Stability analysis of two-degree-of-freedom control via constructing a linear continuous-time periodic system: Application to rotary inverted pendulum

In this paper, we introduce a stability analysis approach for systems with two-degree-of-freedom control by converting the linear continuous time-varying system into a periodic one and applying the stability criterion of linear continuous-time periodic (LCP) systems. We illustrate this approach with swing-up task of a rotary inverted pendulum system. It includes three steps. First, we design a periodic orbit by adding a (virtual) swing-down phase. Second, we design a periodic controller which alternates between the swing-up and swing-down controllers in a piecewise manner. In the swing-up phase, the swing-up controller comprises both feedforward and feedback controllers. The feedforward controller and the corresponding nominal trajectory are developed by solving a two-point boundary value problem. The feedback controller is then designed by linearizing around the nominal trajectory and utilizing the LQR state feedback and zero-order hold methods. In the swing-down phase, the swing-down controller is designed using the similar process of the swing-up controller. Third, we directly analyze the stability of the periodic system by applying the stability criterion for LCP systems. The stability of the swing-up phase is naturally ensured by the stability of the periodic system, as the swing-up and swing-down phases together form the periodic cycle. This idea eliminates the numerical calculations of the state transition matrices of LCP systems. Experimental results validate the effectiveness of the proposed periodic controller and the stability of the proposed periodic system.