This work is dedicated to loan Landau on his 60th birthday for his continuing support in the battle for world passivation.Summary. The objective of this paper is to compare in a simple example three controller design techniques for stabilization of nonlinear systems which have emerged in the last few years: 1) Feedback linearization, which as the name says aims at achieving a linear system in closed-loop via nonlinearities cancellation, is probably the best well known and easier to understand; 2) Integrator backstepping, which is another widely publicized technique that has proven very succesful for systems with special triangular structures; 3) Passivity-based control, which has traditionally received a wide acceptance among practicioning engineers, achieves system stabilization solving an, apparently more natural, passivation problem. Application of these, seemingly unrelated, methodologies will typically lead to the definition of different control schemes. For some specific examples these differences blurr and some interesting connections and similarities between the controller design techniques emerge. The careful study of such cases will improve our understanding of their common ground fostering cross-fertilization. In this paper we investigate these questions, both analytically and via simulations, for the simple problem of stabilization of a magnetically levitated ball. Our motivation in choosing this particular example stems, not just from the fact that due to its simplicity the connections between the controllers are best revealed, but also that such an equipment is available D. Normand-Cyrot (ed.), Perspectives in Control © Springer-Verlag London Limited 1998 98 M. Torres, R. Ortega in many engineering schools, hence experimental work can be easily carried out to complement our studies.

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The risk-sensitive filtering design problem with respect to the exponential mean-square cost criterion is con-sidered for stochastic Gaussian systems with polynomial of second and third degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal fil-tering equations are obtained using quadratic value functions as solutions to the corresponding Focker- Plank-Kolmogorov equation. The performance of the obtained risk-sensitive filtering equations for stochastic polynomial systems of second and third degree is verified in a numerical example against the optimal po-lynomial filtering equations (and extended Kalman-Bucy for system polynomial of second degree), through comparing the exponential mean-square cost criterion values. The simulation results reveal strong advan-tages in favor of the designed risk-sensitive equations for some values of the intensity parameters

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