Approximate output regulation of spherical inverted pendulum by neural network enhanced design

The spherical inverted pendulum is a fairly complex nonlinear system with two-input, two-output, eight states and an unstable zero dynamics. Recently, some attempts have been made to study the output regulation problem of this system subject to a neutrally stable exosystem. The existing approaches have made use of the approximate solution of the regulator equations based on polynomial method or neural network method. However, since the regulator equations of the system are governed by ten nonlinear partial differential and algebraic equations, it is quite tedious to obtain the approximate solution of the regulator equations. In this paper, we will adopt a scheme without solving the regulator equations approximately. This scheme consists of three steps. First, define a feedforward function of dimension two determined by the regulator equations of the system. Second, find a three-layer neural network approximation of the feedforward function by a parameter optimization method. Finally, synthesize a control law based on the approximate solution of the feedforward function. Since the dimension of the feedforward function is only equal to two, the computational complexity of this new scheme is much simpler than the existing approaches. Moreover, when all the states are available, our design offers certain robustness to plant parameter variations and leads to good tracking performance.