Control of pendulum tracking (including swinging up) of IPC system using zeroing-gradient method

The pendulum control of the inverted-pendulum-on-a-cart (IPC) system is one of the most important issues in nonlinear control theory and has been widely investigated. Nevertheless, the control of pendulum tracking and swinging up has often been addressed separately. In this paper, by combining the zeroing dynamics and the conventional gradient dynamics, two concise zeroing-gradient (ZG) controllers (termed, z2g0 controller and z2g1 controller, respectively) are constructed for the IPC system. Importantly, the proposed z2g1 controller not only realizes the simultaneous control of pendulum swinging up and pendulum angle tracking, but also solves the singularity problem elegantly without using any switching strategy. Besides, the ZG method is compared with the optimal control method and the back-stepping method. The theoretical analyses about the convergence performance of z2g0 and z2g1 controllers are further presented. Moreover, the boundedness of both control input u and its derivative (u)over dot of the z2g1 controller is proved. Three illustrative examples are carried out to demonstrate the tracking performance of z2g0 and z2g1 controllers for the pendulum tracking control. In particular, the efficacy and superiority of z2g1 controller for the control of pendulum tracking (including swinging up) of the IPC system in conquering the singularity problem are substantiated by comparative results. Furthermore, this paper investigates the robustness of the proposed ZG controllers (as well as the ZGdesign method) in the situations of time delay and disturbance.