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

Locally optimal controllers and globally inverse optimal controllers

In this paper we consider the problem of global asymptotic stabilization with prescribed local behavior. We show that this problem can be formulated in terms of control Lyapunov functions. Moreover, we show that if the local control law has been synthesized employing a LQ approach, then the associated Lyapunov function can be seen as the value function of an optimal problem with some specific local properties. We illustrate these results on two specific classes of systems: backstepping and feedforward systems. Finally, we show how this framework can be employed when considering the orbital transfer problem

Query management in a sensor environment

Traditional sensor network deployments consisted of fixed infrastructures and were relatively small in size. More and more, we see the deployment of ad-hoc sensor networks with heterogeneous devices on a larger scale, posing new challenges for device management and query processing. In this paper, we present our design and prototype implementation of XSense, an architecture supporting metadata and query services for an underlying large scale dynamic P2P sensor network. We cluster sensor devices into manageable groupings to optimise the query process and automatically locate appropriate clusters based on keyword abstraction from queries. We present experimental analysis to show the benefits of our approach and demonstrate improved query performance and scalability

A Region-Dependent Gain Condition for Asymptotic Stability

A sufficient condition for the stability of a system resulting from the interconnection of dynamical systems is given by the small gain theorem. Roughly speaking, to apply this theorem, it is required that the gains composition is continuous, increasing and upper bounded by the identity function. In this work, an alternative sufficient condition is presented for the case in which this criterion fails due to either lack of continuity or the bound of the composed gain is larger than the identity function. More precisely, the local (resp. non-local) asymptotic stability of the origin (resp. global attractivity of a compact set) is ensured by a region-dependent small gain condition. Under an additional condition that implies convergence of solutions for almost all initial conditions in a suitable domain, the almost global asymptotic stability of the origin is ensured. Two examples illustrate and motivate this approach