Identification Based Generation of Self-excited Oscillations for Underactuated Mechanical Systems via Two-relay Algorithm Luis T. Aguilar, Igor Boiko, Leonid Fridman and Alejandra Ferreira Abstract—A tool for generating a self-excited oscillations for underactuated mechanical systems by means of a variable struc- ture controller is proposed. The design procedure allows to find the explicit expressions of the controller gain parameters in terms of the desired frequency and amplitude through Describing Func- tion (DF) method. Necessary conditions for orbital asymptotic stability of linear systems are derived via a modification of the Loeb criteria. The identification and compensation method yields an exact linearization of the original nonlinear system avoiding imperfections on oscillations characteristics. Performance issue of a self-excited system, applied to a Furuta pendulum, is illustrated by simulations. I. I NTRODUCTION Motivation. Traditionally, control systems are categorized into: regulators, which are supposed to maintain a certain process variable at a desired set point value, and servo systems that are supposed to track external inputs as precisely as possible. However, in practice there are some other control tasks that fall into neither of the above categories. One of those tasks is generating a functional motion: the motion having some properties important to the functionality of a certain system without involvement of set point tracking and specification of other properties of the motion. In this paper, we consider the control of one of the simplest types of a functional motion: generation of a periodic motion in non-minimum-phase underactuated mechanical systems. Cu- rrent representative works on periodic motions in an orbital stabilization of underactuated systems involve finding and using a reference model as a generator of limit cycles, thus considering the problem of obtaining a periodic motion as a servo problem. Orbital stabilization of underactuated systems finds applications in the coordinated motion of biped robots, gymnastic robots, and others. This paper addresses the problem of generating periodic mo- tion in non-minimum phase underactuated mechanical systems without tracking a pre-computed set of reference trajectories. A novel approach is proposed to produce a limit cycle by introducing a variable structure controller in the closed-loop system where the parameters of the controller are used to tune the frequency and amplitude of oscillation. L.T. Aguilar is with Instituto Polit´ ecnico Nacional, Centro de Investigaci´ on y Desarrollo de Tecnolog´ ıa Digital, PMB 88, P.O. BOX 439016 San Ysidro CA USA 92143-9016; (e-mail: luis.aguilar@ieee.org). I. Boiko is with University of Calgary, 2500 University Dr. N.W., Calgary, Alberta, Canada (e-mail: i.boiko@ieee.org). L. Fridman and A. Ferreira are with Universidad Nacional Aut´ onoma de M´ exico (UNAM), Department of Control, Engineering Faculty. C.P. 04510. M´ exico D.F. (e-mail: lfridman@servidor.unam.mx). Contribution. The describing function method and its application to the non-minimum-phase underactuated mechan- ical system is emphasized. The contributions of the paper are: • The modification of the control algorithm proposed in [1]-[3] allowing for both constants to have arbitrary signs ensuring the intersection of the describing function plot with the Nyquist plot in any quadrant of the complex plane. • Necessary conditions for orbital asymptotic stability are derived via a modification of the Loeb criteria [12]. • A robust compensation control is designed based on the exact identification of the difference between the real system and the linearized one by means of the super- twisting algorithm [11]. The robust control compensates exactly the unknown inputs yielding a linear system equal to the linear approximation of the original one. Methodology. When variable structure controllers are used as stabilizing control laws chattering appears resulting in an undesirable component of the system motion. In [1]-[3] was reported the property of second order sliding modes (SOSM) for the purpose of generating a relatively slow motion with a significantly higher amplitude and lower frequency than respectively the amplitude and frequency of chattering in a linearized underactuated system. Nevertheless, the analy- sis were done considering a linearized system, it causes a lack of robustness in the output oscillations. In [1] were reported imperfections on oscillations characteristics attributed to Coulomb friction forces, dead zone, mechanical vibrations, etc. To avoid these phenomena a robust compensation control is added in order to compensate the difference between the real and the linearized system. The compensator is based on the identification and compensation of this difference. The super- twisting algorithm is used for identification. The theoretical results are validated by simulations. The periodic motion is generated around the upright position (which gives the non-minimum phase system case). Outline. The paper is organized as follows: The tracking control problem of a 2-DOF underactuated system and its state equation are introduced in Section II. In Section III, the robust compensation controller is synthesized. In Section IV, the parameter design formulas for the two-relay controller to obtain the desired frequency and amplitude are derived from DF method. The necessary conditions to orbital asymptotic stability are given using a modification of the Loeb criteria [12]. Performance issue of the robust two-relay controller is 978-1-4244-2200-5/08/$25.00 ©2008 IEEE 41 Authorized licensed use limited to: Universidad Nacional Autonoma de Mexico. Downloaded on July 6, 2009 at 13:23 from IEEE Xplore. Restrictions apply.