Limit cycle elimination in inverted pendulums: Furuta pendulum and pendubot

This paper presents the design of a linear state feedback controller for the stabilization of two inverted pendulums, namely, Furuta pendulum and pendubot. Such a controller design allows eliminating the limit cycle that appears in the systems due to the effect of a nonlinearity, that is, dead-zone, which is induced by static friction at the motor shaft. To do this, the differential flatness approach is applied to the linear approximate model of the inverted pendulums under study. Then, the resulting flat systems are translated to the frequency domain for which a control scheme is proposed. Subsequently, the dead-zone nonlinearity is treated off-line as an approximation obtained through the describing function method. Since this is an approach intrinsically based on frequency response, the frequency response-based approach is suitable for tuning the gains of the proposed control scheme. An advantage of using the frequency response-based approach along with the describing function method is that they allow obtaining precise formulas that simplify the tuning of the proposed control scheme, so that the limit cycle caused by the dead-zone is eliminated. This must be contrasted with a time response-based approach, proposed recently by the authors, where precise formulas were not obtained and intuitive ideas have to be used to eliminate limit cycle. Finally, the procedure of the controller design herein proposed is verified via experimental tests.