Nonlinear Control of a Flexible Rotor Magnetic Bearing System: Robustness and the Indefinite Model

Previously published control strategies for magnetic bearings primarily focus on linear optimal control techniques. Whlle these methods afford many advantages, conspicuously absent from the literature are detailed attempts at nonlinear control. Here, we obtain the equations of motion of an overhung flexible rotor supported in magnetic bearings with two different levels of model sophistication. We derive a generic nonlinear controller in the manner of feedback linearization, and compare the eigenanalysis and transient response of the two rotor models under the action of this "perfect model" controller. We then proceed to obtain a robust nonlinear controller through the sliding mode technique and demonstrate that robustness by implementing it on an uncertain model. INTRODUCTION Magnetic bearings have been receiving increasing attention recently, and a wide spectrum of literature exists in the field (Geary, 1963, Humphris, 1985). Much of the literature which discusses control topics concentrates on linear optimal control techniques (references [7], [9], [10], [14], [ 18]). Linearizing the dynamics of magnetic bearing systems about the bearing center at a nominal speed affords opportunities for linear quadratic Gaussian optimal control. Since much of contemporary engineering activity appears to be directed at synthesis and optimization of the design process, linear techniques are justifiably popular. Maslen ( 1991) provides an excellent discussion of this synthesis in magnetic bearing design. Indeed, Burrows et. al. (1988) posit that linearized systems" ... can be justified on the basis that there is a large body of knowledge to aid in the design of linear control systems but the design of nonlinear systems is still less well-defmed." As a practical matter, the issue appears to be not so much concerned with the possibility of a viable nonlinear controller, as with an adequate nonlinear observer. In general, nonlinear control laws frequently require full state feedback. For real systems, this is often a great hindrance, as many of the states either cannot effectively be measured, or can only be measured at great cost and inconvenience. Thus, the probThis work was supported by the Center for Electromechanics (UTAustin), and funded through an ARO grant under the AASERT program. tern evolves into one of designing an effective controller/observer combination, and this is elegantly provided by linear control theory. The well known separation principle of linear systems provides for closed loop system stability when the poles of the controller and the observer are independently stable (Kailath, 1980). This, of course, means that the observer and the controller can be designed independently, and the combination is assured to be stable. This principle, however, is predicated on linear dynamics, and is not necessarily applicable to nonlinear systems, in the general case. Nevertheless, much research has been done to address nonlinear observer design in an effort to make nonlinear compensators applicable to a wider class of problems (Slotine et. al. 1987, Raghavan, 1992, Raghavan and Hedrick, 1990). Still, one wonders what advantages (or, disadvantages) might result from implementing nonlinear controllers on these increasingly popular bearing systems. Pradeep and Gurumoorthy ( 1993) have discussed the issue at one level, but we hope to provide a more explicit examination here. Specifically, we focus on the robustness of nonlinear control as provided by the sliding mode technique. Flexible system modelling, of course, may be done at different levels of sophistication, guided in part by the frequency range within which one expects the model to be accurate or useful. Real systems, behaving as a continuum, exhibit resonant phenomena beyond the highest mode modelled as the range of validity of a given model is exceeded. Thus, unmodelled dynamics are a key source of uncertainty with which control systems must contend. We will pursue nonlinear controller robustness in magnetic bearing systems by postulating a fairly simple flexible rotor system. This rotor will be modelled at increasing levels of sophistication, and nonlinear controllers will be designed to maintain the shaft centered in the bearings. We will then compare the eigenanalysis and transient response of this flexible rotor under the assumption of a "perfect model" (i.e., no uncertainty). Finally, in an effort to address this uncertainty, we will derive a robust nonlinear controller based on the crude rotor model, but implement that controller on the more sophisticated model. Thus, we hope to gain insight regarding the performance of a given nonlinear controller on a real system. DYNAMIC MODEL A generic flexible horizontal rotor supported in controlled de electromagnetic bearings is shown in Figure I. For the present discussion, we will assume a four pole bearing structure as indicated. More importantly, we will also assume that these magnetic actuators are current-