With wider investigations and applications of autonomous robotics and intelligent vehicles, the inverted pendulum on a cart (IPC) system has become more attractive for numerous researchers due to its concise and representative structure. In this article, the tracking-control problem of the IPC system is considered and investigated. Based on Zhang dynamics (ZD) and gradient dynamics (GD), a novel kind of ZG controllers are developed and investigated for achieving the tracking-control purpose, which contains controllers of z2g0 and z2g1 types according to the number of times of using the ZD and GD methods. Besides, theoretical analyses are presented to guarantee the global and exponential convergence performance of both z2g0 and z2g1 controllers. Computer simulations are further performed to substantiate the feasibility and effectiveness of ZG controllers. More importantly, comparative simulation results demonstrate that controllers of z2g1 type can conquer the singularity problem (i.e. the division-by-zero problem).