able to address the singularity difficulties of various control systems.…”

“…The combination of zeroing dynamics and gradient dynamics (GD), called zeroing GD, is able to address the singularity difficulties of various control systems. () This motivates us to consider the combination of optimal control theory and zeroing dynamics to construct a hybrid controller for robot manipulators. In this paper, we incorporate optimal control theory in the design of zeroing dynamics and propose a new motion control method called optimal zeroing dynamics (OZD), which can obtain high convergence accuracy and low sensitivity to disturbances simultaneously.…”