Design and illustration of ZG controllers for linear and nonlinear tracking control of double-integrator system

ZG (Zhang-gradient) method is a combination of Zhang dynamics (ZD) and gradient dynamics (GD) methods. Note that ZD is a powerful method for online time-varying problems solving; and that GD is a typical method for time-invariant problems solving and now generalized for time-varying problems solving. In this paper, ZG method is exploited for the design of ZG controllers for the double-integrator (DI) system, with design processes detailed. In particular, the ZG controllers with GD included can conquer the notorious division-by-zero problem. Furthermore, computer simulations are conducted for linear [e.g., y(t) = x
₁
(t) or y(t) = x
₂
(t)] as well as nonlinear [e.g., y(t) = x
₁
(t)x
₂
(t) or y(t) = x
²
/
₁
+ x
²
/
₂
(t)] tracking control of the DI system. Simulation results demonstrate well the feasibility and effectiveness of ZG controllers for tracking-control problems solving.