Nonl inear Dis turbance Reject ion for Magnet ic Levitat ion

1 I n t r o d u c t i o n With ever increasing demands placed on precision and reliability within manufacturing and research environments, magnetic levitation systems are finding increased utilization within such applications as machine tooling due to their non-contact (i .e. , low friction) force exertion and their capability for active at tenuation of mechanical vibrations. However, magnetic levitation systems suffer from various control complexities such as: i.) open-loop instability, ii.) inherent nonlinearities within the system model, iii.) a unidirectional force input, and iv.) continuous biasing. In response to these difficulties, researchers have employed various modelling and control development techniques to address the aspects of i.) through iii.). For example, Charara et al. [2] constructed a nonlinear model of an inertial wheel supported by active magnetic bearings, for which a sliding mode controller was then designed to stabilize the system. In [7], L@vine et al. proposed a nonlinear feedback control law for the positioning of a shaft based on the current complementarity or current almost complementarity condition. Queiroz et al. [4] utilized a nonlinear model of a planar rotor disk, active magnetic bearing system to develop a global exponential position tracking controller for the full-order electromechanical system. In [8], Mohamed et al. demonstrated that the Q-parameterization theory could be utilized to autobalance the rotor of a vertical shaft active magnetic bearing system. Recently, much effort has been directed toward the area of disturbance suppression within magnetic levitation sys1This work is supported in part by the U.S. NSF Grants DMI9457967, ONR Grant N00014-99-1-0589, a DOC Grant, and an ARO Automotive Center Grant. tems. For example, in [3], Costic et al. introduced a learning based controller to asymptotically regulate a magnetic bearing system while compensating for periodic, exogenous disturbances. Rodrigues et al. [9] proposed an interconnection and damping assignment Passivity-Based controller to yield a smooth stabilizing controller for the active magnetic bearing system. In [6], Gentili utilized a model-based regulation approach to achieve set point regulation of the target with the disturbance input being modeled as the sum of a finite number of sinusoids. Similarly, Behal et al. [1] designed a set of linear, bounded-input bounded-output filters to facilitate the utilization of s tandard adaptive techniques that compensated for an unknown sinusoidal disturbance signal. Though not directly targeted at magnetic levitation systems, Xian et al. [11] proposed an adaptive disturbance rejection approach for single-input single-output, linear time invariant, uncertain systems subjected to sinusoidal disturbances with unknown amplitude and frequency. The approach of [11] utilizes a state estimate observer in a back stepping fashion with only output measurements to achieve asymptotic disturbance rejection. In this paper, the topic of disturbance rejection within the magnetic levitation area is furthered pursued by designing a saturated force control input that achieves asymptotic target position regulation despite the presence of a nonl inear, bounded, periodic disturbance. The control development differs from previous disturbance rejection controllers in that restrictions on the explicit s tructure of the disturbance signal are not required (i .e. , the disturbance force need only be bounded and periodic). As with previously designed magnetic levitation controllers, the proposed control structure must contend with the constraint that the actuation force is unidirectional. That is, the magnetic actuator can only exert an attractive force on the target mass (the earth 's gravitational field is utilized to produce the repulsive action). The remainder of the paper is organized as follows. The model of a magnetic levitation system actuating on a target mass suspended in the gravitation field is presented in Section II. Section III identifies the control objectives and constraints of the control development. The saturated control force input and the learning based disturbance estimator are presented in Section IV. A Lyapunov stability analysis is utilized to illustrate the asymptotic regulation of the target position. Simulation results are presented in Section V. 0-7803-7891-1/03/$17.00 © 2003 IEEE 58 2 S y s t e m M o d e l A magnetic levitat ion system consisting of a target mass suspended vertically in the gravity field subjected to a nonlinear periodic dis turbance force can be modelled by the following dynamics [6] 5~ ---g + u ~ + 6 (t) (1) where x( t ) , & (t), 5~ (t) C ~1 represent the target mass position, velocity, and acceleration signals, respectively, g = 9.81 m / s 2 C ~1 denotes the gravi tat ional acceleration constant, u 2 (t) C ~1 represents the control force input 1 , and 6 (t) c ~1 denotes the nonlinear, periodic dis turbance force (note tha t the system of (1) has been normalized with respect to the target mass m). / u -~/g -kl t anh (k3r) 6 (t) (5) where 6 (t) C ~1 represents a learning based es t imate for 6 (t) tha t is generated on-line via the following expression (t) ----sat~ o (6 (t T)) + k2 t anh (k3r) (6) where the scalar function sat~ o (-) is defined in the following manner ( e ) = / ¢ for [e[_~5o satso sgn(e) 6o for [e I > 6 o (7) k with e C ~1 representing an arb i t rary scalar argument, sgn (-) denoting the s tandard signum function, and kl, k2, k3 E ~1 denoting positive scalar control gains with kl and k2 selected in the following manner kl + k2 < g 6o (8) R e m a r k 1 The disturbance force 6 (t) is assumed to be periodic and bounded as given by the following 6 (t) = 6 (t T ) , [6 (t)[ _< 50 <_ g, (2) where T E ~1 denotes the known period and 6o represents a positive bounding constant.