Linear active disturbance rejection control of underactuated systems: The case of the Furuta pendulum $ M. Ramírez-Neria a,1 , H. Sira-Ramírez a,1 , R. Garrido-Moctezuma a,1 , A. Luviano-Juárez b,n a Centro de Investigación y de Estudios Avanzados del I.P.N. (Cinvestav-IPN), Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, C.P. 07360 México, D.F. Apartado Postal,14-740, 07000 México, D.F., Mexico b Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas IPN, Av. Instituto Politécnico Nacional 2580 Col. Barrio La Laguna Ticomán C.P. 07340 México, D.F., Mexico article info Article history: Received 20 May 2013 Received in revised form 11 September 2013 Accepted 30 September 2013 Available online 16 November 2013 This paper was recommended for publication by Dr. Jeff Pieper Keywords: Furuta pendulum Differentially flat systems Active disturbance rejection control GPI observers abstract An Active Disturbance Rejection Control (ADRC) scheme is proposed for a trajectory tracking problem defined on a nonfeedback linearizable Furuta Pendulum example. A desired rest to rest angular position reference trajectory is to be tracked by the horizontal arm while the unactuated vertical pendulum arm stays around its unstable vertical position without falling down during the entire maneuver and long after it concludes. A linear observer-based linear controller of the ADRC type is designed on the basis of the flat tangent linearization of the system around an arbitrary equilibrium. The advantageous combination of flatness and the ADRC method makes it possible to on-line estimate and cancels the undesirable effects of the higher order nonlinearities disregarded by the linearization. These effects are triggered by fast horizontal arm tracking maneuvers driving the pendulum substantially away from the initial equilibrium point. Convincing experimental results, including a comparative test with a sliding mode controller, are presented. & 2013 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction The control of underactuated systems represents a difficult and challenging problem, specially when experimental implementations of synthesized control solutions are required. This is due, aside of the effect of unmodeled dynamics and external forces, to the associated restrictions on the behavior of the non directly actuated variables [27] and the natural obstacle to linearizability exhibited by a large subclass of these systems. Underactuated systems are becoming popular in many sophisticated control applications, such as spacecraft, aerial robotic systems, underwater vehicles, locomo- tive systems, flexible robotics. Some possible advantages associated to such systems are cost reduction, lighter structures, smaller dimensions, among others (see [24] for a comprehensive treatment of this class of systems). The Furuta pendulum [12], also called the rotational pendulum, is one of the most popular underactuated systems in academic laboratories around the world. The system is provided with one control input and it has two mechanical degrees of freedom. It consists of an actuated arm, which rotates in the horizontal plane; the actuated arm is joined to a non actuated pendulum which rotates loosely in a vertical plane perpendicular at the tip of the horizontal rotating arm. The system is quite nonlinear due to the gravitational forces, the Coriolis and centripetal forces [4] and the acceleration couplings. In addition, it is nonfeedback linearizable and it exhibits a lack of controllability in certain configurations [7]. The system represents a suitable platform for testing diverse linear and nonlinear control laws. Traditional control problems associated with the Furuta pendu- lum are mainly of two kinds: (1) the problem of balancing up the vertical pendulum to the upper, unstable, position (swinging up) and (2) the stabilization around this position. Several methodologies have been proposed to solve the problem of swinging up and balancing the Furuta pendulum, these include the energy based swinging up control [2], passivity-based control [26], adaptive attractive ellipsoid methods [25], friction compensation controllers [34], extended state observer-based controllers [3], among others. In the study reported in [1], some different controllers for the Furuta pendulum were tested and compared to point out the principal advantages and drawbacks of the diverse control schemes. The study also considered the main physical limitations associated with the control of the pendulum, where among others, the possibility of control input saturations was specifically treated. Most of the Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.09.023 ☆ This work was supported by the Centro de Investigación y de Estudios Avanzados del I.P.N. (Cinvestav-IPN), México, D.F., México. n Corresponding author. Tel.: þ52 57296000. E-mail addresses: mramirezn@ctrl.cinvestav.mx (M. Ramírez-Neria), hsira@cinvestav.mx (H. Sira-Ramírez), rgarrido@ctrl.cinvestav.mx (R. Garrido-Moctezuma), aluvianoj@ipn.mx, alberto.luviano@gmail.com (A. Luviano-Juárez). 1 Tel.: þ52 5747 3800. ISA Transactions 53 (2014) 920–928