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

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

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

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

Note on diffeomorphism extension for observer design

An often encountered way of designing an observer is to use coordinates different from the given ones (or the ones of interest). This is done via the construction of a diffeomorphism (maybe obtained following the extension of an injective immersion) and an extended vector field on which is designed the observer. The estimated state is then obtained by employing the left inverse of the diffeomorphism (which may be difficult to obtained). A possible solution to overcome this difficulty is to use a diffeomorphism extension. A first preliminary result is given in this note

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

Remarks on the existence of a Kazantzis-Kravaris/Luenberger observer

International audienceWe state sufficient conditions for the existence, on a given open set, of the extension, to non linear systems, of the Luenberger observer as it has been proposed by Kazantzis and Kravaris. To weaken these conditions, the observer is modified in a way which induces a time rescaling and which follows from a forward unboundedness observability property. Also, we state it is sufficient to choose the dimension of the dynamic system, giving the observer, less than or equal to 2 + twice the dimension of the state to be observed. Finally we show how approximation is allowed and we establish a link with high gain observers

Uniform Practical Nonlinear Output Regulation

International audienceIn this paper, we present a solution to the problem of asymptotic and practical semiglobal regulation by output feedback for nonlinear systems. A key feature of the proposed approach is that practical regulation is achieved uniformly with respect to the dimension of the internal model and to the gain of the stabilizer near the zero error manifold. This property renders the approach interesting for a number of real cases by bridging the gap between output regulation theory and advanced engineering applications. Simulation results regarding meaningful control problems are also presented