Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism

When designing observers for nonlinear systems, the dynamics of the given system and of the designed observer are usually not expressed in the same coordinates or even have states evolving in different spaces. In general, the function, denoted $\tau$ (or its inverse, denoted $\tau^*$) giving one state in terms of the other is not explicitly known and this creates implementation issues. We propose to round this problem by expressing the observer dynamics in the the same coordinates as the given system. But this may impose to add extra coordinates, problem that we call augmentation. This may also impose to modify the domain or the range of the augmented" $\tau$ or $\tau^*$, problem that we call extension. We show that the augmentation problem can be solved partly by a continuous completion of a free family of vectors and that the extension problem can be solved by a function extension making the image of the extended function the whole space. We also show how augmentation and extension can be done without modifying the observer dynamics and therefore with maintaining convergence.Several examples illustrate our results.Comment: Submitted for publication in SIAM Journal of Control and Optimizatio

Robust Asymptotic Stabilization of Nonlinear Systems with Non-Hyperbolic Zero Dynamics

In this paper we present a general tool to handle the presence of zero dynamics which are asymptotically but not locally exponentially stable in problems of robust nonlinear stabilization by output feedback. We show how it is possible to design locally Lipschitz stabilizers under conditions which only rely upon a partial detectability assumption on the controlled plant, by obtaining a robust stabilizing paradigm which is not based on design of observers and separation principles. The main design idea comes from recent achievements in the field of output regulation and specifically in the design of nonlinear internal models.Comment: 30 pages. Preliminary versions accepted at the 47th IEEE Conference on Decision and Control, 200

Transverse exponential stability and applications

We investigate how the following properties are related to each other: i)-A manifold is "transversally" exponentially stable; ii)-The "transverse" linearization along any solution in the manifold is exponentially stable; iii)-There exists a field of positive definite quadratic forms whose restrictions to the directions transversal to the manifold are decreasing along the flow. We illustrate their relevance with the study of exponential incremental stability. Finally, we apply these results to two control design problems, nonlinear observer design and synchronization. In particular, we provide necessary and sufficient conditions for the design of nonlinear observer and of nonlinear synchronizer with exponential convergence property

Nonlinear Rescaling of Control Laws with Application to Stabilization in the Presence of Magnitude Saturation

Motivated by some recent results on the stabilization of homogeneous systems, we present a gain-scheduling approach for the stabilization of non-linear systems. Given a one-parameter family of stabilizing feedbacks and associated Lyapunov functions, we show how the parameter can be rescaled as a function of the state to give a new stabilizing controller. In the case of homogeneous systems, we obtain generalizations of some existing results. We show that this approach can also be applied to nonhomogeneous systems. In particular, the main application considered in this paper is to the problem of stabilization with magnitude limitations. For this problem, we develop a design method for single-input controllable systems with eigenvalues in the left closed plane

Normalisation en commande linéaire adaptative de B. Egardt à L. Praly

Notre contribution sur la technique de normalisation utilisée en commande linéaire adaptative a été controversée. Nous donnons ici notre point de vue en situant notre apport dans la perspective de l'état de l'art lorsque nous avons abordé ce problème en 1981

An asymptotic analysis of the so called intelligent PID controller

This note takes its origin from reading a paper by M. Fliess and C. Join, whose latest version is entitled Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control? that can be obtained at http://arxiv.org/abs/0904.0322 Unfortunately in that paper, the authors write that . . . it is impossible of course to give a complete description of it and that the usual mathematical criteria for robust control become . . . irrelevant. In the following, to obtain an usable description of a controller and to give some guarantees that it performs correctly, we interpret and materialize the ideas proposed in that paper. Of course, this is at the price of trivializing and likely also degrading them. Nevertheless, this allows us to present an elementary analysis using standard mathematical criteria although we know the authors claimed that this is irrelevant

On the Existence of a Kazantzis-Kravaris/Luenberger Observer

We state sufficient conditions for the existence, on a given open set, of the extension, to nonlinear systems, of the Luenberger observer as it has been proposed by Kazantzis and Kravaris. We prove it is sufficient to choose the dimension of the system, giving the observer, less than or equal to 2 + twice the dimension of the state to be observed. We show that it is sufficient to know only an approximation of the solution of a PDE, needed for the implementation. We establish a link with high gain observers. Finally we extend our results to systems satisfying an unboundedness observability property

Robust asymptotic stabilization of nonlinear systems by state feedback

International audienceWe show that regulation to a constant value of the output of a process can be achieved robustly by designing a stabilizer for a model augmented with an integrator of the output and by having the model dynamics close enough to the process ones. This is nothing but the PI controller paradigm extended to the case of nonlinear systems. We recall also that the forwarding technique is well suited for this particular stabilizer design. Finally we illustrate our result with solving the problem of regulating the flight path angle of the longitudinal mode of a plane

A Technical Result for the Study of High-gain Observers with Sign-indefinite Gain Adaptation

International audienceWe address the problem of state observation for a system whose dynamics may involve poorly known, perhaps even nonlocally Lipschitz functions and whose output measurement may be corrupted by noise. It is known that one way to cope with all these uncertainties and noise is to use a high-gain observer with a gain adapted on-line. As a difference from most previous results, we study such a solution with an adaptation law allowing both increase and decrease of the gain. The proposed method, while presented for a particular case, relies on a “generic” analysis tool based on the study of differential inequalities involving quadratic functions of the error system in two coordinate frames plus the gain adaptation law. We establish that, for bounded system solutions, the estimated state and the gain are bounded. Moreover, we provide an upper bound for the mean value of the error signals as a function of the observer parameters