PII: SOOO5-1098(97)00155-6 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ Automat~ca, Vol. 34, No. 1, pp. 1255131, 1998 ‘A 1998 Elsevier Science Ltd. All rights reserved Printed in G reat Bratain ooos-1098/98 $19.00 + 0.00 Technical Communique Regulation and Tracking of the Nonholonomic Double Integrator: A Field-oriented Control Approach* G. ESCOBAR?, R. ORTEGA? and M. REYHANOGLUS Key Words-Nonlinear systems; nonholonomic systems; induction motor. Abstract-In this paper we show how a slight modification to the well-known field-oriented control of induction motors allows us to provide simple solutions to the problems of global regulation and tracking for the nonholonomic double integ- rator. Two controllers are proposed, one motivated by the well-known direct field-oriented motor control, which is a simple zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA continuous static-state feedback that guarantees exponential convergence for all initial conditions in W3 - {xi = x2 = 0). To overcome the latter restriction we add a dy- namic extension to transfer this (initial conditions) singularity to the controller state, hence making the stability global. This controller is a direct outgrowth of indirect field oriented motor control. In view of the well-known Brockett’s condition we obviously do not ensure Lyapunov stability. 0 1998 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION In the past few years, considerable attention has been paid to the problem of stabilizing non- holonomic systems. It is well known that non- holonomic systems constitute a remarkable class of controllable nonlinear systems which fail Brockett’s necessary condition for the existence of asymptotically stabilizing time-invariant continu- ous (static or dynamic) state feedback. As a conse- quence, research on feedback stabilization of non- holonomic systems has been directed toward the design of time-varying smooth feedback control laws (Samson, 1995; Pomet, 1992), time-varying nonsmooth feedback control laws (McCloskey and Murray, 1997), and discontinuous feedback control laws (Astolfi, 1996; Bloch and Drakunov, 1994; Bloch zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA et al., 1992; Hespanha, 1996; Kolmanovsky et al., 1994). A benchmark problem to study nonholonomic systems, which was the first example of a locally *Corresponding author. Professor R. Ortega. E-mail: ror- tega@lss.supelec.fr. TLaboratoire des Signaux et Systemes, CNRS-SUPELEC, UMR CNRS 16, 91192, Gif-stir-Yvette, France. $ Automation Robotics Technology, TSTC, 300 College Drive, Sweetwater, TX 79556, U.K. 125 controllable nonlinear system which is not smoothly stabilizable, is the so-called non- holonomic double integrator 1, = ui, (I) 12 = u2, (2) & = xiuz - xzui. (3) A well-known physical example which can be modeled using equations above is the Heisenberg flywheel. This system consists of a point-mass con- nected by a massless rod to a flywheel which has a moment of inertia. The flywheel is in turn at- tached to a table by a joint on which the wheel can spin freely. The joint is frictionless so that it exerts no torque on the assembly. It is easy to show, though apparently not very well known, that this model describes as well (after a suitable change of coordinates) the behavior of current-fed induction motors (Neimark and Fufaev, 1972; Ortega et al., 1996). Control of induc- tion motors has been extensively studied in the literature, and some algorithms, in particular, the well-known field-oriented controller (FOC), has demonstrated remarkable performances, both practically (Kim et al., 1997; Leonhard, 1996; Taoutaou et al., 1997) and theoretically (de Wit et al., 1996; Ortega et al., 1996). Our objective in this short note is to show how the principles of FOC can be adapted to provide simple solutions to the problems of global regula- tion and tracking for the nonholonomic double integrator. The remaining of the paper is organized as fol- lows. In Section 2 we briefly review FOC. Then, in Section 3 we show how FOC applies verbatim to the nonholonomic double integrator. Interestingly, a special precaution must be taken only in the case when we want to regulate the state to zero, but other than that the controller is identical to the FOC above. In Section 4 we illustrate the perfor- mance of the proposed controllers with some